Research ArticleA Heterogeneous Multiattribute Group Decision-MakingMethod Based on Intuitionistic Triangular Fuzzy Information

Jun Xu 12 Jiu-Ying Dong2 Shu-PingWan 3 De-Yan Yang 4 and Yi-Feng Zeng5

1College of Modern Economics amp Management Jiangxi University of Finance and Economics Nanchang 330013 China2School of Statistics Jiangxi University of Finance and Economics Nanchang 330013 China3College of Information Technology Jiangxi University of Finance and Economics Nanchang 330013 China4Business School Jiangsu Normal University Xuzhou 221116 China5Department of Computer Science and Information Systems Teesside University Middlesbrough Ts13ba UK

Correspondence should be addressed to Shu-PingWan shupingwan163com and De-Yan Yang yangdeyan027foxmailcom

Received 4 April 2019 Accepted 16 July 2019 Published 7 August 2019

Academic Editor Chittaranjan Hens

Copyright copy 2019 Jun Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

How to aggregate decision information in heterogeneous multiattribute group decision making (HMAGDM) is vital The aimof this paper is to develop an approach to aggregating decision data into intuitionistic triangular fuzzy numbers (ITFNs) forheterogeneousMAGDMproblems with real numbers (RNs) interval numbers (INs) triangular fuzzy numbers (TFNs) trapezoidalfuzzy numbers (TrFNs) and triangular intuitionistic fuzzy number (TIFNs) Using the relative closeness of technique for orderpreference by similarity to ideal solution (TOPSIS) and geometry entropymethod we first present a general approach to aggregatingheterogeneous information into ITFNs which takes the group consistency of experts into account Based on the collectiveintuitionistic triangular fuzzy decision matrix and extended TOPSIS a multiple objective mathematical program is constructedto determine the optimal attribute weights Subsequently a new method to solve HMAGDM problems is presented based on theaforementioned discussion A trustworthy service selection example is provided to verify the practicality and effectiveness of theproposed method

1 Introduction

Group decision making (GDM) has been known as a popularmethod for finding the best alternative from a set of alter-natives through aggregating decision information given in agroup of experts in which the evaluation of alternatives mayinvolve multiple attributes including objective and subjectiveinformation [1ndash3] Due to the limited cognition and prefer-ence of decision maker it is hard for different attributes to usethe same information format to express the evaluation Forinstance in an online seller evaluation the service attitudeof the seller is suited to be described by triangular fuzzynumbers (TFNs) since the service of the seller is generallystable but sometimes it is excellent and sometimes bad Itis convenient to describe the shipping speed of seller withinterval numbers (INs) since it is not fixed but fluctuatesin a certain range These types of GDM problems withmultiple conflicting attributes whose values are given by

decision makers (DMs) may be represented in the form ofmultiple formats such as real numbers (RNs) INs TFNstrapezoidal fuzzy numbers (TrFNs) and linguistic values(LVs) called heterogeneous multiattribute group decisionmaking (HMAGDM) problems [4]

In this recent research HMAGDM methods have beensuccessfully applied to various fields such as supply chaincoordination [5] business processes [6] and software qualityevaluation [7ndash9] The key to tackling such problems is howto fuse various types of attribute values [10] So far manyuseful and valuable methods have been developed to studythe fusion process of heterogeneous information which canbe roughly classified into three main categories [4 10] (1)The indirect approaches [11ndash14] in which the heteroge-neous decision information given by DMs is converted intouniformed information by transformation methods Wangand Cai [13] developed a generic distance-based VIKORwhich can use aggregation function to convert heterogeneous

HindawiComplexityVolume 2019 Article ID 9846582 18 pageshttpsdoiorg10115520199846582

2 Complexity

information into a uniform nonfuzzy degree and applied it todeal with emergency supplier selectionUsing transformationfunction Zhang et al [14] transformed the multigranularlinguistic decision matrices (LDMs) into uniform LDMsThen a new optimization consensus model was constructedfor 2-Rank multigranular linguistic MAGDM problems (2)The optimization-based approaches [5 15ndash17] in which theheterogeneous information is integrated by constructing dif-ferentmultiple objective optimizationmodels Dong et al [15]proposed a new complex and dynamicHMAGDMmethod todeal with the differences between individual sets of attributesand heterogeneous information Zhang et al [16] developed aHMAGDMmethod with aspirations information by combin-ing the prospect theory and a biobjective intuitionistic fuzzyprogramming model Yu et al [17] incorporated risk attitudeand preference deviation of experts into the mathematicalprogramming models to solve the HMAGDM problems withRNs INs Ifs LVs and TFNs (3) The direct approaches [1018ndash21] In the direct approach the collective decision infor-mation is obtained by aggregating standardized individualdecision information Then the heterogeneous informationis transformed into some comparable preference informa-tion Yue and Jia [10] introduced a projection measure toaggregate decision information including IFNs and IVIFsYue [18] proposed a direct projection-based group decision-making methodology with RNs and INs To overcome theirrationality of the classical projection formulae in RN andIN vector settings Yue [19] presented a normalized projec-tion measure and applied it to solve HMAGDM problemswith RNs and INs In order to integrate heterogeneouslyinterrelated attributes in the HMAGDM problem Das etal [20] develop an Atanassovrsquos intuitionistic fuzzy extendedBonferroni mean based on a strict t-conorm Li et al [21]proposed a new HMAGDM method using weighted poweraverage operator to integrate the heterogeneous decisiondata

These achievements have provided the foundation ofthe HMAGDM problems It is noticed that methods [1 5ndash8 15ndash21] are on the basis of the hypothesis that the ratingsprovided by DMs are completely affirmative neglecting thejudgment subjectivity thus the impreciseness and uncer-tainty of original decision information cannot be capturedMethods [10ndash14] turned the heterogeneous information intoa unified form of linguistic terms which are subjective andcannot measure quantitatively and intuitively the uncertaintyof attribute values Intuitionistic fuzzy (IF) sets (IFSs) [22 23]and interval-valued intuitionistic fuzzy sets (IVIFSs) [24] canbe viewed as an effective tool to describe the uncertainty andambiguity which has led to the wide applications of IFSs andIVIFSs [25ndash27] To fill these research gaps many practicalstudies have been proposed to aggregate decision data intoIFSs [22ndash24] which can be divided into two categories (1) themethods for aggregatingRNs into intuitionistic fuzzy number(IFN) based on Golden Section idea [28 29] MinimaxCriterion [30] and statistical theory [31] (2) the methodsfor aggregating RNs or INs into interval-valued intuitionisticfuzzy number (IVIFN) [1] based on Minimax Criterion [32]linear transformation [33] and mean and standard deviation[34] However these methods [28ndash34] cannot be suitable for

HMAGDM problems More recently Xu et al [35] presenteda general method to aggregate decision information into IFNand applied it to select cloud computing service providerswherein the assessments take the form of RNs INs TFNsTrFNs and LVs Combined with the relative closeness intechnique for order preference by similarity to ideal solution(TOPSIS) and statistical theory [36] Wan et al [37] devel-oped a new general method to aggregate the attribute valuevector into IVIFNs and used it for HMAGDM problem withRNs INs TFNs and TrFNs

The aforementioned methods [35 37] have made deepdiscussions to HMAGDM problems based on aggregatingdecision data into IF information but these aggregationtechniques [28ndash35 37] still suffer from some deficiencies (1)They cannot deal with more complicated attribute values rep-resented by triangular intuitionistic fuzzy number (TIFNs)and trapezoidal intuitionistic fuzzy numbers (TrIFNs) (2)They ignore the influence of different experts in aggregationprocess which may lead to unreasonable results (3) Themembership degree and nonmembership degree of inte-grated value in [28ndash35 37] cannot reflect the distributioncharacteristics of the data like normal distribution In manyreal decision situations the evaluations of decision makerare based on a number of historical feedbacks on the cor-responding attribute Studies showed that the distribution ofhistorical feedbacks is generally close to a normal distributionwhen the number of feedbacks is larger It may be effective touse TFNs to model the integrated value instead of crisp valueand interval-value since TFNs contain more information andare more consistent with normal distribution characteristicsThus when an assessment vector is aggregated into anIFN and IVIFN the loss of information is likely to occurIntuitionistic triangular fuzzy numbers (ITFNs) introducedby Liu and Yuan [38] as an extension of IFSs can expressmore information from different dimension decision infor-mation [39] than IFNs and IVIFNs since its prominentcharacteristic is that the corresponding membership degreeand nonmembership degree are described by TFNs [40]Thus ITFNs can not only depict the fuzzy concept ofldquogoodrdquo or ldquoexcellentrdquo but also outstand the satisfaction anddissatisfaction information with the maximum probabilityand also recoup the deficiency due to the loss of the centerof gravity in IVIFNs [41ndash43] For instance in a trustworthyseller selection example the service attitudemay be expressedby an ITFN ((040607)(010203)) which contains twoaspects of implication in the historical ratings of a sellerone is that usersrsquo satisfactory degree is between 04 and 07the most possible satisfactory degree is 06 the other is thatusersrsquo dissatisfactory degree is between 01 and 03 the mostpossible dissatisfactory degree is 02 Some theories andGDMmethods based on ITFNs have been developed Wang [40]defined score function and accuracy function to compare theITFNs and developed several ITFN geometric aggregationoperators Wei [41] proposed the ITFN weighted averagingoperator and ITFN ordered weighted averaging operatorand applied them to solve GDM problems To consider theinteraction among attributes Gao et al [42] presented someITFN aggregation operators with interaction Yu and Xu [43]investigated a series of intuitionistic multiplicative triangular

Complexity 3

fuzzy aggregation operators Although these studies [38ndash43] focused on different aspects of ITFNs it can onlyaggregate ITFNs Therefore to push ahead with the applica-tion of the above aggregations it is necessary to aggregatemultiple types of decision information into ITFNs whichis very interesting yet relatively sophisticated to disposeof

To do that this paper aims to propose a novel HMAGDMmethod based on ITFNs The primary contributions of thispaper can be illuminated briefly as follows

(1) We first present an aggregation technology to aggre-gate heterogeneous information into TIFNs Compared withexisting methods [28ndash35 37] the proposed aggregate tech-nology has the following advantages For more details referto Section 52

(i) A new elicitation of the support opposite and uncer-tain information based on distance is introducedwhich can accommodate more complicated attributevalues including TIFNs and TrIFNs since it just needsto calculate the distance from decision data to themaximum and minimum grade

(ii) A new construction approach of ITFN is presented bygroup consistency which not only takes into accountexpertrsquos weight but can overcome the shortcoming ofthe hypothesis of the normal distribution

(iii) It can not only effectively avoid the loss of originalinformation but also reflect the distribution charac-teristics of the original decision data

(2) A new similarity measure of ITFNs is developedand applied to construct a multiple objective linear pro-gramming to determine attribute weights in ITFN envi-ronment with incomplete information The determinationmethod of attribute weights can effectively avoid the sub-jectivity brought by the given attribute weights in ad-vance

(3) Based on the aforesaid provision a new method todeal with HMAGDM problems with RNs INs TFNs TrFNsand TIFNs is proposed The comprehensive evaluation valueof the alternative is an ITFN which preserves more usefulinformation

The remainder of this paper is set out as followsSection 2 briefly introduces related basic concepts Section 3presents an approach to aggregating heterogeneous deci-sion data into ITFNs Section 4 builds a multiple objectivelinear programming model to determine attribute weightsand propose a HMAGDM method Section 5 provides anumerical example to illustrate the feasibility and reasonable-ness of the proposed method Section 6 makes our conclu-sions

2 Preliminary

In this section some basic concepts of ITFN and distancemeasures are briefly described below

21 Intuitionistic Triangular Fuzzy Number

Definition 1 (see [38]) A triangular fuzzy number (TFN) Ais a special fuzzy set on a real number set R its membershipfunction is defined by

119865119860 (119909) = 119909 minus 119905119897119905119898 minus 119905119897 if 119905119897 le 119909 le 119905119898119905ℎ minus 119909119905ℎ minus 119905119898 if 119905119898 le 119909 le 119905ℎ0 if otherwise (1)

where 0 le 119905119897 le 119905119898 le 119905ℎ le 1 119905119897 and 119905ℎ present the lower limitand upper limit of A respectively and 119905119898 is the mode whichcan be denoted as a triplet (119905119897 119905119898 119905ℎ)Definition 2 (see [38]) Let X be a fixed set 120583119860(119909) =(119905119897119860(119909) 119905119898119860(119909) 119905ℎ119860(119909)) and V119860(119909) = (119891119897119860(119909) 119891119898119860 (119909) 119891ℎ119860(119909)) areTFNs defined on the unit interval [0 1] then an intuitionistictriangular fuzzy set 119860 over X is defined as 119860 = (119909 lt120583119860(119909) V119860(119909) gt) | 119909 isin 119883 where the parameters 120583119860(119909)and V119860(119909) indicate respectively the membership degree andnonmembership degree of the element x in 119860 with theconditions 0 le 119905ℎ119860(119909) + 119891ℎ119860(119909) le 1

For convenience we call = ((119905119897119860 119905119898119860 119905ℎ119860) (119891119897119860 119891119898119860 119891ℎ119860))an intuitionistic triangular fuzzy number (ITFN) where

119905119897119860 119905119898119860 119905ℎ119860 isin [0 1] 119891119897119860 119891119898119860 119891ℎ119860 isin [0 1] 119905ℎ119860 + 119891ℎ119860 isin [0 1] (2)

It is clear that the largest and smallest ITFN are 120572+ =((1 1 1) (0 0 0)) and 120572minus = ((0 0 0) (1 1 1)) respectivelyDefinition 3 (see [38]) Let 1 = ((1199051198971 1199051198981 119905ℎ1) (1198911198971 1198911198981 119891ℎ1 ))and 2 = ((1199051198972 1199051198982 119905ℎ2) (1198911198972 1198911198982 119891ℎ2 )) be two ITFNs then thecontainment is1 sube 2iff 1199051198971 le 1199051198972 1199051198981 le 1199051198982 119905ℎ1 le 119905ℎ2 1198911198971 ge 1198911198972 1198911198981 ge 1198911198982 and 119891ℎ1 ge 119891ℎ2 (3)

Some arithmetic operations between ITFNs 1 and 2 areshown as below [40]

(1) 1 + 2 = ((1199051198971 + 1199051198972 minus 11990511989711199051198972 1199051198981 + 1199051198982 minus 1199051198981 1199051198982 119905ℎ1 + 119905ℎ2 minus 119905ℎ1119905ℎ2)(11989111989711198911198972 1198911198981 1198911198982 119891ℎ1 119891ℎ2 ))(2) 120582 = ((1 minus (1 minus 1199051198971)120582 1 minus (1 minus 1199051198981 )120582 1 minus (1 minus 119905ℎ1)120582)((1198911198971)120582 (1198911198981 )120582 (119891ℎ1 )120582)) 120582 gt 0

