Research Article MADM Problems with Correlation Coefficient of … · 2019. 5. 9. · Research...

Research ArticleMADM Problems with Correlation Coefficient of TrapezoidalFuzzy Intuitionistic Fuzzy Sets

John Robinson and Henry Amirtharaj

Bishop Heber College, Tiruchirappalli, Tamil Nadu 620017, India

Correspondence should be addressed to John Robinson; [email protected]

Received 4 May 2014; Accepted 8 July 2014; Published 4 August 2014

Academic Editor: Panagote M. Pardalos

Copyright © 2014 J. Robinson and H. Amirtharaj. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper discusses on the notion of trapezoidal fuzzy intuitionistic fuzzy sets (TzFIFSs) and some of the arithmetic operationsof the same. Correlation coefficient of TzFIFS is proposed based on the membership, nonmembership, and hesitation degrees. Theweighted averaging (WA) operator and the weighted geometric (WG) operator are proposed for TzFIFSs. Based on these operatorsand the correlation coefficient defined for the TzFIFS, new multiattribute decision making (MADM) models are proposed andnumerical illustration is given.

1. Introduction

Intuitionistic fuzzy sets (IFSs) proposed by Atanassov [1–3]are a generalization of the concept of fuzzy sets. Atanassovand Gargov [4] expanded the IFSs, using interval value toexpress membership and nonmembership function of IFSs.Liu and Yuan [5] introduced the concept of fuzzy numberIFSs as a further generalization of IFSs. Among the worksdone in IFSs, Szmidt and Kacprzyk [6–8] can be mentioned.To the best of our knowledge, Burillo et al. [9] proposed thedefinition of intuitionistic fuzzy number (IFN) and studiedthe perturbations of IFN and the first properties of the corre-lation between these numbers.Many researchers have appliedthe IFS theory to the field of decision making. Recentlysome researches [5, 10–14] showed great interest in the fuzzynumber IFSs and applied it to the field of decision making.Based on the arithmetic aggregation operators, Xu and Yager[15], Xu and Chen [16, 17], and Wang [13] developed somenew geometric aggregation operators and intuitionistic fuzzyordered weighted averaging (IFOWA) operator. Szmidt andKacprzyk [7] proposed some solution concepts in group deci-sion making with intuitionistic (individual and social) fuzzypreference relations. Szmidt and Kacprzyk [8] investigatedthe consensus-reaching process in group decision makingbased on individual intuitionistic fuzzy preference relations.Herrera et al. [18] developed an aggregation process for

combining numerical, interval valued, and linguistic infor-mation and then proposed different extensions of this processto deal with contexts in which information such as IFSsor multigranular linguistic information can appear. Xu andYager [15] developed some geometric aggregation operatorsfor MADM problems. Li [19] investigated MADM problemswith intuitionistic fuzzy information and constructed severallinear programming models to generate optimal weights forattributes.

Multiattribute decisionmaking (MADM) problems are ofimportance in most kinds of fields such as engineering, eco-nomics, andmanagement. It is obvious that much knowledgein the real world is fuzzy rather than precise. Imprecisioncomes from a variety of sources such as unquantifiableinformation [20]. In many situations decision makers haveimprecise/vague information about alternatives with respectto attributes. It is well known that the conventional decisionmaking analysis using different techniques and tools has beenfound to be inadequate to handle uncertainty of fuzzy data.To overcome this problem, the concept of fuzzy approach hasbeen used in the evaluation of decision making systems. Fora long period of time, efforts have been made in designingvarious decision making systems suitable for the arising day-to-day problems. MADM problems are widespread in real-life decision making situations and the problem is to find a

Hindawi Publishing CorporationAdvances in Decision SciencesVolume 2014, Article ID 159126, 10 pageshttp://dx.doi.org/10.1155/2014/159126

2 Advances in Decision Sciences

desirable solution from a finite number of feasible alterna-tives assessed on multiple attributes, both quantitative andqualitative [21]. In order to choose a desirable solution, thedecisionmaker oftenprovides his/her preference informationwhich takes the form of numerical values, such as exactvalues, interval number values, and fuzzy numbers. However,under many conditions, numerical values are inadequateor insufficient to model real-life decision problems. Indeed,human judgments including preference information maybe stated in intuitionistic fuzzy information, especially intrapezoidal fuzzy intuitionistic fuzzy information. Hence,MADM problems under intuitionistic fuzzy or trapezoidalfuzzy intuitionistic fuzzy environment are an interesting areaof study for researchers in the recent days.