4 Complexity

Definition 4 (see [41]) For a set of ITFNs 119894 (119894 = 1 2 119899)that have associated an importance weight vector 119908 =(1199081 1199082 119908119899)119879 with 119908119894 isin [0 1] and sum119899119894=1119908119894 = 1 We call

119868119879119865119882119860(1198861 1198862 119886119899) = 119899sum119894=1

119908119894119886119894 = ((1minus 119899prod119894=1

(1 minus 119905119897119894)119908119894 1 minus 119899prod119894=1

(1 minus 119905119898119894 )119908119894 1minus 119899prod119894=1

(1 minus 119905ℎ119894 )119908119894) ( 119899prod119894=1

(119891119897119894 )119908119894 119899prod119894=1

(119891119898119894 )119908119894 119899prod119894=1

(119891ℎ119894 )119908119894))(4)

an intuitionistic triangular fuzzy weighted average operator(ITFWA)

Definition 5 Let 1 = ((1199051198971 1199051198981 119905ℎ1) (1198911198971 1198911198981 119891ℎ1 )) and 2 =((1199051198972 1199051198982 119905ℎ2) (1198911198972 1198911198982 119891ℎ2 )) be two ITFNs A similarity measure120599(1 2) between the ITFNs 1 and 2 is defined as follows120599 (1 2) = 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12sdotmax (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)](5)

Theorem 6 e similarity measure 120599(1 2) satisfies thefollowing properties

(i) 0 le 120599(1 2) le 1(ii) 120599(1 2) = 1 if and only if 1 = 2(iii) 120599(1 2) = 120599(2 1)(iv) If 3 is a ITFN and 1 sube 2 sube 3 then 120599(1 3) le120599(1 2) and 120599(1 3) le 120599(2 3)

Proof It is easy to see that the proposed similarity measure120599(1 2) meets the third property of Theorem 6 We onlyneed to prove (i) (ii) and (iv)

For (i) By (2) we have0 le 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 10 le 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10 le 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 1

0 le 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 1(6)

It is easy to see that0 le 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 120 le 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 12(7)

Thus we get0 le 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]le 1(8)

And then the inequality 0 le 120599(1 2) le 1 is establishedFor (ii) When 120599(1 2) = 1 if and only if112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 012 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 0

(9)

Apparently it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 = 010038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 = 010038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 = 0100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 = 010038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 = 010038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 = 0(10)

Thus we get 1199051198971 = 1199051198972 1199051198981 = 1199051198982 119905ℎ1 = 119905ℎ2 1198911198971 = 1198911198972 1198911198981 = 1198911198982 119891ℎ1 minus 119891ℎ2 And then 1 = 2

Complexity 5

For (iv) Since 1199051198971 le 1199051198972 le 11990511989731199051198981 le 1199051198982 le 1199051198983 119905ℎ1 le 119905ℎ2 le 119905ℎ3 1198911198971 ge 1198911198972 ge 11989111989731198911198981 ge 1198911198982 ge 1198911198983 119891ℎ1 ge 119891ℎ2 ge 119891ℎ3 (11)

we get 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816 (12)

Based on the above inequalities it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816

max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le max (100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816) (13)

Thus it holds that1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]ge 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)] (14)

Thus 120599(1 3) le 120599(1 2) In the same way it is provedthat 120599(1 3) le 120599(2 3)22 Distance Measures Hamming distance is easily pro-cessed and commonly used in the process of heterogeneousinformation processing For INs 119886 = [119886119897 119886119906] and 119887 =[119887119897 119887119906] (or TFNs 119886 = (1198861 1198862 1198863) and 119887 = (1198871 1198872 1198873) TrFNs119886 = (1198861 1198862 1198863 1198864) and 119887 = (1198871 1198872 1198873 1198874) and TIFNs 119886 =((1198861 1198862 1198863) 119906119886 V119886) and 119887 = ((1198871 1198872 1198873) 119906119887 V119887)) the distancemeasures can be defined as follows [35 44]

119889 (119886 119887)=

12 (10038161003816100381610038161003816119886119897 minus 11988711989710038161003816100381610038161003816 + 1003816100381610038161003816119886119906 minus 1198871199061003816100381610038161003816) if 119886 119887 are INs13 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816) if 119886 119887 are TFNs14 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 + 10038161003816100381610038161198864 minus 11988741003816100381610038161003816) if 119886 119887 are TrFNs16 (10038161003816100381610038161199061198861198861 minus 11990611988711988711003816100381610038161003816 + 10038161003816100381610038161199061198861198862 minus 11990611988711988721003816100381610038161003816 + 10038161003816100381610038161199061198861198863 minus 11990611988711988731003816100381610038161003816) + 1003816100381610038161003816V1198861198861 minus V11988711988711003816100381610038161003816 + 1003816100381610038161003816V1198861198862 minus V11988711988721003816100381610038161003816 + 1003816100381610038161003816V1198861198863 minus V11988711988731003816100381610038161003816 + 10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 ) if 119886 119887 are TIFNs(15)

3 A New Method for HeterogeneousMAGDM Problems

In this section the presentation of heterogeneous MAGDMproblems is given first Then an approach to aggregatingheterogeneous information into ITFNs is developed

31 Heterogeneous MAGDM Problems For the sake of con-venience some symbols are introduced to characterize theheterogeneous MAGDM problems as follows

(1) The group of DMs119863119894 (119894 isin 119872 = 1 2 119898)(2)The set of attributes119860119895 (119895 isin 119873 = 1 2 119899) Denote

the attribute weight vector by119908 = (1199081 1199082 119908119899) where119908119895

represents the weight of119860119895 such that 119908119895 isin [0 1] (119895 isin 119873) andsum119899119895=1119908119895 = 1(3) The set of alternatives 119878119896 (119896 isin 119875 = 1 2 119901)Since there are multiple formats of rating values the

attribute set A = 1198601 1198602 119860119899 is divided into four subsets1198601 = 1198601 1198602 1198601198951 1198602 = 1198601198951+1 1198601198951+2 1198601198952 1198603 =1198601198952+1 1198601198952+2 1198601198953 1198604 = 1198601198953+1 1198601198953+2 1198601198954 and1198605 = 1198601198954+1 1198601198954+2 1198601198955 where 1 le 1198951 le 1198952 le 1198953 le1198954 le 1198955 le 119899 119860 119905 cap 119860119896 = 0 (119905 119896 = 1 2 3 4 5 119905 = 119896)and ⋃5119905=1 119860 119905 = 119860 0 is an empty set The rating values inthe subsets 119860119890 (119890 = 1 2 3 4 5) are in the form of RNs INsTFNs TrFNs and TIFNs respectively Denote the subscript

6 Complexity

TOPSIS

Opposition set

A

Uncertain setSupport set

Group consistency

Geometry entropy

Linear transformation

Mode of support set Mode of opposition set

Min-Max

Mean of uncertain set

Mean method

Quasi-satisfactory triangular Quasi-dissatisfactory triangular

Elicit Rsd Rdd Rud

Calculate mode

Construct Qst Qdt

Induce an ITFN

kj = k1j k2j middot middot middot

kmj

m(kj) mean(kj)

kj = ((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))

kj = (ＧＣＨ kj m (kj) ＧＲ kj) kj = (ＧＣＨ kj m ( kj) ＧＲ kj)

m( kj)

kj = k1j k2j middot middot middot

kmj kj = k

1jk2j middot middot middot

kmj

kj = (xk1j x

k2j middot middot middot x

kmj)

Figure 1 Framework for aggregating ITFN

sets for subsets 119860119890 (119890 = 1 2 3 4 5) by 1198731 = 1 2 11989511198732 = 1198951 + 1 1198951 + 2 1198952 1198733 = 1198952 + 1 1198952 + 2 11989531198734 = 1198953 + 1 1198953 + 2 1198954 and1198735 = 1198954 + 1 1198954 + 2 1198955respectively

(4) The group decision matrixSuppose that the rating of alternative 119878119896 with respect to

the attribute 119860119895 given by DM119863119894 is denoted by 119909119896119894119895 (119894 isin 119872 119895 isin119873 119896 isin 119875) If 119895 isin 1198731 then 119909119896119894119895 is a RN If 119895 isin 1198732 then 119909119896119894119895 =[119909119896119894119895 119909119896119894119895] is an IN If 119895 isin 1198733 then 119909119896119894119895 = (119886119896119894119895 119887119896119894119895 119888119896119894119895) is a TFNIf 119895 isin 1198734 then 119909119896119894119895 = (119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) is a TrFN If 119895 isin 1198735then 119909119896119894119895 = ((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) Namely 119909119896119894119895 can be unified asfollows

119909119896119894119895 =

119909119896119894119895 if 119895 isin 1198731[119909119896119894119895 119909119896119894119895] if 119895 isin 1198732(119886119896119894119895 119887119896119894119895 119888119896119894119895) if 119895 isin 1198733(119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) if 119895 isin 1198734((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) if 119895 isin 1198735(16)

Hence a group decision matrix of alternative 119878119896 can beexpressed as

119883119896 = (119909119896119894119895)119898times119899 =

1198601 1198602 sdot sdot sdot 11986011989911986311198632119863119898((

11990911989611 11990911989612 sdot sdot sdot 119909119896111989911990911989621 11990911989622 sdot sdot sdot 1199091198962119899 1199091198961198981 1199091198961198982 sdot sdot sdot 119909119896119898119899))(119896 isin 119875)

(17)

To reduce information loss and simplify the focusedproblems the group decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) can be integrated into a collective ITFN decisionmatrix The key to addressing this issue lies in an effectiveapproach for constructing ITFNs based on the expertsrsquoassessment expressed in different types of data

32 An Approach to Aggregating Heterogeneous Informationinto ITFNs To facilitate the calculation denote the jthcolumn vector in the matrix 119883119896 as

119860119896119895 = (1199091198961119895 1199091198962119895 119909119896119898119895) (119896 isin 119875 119895 isin 119873) (18)

which is the normalized assessment vector of alternative 119878119896 onattribute 119860119895 given by all DMs 119863119894 (119894 = 1 2 119898) Let 119860max

119895

and 119860min119895 be the largest grade and smallest grade employed

in the rating system For example if the assessments in 119860119895are TFNs then 119860max

119895 = (1 1 1) and 119860min119895 = (0 0 0) if the

assessments in 119860119895 are INs then 119860max119895 = [1 1] and 119860min

119895 =[0 0] To integrate the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective ITFN decision matrix all theelements in vector 119860119896119895 need to be aggregated into an ITFNThe implementation of the aggregation approach involves afour-stage framework (see Figure 1) (1) Elicit Rsd Rdd andRud In this process we use the TOPSIS method to obtainthe rating satisfactory degree (Rsd) and rating dissatisfactorydegree (Rdd) of 119909119896119894119895 and construct the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 The rating uncertain degree (Rud)of 119909119896119894119895 and the corresponding uncertain set 120578119896119895 are derived bygeometry entropy (2) Calculate mode Combining the groupconsistency and mean method the modes of the above sets120585119896119895 120577119896119895 and 120578119896119895 are computed in this stage (3) ConstructQst and Qdt According to the Min-Max method the quasi-satisfactory triangular (Qst) and quasi-dissatisfactory trian-gular (Qdt) of119860119896119895 can be built (4) Induce an ITFNThe ITFN

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Complexity 3

fuzzy aggregation operators Although these studies [38ndash43] focused on different aspects of ITFNs it can onlyaggregate ITFNs Therefore to push ahead with the applica-tion of the above aggregations it is necessary to aggregatemultiple types of decision information into ITFNs whichis very interesting yet relatively sophisticated to disposeof

To do that this paper aims to propose a novel HMAGDMmethod based on ITFNs The primary contributions of thispaper can be illuminated briefly as follows

(1) We first present an aggregation technology to aggre-gate heterogeneous information into TIFNs Compared withexisting methods [28ndash35 37] the proposed aggregate tech-nology has the following advantages For more details referto Section 52

(i) A new elicitation of the support opposite and uncer-tain information based on distance is introducedwhich can accommodate more complicated attributevalues including TIFNs and TrIFNs since it just needsto calculate the distance from decision data to themaximum and minimum grade

(ii) A new construction approach of ITFN is presented bygroup consistency which not only takes into accountexpertrsquos weight but can overcome the shortcoming ofthe hypothesis of the normal distribution

(iii) It can not only effectively avoid the loss of originalinformation but also reflect the distribution charac-teristics of the original decision data

(2) A new similarity measure of ITFNs is developedand applied to construct a multiple objective linear pro-gramming to determine attribute weights in ITFN envi-ronment with incomplete information The determinationmethod of attribute weights can effectively avoid the sub-jectivity brought by the given attribute weights in ad-vance

(3) Based on the aforesaid provision a new method todeal with HMAGDM problems with RNs INs TFNs TrFNsand TIFNs is proposed The comprehensive evaluation valueof the alternative is an ITFN which preserves more usefulinformation

The remainder of this paper is set out as followsSection 2 briefly introduces related basic concepts Section 3presents an approach to aggregating heterogeneous deci-sion data into ITFNs Section 4 builds a multiple objectivelinear programming model to determine attribute weightsand propose a HMAGDM method Section 5 provides anumerical example to illustrate the feasibility and reasonable-ness of the proposed method Section 6 makes our conclu-sions

2 Preliminary

In this section some basic concepts of ITFN and distancemeasures are briefly described below

21 Intuitionistic Triangular Fuzzy Number

Definition 1 (see [38]) A triangular fuzzy number (TFN) Ais a special fuzzy set on a real number set R its membershipfunction is defined by

119865119860 (119909) = 119909 minus 119905119897119905119898 minus 119905119897 if 119905119897 le 119909 le 119905119898119905ℎ minus 119909119905ℎ minus 119905119898 if 119905119898 le 119909 le 119905ℎ0 if otherwise (1)

where 0 le 119905119897 le 119905119898 le 119905ℎ le 1 119905119897 and 119905ℎ present the lower limitand upper limit of A respectively and 119905119898 is the mode whichcan be denoted as a triplet (119905119897 119905119898 119905ℎ)Definition 2 (see [38]) Let X be a fixed set 120583119860(119909) =(119905119897119860(119909) 119905119898119860(119909) 119905ℎ119860(119909)) and V119860(119909) = (119891119897119860(119909) 119891119898119860 (119909) 119891ℎ119860(119909)) areTFNs defined on the unit interval [0 1] then an intuitionistictriangular fuzzy set 119860 over X is defined as 119860 = (119909 lt120583119860(119909) V119860(119909) gt) | 119909 isin 119883 where the parameters 120583119860(119909)and V119860(119909) indicate respectively the membership degree andnonmembership degree of the element x in 119860 with theconditions 0 le 119905ℎ119860(119909) + 119891ℎ119860(119909) le 1