It is well known that the conventional correlation analysisusing probabilities and statistics has been found to be inad-equate to handle uncertainty of failure data and modeling.Themethod tomeasure the correlation between two variablesinvolving fuzziness is a challenge to classical statistical theory.Park et al. [22] proposed the correlation coefficient of intervalvalued intuitionistic fuzzy sets. Robinson and Amirtharaj[23–26] proposed correlation coefficient for vague sets,interval vague sets, triangular and trapezoidal IFSs, and arevised correlation coefficient for triangular and trapezoidalIFSs using graded mean integration representation. In thispaper, a novelmethod of correlation coefficient of trapezoidalfuzzy intuitionistic fuzzy sets (TzFIFSs) is proposed anddeveloped by taking into account themembership, nonmem-bership, and the hesitation degrees of TzFIFSs. The weightedaveraging (WA) and weighted geometric (WG) operatorsfor TzFIFSs are proposed for MADM problems. Based onthese operators, an approach is suggested to solve uncertainmultiple attribute group decision making problems, wherethe attribute values are trapezoidal fuzzy intuitionistic fuzzynumbers. A new algorithm is developed to solve the MADMproblems in which the correlation coefficient of TzFIFSs isused for ranking alternatives.

2. Arithmetic Operations for TzFIFS

Arithmetic operations of TzFIFNs are based on the arith-metic operations of TzFNs. In TzFIFN the membership andnonmembership degrees take the form of trapezoidal fuzzynumber. The basic concepts related to TzFIFNs are presentedin the following.

Definition 1 (trapezoidal fuzzy number (TzFN)). 𝐴 =(𝑎, 𝑏, 𝑐, 𝑑) is called a trapezoidal fuzzy number, if the mem-bership function 𝜇

𝐴: 𝑅 → [0, 1] is expressed as

𝜇𝐴 (𝑥) =

{{{{{{

{{{{{{

{

𝑥 − 𝑎

𝑏 − 𝑎for 𝑎 ≤ 𝑥 < 𝑏

1 for 𝑏 ≤ 𝑥 < 𝑐𝑐 − 𝑥

𝑑 − 𝑐for 𝑐 ≤ 𝑥 < 𝑑

0 otherwise,

(1)

where 𝑥 ∈ 𝑅, 0 ≤ 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑 ≤ 1.

Definition 2 (trapezoidal fuzzy intuitionistic fuzzy num-ber (TzFIFN)). Let 𝑋 be a nonempty set. Then 𝐴 ={⟨𝑥, 𝜇

𝐴(𝑥), 𝛾𝐴(𝑥)⟩/𝑥 ∈ 𝑋} is called a trapezoidal fuzzy

intuitionistic fuzzy number if 𝜇𝐴(𝑥) = (𝜇

𝐴1

(𝑥), 𝜇𝐴2

(𝑥),

𝜇𝐴3

(𝑥), 𝜇𝐴4

(𝑥)) and 𝛾𝐴(𝑥) = (𝛾

𝐴1

(𝑥), 𝛾𝐴2

(𝑥), 𝛾𝐴3

(𝑥), 𝛾𝐴4

(𝑥))

are trapezoidal fuzzy numbers, which can express the mem-bership degree and the nonmembership degree of 𝑥 in𝑋 andfulfill 0 ≤ 𝜇

𝐴4

(𝑥) + 𝛾𝐴4

(𝑥) ≤ 1, for all 𝑥 ∈ 𝑋.An intuitionistic fuzzy number expressed on the basis

of trapezoidal fuzzy number is called trapezoidal fuzzyintuitionistic fuzzy number.

𝜋𝐴(𝑥) = (𝜋

𝐴1

(𝑥), 𝜋𝐴2

(𝑥), 𝜋𝐴3

(𝑥), 𝜋𝐴4

(𝑥)) is called thehesitation degree of the given trapezoidal fuzzy intuitionisticfuzzy set. Also we have 0 ≤ 𝜋

𝐴1

(𝑥), 𝜋𝐴2

(𝑥), 𝜋𝐴3

(𝑥), 𝜋𝐴4

(𝑥) ≤

1, for all 𝑥 ∈ 𝑋.Suppose

𝐴 = {⟨𝑥, [𝜇𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,

[𝛾𝐴1(𝑥) , 𝛾𝐴

2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)]⟩ × (𝑥 ∈ 𝑋)

−1} ,

𝐵 = {⟨𝑥, [𝜇𝐵1(𝑥) , 𝜇𝐵

2(𝑥) , 𝜇𝐵

3(𝑥) , 𝜇𝐵

4(𝑥)] ,

[𝛾𝐵1(𝑥) , 𝛾𝐵

2(𝑥) , 𝛾𝐵

3(𝑥) , 𝛾𝐵

4(𝑥)]⟩ × (𝑥 ∈ 𝑋)

−1}

(2)

are two TzFIFNs; then according to the above operation rulesof intuitionistic fuzzy numbers and the operation rules oftrapezoidal fuzzy numbers, the operational rules of TzFIFNsare as follows:

𝐴 + 𝐵 = {⟨[𝜇𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,


2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)]⟩

+ ⟨[𝜇𝐵1(𝑥) , 𝜇𝐵

2(𝑥) , 𝜇𝐵

4(𝑥)] ,


2(𝑥) , 𝛾𝐵

3(𝑥) , 𝛾𝐵

4(𝑥)]⟩}

= {⟨[𝜇𝐴1(𝑥) + 𝜇𝐵

1(𝑥) − 𝜇𝐴

1(𝑥) ⋅ 𝜇𝐵

1(𝑥) ,

𝜇𝐴2(𝑥) + 𝜇𝐵

2(𝑥) − 𝜇𝐴

2(𝑥) ⋅ 𝜇𝐵

2(𝑥) ,


3(𝑥) − 𝜇𝐴

3(𝑥) ⋅ 𝜇𝐵

3(𝑥) ,


4(𝑥) − 𝜇𝐴

4(𝑥) ⋅ 𝜇𝐵

4(𝑥)] ,

[𝛾𝐴1(𝑥) ⋅ 𝛾𝐵

1(𝑥) , 𝛾𝐴

2(𝑥) ⋅ 𝛾𝐵

2(𝑥) ,

𝛾𝐴3(𝑥) ⋅ 𝛾𝐵

3(𝑥) , 𝛾𝐴

4(𝑥) ⋅ 𝛾𝐵

4(𝑥)]⟩} ,

(3)

Advances in Decision Sciences 3

𝐴 ⋅ 𝐵 = {⟨[𝜇𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,


2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)]⟩

⋅ ⟨[𝜇𝐵1(𝑥) , 𝜇𝐵

2(𝑥) , 𝜇𝐵

4(𝑥)] ,


2(𝑥) , 𝛾𝐵

3(𝑥) , 𝛾𝐵

4(𝑥)]⟩}

= {⟨[𝜇𝐴1(𝑥) ⋅ 𝜇𝐵

1(𝑥) , 𝜇𝐴

2(𝑥) ⋅ 𝜇𝐵

2(𝑥) ,

𝜇𝐴3(𝑥) ⋅ 𝜇𝐵

3(𝑥) , 𝜇𝐴

4(𝑥) ⋅ 𝜇𝐵

4(𝑥)] ,

[𝛾𝐴1(𝑥) + 𝛾𝐵

1(𝑥) − 𝛾𝐴

1(𝑥) ⋅ 𝛾𝐵

1(𝑥) ,

𝛾𝐴2(𝑥) + 𝛾𝐵

2(𝑥) − 𝛾𝐴

2(𝑥) ⋅ 𝛾𝐵

2(𝑥) ,


3(𝑥) − 𝛾𝐴

3(𝑥) ⋅ 𝛾𝐵

3(𝑥) ,


4(𝑥) − 𝛾𝐴

4(𝑥) ⋅ 𝛾𝐵

4(𝑥)]⟩} ,

(4)

𝜆𝐴 = 𝜆 ([𝜇𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,


2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)])