For convenience we call = ((119905119897119860 119905119898119860 119905ℎ119860) (119891119897119860 119891119898119860 119891ℎ119860))an intuitionistic triangular fuzzy number (ITFN) where

119905119897119860 119905119898119860 119905ℎ119860 isin [0 1] 119891119897119860 119891119898119860 119891ℎ119860 isin [0 1] 119905ℎ119860 + 119891ℎ119860 isin [0 1] (2)

It is clear that the largest and smallest ITFN are 120572+ =((1 1 1) (0 0 0)) and 120572minus = ((0 0 0) (1 1 1)) respectivelyDefinition 3 (see [38]) Let 1 = ((1199051198971 1199051198981 119905ℎ1) (1198911198971 1198911198981 119891ℎ1 ))and 2 = ((1199051198972 1199051198982 119905ℎ2) (1198911198972 1198911198982 119891ℎ2 )) be two ITFNs then thecontainment is1 sube 2iff 1199051198971 le 1199051198972 1199051198981 le 1199051198982 119905ℎ1 le 119905ℎ2 1198911198971 ge 1198911198972 1198911198981 ge 1198911198982 and 119891ℎ1 ge 119891ℎ2 (3)

Some arithmetic operations between ITFNs 1 and 2 areshown as below [40]

(1) 1 + 2 = ((1199051198971 + 1199051198972 minus 11990511989711199051198972 1199051198981 + 1199051198982 minus 1199051198981 1199051198982 119905ℎ1 + 119905ℎ2 minus 119905ℎ1119905ℎ2)(11989111989711198911198972 1198911198981 1198911198982 119891ℎ1 119891ℎ2 ))(2) 120582 = ((1 minus (1 minus 1199051198971)120582 1 minus (1 minus 1199051198981 )120582 1 minus (1 minus 119905ℎ1)120582)((1198911198971)120582 (1198911198981 )120582 (119891ℎ1 )120582)) 120582 gt 0

4 Complexity

Definition 4 (see [41]) For a set of ITFNs 119894 (119894 = 1 2 119899)that have associated an importance weight vector 119908 =(1199081 1199082 119908119899)119879 with 119908119894 isin [0 1] and sum119899119894=1119908119894 = 1 We call

119868119879119865119882119860(1198861 1198862 119886119899) = 119899sum119894=1

119908119894119886119894 = ((1minus 119899prod119894=1

(1 minus 119905119897119894)119908119894 1 minus 119899prod119894=1

(1 minus 119905119898119894 )119908119894 1minus 119899prod119894=1

(1 minus 119905ℎ119894 )119908119894) ( 119899prod119894=1

(119891119897119894 )119908119894 119899prod119894=1

(119891119898119894 )119908119894 119899prod119894=1

(119891ℎ119894 )119908119894))(4)

an intuitionistic triangular fuzzy weighted average operator(ITFWA)

Definition 5 Let 1 = ((1199051198971 1199051198981 119905ℎ1) (1198911198971 1198911198981 119891ℎ1 )) and 2 =((1199051198972 1199051198982 119905ℎ2) (1198911198972 1198911198982 119891ℎ2 )) be two ITFNs A similarity measure120599(1 2) between the ITFNs 1 and 2 is defined as follows120599 (1 2) = 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12sdotmax (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)](5)

Theorem 6 e similarity measure 120599(1 2) satisfies thefollowing properties

(i) 0 le 120599(1 2) le 1(ii) 120599(1 2) = 1 if and only if 1 = 2(iii) 120599(1 2) = 120599(2 1)(iv) If 3 is a ITFN and 1 sube 2 sube 3 then 120599(1 3) le120599(1 2) and 120599(1 3) le 120599(2 3)

Proof It is easy to see that the proposed similarity measure120599(1 2) meets the third property of Theorem 6 We onlyneed to prove (i) (ii) and (iv)

For (i) By (2) we have0 le 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 10 le 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10 le 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 1

0 le 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 1(6)

It is easy to see that0 le 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 120 le 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 12(7)

Thus we get0 le 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]le 1(8)

And then the inequality 0 le 120599(1 2) le 1 is establishedFor (ii) When 120599(1 2) = 1 if and only if112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 012 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 0

(9)

Apparently it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 = 010038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 = 010038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 = 0100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 = 010038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 = 010038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 = 0(10)

Thus we get 1199051198971 = 1199051198972 1199051198981 = 1199051198982 119905ℎ1 = 119905ℎ2 1198911198971 = 1198911198972 1198911198981 = 1198911198982 119891ℎ1 minus 119891ℎ2 And then 1 = 2

Complexity 5

For (iv) Since 1199051198971 le 1199051198972 le 11990511989731199051198981 le 1199051198982 le 1199051198983 119905ℎ1 le 119905ℎ2 le 119905ℎ3 1198911198971 ge 1198911198972 ge 11989111989731198911198981 ge 1198911198982 ge 1198911198983 119891ℎ1 ge 119891ℎ2 ge 119891ℎ3 (11)

we get 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816 (12)

Based on the above inequalities it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816

max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le max (100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816) (13)

Thus it holds that1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]ge 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)] (14)

Thus 120599(1 3) le 120599(1 2) In the same way it is provedthat 120599(1 3) le 120599(2 3)22 Distance Measures Hamming distance is easily pro-cessed and commonly used in the process of heterogeneousinformation processing For INs 119886 = [119886119897 119886119906] and 119887 =[119887119897 119887119906] (or TFNs 119886 = (1198861 1198862 1198863) and 119887 = (1198871 1198872 1198873) TrFNs119886 = (1198861 1198862 1198863 1198864) and 119887 = (1198871 1198872 1198873 1198874) and TIFNs 119886 =((1198861 1198862 1198863) 119906119886 V119886) and 119887 = ((1198871 1198872 1198873) 119906119887 V119887)) the distancemeasures can be defined as follows [35 44]

119889 (119886 119887)=

12 (10038161003816100381610038161003816119886119897 minus 11988711989710038161003816100381610038161003816 + 1003816100381610038161003816119886119906 minus 1198871199061003816100381610038161003816) if 119886 119887 are INs13 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816) if 119886 119887 are TFNs14 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 + 10038161003816100381610038161198864 minus 11988741003816100381610038161003816) if 119886 119887 are TrFNs16 (10038161003816100381610038161199061198861198861 minus 11990611988711988711003816100381610038161003816 + 10038161003816100381610038161199061198861198862 minus 11990611988711988721003816100381610038161003816 + 10038161003816100381610038161199061198861198863 minus 11990611988711988731003816100381610038161003816) + 1003816100381610038161003816V1198861198861 minus V11988711988711003816100381610038161003816 + 1003816100381610038161003816V1198861198862 minus V11988711988721003816100381610038161003816 + 1003816100381610038161003816V1198861198863 minus V11988711988731003816100381610038161003816 + 10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 ) if 119886 119887 are TIFNs(15)

3 A New Method for HeterogeneousMAGDM Problems

In this section the presentation of heterogeneous MAGDMproblems is given first Then an approach to aggregatingheterogeneous information into ITFNs is developed

31 Heterogeneous MAGDM Problems For the sake of con-venience some symbols are introduced to characterize theheterogeneous MAGDM problems as follows

(1) The group of DMs119863119894 (119894 isin 119872 = 1 2 119898)(2)The set of attributes119860119895 (119895 isin 119873 = 1 2 119899) Denote

the attribute weight vector by119908 = (1199081 1199082 119908119899) where119908119895

represents the weight of119860119895 such that 119908119895 isin [0 1] (119895 isin 119873) andsum119899119895=1119908119895 = 1(3) The set of alternatives 119878119896 (119896 isin 119875 = 1 2 119901)Since there are multiple formats of rating values the

attribute set A = 1198601 1198602 119860119899 is divided into four subsets1198601 = 1198601 1198602 1198601198951 1198602 = 1198601198951+1 1198601198951+2 1198601198952 1198603 =1198601198952+1 1198601198952+2 1198601198953 1198604 = 1198601198953+1 1198601198953+2 1198601198954 and1198605 = 1198601198954+1 1198601198954+2 1198601198955 where 1 le 1198951 le 1198952 le 1198953 le1198954 le 1198955 le 119899 119860 119905 cap 119860119896 = 0 (119905 119896 = 1 2 3 4 5 119905 = 119896)and ⋃5119905=1 119860 119905 = 119860 0 is an empty set The rating values inthe subsets 119860119890 (119890 = 1 2 3 4 5) are in the form of RNs INsTFNs TrFNs and TIFNs respectively Denote the subscript

6 Complexity

TOPSIS

Opposition set

A

Uncertain setSupport set

Group consistency

Geometry entropy

Linear transformation

Mode of support set Mode of opposition set

Min-Max

Mean of uncertain set

Mean method

Quasi-satisfactory triangular Quasi-dissatisfactory triangular

Elicit Rsd Rdd Rud

Calculate mode

Construct Qst Qdt

Induce an ITFN

kj = k1j k2j middot middot middot

kmj

m(kj) mean(kj)

kj = ((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))

kj = (ＧＣＨ kj m (kj) ＧＲ kj) kj = (ＧＣＨ kj m ( kj) ＧＲ kj)

m( kj)

kj = k1j k2j middot middot middot

kmj kj = k

1jk2j middot middot middot

kmj

kj = (xk1j x

k2j middot middot middot x

kmj)

Figure 1 Framework for aggregating ITFN

sets for subsets 119860119890 (119890 = 1 2 3 4 5) by 1198731 = 1 2 11989511198732 = 1198951 + 1 1198951 + 2 1198952 1198733 = 1198952 + 1 1198952 + 2 11989531198734 = 1198953 + 1 1198953 + 2 1198954 and1198735 = 1198954 + 1 1198954 + 2 1198955respectively

(4) The group decision matrixSuppose that the rating of alternative 119878119896 with respect to

the attribute 119860119895 given by DM119863119894 is denoted by 119909119896119894119895 (119894 isin 119872 119895 isin119873 119896 isin 119875) If 119895 isin 1198731 then 119909119896119894119895 is a RN If 119895 isin 1198732 then 119909119896119894119895 =[119909119896119894119895 119909119896119894119895] is an IN If 119895 isin 1198733 then 119909119896119894119895 = (119886119896119894119895 119887119896119894119895 119888119896119894119895) is a TFNIf 119895 isin 1198734 then 119909119896119894119895 = (119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) is a TrFN If 119895 isin 1198735then 119909119896119894119895 = ((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) Namely 119909119896119894119895 can be unified asfollows

119909119896119894119895 =

119909119896119894119895 if 119895 isin 1198731[119909119896119894119895 119909119896119894119895] if 119895 isin 1198732(119886119896119894119895 119887119896119894119895 119888119896119894119895) if 119895 isin 1198733(119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) if 119895 isin 1198734((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) if 119895 isin 1198735(16)

Hence a group decision matrix of alternative 119878119896 can beexpressed as

119883119896 = (119909119896119894119895)119898times119899 =

1198601 1198602 sdot sdot sdot 11986011989911986311198632119863119898((

11990911989611 11990911989612 sdot sdot sdot 119909119896111989911990911989621 11990911989622 sdot sdot sdot 1199091198962119899 1199091198961198981 1199091198961198982 sdot sdot sdot 119909119896119898119899))(119896 isin 119875)

(17)

To reduce information loss and simplify the focusedproblems the group decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) can be integrated into a collective ITFN decisionmatrix The key to addressing this issue lies in an effectiveapproach for constructing ITFNs based on the expertsrsquoassessment expressed in different types of data

32 An Approach to Aggregating Heterogeneous Informationinto ITFNs To facilitate the calculation denote the jthcolumn vector in the matrix 119883119896 as

119860119896119895 = (1199091198961119895 1199091198962119895 119909119896119898119895) (119896 isin 119875 119895 isin 119873) (18)

which is the normalized assessment vector of alternative 119878119896 onattribute 119860119895 given by all DMs 119863119894 (119894 = 1 2 119898) Let 119860max

119895

and 119860min119895 be the largest grade and smallest grade employed

in the rating system For example if the assessments in 119860119895are TFNs then 119860max

119895 = (1 1 1) and 119860min119895 = (0 0 0) if the

assessments in 119860119895 are INs then 119860max119895 = [1 1] and 119860min

119895 =[0 0] To integrate the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective ITFN decision matrix all theelements in vector 119860119896119895 need to be aggregated into an ITFNThe implementation of the aggregation approach involves afour-stage framework (see Figure 1) (1) Elicit Rsd Rdd andRud In this process we use the TOPSIS method to obtainthe rating satisfactory degree (Rsd) and rating dissatisfactorydegree (Rdd) of 119909119896119894119895 and construct the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 The rating uncertain degree (Rud)of 119909119896119894119895 and the corresponding uncertain set 120578119896119895 are derived bygeometry entropy (2) Calculate mode Combining the groupconsistency and mean method the modes of the above sets120585119896119895 120577119896119895 and 120578119896119895 are computed in this stage (3) ConstructQst and Qdt According to the Min-Max method the quasi-satisfactory triangular (Qst) and quasi-dissatisfactory trian-gular (Qdt) of119860119896119895 can be built (4) Induce an ITFNThe ITFN

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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4 Complexity

Definition 4 (see [41]) For a set of ITFNs 119894 (119894 = 1 2 119899)that have associated an importance weight vector 119908 =(1199081 1199082 119908119899)119879 with 119908119894 isin [0 1] and sum119899119894=1119908119894 = 1 We call

119868119879119865119882119860(1198861 1198862 119886119899) = 119899sum119894=1

119908119894119886119894 = ((1minus 119899prod119894=1

(1 minus 119905119897119894)119908119894 1 minus 119899prod119894=1

(1 minus 119905119898119894 )119908119894 1minus 119899prod119894=1

(1 minus 119905ℎ119894 )119908119894) ( 119899prod119894=1

(119891119897119894 )119908119894 119899prod119894=1

(119891119898119894 )119908119894 119899prod119894=1

(119891ℎ119894 )119908119894))(4)

an intuitionistic triangular fuzzy weighted average operator(ITFWA)

Definition 5 Let 1 = ((1199051198971 1199051198981 119905ℎ1) (1198911198971 1198911198981 119891ℎ1 )) and 2 =((1199051198972 1199051198982 119905ℎ2) (1198911198972 1198911198982 119891ℎ2 )) be two ITFNs A similarity measure120599(1 2) between the ITFNs 1 and 2 is defined as follows120599 (1 2) = 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12sdotmax (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)](5)

Theorem 6 e similarity measure 120599(1 2) satisfies thefollowing properties

(i) 0 le 120599(1 2) le 1(ii) 120599(1 2) = 1 if and only if 1 = 2(iii) 120599(1 2) = 120599(2 1)(iv) If 3 is a ITFN and 1 sube 2 sube 3 then 120599(1 3) le120599(1 2) and 120599(1 3) le 120599(2 3)

Proof It is easy to see that the proposed similarity measure120599(1 2) meets the third property of Theorem 6 We onlyneed to prove (i) (ii) and (iv)