= {⟨[1 − (1 − 𝜇𝐴1(𝑥))𝜆

, 1 − (1 − 𝜇𝐴2(𝑥))𝜆

,

1 − (1 − 𝜇𝐴3(𝑥))𝜆

, 1 − (1 − 𝜇𝐴4(𝑥))𝜆

] ,

[(𝛾𝐴1(𝑥))𝑛

, (𝛾𝐴2(𝑥))𝑛

,

(𝛾𝐴3(𝑥))𝑛

, (𝛾𝐴4(𝑥))𝑛

]⟩} ,

(5)

𝐴𝜆= ([𝜇

𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,


2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)])𝜆

= {⟨[(𝜇𝐴1(𝑥))𝑛

, (𝜇𝐴2(𝑥))𝑛

,

(𝜇𝐴3(𝑥))𝑛

, (𝜇𝐴4(𝑥))𝑛

] ,

[1 − (1 − 𝛾𝐴1(𝑥))𝜆

, 1 − (1 − 𝛾𝐴2(𝑥))𝜆

,

1 − (1 − 𝛾𝐴3(𝑥))𝜆

, 1 − (1 − 𝛾𝐴4(𝑥))𝜆

]⟩} ,

𝜆 ≥ 0.

(6)

For the above operation rules, the following are true:

(i) 𝐴 + 𝐵 = 𝐵 + 𝐴,

(ii) 𝐴 ⋅ 𝐵 = 𝐵 ⋅ 𝐴,

(iii) 𝜆(𝐴 + 𝐵) = 𝜆𝐴 + 𝜆𝐵, 𝜆 ≥ 0,

(iv) 𝜆1𝐴 + 𝜆

2𝐴 = (𝜆

1+ 𝜆2)𝐴, 𝜆1, 𝜆2≥ 0,

(v) 𝐴𝜆1 ⋅ 𝐴𝜆2 = (𝐴)𝜆1+𝜆2 , 𝜆1, 𝜆2≥ 0.

3. Correlation Coefficient of TzFIFS

Let 𝑋 = {𝑥1, 𝑥2, . . . , 𝑥

𝑛} be the finite universal set and let 𝐴,

𝐵 ∈ TzFIFS(𝑋) be given by

𝐴 = {⟨𝑥, [𝜇𝐴1(𝑥) , 𝜇𝐴

2(𝑥) , 𝜇𝐴

3(𝑥) , 𝜇𝐴

4(𝑥)] ,


2(𝑥) , 𝛾𝐴

3(𝑥) , 𝛾𝐴

4(𝑥)]⟩ × (𝑥 ∈ 𝑋)

−1} ,

𝐵 = {⟨𝑥, [𝜇𝐵1(𝑥) , 𝜇𝐵

2(𝑥) , 𝜇𝐵

3(𝑥) , 𝜇𝐵

4(𝑥)] ,


2(𝑥) , 𝛾𝐵

3(𝑥) , 𝛾𝐵

4(𝑥)]⟩ × (𝑥 ∈ 𝑋)

−1} ,

(7)

which are two trapezoidal fuzzy intuitionistic fuzzy numbers.Then the correlation of trapezoidal fuzzy intuitionistic

fuzzy numbers (TzFIFNs) is defined as follows.

Definition 3. Let 𝐴 = {⟨𝑥, 𝜇𝐴(𝑥), 𝛾𝐴(𝑥)⟩/𝑥 ∈ 𝑋} be a

trapezoidal fuzzy intuitionistic fuzzy number, where 𝜇𝐴(𝑥) =

(𝜇𝐴1

(𝑥), 𝜇𝐴2

(𝑥), 𝜇𝐴3

(𝑥), 𝜇𝐴4

(𝑥)) is the membership degree,𝛾𝐴(𝑥) = (𝛾

𝐴1

(𝑥), 𝛾𝐴2

(𝑥), 𝛾𝐴3

(𝑥), 𝛾𝐴4

(𝑥)) is the nonmember-ship degree, and 𝜋

𝐴(𝑥) = (𝜋

𝐴1

(𝑥), 𝜋𝐴2

(𝑥), 𝜋𝐴3

(𝑥), 𝜋𝐴4

(𝑥)) isthe hesitation degree. Then the trapezoidal fuzzy intuitionis-tic energy of the sets 𝐴 and 𝐵 is defined as

𝐸TzFIFS (𝐴)

=1

4

𝑛

∑

𝑖=1

[𝜇2

𝐴(𝑥𝑖) + 𝛾2

𝐴(𝑥𝑖) + 𝜋2

𝐴(𝑥𝑖)]

=1

4

𝑛

∑

𝑖=1

{[(𝜇2

𝐴1

(𝑥𝑖) + 𝜇2

𝐴2

(𝑥𝑖) + 𝜇2

𝐴3

(𝑥𝑖) + 𝜇2

𝐴4

(𝑥𝑖))]

+ [(𝛾2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖))]

+ [(𝜋2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖))]} ,

(8)

𝐸TzFIFS (𝐵)

=1

4

𝑛

∑

𝑖=1

[𝜇2

𝐵(𝑥𝑖) + 𝛾2

𝐵(𝑥𝑖) + 𝜋2

𝐵(𝑥𝑖)]

=1

4

𝑛

∑

𝑖=1

{[(𝜇2

𝐵1

(𝑥𝑖) + 𝜇2

𝐵2

(𝑥𝑖) + 𝜇2

𝐵3

(𝑥𝑖) + 𝜇2

𝐵4

(𝑥𝑖))]

+ [(𝛾2

𝐵1

(𝑥𝑖) + 𝛾2

𝐵2

(𝑥𝑖) + 𝛾2

𝐵3

(𝑥𝑖) + 𝛾2

𝐵4

(𝑥𝑖))]

+ [(𝜋2

𝐵1

(𝑥𝑖) + 𝜋2

𝐵2

(𝑥𝑖) + 𝜋2

𝐵3

(𝑥𝑖) + 𝜋2

𝐵4

(𝑥𝑖))]} .

(9)

Definition 4. Let 𝐴 = {⟨𝑥, 𝜇𝐴(𝑥), 𝛾𝐴(𝑥)⟩/𝑥 ∈ 𝑋} be a

trapezoidal fuzzy intuitionistic fuzzy number, where 𝜇𝐴(𝑥) =

(𝜇𝐴1

(𝑥), 𝜇𝐴2

(𝑥), 𝜇𝐴3

(𝑥), 𝜇𝐴4

(𝑥)) is the membership degree,𝛾𝐴(𝑥) = (𝛾

𝐴1

(𝑥), 𝛾𝐴2

(𝑥), 𝛾𝐴3

(𝑥), 𝛾𝐴4

(𝑥)) is the nonmember-ship degree, and 𝜋

𝐴(𝑥) = (𝜋

𝐴1

(𝑥), 𝜋𝐴2

(𝑥), 𝜋𝐴3

(𝑥), 𝜋𝐴4

(𝑥)) is


the hesitation degree. Then the correlation between the sets𝐴 and 𝐵 is defined as

𝐶TzFIFS (𝐴, 𝐵)

=1

4

𝑛

∑

𝑖=1

[𝜇𝐴(𝑥𝑖) 𝜇𝐵(𝑥𝑖) + 𝛾𝐴(𝑥𝑖) 𝛾𝐵(𝑥𝑖) + 𝜋𝐴(𝑥𝑖) 𝜋𝐵(𝑥𝑖)]

=1

4

𝑛

∑

𝑖=1

{[(𝜇𝐴1

(𝑥𝑖) 𝜇𝐵1

(𝑥𝑖) + 𝜇𝐴2


(𝑥𝑖)

+𝜇𝐴3




(𝑥𝑖))]

+ [(𝛾𝐴1

(𝑥𝑖) 𝛾𝐵1

(𝑥𝑖) + 𝛾𝐴2


(𝑥𝑖)

+𝛾𝐴3




(𝑥𝑖))]

+ [(𝜋𝐴1

(𝑥𝑖) 𝜋𝐵1

(𝑥𝑖) + 𝜋𝐴2


(𝑥𝑖)

+𝜋𝐴3




(𝑥𝑖))]} .

(10)

The correlation coefficient is given as

𝐾TzFIFS (𝐴, 𝐵) =𝐶TzFIFS (𝐴, 𝐵)

√𝐸TzFIFS (𝐴) ⋅ 𝐸TzFIFS (𝐵),

0 ≤ 𝐾TzFIFS (𝐴, 𝐵) ≤ 1.