For (i) By (2) we have0 le 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 10 le 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10 le 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 1

0 le 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 1(6)

It is easy to see that0 le 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 120 le 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le 12(7)

Thus we get0 le 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]le 1(8)

And then the inequality 0 le 120599(1 2) le 1 is establishedFor (ii) When 120599(1 2) = 1 if and only if112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 012 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) = 0

(9)

Apparently it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 = 010038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 = 010038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 = 0100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 = 010038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 = 010038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 = 0(10)

Thus we get 1199051198971 = 1199051198972 1199051198981 = 1199051198982 119905ℎ1 = 119905ℎ2 1198911198971 = 1198911198972 1198911198981 = 1198911198982 119891ℎ1 minus 119891ℎ2 And then 1 = 2

Complexity 5

For (iv) Since 1199051198971 le 1199051198972 le 11990511989731199051198981 le 1199051198982 le 1199051198983 119905ℎ1 le 119905ℎ2 le 119905ℎ3 1198911198971 ge 1198911198972 ge 11989111989731198911198981 ge 1198911198982 ge 1198911198983 119891ℎ1 ge 119891ℎ2 ge 119891ℎ3 (11)

we get 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816 (12)

Based on the above inequalities it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816

max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le max (100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816) (13)

Thus it holds that1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]ge 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)] (14)

Thus 120599(1 3) le 120599(1 2) In the same way it is provedthat 120599(1 3) le 120599(2 3)22 Distance Measures Hamming distance is easily pro-cessed and commonly used in the process of heterogeneousinformation processing For INs 119886 = [119886119897 119886119906] and 119887 =[119887119897 119887119906] (or TFNs 119886 = (1198861 1198862 1198863) and 119887 = (1198871 1198872 1198873) TrFNs119886 = (1198861 1198862 1198863 1198864) and 119887 = (1198871 1198872 1198873 1198874) and TIFNs 119886 =((1198861 1198862 1198863) 119906119886 V119886) and 119887 = ((1198871 1198872 1198873) 119906119887 V119887)) the distancemeasures can be defined as follows [35 44]

119889 (119886 119887)=

12 (10038161003816100381610038161003816119886119897 minus 11988711989710038161003816100381610038161003816 + 1003816100381610038161003816119886119906 minus 1198871199061003816100381610038161003816) if 119886 119887 are INs13 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816) if 119886 119887 are TFNs14 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 + 10038161003816100381610038161198864 minus 11988741003816100381610038161003816) if 119886 119887 are TrFNs16 (10038161003816100381610038161199061198861198861 minus 11990611988711988711003816100381610038161003816 + 10038161003816100381610038161199061198861198862 minus 11990611988711988721003816100381610038161003816 + 10038161003816100381610038161199061198861198863 minus 11990611988711988731003816100381610038161003816) + 1003816100381610038161003816V1198861198861 minus V11988711988711003816100381610038161003816 + 1003816100381610038161003816V1198861198862 minus V11988711988721003816100381610038161003816 + 1003816100381610038161003816V1198861198863 minus V11988711988731003816100381610038161003816 + 10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 ) if 119886 119887 are TIFNs(15)

3 A New Method for HeterogeneousMAGDM Problems

In this section the presentation of heterogeneous MAGDMproblems is given first Then an approach to aggregatingheterogeneous information into ITFNs is developed

31 Heterogeneous MAGDM Problems For the sake of con-venience some symbols are introduced to characterize theheterogeneous MAGDM problems as follows

(1) The group of DMs119863119894 (119894 isin 119872 = 1 2 119898)(2)The set of attributes119860119895 (119895 isin 119873 = 1 2 119899) Denote

the attribute weight vector by119908 = (1199081 1199082 119908119899) where119908119895

represents the weight of119860119895 such that 119908119895 isin [0 1] (119895 isin 119873) andsum119899119895=1119908119895 = 1(3) The set of alternatives 119878119896 (119896 isin 119875 = 1 2 119901)Since there are multiple formats of rating values the

attribute set A = 1198601 1198602 119860119899 is divided into four subsets1198601 = 1198601 1198602 1198601198951 1198602 = 1198601198951+1 1198601198951+2 1198601198952 1198603 =1198601198952+1 1198601198952+2 1198601198953 1198604 = 1198601198953+1 1198601198953+2 1198601198954 and1198605 = 1198601198954+1 1198601198954+2 1198601198955 where 1 le 1198951 le 1198952 le 1198953 le1198954 le 1198955 le 119899 119860 119905 cap 119860119896 = 0 (119905 119896 = 1 2 3 4 5 119905 = 119896)and ⋃5119905=1 119860 119905 = 119860 0 is an empty set The rating values inthe subsets 119860119890 (119890 = 1 2 3 4 5) are in the form of RNs INsTFNs TrFNs and TIFNs respectively Denote the subscript

6 Complexity

TOPSIS

Opposition set

A

Uncertain setSupport set

Group consistency

Geometry entropy

Linear transformation

Mode of support set Mode of opposition set

Min-Max

Mean of uncertain set

Mean method

Quasi-satisfactory triangular Quasi-dissatisfactory triangular

Elicit Rsd Rdd Rud

Calculate mode

Construct Qst Qdt

Induce an ITFN

kj = k1j k2j middot middot middot

kmj

m(kj) mean(kj)

kj = ((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))

kj = (ＧＣＨ kj m (kj) ＧＲ kj) kj = (ＧＣＨ kj m ( kj) ＧＲ kj)

m( kj)

kj = k1j k2j middot middot middot

kmj kj = k

1jk2j middot middot middot

kmj

kj = (xk1j x

k2j middot middot middot x

kmj)

Figure 1 Framework for aggregating ITFN

sets for subsets 119860119890 (119890 = 1 2 3 4 5) by 1198731 = 1 2 11989511198732 = 1198951 + 1 1198951 + 2 1198952 1198733 = 1198952 + 1 1198952 + 2 11989531198734 = 1198953 + 1 1198953 + 2 1198954 and1198735 = 1198954 + 1 1198954 + 2 1198955respectively

(4) The group decision matrixSuppose that the rating of alternative 119878119896 with respect to

the attribute 119860119895 given by DM119863119894 is denoted by 119909119896119894119895 (119894 isin 119872 119895 isin119873 119896 isin 119875) If 119895 isin 1198731 then 119909119896119894119895 is a RN If 119895 isin 1198732 then 119909119896119894119895 =[119909119896119894119895 119909119896119894119895] is an IN If 119895 isin 1198733 then 119909119896119894119895 = (119886119896119894119895 119887119896119894119895 119888119896119894119895) is a TFNIf 119895 isin 1198734 then 119909119896119894119895 = (119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) is a TrFN If 119895 isin 1198735then 119909119896119894119895 = ((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) Namely 119909119896119894119895 can be unified asfollows

119909119896119894119895 =

119909119896119894119895 if 119895 isin 1198731[119909119896119894119895 119909119896119894119895] if 119895 isin 1198732(119886119896119894119895 119887119896119894119895 119888119896119894119895) if 119895 isin 1198733(119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) if 119895 isin 1198734((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) if 119895 isin 1198735(16)

Hence a group decision matrix of alternative 119878119896 can beexpressed as

119883119896 = (119909119896119894119895)119898times119899 =

1198601 1198602 sdot sdot sdot 11986011989911986311198632119863119898((

11990911989611 11990911989612 sdot sdot sdot 119909119896111989911990911989621 11990911989622 sdot sdot sdot 1199091198962119899 1199091198961198981 1199091198961198982 sdot sdot sdot 119909119896119898119899))(119896 isin 119875)

(17)

To reduce information loss and simplify the focusedproblems the group decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) can be integrated into a collective ITFN decisionmatrix The key to addressing this issue lies in an effectiveapproach for constructing ITFNs based on the expertsrsquoassessment expressed in different types of data

32 An Approach to Aggregating Heterogeneous Informationinto ITFNs To facilitate the calculation denote the jthcolumn vector in the matrix 119883119896 as

119860119896119895 = (1199091198961119895 1199091198962119895 119909119896119898119895) (119896 isin 119875 119895 isin 119873) (18)

which is the normalized assessment vector of alternative 119878119896 onattribute 119860119895 given by all DMs 119863119894 (119894 = 1 2 119898) Let 119860max

119895

and 119860min119895 be the largest grade and smallest grade employed

in the rating system For example if the assessments in 119860119895are TFNs then 119860max

119895 = (1 1 1) and 119860min119895 = (0 0 0) if the

assessments in 119860119895 are INs then 119860max119895 = [1 1] and 119860min

119895 =[0 0] To integrate the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective ITFN decision matrix all theelements in vector 119860119896119895 need to be aggregated into an ITFNThe implementation of the aggregation approach involves afour-stage framework (see Figure 1) (1) Elicit Rsd Rdd andRud In this process we use the TOPSIS method to obtainthe rating satisfactory degree (Rsd) and rating dissatisfactorydegree (Rdd) of 119909119896119894119895 and construct the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 The rating uncertain degree (Rud)of 119909119896119894119895 and the corresponding uncertain set 120578119896119895 are derived bygeometry entropy (2) Calculate mode Combining the groupconsistency and mean method the modes of the above sets120585119896119895 120577119896119895 and 120578119896119895 are computed in this stage (3) ConstructQst and Qdt According to the Min-Max method the quasi-satisfactory triangular (Qst) and quasi-dissatisfactory trian-gular (Qdt) of119860119896119895 can be built (4) Induce an ITFNThe ITFN

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Complexity 5

For (iv) Since 1199051198971 le 1199051198972 le 11990511989731199051198981 le 1199051198982 le 1199051198983 119905ℎ1 le 119905ℎ2 le 119905ℎ3 1198911198971 ge 1198911198972 ge 11989111989731198911198981 ge 1198911198982 ge 1198911198983 119891ℎ1 ge 119891ℎ2 ge 119891ℎ3 (11)

we get 100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 le 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 le 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 le 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816 (12)

Based on the above inequalities it is easy to derive100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816 le 100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816+ 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816

max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) le max (100381610038161003816100381610038161199051198971 minus 119905119897310038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198983 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ3 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897310038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198983 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ3 10038161003816100381610038161003816) (13)

Thus it holds that1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816+ 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)]ge 1 minus [ 112 (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 + 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 + 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 + 100381610038161003816100381610038161198911198971minus 119891119897210038161003816100381610038161003816 + 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 + 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816) + 12 max (100381610038161003816100381610038161199051198971 minus 119905119897210038161003816100381610038161003816 10038161003816100381610038161199051198981 minus 1199051198982 1003816100381610038161003816 10038161003816100381610038161003816119905ℎ1 minus 119905ℎ2 10038161003816100381610038161003816 100381610038161003816100381610038161198911198971 minus 119891119897210038161003816100381610038161003816 10038161003816100381610038161198911198981 minus 1198911198982 1003816100381610038161003816 10038161003816100381610038161003816119891ℎ1 minus 119891ℎ2 10038161003816100381610038161003816)] (14)

Thus 120599(1 3) le 120599(1 2) In the same way it is provedthat 120599(1 3) le 120599(2 3)22 Distance Measures Hamming distance is easily pro-cessed and commonly used in the process of heterogeneousinformation processing For INs 119886 = [119886119897 119886119906] and 119887 =[119887119897 119887119906] (or TFNs 119886 = (1198861 1198862 1198863) and 119887 = (1198871 1198872 1198873) TrFNs119886 = (1198861 1198862 1198863 1198864) and 119887 = (1198871 1198872 1198873 1198874) and TIFNs 119886 =((1198861 1198862 1198863) 119906119886 V119886) and 119887 = ((1198871 1198872 1198873) 119906119887 V119887)) the distancemeasures can be defined as follows [35 44]

119889 (119886 119887)=

12 (10038161003816100381610038161003816119886119897 minus 11988711989710038161003816100381610038161003816 + 1003816100381610038161003816119886119906 minus 1198871199061003816100381610038161003816) if 119886 119887 are INs13 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816) if 119886 119887 are TFNs14 (10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 + 10038161003816100381610038161198864 minus 11988741003816100381610038161003816) if 119886 119887 are TrFNs16 (10038161003816100381610038161199061198861198861 minus 11990611988711988711003816100381610038161003816 + 10038161003816100381610038161199061198861198862 minus 11990611988711988721003816100381610038161003816 + 10038161003816100381610038161199061198861198863 minus 11990611988711988731003816100381610038161003816) + 1003816100381610038161003816V1198861198861 minus V11988711988711003816100381610038161003816 + 1003816100381610038161003816V1198861198862 minus V11988711988721003816100381610038161003816 + 1003816100381610038161003816V1198861198863 minus V11988711988731003816100381610038161003816 + 10038161003816100381610038161198861 minus 11988711003816100381610038161003816 + 10038161003816100381610038161198862 minus 11988721003816100381610038161003816 + 10038161003816100381610038161198863 minus 11988731003816100381610038161003816 ) if 119886 119887 are TIFNs(15)

3 A New Method for HeterogeneousMAGDM Problems

In this section the presentation of heterogeneous MAGDMproblems is given first Then an approach to aggregatingheterogeneous information into ITFNs is developed

31 Heterogeneous MAGDM Problems For the sake of con-venience some symbols are introduced to characterize theheterogeneous MAGDM problems as follows

(1) The group of DMs119863119894 (119894 isin 119872 = 1 2 119898)(2)The set of attributes119860119895 (119895 isin 119873 = 1 2 119899) Denote

the attribute weight vector by119908 = (1199081 1199082 119908119899) where119908119895

represents the weight of119860119895 such that 119908119895 isin [0 1] (119895 isin 119873) andsum119899119895=1119908119895 = 1(3) The set of alternatives 119878119896 (119896 isin 119875 = 1 2 119901)Since there are multiple formats of rating values the

attribute set A = 1198601 1198602 119860119899 is divided into four subsets1198601 = 1198601 1198602 1198601198951 1198602 = 1198601198951+1 1198601198951+2 1198601198952 1198603 =1198601198952+1 1198601198952+2 1198601198953 1198604 = 1198601198953+1 1198601198953+2 1198601198954 and1198605 = 1198601198954+1 1198601198954+2 1198601198955 where 1 le 1198951 le 1198952 le 1198953 le1198954 le 1198955 le 119899 119860 119905 cap 119860119896 = 0 (119905 119896 = 1 2 3 4 5 119905 = 119896)and ⋃5119905=1 119860 119905 = 119860 0 is an empty set The rating values inthe subsets 119860119890 (119890 = 1 2 3 4 5) are in the form of RNs INsTFNs TrFNs and TIFNs respectively Denote the subscript