(11)

Proposition 5. For 𝐴, 𝐵 ∈ TzFIFS(𝑋) the following are true:

(1) 0 ≤ 𝐾TzFIFS(𝐴, 𝐵) ≤ 1,

(2) 𝐶TzFIFS(𝐴, 𝐵) = 𝐶TzFIFS(𝐵, 𝐴),

(3) 𝐾TzFIFS(𝐴, 𝐵) = 𝐾TzFIFS(𝐵, 𝐴).

Theorem 6. For 𝐴, 𝐵 ∈ TzFIFS(𝑋), then 0 ≤ 𝐾TzFIFS(𝐴, 𝐵) ≤1.

Proof. Since 𝐶TzFIFS(𝐴, 𝐵) ≥ 0, it can be proved that𝐾TzFIFS(𝐴, 𝐵) ≤ 1.

For any arbitrary real number 𝜉, the following relation istrue:

0 ≤

𝑛

∑

𝑖=1

{(𝜇𝐴1

(𝑥𝑖) − 𝜉𝜇

𝐵1

(𝑥𝑖))2

+ (𝜇𝐴2


𝐵2

(𝑥𝑖))2

+ (𝜇𝐴3


𝐵3

(𝑥𝑖))2

+ (𝜇𝐴4


𝐵4

(𝑥𝑖))2

+ (𝛾𝐴1

(𝑥𝑖) − 𝜉𝛾

𝐵1

(𝑥𝑖))2

+ (𝛾𝐴2


𝐵2

(𝑥𝑖))2

+ (𝛾𝐴3


𝐵3

(𝑥𝑖))2

+ (𝛾𝐴4


𝐵4

(𝑥𝑖))2

+ (𝜋𝐴1

(𝑥𝑖) − 𝜉𝜋

𝐵1

(𝑥𝑖))2

+ (𝜋𝐴2


𝐵2

(𝑥𝑖))2

+ (𝜋𝐴3


𝐵3

(𝑥𝑖))2

+ (𝜋𝐴4


𝐵4

(𝑥𝑖))2

}

=

𝑛

∑

𝑖=1

{(𝜇2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴1

(𝑥𝑖))

+ (𝜇2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖))

+ (𝜇2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖))

+ (𝜇2

𝐴4

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖))

− 2𝜉 (𝜇𝐴1




(𝑥𝑖)

+𝜋𝐴1


(𝑥𝑖))

− 2𝜉 (𝜇𝐴2




(𝑥𝑖)

+𝜋𝐴2


(𝑥𝑖))

− 2𝜉 (𝜇𝐴3




(𝑥𝑖)

+𝜋𝐴3


(𝑥𝑖))

− 2𝜉 (𝜇𝐴4




(𝑥𝑖)

+𝜋𝐴4


(𝑥𝑖))

+ 𝜉2(𝜇2

𝐵1

(𝑥𝑖) + 𝛾2

𝐵1

(𝑥𝑖) + 𝜋2

𝐵1

(𝑥𝑖))

+ 𝜉2(𝜇2

𝐵2

(𝑥𝑖) + 𝛾2

𝐵2

(𝑥𝑖) + 𝜋2

𝐵2

(𝑥𝑖))

+ 𝜉2(𝜇2

𝐵3

(𝑥𝑖) + 𝛾2

𝐵3

(𝑥𝑖) + 𝜋2

𝐵3

(𝑥𝑖))

+𝜉2(𝜇2

𝐵4

(𝑥𝑖) + 𝛾2

𝐵4

(𝑥𝑖) + 𝜋2

𝐵4

(𝑥𝑖))} .