6 Complexity

TOPSIS

Opposition set

A

Uncertain setSupport set

Group consistency

Geometry entropy

Linear transformation

Mode of support set Mode of opposition set

Min-Max

Mean of uncertain set

Mean method

Quasi-satisfactory triangular Quasi-dissatisfactory triangular

Elicit Rsd Rdd Rud

Calculate mode

Construct Qst Qdt

Induce an ITFN

kj = k1j k2j middot middot middot

kmj

m(kj) mean(kj)

kj = ((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))

kj = (ＧＣＨ kj m (kj) ＧＲ kj) kj = (ＧＣＨ kj m ( kj) ＧＲ kj)

m( kj)

kj = k1j k2j middot middot middot

kmj kj = k

1jk2j middot middot middot

kmj

kj = (xk1j x

k2j middot middot middot x

kmj)

Figure 1 Framework for aggregating ITFN

sets for subsets 119860119890 (119890 = 1 2 3 4 5) by 1198731 = 1 2 11989511198732 = 1198951 + 1 1198951 + 2 1198952 1198733 = 1198952 + 1 1198952 + 2 11989531198734 = 1198953 + 1 1198953 + 2 1198954 and1198735 = 1198954 + 1 1198954 + 2 1198955respectively

(4) The group decision matrixSuppose that the rating of alternative 119878119896 with respect to

the attribute 119860119895 given by DM119863119894 is denoted by 119909119896119894119895 (119894 isin 119872 119895 isin119873 119896 isin 119875) If 119895 isin 1198731 then 119909119896119894119895 is a RN If 119895 isin 1198732 then 119909119896119894119895 =[119909119896119894119895 119909119896119894119895] is an IN If 119895 isin 1198733 then 119909119896119894119895 = (119886119896119894119895 119887119896119894119895 119888119896119894119895) is a TFNIf 119895 isin 1198734 then 119909119896119894119895 = (119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) is a TrFN If 119895 isin 1198735then 119909119896119894119895 = ((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) Namely 119909119896119894119895 can be unified asfollows

119909119896119894119895 =

119909119896119894119895 if 119895 isin 1198731[119909119896119894119895 119909119896119894119895] if 119895 isin 1198732(119886119896119894119895 119887119896119894119895 119888119896119894119895) if 119895 isin 1198733(119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) if 119895 isin 1198734((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) if 119895 isin 1198735(16)

Hence a group decision matrix of alternative 119878119896 can beexpressed as

119883119896 = (119909119896119894119895)119898times119899 =

1198601 1198602 sdot sdot sdot 11986011989911986311198632119863119898((

11990911989611 11990911989612 sdot sdot sdot 119909119896111989911990911989621 11990911989622 sdot sdot sdot 1199091198962119899 1199091198961198981 1199091198961198982 sdot sdot sdot 119909119896119898119899))(119896 isin 119875)

(17)

To reduce information loss and simplify the focusedproblems the group decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) can be integrated into a collective ITFN decisionmatrix The key to addressing this issue lies in an effectiveapproach for constructing ITFNs based on the expertsrsquoassessment expressed in different types of data

32 An Approach to Aggregating Heterogeneous Informationinto ITFNs To facilitate the calculation denote the jthcolumn vector in the matrix 119883119896 as

119860119896119895 = (1199091198961119895 1199091198962119895 119909119896119898119895) (119896 isin 119875 119895 isin 119873) (18)

which is the normalized assessment vector of alternative 119878119896 onattribute 119860119895 given by all DMs 119863119894 (119894 = 1 2 119898) Let 119860max

119895

and 119860min119895 be the largest grade and smallest grade employed

in the rating system For example if the assessments in 119860119895are TFNs then 119860max

119895 = (1 1 1) and 119860min119895 = (0 0 0) if the

assessments in 119860119895 are INs then 119860max119895 = [1 1] and 119860min

119895 =[0 0] To integrate the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective ITFN decision matrix all theelements in vector 119860119896119895 need to be aggregated into an ITFNThe implementation of the aggregation approach involves afour-stage framework (see Figure 1) (1) Elicit Rsd Rdd andRud In this process we use the TOPSIS method to obtainthe rating satisfactory degree (Rsd) and rating dissatisfactorydegree (Rdd) of 119909119896119894119895 and construct the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 The rating uncertain degree (Rud)of 119909119896119894119895 and the corresponding uncertain set 120578119896119895 are derived bygeometry entropy (2) Calculate mode Combining the groupconsistency and mean method the modes of the above sets120585119896119895 120577119896119895 and 120578119896119895 are computed in this stage (3) ConstructQst and Qdt According to the Min-Max method the quasi-satisfactory triangular (Qst) and quasi-dissatisfactory trian-gular (Qdt) of119860119896119895 can be built (4) Induce an ITFNThe ITFN

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

6 Complexity

TOPSIS

Opposition set

A

Uncertain setSupport set

Group consistency

Geometry entropy

Linear transformation

Mode of support set Mode of opposition set

Min-Max

Mean of uncertain set

Mean method

Quasi-satisfactory triangular Quasi-dissatisfactory triangular

Elicit Rsd Rdd Rud

Calculate mode

Construct Qst Qdt

Induce an ITFN

kj = k1j k2j middot middot middot

kmj

m(kj) mean(kj)

kj = ((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))

kj = (ＧＣＨ kj m (kj) ＧＲ kj) kj = (ＧＣＨ kj m ( kj) ＧＲ kj)

m( kj)

kj = k1j k2j middot middot middot

kmj kj = k

1jk2j middot middot middot

kmj

kj = (xk1j x

k2j middot middot middot x

kmj)

Figure 1 Framework for aggregating ITFN

sets for subsets 119860119890 (119890 = 1 2 3 4 5) by 1198731 = 1 2 11989511198732 = 1198951 + 1 1198951 + 2 1198952 1198733 = 1198952 + 1 1198952 + 2 11989531198734 = 1198953 + 1 1198953 + 2 1198954 and1198735 = 1198954 + 1 1198954 + 2 1198955respectively

(4) The group decision matrixSuppose that the rating of alternative 119878119896 with respect to

the attribute 119860119895 given by DM119863119894 is denoted by 119909119896119894119895 (119894 isin 119872 119895 isin119873 119896 isin 119875) If 119895 isin 1198731 then 119909119896119894119895 is a RN If 119895 isin 1198732 then 119909119896119894119895 =[119909119896119894119895 119909119896119894119895] is an IN If 119895 isin 1198733 then 119909119896119894119895 = (119886119896119894119895 119887119896119894119895 119888119896119894119895) is a TFNIf 119895 isin 1198734 then 119909119896119894119895 = (119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) is a TrFN If 119895 isin 1198735then 119909119896119894119895 = ((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) Namely 119909119896119894119895 can be unified asfollows

119909119896119894119895 =

119909119896119894119895 if 119895 isin 1198731[119909119896119894119895 119909119896119894119895] if 119895 isin 1198732(119886119896119894119895 119887119896119894119895 119888119896119894119895) if 119895 isin 1198733(119890119896119894119895 119891119896119894119895 119892119896119894119895 ℎ119896119894119895) if 119895 isin 1198734((119905119896119894119895 119905119896119894119895 119905119896119894119895) 119906119896119894119895 V119896119894119895) if 119895 isin 1198735(16)

Hence a group decision matrix of alternative 119878119896 can beexpressed as

119883119896 = (119909119896119894119895)119898times119899 =

1198601 1198602 sdot sdot sdot 11986011989911986311198632119863119898((

11990911989611 11990911989612 sdot sdot sdot 119909119896111989911990911989621 11990911989622 sdot sdot sdot 1199091198962119899 1199091198961198981 1199091198961198982 sdot sdot sdot 119909119896119898119899))(119896 isin 119875)

(17)

To reduce information loss and simplify the focusedproblems the group decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) can be integrated into a collective ITFN decisionmatrix The key to addressing this issue lies in an effectiveapproach for constructing ITFNs based on the expertsrsquoassessment expressed in different types of data

32 An Approach to Aggregating Heterogeneous Informationinto ITFNs To facilitate the calculation denote the jthcolumn vector in the matrix 119883119896 as

119860119896119895 = (1199091198961119895 1199091198962119895 119909119896119898119895) (119896 isin 119875 119895 isin 119873) (18)

which is the normalized assessment vector of alternative 119878119896 onattribute 119860119895 given by all DMs 119863119894 (119894 = 1 2 119898) Let 119860max

119895

and 119860min119895 be the largest grade and smallest grade employed

in the rating system For example if the assessments in 119860119895are TFNs then 119860max

119895 = (1 1 1) and 119860min119895 = (0 0 0) if the

assessments in 119860119895 are INs then 119860max119895 = [1 1] and 119860min

119895 =[0 0] To integrate the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective ITFN decision matrix all theelements in vector 119860119896119895 need to be aggregated into an ITFNThe implementation of the aggregation approach involves afour-stage framework (see Figure 1) (1) Elicit Rsd Rdd andRud In this process we use the TOPSIS method to obtainthe rating satisfactory degree (Rsd) and rating dissatisfactorydegree (Rdd) of 119909119896119894119895 and construct the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 The rating uncertain degree (Rud)of 119909119896119894119895 and the corresponding uncertain set 120578119896119895 are derived bygeometry entropy (2) Calculate mode Combining the groupconsistency and mean method the modes of the above sets120585119896119895 120577119896119895 and 120578119896119895 are computed in this stage (3) ConstructQst and Qdt According to the Min-Max method the quasi-satisfactory triangular (Qst) and quasi-dissatisfactory trian-gular (Qdt) of119860119896119895 can be built (4) Induce an ITFNThe ITFN

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

Complexity 7

O

d(xkij A

ＧＣＨj ) d(xk

ij AＧＲj )

M(AＧＣＨj ) N(AＧＲ

j )xkij

Figure 2 Distances ratio-based rating uncertain degree for 119909119896119894119895of119860119896119895 can be obtained through a linear transformation in thisprocess

321 Elicit the Rsd Rdd and Rud Consider that (1) therelative closeness [45] from119909119896119894119895 to119860max

119895 implies the satisfactionof DM (2) the relative closeness from 119909119896119894119895 to 119860min

119895 impliesthe dissatisfaction of DM and (3) according to the ratio-based measure of fuzziness [46 47] the ratio of distancesfrom 119909119896119894119895 to 119860min

119895 and from 119909119896119894119895 to 119860max119895 can also express the

fuzziness degree of 119909119896119894119895Thus combining the relative closenessof TOPSIS [36] and geometry entropy method [46] the RsdRdd and Rud of 119909119896119894119895 can be elicited as follows

Definition 7 Let 119860119896119895 be a benefit attribute vector and let 119909119896119894119895be an arbitrary element in 119860119896119895 The Rsd Rdd and Rud of 119909119896119894119895are defined as120585119896119894119895 = 119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (19)

120577119896119894119895 = 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (20)

120578119896119894119895 = min119889(119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860min119895 ) (21)

respectively where 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 denote the Rsd Rddand Rud given by 119889119894 on attribute 119860119895 in the alternative 119878119896119889(119909119896119894119895 119860max

119895 ) is the distance between 119909119896119894119895 and the largest grade119860max119895 of the attribute 119860119895 and 119889(119909119896119894119895 119860min

119895 ) is the distancebetween 119909119896119894119895 and the smallest grade 119860min

119895 of the attribute 119860119895For Example 1 Consider that 119909111 = (46 72 83) is a TFNin the ten-mark system then 119860min

119895 = (0 0 0) 119860max119895 =(10 10 10) According to (19)-(21) the Rsd Rdd and Rud of119909111 are calculated respectively as 120585111 = 067 120577111 = 033 and120578111 = 049

Theorem 8 Rud 120578119896119894119895 of 119909119896119894119895 in 119860119896119895 has the following properties(EP1) If 119909119896119894119895 = 119860max

119895 or 119909119896119894119895 = 119860min119895 then 120578119896119894119895 = 0 which

means that 119860min119895 and 119860max

119895 are not fuzzy since 119860min119895 and 119860max

119895

are crisp sets(EP2) If 119889(119909119896119894119895 119860max

119895 ) = 119889(119909119896119894119895 119860min119895 ) (namely 119909119896119894119895 is the

middle point) then 120578119896119894119895 = 1 which means that 119909119896119894119895 is the fuzziestelement

(EP3) 120578119896119894119895 le 120578119896119905119895 (119894 119905 isin 119872 = 1 2 119898) if 120578119896119894119895 is less fuzzythan 120578119896119894119895 ie119889 (119909119896119894119895 119860min

119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860min

119895 )119889 (119909119896119905119895 119860max119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 ) 119889 (119909119896119894119895 119860max119895 )119889 (119909119896119894119895 119860min119895 ) le 119889 (119909119896119905119895 119860max

119895 )119889 (119909119896119905119895 119860min119895 )

for 119889 (119909119896119894119895 119860min119895 ) ge 119889 (119909119896119894119895 119860max

119895 ) 119889 (119909119896119905119895 119860min119895 ) ge 119889 (119909119896119905119895 119860max

119895 )

(22)

It is easy to prove that Rud 120578119896119894119895 meets the properties (EP1)-(EP3) which are consistent with the axioms (1)-(3) of fuzzyentropy based on distance in [48] Froma geometric standpoint119860max119895 and 119860min

119895 in the rating system are nonfuzzy which cancorrespond to the position M and position N (Figure 2) Whena fuzzy number 119909119896119894119895 is moved from position M (or N) towardsmiddle position O the distance from 119909119896119894119895 to M is close to NMeanwhile119909119896119894119895 become more and more vague ie 120578119896119894119895 is gettingbigger and bigger Particularly when 119909119896119894119895 is in position P thedistance from 119909119896119894119895 to M is equal to N So the fuzzy number 119909119896119894119895is the fuzziest ie 120578119896119894119895 = 1 According to the analysis above it isreasonable that Rud measures the uncertainty of the originalassessment It is worth mentioning that (21) is suitable fordifferent forms of decision data such as INs TFNs TIFNs andTrIFNs

Remark 9 From (19)-(21) the extraction method in thispaper relies on just the largest grade 119860max

119895 and the smallestgrade119860min

119895 of the attribute 119860119895 while the methods [35 37] relyon 119860max119895 and 119860min

119895 as well as the middle grade 119860mid119895 of the

attribute 119860119895 However it is hard to determine the middlegrade for some sets of TIFNs [43] and TrIFNs [49 50]Hence the proposed extraction method is more effective andsimple

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

8 Complexity

Remark 10 When119860119896119895 is a cost attribute vector the Rud of 119909119896119894119895can be derived by (21)TheRsd andRdd of119909119896119894119895 can be rewrittenas 120585119896119894119895 = 119889 (119909119896119894119895 119860max

119895 )119889 (119909119896119894119895 119860max119895 ) + 119889 (119909119896119894119895 119860min

119895 ) (23)

120577119896119894119895 = 119889 (119909119896119894119895 119860min119895 )119889 (119909119896119894119895 119860max

119895 ) + 119889 (119909119896119894119895 119860min119895 ) (24)

Remark 11 For the benefit attribute vector 119860119896119895 = (11990911989611198951199091198962119895 119909119896119898119895) 120585119896119894119895 120577119896119894119895 and 120578119896119894119895 of each element are composedof support set 120585119896119895 opposition set 120577119896119895 and uncertain set 120578119896119895 of119860119896119895 which can be defined as 120585119896119895 = 1205851198961119895 1205851198962119895 120585119896119898119895 and 120577119896119895 =1205771198961119895 1205771198962119895 120577119896119898119895 and 120578119896119895 = 1205781198961119895 1205781198962119895 120578119896119898119895 respectively322 Calculate the Mode For the support set 120585119896119895 = 12058511989611198951205851198962119895 120585119896119898119895 the mode of the TFN is located in the centeraround which the Rsd 120585119896119894119895 (119894 = 1 2 119898) gather Inspiredby the literature [49 50] the more consistent it is with therest of 120585119896119895 the greater the importance of the Rsd 120585119896119894119895 given byDM 119863119894 That is to say the weighted average of the collection120585119896119895 can be regarded as its mode Here we utilize the distancebetween 120585119896119894119895 and 120585119896119890119895 to define the consistency degree of 119863119894 onsupport set 120585119896119895 to the rest of experts which can be obtainedby 119862119868119896119894119895 = 1119898 minus 1 119898sum119894=1119894 =119895 (1 minus 119889 (120585119896119894119895 120585119896119890119895)) (25)

where 119889(sdot) is the distance between 120585119896119894119895 and other Rsds in 120585119896119895Clearly 0 le 119878119896 le 1

Generally an expertrsquos Rsd is more important if heshe ismore similar to the grouprsquos Rsd In other words the larger thevalue of 119862119868119896119894119895 is the more important 120585119896119895 is Thus the weight of119863119894 on 120585119896119895 can be obtained by

119906119896119894119895 = 119862119868119896119894119895sum119898119894=1 119862119868119896119894119895 (119894 = 1 2 119898) (26)

Then the mode of 120575119896119895 for support set 120585119896119895 is derived as119898(120585119896119895) = 119898sum119894=1

119906119896119894119895120585119896119894119895 (27)

Similarly we have the mode119898(120577119896119895) of opposition set 120577119896119895323 Construct Qst and Qdt Note that the membershipdegree and nonmembership degree of aTIFN areTFNs ratherthan real numbers Moreover 120585119896119894119895 and 120577119896119894119895 are real numberswhich are difficult to express the imprecise and vague expertsrsquosubjective judgment By doing this the TFNs of 120585119896119894119895 and 120577119896119894119895

are commonly used to represent Qsd and Qdd of 119860119896119895 sinceTFN is characterized by a membership function Thus it isnecessary to construct the Qst and Qdt of 119860119896119895 As per thedefinition of TFN the corresponding TFNs of 120585119896119895 and 120577119896119895 canbe constructed as follows

Definition 12 For the attribute vector119860119896119895 theQst120575119896119895 andQdt120574119896119895 of alternative 119878119896 on attribute 119860119895 are defined as120575119896119895 = (min 120585119896119895 119898 (120585119896119895) max 120585119896119895) (119896 isin 119875 119895 isin 119873) (28)

120574119896119895 = (min 120577119896119895 119898 (120577119896119895) max 120577119896119895) (119896 isin 119875 119895 isin 119873) (29)

where the min120585119896119895 and max120585119896119895 are the minimum valueand maximum value of the support set 120585119896119895 and min120577119896119895 andmax120577119896119895 are the minimum value and maximum value ofopposition set 120577119896119895 For the convenience of discussion the pair(120575119896119895 120574119896119895) is called a quasi-ITFN

Remark 13 To calculate the mode of triangular fuzzy num-bers 120575119896119895 and 120574119896119895 this paper employs the weighted averagingvalue that considers the distribution of ratings whereas someworks used the mean value method The essential differenceis that the current method takes the consistency of the groupinto account while the mean value method is based onstatistical assumptions

324 Inducing an ITFN Finally an ITFN is inducedfrom the Qst and Qdt of alternative 119878119896 on the attribute119860119895 by the following normalized method Let 120572119896119895 =((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) be the inducedITFN by the attribute vector 119860119896119895 and mean(120578119896119895) =(1119898)sum119898119894=1 120578119896119894119895 is the uncertain degree of 119860119896119895 To satisfy theconditions in (3) and consider the influence of uncertaindegree the values of 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 can becomputed as follows

119905119897119896119895 = min 120585119896119895120595119896119895 119905119898119896119895 = mean (120585119896119895)120595119896119895 119905ℎ119896119895 = max 120585119896119895120595119896119895 119891119897119896119895 = min 120577119896119895120595119896119895 119891119898119896119895 = mean (120577119896119895)120595119896119895

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

Complexity 9

119891ℎ119896119895 = max 120577119896119895120595119896119895 (119896 isin 119875 119895 isin 119873)(30)

respectively where120595119896119895 = (15)(min120585119896119895+119898(120585119896119895)+min120577119896119895+119898(120577119896119895) +mean(120578119896119895)) +max120585119896119895 +max120577119896119895Apparently 119905119897119896119895 119905119898119896119895 119905ℎ119896119895 119891119897119896119895 119891119898119896119895 and 119891ℎ119896119895 satisfy (3) Thus120572119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) is an ITFN Namely all the

attribute values in the vector 119860119896119895 can be aggregated into anITFN 120572119896119895For Example 2 Consider that 119860119896119895 = ((46 72 83)(34 46 67) (56 62 82) (45 65 10) (37 65 72)) is aTFN vector in the ten-mark system then 119860min

119895 = (0 0 0)

119860max119895 = (10 10 10) According to (19)-(21) the support

set opposition set and uncertain set of 119860119896119895 are calculatedrespectively as 120585119896119895 = 067 049 070 065 058 120577119896119895 = 033051 030 035 042 and 120578119896119895 = 049 039 052 05 044Subsequently the mode of the sets 120585119896119895 and 120577119896119895 is calculatedfrom (25)-(27) to be 119898(120585119896119895) = 062 and 119898(120577119896119895) = 038It follows from (28) and (29) that 120575119896119895 = (049 062 070)and 120574119896119895 = (030 038 051) Finally by using (29) theinduced ITFN associated with 119860119896119895 is derived to be 120572119896119895 =((0349 0445 0499) (0214 0268 0364))4 A Novel Approach for HeterogeneousMAGDM Problems

According to the proposed aggregationmethod the collectivedecision matrix 119883 = (119903119896119895)119901times119899 is aggregated as follows

1198601 1198602 sdot sdot sdot 11986011989911987811198782119878119901 ( ((11990511989711 11990511989811 119905ℎ11) (11989111989711 11989111989811 119891ℎ11))((11990511989721 11990511989821 119905ℎ21) (11989111989721 11989111989821 119891ℎ21))((1199051198971199011 1199051198981199011 119905ℎ1199011) (1198911198971199011 1198911198981199011 119891ℎ1199011))((11990511989712 11990511989812 119905ℎ12) (11989111989712 11989111989812 119891ℎ12))((11990511989722 11990511989822 119905ℎ22) (11989111989722 11989111989822 119891ℎ22))((1199051198971199012 1199051198981199012 119905ℎ1199012) (1198911198971199012 1198911198981199012 119891ℎ1199012))

sdot sdot sdotsdot sdot sdotsdot sdot sdot((1199051198971119899 1199051198981119899 119905ℎ1119899) (1198911198971119899 1198911198981119899 119891ℎ1119899))((1199051198972119899 1199051198982119899 119905ℎ2119899) (1198911198972119899 1198911198982119899 119891ℎ2119899))((119905119897119901119899 119905119898119901119899 119905ℎ119901119899) (119891119897119901119899 119891119898119901119899 119891ℎ119901119899)) )

(31)

where 119903119896119895 = ((119905119897119896119895 119905119898119896119895 119905ℎ119896119895) (119891119897119896119895 119891119898119896119895 119891ℎ119896119895)) (119896 isin 119875 119895 isin 119873) areITFNs aggregated by the attribute vector 119860119896119895 in119883

119896Assume that the weight vector 119908 = (1199081 1199082 119908119899) of

attributes is fully known by (4) we can easily obtain thecomprehensive rating 119903119896 = ((119905119897119896 119905119898119896 119905ℎ119896) (119891119897119896 119891119898119896 119891ℎ119896 )) of thealternative 119878119896When119908 is incompletely known wemay utilizethe following programming model to establish the attributeweights

41 Determination of Attribute Weights As all known it isreasonable to determine the attribute weights by makingalternative similar to the positive ideal solution (PIS) andat the same time far away from the negative ideal solution(NIS) as far as possible Let 119878+ = 119903+1 119903+2 119903+119899 and 119878minus =119903minus1 119903minus2 119903minus119899 be the intuitionistic triangular fuzzy PIS andNIS respectively where119903+119895 = ((max

119896119905119897119896119895max119896119905119898119896119895max119896119905ℎ119896119895) (min

119896119891119897119896119895min119896119891119898119896119895 min

119896119891ℎ119896119895)= ((119905119897119895+ 119905119898119895 + 119905ℎ119895 +) (119891119897119895+ 119891119898119895 + 119891ℎ119895 +)) (32)

119903minus119895 = ((min119896119905119897119896119895min119896119905119898119896119895min119896119905ℎ119896119895) (max

119896119891119897119896119895max

119896119891119898119896119895 max

119896119891ℎ119896119895)= ((119905119897119895minus 119905119898119895 minus 119905ℎ119895 minus) (119891119897119895minus 119891119898119895 minus 119891ℎ119895 minus))

(33)

(119895 isin 119873 = 1 2 119899) Based on (5) we can define the sim-ilarity degree between alternative 119878119896 (119896 isin 119875 = 1 2 119901)and intuitionistic triangular fuzzy ideal solution shown asfollows120599+119896 = 1 minus 119901sum

119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 +100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 +100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895+100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 +10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 +100381610038161003816100381610038161003816)](34)

120599minus119896 = 1 minus 119901sum119895=1

119908119895 lowast [ 112 (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119905ℎ119896119895minus 119905ℎ119895 minus100381610038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 + 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 + 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)+ 12 max (100381610038161003816100381610038161003816119905119897119896119895 minus 119905119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119905119898119896119895 minus 119905119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119905ℎ119896119895 minus 119905ℎ119895 minus100381610038161003816100381610038161003816 100381610038161003816100381610038161003816119891119897119896119895 minus 119891119897119895minus100381610038161003816100381610038161003816 10038161003816100381610038161003816119891119898119896119895 minus 119891119898119895 minus10038161003816100381610038161003816 100381610038161003816100381610038161003816119891ℎ119896119895 minus 119891ℎ119895 minus100381610038161003816100381610038161003816)](35)

Then a multiple objective linear mathematical programmingmodel is constructed as follows

max 120599+119896 minus 120599minus119896 (119896 isin 119875)119904119905 119908 isin Ω (36)

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

10 Complexity

For a multiple objective programming there are severalsolution methods Here we apply the Max-Min method Let120588 = min120599+119896 minus120599minus119896 By using theMax-Minmethod (36) can besolved by the following single objective linear programmingmodel

max 120588119904119905 120599+119896 minus 120599minus119896 ge 120588 (119896 isin 119875)119908 isin Ω (37)

By plugging (34) and (35) in (37) it is easily seen that wecan derive the attribute weights since the optimal solution of(37) is a Pareto optimal solution of (36)

Remark 14 To obtain the weights of attributes in the intu-itionistic triangular fuzzy environment the current method-ology first combines a multiple objective mathematical pro-gramming model with TOPSIS idea based on the abovecollective decision matrix Then this model can be solvedby Max-Min method which is relatively simple However Liand Chen [49] determined the attribute weights by expectedweight value that involved options of decision makersShan and Xu [42] gave the attribute weights in advanceTherefore our method could be more reasonable and objec-tive

Thus the ranking order of the alternative 119878119896 can beconducted by the following relative closeness coefficient(RCC) 119877119862119862119896 = 120599+119896120599+

119896+ 120599minus119896

(38)

where 0 le 119877119862119862119896 le 1 (119896 isin 119875) It is obvious that the larger119877119862119862119896 the better the alternative 11987811989642 Procedure of the Proposed Aggregation Method for Hetero-geneous MAGDM On the basis of the above analysis a newapproach to heterogeneous MAGDM problems involves thefollowing primary steps

Step 1 Obtain the decision matrix of alternative 119878119896 (119896 =1 2 119901) taking the form of 119884119896 = (119910119896119894119895)119898times119899 by (18)Step 2 Convert the decision matrices 119883119896 = (119909119896119894119895)119898times119899 (119896 =1 2 119901) into a collective decision matrix 119883 = (119903119896119895)119901times119899through the aggregation method developed in Section 3Based on each column vector 119860119896119895 in the matrix 119883119896 conductthe following substeps

Step 21 Compute theRsd Rdd andRud of119909119896119894119895 by plugging theHamming distance of different attribute types into (19) (20)(21) (23) and (24) and construct support set 120585119896119895 oppositionset 120577119896119895 and uncertain set 120578119896119895 of 119860119896119895Step 22 Calculate the mode of the support set 120585119896119895 andopposition set 120577119896119895 of 119860119896119895 by (25)-(27)

Step 23 Construct the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895 by (28) and(29)

Step 24 Induce the corresponding ITFN of 119860119896119895 using (30)

Step 3 Determine the attribute weights by constructing amultiple objective programming model The detailed stepsare as follows

Step 31 Define the intuitionistic triangular fuzzy PIS 119878+ =119903+1 119903+2 119903+119899 and NIS 119878minus = 119903minus1 119903minus2 119903minus119899 by (32) and (33)respectively

Step 32 Compute the similarity degrees 120599+119896 and 120599minus119896 from theelements at the kth row of the collective decision matrix 119883 =(119903119896119895)119901times119899 to PIS 119878+ and NIS 119878minus by (34) and (35) respective-ly

Step 33 Construct a multiple objective programming modelbased on (36)

Step 34 Convert the above model into a single objectiveprogramming model by (37)

Step 35 Obtain the optimal weights of attributes by solvingthe linear programming model

Step 4 Calculate the similarity degrees 120599+119896 and 120599minus119896 of alterna-tive 119878119896 (119896 isin 119875) by the obtained attribute weights and (34) and(35)

Step 5 Calculate the RCC of alternative 119878119896 (119896 isin 119875) by (38)Step 6 Rank the alternatives according to the RCC and selectthe best one

The decision procedure of the proposed algorithmmay bedepicted in Figure 3

5 A Trustworthy Seller Selection Problem andComparison Analyses

To demonstrate the efficacy of the proposed HMAGDMmethod this section gives a trustworthy seller selectionexample and conducts comparison analyses with the ones ofthe existing methods [34 35 37]