(12)

Hence,

{

𝑛

∑

𝑖=1

[(𝜇𝐴1




(𝑥𝑖)

+ 𝜇𝐴3




(𝑥𝑖)

+ 𝛾𝐴1




(𝑥𝑖)

+ 𝛾𝐴3




(𝑥𝑖)

+ 𝜋𝐴1




(𝑥𝑖)

+𝜋𝐴3




(𝑥𝑖))]}

2

≤ (

𝑛

∑

𝑖=1

{(𝜇2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴1

(𝑥𝑖))

+ (𝜇2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖))

+ (𝜇2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖))

+ (𝜇2

𝐴4

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖))}


×

𝑛

∑

𝑖=1

{(𝜇2

𝐵1

(𝑥𝑖) + 𝛾2

𝐵1

(𝑥𝑖) + 𝜋2

𝐵1

(𝑥𝑖))

+ (𝜇2

𝐵2

(𝑥𝑖) + 𝛾2

𝐵2

(𝑥𝑖) + 𝜋2

𝐵2

(𝑥𝑖))

+ (𝜇2

𝐵3

(𝑥𝑖) + 𝛾2

𝐵3

(𝑥𝑖) + 𝜋2

𝐵3

(𝑥𝑖))}

+ (𝜇2

𝐵4

(𝑥𝑖) + 𝛾2

𝐵4

(𝑥𝑖) + 𝜋2

𝐵4

(𝑥𝑖))) .

(13)

The above inequality can be written as

({

𝑛

∑

𝑖=1

[𝜇𝐴1




(𝑥𝑖)

+ 𝜇𝐴3




(𝑥𝑖)

+ 𝛾𝐴1




(𝑥𝑖)

+ 𝛾𝐴3




(𝑥𝑖)

+ 𝜋𝐴1




(𝑥𝑖)

+𝜋𝐴3




(𝑥𝑖)]2

}

× ((

𝑛

∑

𝑖=1

{(𝜇2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴1

(𝑥𝑖))

+ (𝜇2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖))

+ (𝜇2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖))

+ (𝜇2

𝐴4

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖))}

×

𝑛

∑

𝑖=1

{(𝜇2

𝐵1

(𝑥𝑖) + 𝛾2

𝐵1

(𝑥𝑖) + 𝜋2

𝐵1

(𝑥𝑖))

+ (𝜇2

𝐵2

(𝑥𝑖) + 𝛾2

𝐵2

(𝑥𝑖) + 𝜋2

𝐵2

(𝑥𝑖))

+ (𝜇2

𝐵3

(𝑥𝑖) + 𝛾2

𝐵3

(𝑥𝑖) + 𝜋2

𝐵3

(𝑥𝑖))

+ (𝜇2

𝐵4

(𝑥𝑖) + 𝛾2

𝐵4

(𝑥𝑖) + 𝜋2

𝐵4

(𝑥𝑖))}))

−1

) .

≤ 1.

(14)

Therefore [𝐶TzFIFS(𝐴, 𝐵)]2/𝐸TzFIFS(𝐴) ⋅ 𝐸TzFIFS(𝐵) ≤ 1; hence

𝐾TzFIFS(𝐴, 𝐵) = 𝐶TzFIFS(𝐴, 𝐵)/√𝐸TzFIFS(𝐴) ⋅ 𝐸TzFIFS(𝐵) ≤ 1.

Theorem 7. Consider KTzFIFS(A,B) = 1 ⇔ A = B.

Proof. Considering the inequality in the proof of Theorem 6,then the equality holds if and only if the following are satisfied

(i) 𝜇𝐴1

(𝑥𝑖) = 𝜉𝜇

𝐵1

(𝑥𝑖), 𝜇𝐴2


𝐵2


(𝑥𝑖) =

𝜉𝜇𝐵3



𝐵4

(𝑥𝑖),

(ii) 𝛾𝐴1

(𝑥𝑖) = 𝜉𝛾

𝐵1

(𝑥𝑖), 𝛾𝐴2


𝐵2


(𝑥𝑖) =

𝜉𝛾𝐵3



𝐵4

(𝑥𝑖),

(iii) 𝜋𝐴1

(𝑥𝑖) = 𝜉𝜋

𝐵1

(𝑥𝑖), 𝜋𝐴2


𝐵2


(𝑥𝑖) =

𝜉𝜋𝐵3



𝐵4

(𝑥𝑖),

for some positive real 𝜉.As

𝜇𝐴1



(𝑥𝑖)

= 𝜇𝐵1

(𝑥𝑖) + 𝛾𝐵1

(𝑥𝑖) + 𝜋𝐵1

(𝑥𝑖) = 1,

𝜇𝐴2



(𝑥𝑖)

= 𝜇𝐵2



(𝑥𝑖) = 1,

𝜇𝐴3



(𝑥𝑖)

= 𝜇𝐵3



(𝑥𝑖) = 1,

𝜇𝐴4



(𝑥𝑖)

= 𝜇𝐵4



(𝑥𝑖) = 1,

(15)

then it means 𝜉 = 1, and therefore 𝐴 = 𝐵.

Theorem 8. 𝐶𝑇𝑟𝐹𝐼𝐹𝑆

(𝐴, 𝐵) = 0 ⇔ 𝐴 and 𝐵 are nonfuzzysets and satisfy the condition 𝜇

𝐴(𝑥𝑖) + 𝜇𝐵(𝑥𝑖) = 1 or 𝛾

𝐴(𝑥𝑖) +

𝛾𝐵(𝑥𝑖) = 1 or 𝜋

𝐴(𝑥𝑖) + 𝜋𝐵(𝑥𝑖) = 1, for all 𝑥

𝑖∈ 𝑋.

Proof. For all 𝑥𝑖∈ 𝑋, the following are true:

(𝜇𝐴1




(𝑥𝑖)

+𝜋𝐴1


(𝑥𝑖)) ≥ 0,

(𝜇𝐴2




(𝑥𝑖)

+𝜋𝐴2


(𝑥𝑖)) ≥ 0,

(𝜇𝐴3




(𝑥𝑖)

+𝜋𝐴3


(𝑥𝑖)) ≥ 0,

(𝜇𝐴4




(𝑥𝑖)

+𝜋𝐴4


(𝑥𝑖)) ≥ 0.

(16)


Hence,

{(𝜇𝐴1




(𝑥𝑖)

+𝜋𝐴1


(𝑥𝑖))

+ (𝜇𝐴2




(𝑥𝑖)

+𝜋𝐴2


(𝑥𝑖))

+ (𝜇𝐴3




(𝑥𝑖)

+𝜋𝐴3


(𝑥𝑖))

+ (𝜇𝐴4




(𝑥𝑖)

+𝜋𝐴4


(𝑥𝑖))} ≥ 0,

{𝜇𝐴1




(𝑥𝑖)

+ 𝜇𝐴3




(𝑥𝑖)

+ 𝛾𝐴1




(𝑥𝑖)

+ 𝛾𝐴3




(𝑥𝑖)

+ 𝜋𝐴1




(𝑥𝑖)

+ 𝜋𝐴3




(𝑥𝑖)} ≥ 0.

(17)

If 𝐶TzFIFS(𝐴, 𝐵) = 0 for all 𝑥𝑖 ∈ 𝑋, then the following shouldbe true:

𝜇𝐴1




(𝑥𝑖)

+ 𝜇𝐴3




(𝑥𝑖) = 0,

𝛾𝐴1




(𝑥𝑖)

+ 𝛾𝐴3




(𝑥𝑖) = 0,

𝜋𝐴1




(𝑥𝑖)

+ 𝜋𝐴3




(𝑥𝑖) = 0.

(18)

(i) If 𝜇𝐴1

(𝑥𝑖) = 1 then 𝜇

𝐵1

(𝑥𝑖) = 0 and 𝛾

𝐴1

(𝑥𝑖) =

𝜋𝐴1

(𝑥𝑖) = 0,

(ii) if 𝜇𝐴2


𝐵2


𝐴2

(𝑥𝑖) =

𝜋𝐴2

(𝑥𝑖) = 0,

(iii) if 𝜇𝐴3


𝐵3


𝐴3

(𝑥𝑖) =

𝜋𝐴3

(𝑥𝑖) = 0,

(iv) if 𝜇𝐴4


𝐵4


𝐴4

(𝑥𝑖) =

𝜋𝐴4

(𝑥𝑖) = 0.

Also,

(i) if 𝜇𝐵1


𝐴1


𝐵1

(𝑥𝑖) =

𝜋𝐵1

(𝑥𝑖) = 0,

(ii) if 𝜇𝐵2


𝐴2


𝐵2

(𝑥𝑖) =

𝜋𝐵2

(𝑥𝑖) = 0,

(iii) if 𝜇𝐵3


𝐴3


𝐵3

(𝑥𝑖) =

𝜋𝐵3

(𝑥𝑖) = 0,