51 A Trustworthy Seller Selection Example and Its Solu-tion Procedure Online service trading usually takes placebetween parties who are autonomous in an environmentwhere the buyer often has not enough information aboutthe seller and goods Many scholars think that trust is aprerequisite for successful tradingTherefore it is very impor-tant that buyers can identify the most trustworthy sellerSuppose that a consumer desires to select a trustworthy sellerAfter preliminary screening four candidate sellers 1198781 1198782 1198783and 1198784 remain to be further evaluated Based on detailedseller ratings the decision-making committee assesses the

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Submit your manuscripts atwwwhindawicom

Complexity 11

Obtain group decision matrix

Aggregating decision information into ITFN

Calculate the mode

Construct Qst and Qdt

Induce an ITFN

Determine the weights of attributesEstablish multi-objective

programming model

Convert above model into singleobjective programming model

Obtain the attribute weights by solving the model

Rank the alternatives by relative

DMInterval number

attributeTFN

attributeTrFN

attributeTIFN

attribute

Decision goal

Elicit the Rsd Rdd Rud

closeness coefficient Uk

Calculate the similarity degree +k minus

k

Define the similarity degree +k minus

k((tlkj tmkj t

ℎkj) (f

lkj f

mkj f

ℎkj))ptimesn

S1 Sp

Figure 3 The framework of the proposed heterogeneousMAGDMmethod

four candidate sellers according to the five trust factorsincluding product quality (A1) service attitude (A2) web-site usability (A3) response time (A4) and shipping speed(A5)

Product quality (A1) service attitude (A2) and websiteusability (A3) given by DMs with a one-mark system are allbenefit attributes It is better to use TFNs to assess productquality (A1) For service attitude (A2) the experts like toprovide the lower and upper limits and the most possibleintervals thus the assessments of A2 can be represented by

TrFNs The website usability (A3) is expressed by TIFNswhile response time (A4) and shipping speed (A5) given byDMs with a ten-mark system both are cost attributes Theassessments of the sellers on A4 can be represented by RNsDue to the uncertainty of shipping speed INs are suitablyutilized to represent the assessments of shipping speed (A5)The assessments of four sellers on five attributes given byfive experts are listed in Table 1 The attributesrsquo importance isincomplete and experts give incomplete information on theattributesrsquo importance as follows

Ω = 1199081 ge 01 1199082 minus 1199081 le 005 1199083 minus 1199081 le 005 1199085 minus 1199081 le 0051199081 + 1199083 + 1199085 le 06 1199084 le 02 1199081 + 1199082 + 1199083 + 1199084 + 1199085 = 1 (39)

Obviously the decision problem mentioned above isa heterogeneous MAGDM problem involving five differ-ent formats of data RNs INs TFNs TrFNs and TIFNs

To address this problem we apply the proposed decisionmethod to the selection of the trustworthy sellers be-low

12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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12 Complexity

Table 1 The decision matrix of four alternatives

Sellers DMs A1 A2 A3 A4 A5

S1 D1 (046072083) (02 03 05 10) ((03305067)0503) 34 [2 6]D2 (034046067) (04 05 06 08) ((03305067)0405) 54 [1 3]D3 (056062082) (04 04 05 08) ((03305067)0601) 72 [1 6]D4 (045065100) (02 03 05 09) ((05067083)0702) 81 [1 4]D5 (037065072) (00 01 03 07) ((01703305)0406) 59 [3 8]

S2 D1 (051067073) (02 03 04 07) ((06708310)0603) 42 [3 6]D2 (035064073) (03 04 04 09) ((03305067)0405) 54 [2 5]D3 (066082100) (02 02 04 05) ((06708310)0602) 72 [4 7]D4 (034051077) (05 06 08 08) ((01703305)0702) 100 [1 3]D5 (068082100) (01 01 02 05) ((06708310)0306) 71 [6 7]

S3 D1 (052066081) (05 06 06 08) ((00017033)0503) 78 [2 4]D2 (039066068) (04 05 07 08) ((03305067)0405) 65 [2 4]D3 (062067084) (03 03 05 09) ((00017033)0601) 43 [1 3]D4 (058074100) (04 04 06 08) ((03305067)0603) 83 [3 5]D5 (044050067) (03 03 04 05) ((05067083)0406) 55 [4 7]

S4 D1 (034047067) (02 03 04 05) ((05067083)0404) 49 [6 7]D2 (059066068) (02 03 04 04) ((05067083)0604) 81 [5 6]D3 (047056067) (01 01 02 07) ((03305067)0601) 39 [1 10]D4 (045050069) (04 05 07 08) ((03305067)0703) 58 [2 4]D5 (057074100) (02 03 06 06) ((01703305)0406) 70 [4 6]

Step 1 Thegroup decisionmatrices are obtained as in Table 1

Step 2 Due to the ratings of A1 A2 A3 given by DMs basedon the one-mark system and the ratings of A4 A5 with theten-mark system we have

119860min119895 =

0 if 119895 isin 1198731[0 0] if 119895 isin 1198732(0 0 0) if 119895 isin 1198733(0 0 0 0) if 119895 isin 1198734((0 0 0) 0 1) if 119895 isin 1198735119860max119895 =

10 if 119895 isin 1198731[10 10] if 119895 isin 1198732(1 1 1) if 119895 isin 1198733(1 1 1 1) if 119895 isin 1198734((1 1 1) 1 0) if 119895 isin 1198735

(40)

Obtain the aggregated ITFNs corresponding to theattribute vectors

Step 21 By plugging (15) into (19)-(21) for benefit attributes(plugging (15) into (23) (24) and (21) for cost attributes) wecan compute the Rsd Rdd and Rud of 119909119896119894119895 which are shownin Table 2 Thereby the support set 120585119896119895 and opposition set 120577119896119895of 119860119896119895 can be derived

Step 22 By (25)-(27) we can obtain the modes of 120585119896119895 and 120577119896119895which are listed in Table 3

Step 23 Using (28) and (29) the Qst 120575119896119895 and Qdt 120574119896119895 of 119860119896119895can be constructed and the results are presented in Table 3

Step 24 Based on (30) the aggregated ITFNs of 119860119896119895 are alsoshown in Table 3

Step 3 Determine the attribute weights

Step 31 By (32) and (33) the intuitionistic triangular fuzzyPIS and NIS are defined as follows119878+ = ((041 051 063) (013 024 034)) ((023 033 039) (018 029 039)) ((028 038 046) (019 027 036)) ((010 022 036) (018 032 044)) ((032 046 056) (013 024 035)) 119878minus = ((035 042 050) (021 028 036)) ((013 022 033) (024 034 043)) ((007 018 029) (024 034 045)) ((000 016 029) (023 034 049)) ((022 032 044) (019 031 041))

(41)

Step 32 By (34) and (35) the similarity degrees 120599+119896 and120599minus119896 (119896 = 1 2 3 4) are calculated as follows120599+1 = 1 minus (00971199081 + 00621199082 + 00511199083 + 00041199084+ 00261199085)

Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

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Complexity 13

Table 2 Rsd Rdd and Rud of each attribute value

Providers DM A1 A2 A3 A4 A5

S1 D1 067033049 050050100 042058064 066034052 060040067

D2 049051039 058042074 042058061 046054085 080020025

D3 067033052 053047090 041059068 028072039 065035054

D4 070030050 048052090 059041079 019081024 075025033

D5 058042044 028072038 067033080 041059070 045055082

S2 D1 064036049 040060067 070030058 058042072 055045082

D2 057043043 050050100 042058061 046054085 065035054

D3 082018060 033067068 069031056 028072039 045055082

D4 054046041 068032033 045055076 000100000 080020025

D5 083017060 023077078 063037089 029071041 035065054

S3 D1 066034050 063037060 015085017 022078028 070030043

D2 058042044 060040067 042058061 035065054 070030043

D3 071029055 050050100 014086016 057043075 080020025

D4 077023056 055045082 044056070 017083021 060040067

D5 054046043 038062060 056044091 045055082 045055082

S4 D1 049051039 035065054 053047090 051049096 045055082

D2 064036051 033067048 059041090 019081024 045055082

D3 057043045 028072038 041059068 061039064 045055082

D4 055045044 060040067 047053077 042058072 070030043

D5 077023055 043057074 056044091 030070043 050050100

120599+2 = 1 minus (00021199081 + 00781199082 + 00081199083 + 00781199084+ 00861199085)120599+3 = 1 minus (00401199081 + 00301199082 + 01651199083 + 00431199084+ 00261199085)120599+4 = 1 minus (00721199081 + 00851199082 + 00981199083 + 00181199084+ 01091199085)120599minus1 = 1 minus (00141199081 + 00381199082 + 01411199083 + 00791199084+ 01141199085)120599minus2 = 1 minus (01001199081 + 00431199082 + 01651199083 + 01461199084+ 02671199085)120599minus3 = 1 minus (00601199081 + 00871199082 + 00031199083 + 01661199084+ 03231199085)120599minus4 = 1 minus (00341199081 + 00211199082 + 00771199083 + 01911199084+ 02301199085)(42)

Step 33 By using (36) a multiple objective programmingmodel is expressed as follows

max 120599+1 minus 120599minus1 = minus00841199081 minus 00231199082 + 00891199083+ 00751199084 + 00881199085max 120599+2 minus 120599minus2 = 00981199081 minus 00351199082 + 01561199083+ 00681199084 + 01811199085max 120599+3 minus 120599minus3 = 00201199081 + 00581199082 minus 01621199083+ 01231199084 + 02981199085max 120599+4 minus 120599minus4 = minus00381199081 minus 00641199082 minus 00211199083+ 01721199084 + 01221199085119904119905 119908 isin Ω

(43)

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Complexity

Table 3 Mode119898120585119896119895119898120577119896119895 Qst 120575119896119895 Qdt 120574119896119895 and aggregated ITFNs of attribute vectorsSellers Attributes 119898120585119896119895 120585119896119895 119898120577119896119895 120577119896119895 ITFNsS1 A1 062 (049062070) 038 (030038051) ((035044049)(021027036))

A2 048 (028048058) 052 (043052073) ((016028034)(025031043))A3 059 (050059083) 041 (017041050) ((035041058)(012029035))A4 040 (019040066) 060 (034060081) ((010022036)(019033045))A5 065 (045065080) 035 (020035055) ((032046056)(014024039))

S2 A1 068 (054068083) 032 (017032046) ((040051062)(012024034))A2 042 (023042068) 058 (033058078) ((013024038)(018033044))A3 071 (050071083) 029 (017029050) ((037053062)(012022037))A4 067 (000033058) 067 (042067100) ((000016029)(021033050))A5 044 (035056080) 044 (020044065) ((022035051)(013028041))

S3 A1 065 (054065077) 035 (023035046) ((040048057)(017026034))A2 053 (038053063) 047 (038047063) ((024034040)(024030040)A3 040 (017040067) 060 (033060083) ((009022037)(018033046))A4 035 (017035057) 065 (043065083) ((009019031)(023035045))A5 046 (032046056) 024 (014024039) ((032046056)(014024039))

S4 A1 060 (049060077) 040 (023040051) ((035042054)(016028035))A2 039 (028039060) 061 (040061073) ((016022034)(023035041))A3 054 (033054067) 046 (033046067) ((021034042)(021029042))A4 041 (019041061) 059 (039059081) ((011023034)(022033045))A5 050 (045050070) 040 (030040055) ((029033046)(020032036))

Step 34 By using (37) (43) can be changed into the followingsingle objective linear programming model

max 119909119904119905

minus00841199081 minus 00201199082 + 00891199083 + 00751199084 + 00881199085 ge 11990900981199081 minus 00301199082 + 01541199083 + 00681199084 + 01811199085 ge 11990900201199081 + 00581199082 minus 01621199083 + 01231199084 + 02981199085 ge 119909minus00381199081 minus 00701199082 minus 00211199083 + 01721199084 + 01221199085 ge 119909119908 isin Ω

(44)

Step 35 Applying Lingo 110 we have the attribute weightvector 119908 = (0162 0212 0214 0200 0212)Step 4 By plugging 119908 into (34) and (35) the similaritydegrees 120599+119896 and 120599minus119896 of alternative 119878119896 can be calculated asfollows 120599+1 = 0954120599minus1 = 0919120599+2 = 0948120599minus2 = 0854120599+3 = 0938

120599minus3 = 0869120599+4 = 0923120599minus4 = 0886(45)

Step 5 Based on (37) the RCC of alternative 119878119896 is yielded asfollows 1198771198621198621 = 05091198771198621198622 = 05261198771198621198623 = 05191198771198621198624 = 0510 (46)

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 15

Step 6 Since 1198771198621198622 gt 1198771198621198623 gt 1198771198621198624 gt 1198771198621198621 we can easilyrank the preference order of the sellers as 1198782 ≻ 1198783 ≻ 1198784 ≻ 1198781Hence the best seller is 119878252 Comparison Analyses with Existing Methods AggregatedIF Information Generally the key focus of the heterogeneousMAGDMmethods is how to aggregate the assessments takingthe form of different data types determine the weights ofexperts and attributes and rank the preference order of alter-natives Here we make detailed comparison analyses withsome similar methods [34 35 37] from the abovementionedkey issues which are shown in Table 4 Furthermore thecurrent methodology has the following superiorities

(1) A comparison between the current methodology andYuersquos method [34] is made We consider attribute weightvector 119908 = (04 02 04) and employ the former to evaluatethe massesrsquo satisfaction of three leaders used in [34] theranking of three suppliers is 1198781 ≻ 1198783 ≻ 1198782 which isinconsistent with 1198781 ≻ 1198782 ≻ 1198783 shown in [34] Althoughthe ranking is inconsistent the best satisfactory leader isthe same Thus the current methodology can well adaptto the special decision circumstance in [34] However thelatter cannot deal with the heterogeneousMAGDMproblemssince it is only suitable for interval numbers Moreoverin the former the attribute weights are derived by usinga multiple objective programming model that can avoidsubjective randomness in the latter

(2) A comparison between the current methodology andXu et al method [35] is made Assume that 119886 = (1205831 V1) and119887 = (1205832 V2) are two IFNs 119889(119886 119887) = (12)(|1205831 minus 1205832| + |V1 minus V2|)is the distance of a and b and 120572+ = (1 0) and 120572minus = (0 1) arethe largest and smallest IFN Then the former is applied toevaluate the cloud computing service provider used in [35]the ranking of four suppliers is 1198781 ≻ 1198782 ≻ 1198783 ≻ 1198784 which isinconsistent with 1198783 ≻ 1198781 ≻ 1198782 ≻ 1198784 shown in [35] Thusthe current methodology can well adapt to a special decisioncircumstance in [35] The main reasons for the difference inthe ranking are as follows (i) The method for eliciting thesatisfaction and dissatisfaction of DMs based on the TOPSISidea in the current methodology is more complete than thatin [35] because the former takes into account the two aspectsof satisfaction and dissatisfaction in each element while thelatter considered only one aspect (ii) Besides the rating valuesdenoted by RNs INs TFNs TrFNs and LVs the former candeal with more complicated rating values including TIFNswhereas the latter is only suitable for the HMAGDMproblemwith RNs INs TFNs TrFNs and LVs Thus the latter cannotsolve the abovementioned example (iii) In the former theintegrated information of experts is expressed by a TIFNwhereas that in the latter is an IFN Hence there are moreopportunities resulting in a loss of information

(3) A comparison between the current methodology andWan et al method [37] is made We employ the former toevaluate the IT outsourcing service provider used in [37] theranking of four suppliers is 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 which isconsistent with 1198783 ≻ 1198782 ≻ 1198781 ≻ 1198784 shown in [37] Thusthe current methodology can well adapt to a special decisioncircumstance in [37] However the current methodology issuperior over the later in the following aspects (i) In the

former the integrated information of experts on the sameattribute is a TIFN whereas that in the latter is an IVIFNHence the current methodology can express vaguenessinformation of reality more accurately and abundantly (ii)The former takes into account the weights of experts by theirconsistency degree whereas there is no consideration in thelatter So the current methodology is more reasonable (iii)The same as the above the latter is only suitable for theHMAGDM problem with RNs INs TFNs and TrFNs Thusthe latter cannot solve the abovementioned example

53 Comparison Analyses with Existing HMAGDMMethodsIn this section we compare the proposed method with othertwo methods for HMAGDM problems one is the complexand dynamic MAGDM method developed by Dong et al[15] that is an optimization-based approach and the otheris the GDM method based on integrating heterogeneousinformation introduced by Li [21] that is a direct approachFor simplicity the comparative analysis methods are denotedas CD-GDM and IGI-GDM The highlighted features of theproposed methods can be summarized as follows

(1) During the initial phase CD-GDM and IGI-GDMneed to standardize the decision data whereas there is noneed for standardization in the proposed method which isrelatively simple

(2) The GDMmatrix of the proposed method is an ITFNdecision-making matrix containing ITFNs only which is easyto handle whereas that of IGI-GDMremains a heterogeneousdecision-making matrix which is difficult to deal with

(3) In CD-GDM and IGI-GDM the integrated infor-mation of experts is expressed by original decision datatypes and real number type which contains less informationwhile the integrated information of experts in the proposedmethod is represented by ITFNswhich can express intuitivelyand describe satisfaction dissatisfaction and distribution ofexperts

(4) The weights of attributes in IGI-GDM are given bydecision makers in advance which are subjective CD-GDMdetermine the weights of attributes by nonlinear program-ming model whereas they construct a multiple objectivelinear programming model to establish attributesrsquo weightsThus the proposed method is more objective and effective

6 Conclusions

In this paper we put forward a new aggregation approachto solve such heterogeneous MAGDM problems in whichthe weights of the attributes are incompletely known Thekey features of the proposed method are listed as follows(1) a new similarity measure of ITFNs is proposed (2) anew general approach to aggregating decision informationinto ITFNs is proposed It can not only accommodate morecomplicated data of types including INs TFNs TrFNs andTIFNs but also take importance of experts into account (3) anew multiple objective mathematical programming model isdeveloped for determining the attribute weights objectivelyunder intuitionistic triangular fuzzy environment (4) anew method is presented to solve heterogeneous MAGDMproblems which considers fully the indeterminacy of the

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

16 Complexity

Table4Com

paris

onsb

etwe

enthec

urrent

metho

dandexistingmetho

ds

Characteris

tics

Metho

d[34]

Metho

d[35]

Metho

d[37]

Thec

urrent

metho

dAttributev

alues

Ins

RNsInsTF

NsTrFN

sIFNsLV

sRN

sInsTF

NsTrFN

sRN

sInsTF

NsTrFN

sTIFN

sLV

sAttributen

ormalizing

Need

Need

Need

Noneed

Solvep

roblem

MAG

DM

HMAG

DM

HMAG

DM

HMAG

DM

Satisfactoryinformation

Derived

from[005]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)lt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Dissatisfactory

inform

ation

Derived

from[051]

Basedon

elem

entssatisfy

119889(119909119896 119894119895119860max119895)gt119889(119909119896 119894119895

119860min 119895)Ba

sedon

allelementsby

TOPS

ISBa

sedon

allelementsby

TOPS

IS

Uncertain

inform

ation

Non

eBa

sedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

dista

nces

from

elem

entsto

the

largestmiddlea

ndsm

allestgrade

Basedon

geom

etry

entro

pymetho

d

Expertwe

ight

Non

eNon

eNon

eDerived

from

consistency

degree

Attributew

eigh

tGiven

inadvance

Bymulti-ob

jectiveIFprogramming

Bymulti-ob

jectiveIVIF

programming

Bymulti-ob

jectivelinearp

rogram

ming

Aggregatedinform

ation

IFNs

IFNs

IVIFNs

ITFN

s

Rank

metho

dBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

scorea

ndaccuracy

functio

nBa

sedon

RCC

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 17

DMs in the assessment thus the final decision results derivedby the proposed method are more reasonable Additionallythe proposed method can be also appropriate for the com-plex multiattribute large-group decision-making problems[51] Future research will extend the developed method toheterogeneous MAGDM with complete unknown weightinformation under complex fuzzy environment Meanwhileas the scale of group increases and the decision makershave different backgrounds and levels of knowledge it isdifficult to achieve consensus among decisionmakers [52 53]Therefore it will be very interesting in future studies todiscuss the consensus reaching mechanism in the large-scaleHMAGDM

Data Availability

We state that the data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Nos 61602219 71740021 11861034 and11461030) the Colleges Humanities Social Science ResearchProject of Jiangxi Province of China (No GL162047) and theScience and Technology Project of Jiangxi Province of China(No GJJ188412) the Natural Science Foundation of JiangxiProvince of China (No 20192BAB207012) and the ldquoThirteenFiverdquo ProgrammingProject of Jiangxi Province Social Science(No 18GL13)

References

[1] Z L Yue ldquoTOPSIS-based group decision-makingmethodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277 pp141ndash153 2014

[2] P Liu and F Teng ldquoMultiple criteria decision making methodbased on normal interval-valued intuitionistic fuzzy general-ized aggregation operatorrdquo Complexity vol 21 no 5 pp 277ndash290 2016

[3] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic variablerdquo Complexity vol 21 no 5 pp 20ndash302016

[4] X Chen H Zhang and Y Dong ldquoThe fusion process with het-erogeneous preference structures in group decision making asurveyrdquo Information Fusion vol 24 pp 72ndash83 2015

[5] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[6] C De Maio G Fenza V Loia F Orciuoli and E Herrera-Viedma ldquoA framework for context-aware heterogeneous groupdecision making in business processesrdquo Knowledge-Based Sys-tems vol 102 pp 39ndash50 2016

[7] C Yue ldquoA projection-based approach to software quality eval-uation from the users perspectivesrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash13 2018

[8] C Yue ldquoA normalized projection-based group decision-makingmethod with heterogeneous decision information and appli-cation to software development effort assessmentrdquo AppliedIntelligence 2019

[9] C Yue ldquoNormalization of attribute values with interval infor-mation in group decision-making setting With an applicationto software quality evaluationrdquo Journal of Experimental ampeoretical Artificial Intelligence vol 31 no 3 pp 475ndash492 2019

[10] Z Yue and Y Jia ldquoA group decision making model with hybridintuitionistic fuzzy informationrdquo Computers amp Industrial Engi-neering vol 87 pp 202ndash212 2015

[11] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[12] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[13] X Wang and J Cai ldquoA group decision-making model based ondistance-based VIKOR with incomplete heterogeneous infor-mation and its application to emergency supplier selectionrdquoKybernetes vol 46 no 3 pp 501ndash529 2017

[14] H Zhang Y Dong and X Chen ldquoThe 2-rank consensus reach-ing model in the multigranular linguistic multiple-attributegroup decision-makingrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 48 no 12 pp 2080ndash2094 2018

[15] Y Dong H Zhang and E Herrera-Viedma ldquoConsensus reach-ing model in the complex and dynamic MAGDM problemrdquoKnowledge-Based Systems vol 106 pp 206ndash219 2016

[16] W Zhang Y Ju X Liu and M Giannakis ldquoA mathematicalprogramming-based method for heterogeneous multicriteriagroup decision analysis with aspirations and incomplete pref-erence informationrdquo Computers amp Industrial Engineering vol113 pp 541ndash557 2017

[17] J Dong and S Wan ldquoA PROMETHEE-FLP method for hetero-geneous multi-attributes group decision makingrdquo IEEE Accessvol 6 pp 46656ndash46667 2018

[18] Z Yue and Y Jia ldquoA direct projection-based group decision-making methodology with crisp values and interval datardquo SoComputing vol 21 no 9 pp 2395ndash2405 2017

[19] C Yue ldquoNormalized projection approach to group decision-making with hybrid decision informationrdquo International Jour-nal of Machine Learning and Cybernetics vol 9 no 8 pp 1365ndash1375 2018

[20] S Das D Guha and R Mesiar ldquoExtended bonferroni meanunder intuitionistic fuzzy environment based on a strict t-conormrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2083ndash2099 2017

[21] G Li G Kou and Y Peng ldquoA group decision making model forintegrating heterogeneous informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 48 no 6 pp 982ndash992 2018

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[23] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[24] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

18 Complexity

[25] J Xu Y Zhong and S Wan ldquoIncentive adaptive trust modelbased on integrated intuitionistic fuzzy informationrdquo Journal ofElectronics and Information Technology vol 38 no 4 pp 803ndash810 2016

[26] C Yue ldquoAn interval-valued intuitionistic fuzzy projection-based approach and application to evaluating knowledge trans-fer effectivenessrdquo Neural Computing and Applications 2018

[27] C Yue ldquoAn intuitionistic fuzzy projection-based approach andapplication to software quality evaluationrdquo So Computing2019

[28] Z L Yue ldquoThe comprehensive evaluation of urban environ-mental quality based on intuitionistic fuzzy setrdquo Mathematicsin Practice andeory vol 38 no 8 pp 51ndash58 2008

[29] Z L Yue Y Y Jia and X L Huang ldquoMultiple attribute decisionmakingmethod based on intuitionistic fuzzy setrdquo inProceedingsof the Second International Conference on Intelligent InformationManagement Systems and Technology vol 24 pp 137ndash140 2007

[30] Z L Yue Y Y Jia and G D Ye ldquoAn approach for multipleattribute group decision making based on intuitionistic fuzzyinformationrdquo International Journal of Uncertainty Fuzzinessand Knowledge-Based System vol 17 no 3 pp 317ndash332 2009

[31] Z Yue ldquoAggregating crisp values into intuitionistic fuzzy num-ber for group decision makingrdquo Applied Mathematical Mod-elling Simulation and Computation for Engineering and Envi-ronmental Systems vol 38 no 11-12 pp 2969ndash2982 2014

[32] Z L Yue ldquoAn approach to aggregating interval numbers intointerval-valued intuitionistic fuzzy information for group deci-sion makingrdquo Expert Systems with Applications vol 38 no 5pp 6333ndash6338 2011

[33] Z L Yue and Y Y Jia ldquoA method to aggregate crisp valuesinto interval-valued intuitionistic fuzzy information for groupdecision makingrdquo Applied So Computing vol 13 no 5 pp2304ndash2317 2013

[34] Z Yue ldquoA group decision making approach based on aggre-gating interval data into interval-valued intuitionistic fuzzyinformationrdquoAppliedMathematicalModelling vol 38 no 2 pp683ndash698 2014

[35] J Xu S-PWan and J-Y Dong ldquoAggregating decision informa-tion into Atanassovrsquos intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Applied SoComputing vol 41 pp 331ndash351 2016

[36] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods andApplications Springer Heidelberg Germany 1981

[37] S Wan J Xu and J Dong ldquoAggregating decision informationinto interval-valued intuitionistic fuzzy numbers for hetero-geneous multi-attribute group decision makingrdquo Knowledge-Based Systems vol 113 pp 155ndash170 2016

[38] F Liu and X H Yuan ldquoFuzzy number intuitionistic fuzzy setrdquoFuzzy Systems and Mathematics vol 21 no 1 pp 88ndash91 2007

[39] M H Shu C H Cheng and J R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[40] X FWang ldquoFuzzynumber intuitionistic fuzzy geometric aggre-gation operators and their application to decision makingrdquoControl and Decision vol 10 no 2 pp 104ndash111 2008

[41] G W Wei ldquoInduced fuzzy number intuitionistic fuzzy orderedweighted averaging (I-FIFOWA) operator and its application togroup decision makingrdquo Chinese Journal of Management vol 7no 6 pp 903ndash908 2010

[42] Y Gao D Q Zhou C C Liu and L Zhang ldquoTriangular fuzzynumber intuitionistic fuzzy aggregation operators and theirapplication based on interactionrdquo Systems Engineering eoryamp Practice vol 32 no 9 pp 1964ndash1972 2012

[43] S Yu and Z Xu ldquoAggregation and decision making using intu-itionisticmultiplicative triangular fuzzy informationrdquo Journal ofSystems Science and Systems Engineering vol 23 no 1 pp 20ndash38 2014

[44] Q Qin F Liang L Li Y-W Chen and G-F Yu ldquoA TODIM-based multi-criteria group decision making with triangularintuitionistic fuzzy numbersrdquo Applied So Computing vol 55pp 93ndash107 2017

[45] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[46] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[47] T Y Chen and C H Li ldquoDetermining objective weights withintuitionistic fuzzy entropy measures a comparative analysisrdquoInformation Sciences vol 180 no 21 pp 4207ndash4222 2010

[48] B Farhadinia ldquoA theoretical development on the entropy ofinterval-valued fuzzy sets based on the intuitionistic distanceand its relationship with similarity measurerdquo Knowledge-BasedSystems vol 39 pp 79ndash84 2013

[49] C-B Cheng ldquoGroup opinion aggregationbased on a gradingprocess a method for constructing triangular fuzzy numbersrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1619ndash1632 2004

[50] X Li and X Chen ldquoMulti-criteria group decisionmaking basedon trapezoidal intuitionistic fuzzy informationrdquo Applied SoComputing vol 30 pp 454ndash461 2015

[51] A Labella Y Liu R M Rodrıguez and LMartınez ldquoAnalyzingthe performance of classical consensus models in large scalegroup decision making A comparative studyrdquo Applied SoComputing vol 67 pp 677ndash690 2018

[52] H Zhang Y Dong I Palomares-Carrascosa and H ZhouldquoFailure mode and effect analysis in a linguistic context a con-sensus-based multiattribute group decision-making approachrdquoIEEE Transactions on Reliability vol 68 no 2 pp 566ndash5822018

[53] H Zhang YDong F Chiclana and S Yu ldquoConsensus efficiencyin group decision making A comprehensive comparativestudy and its optimal designrdquo European Journal of OperationalResearch vol 275 no 2 pp 580ndash598 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom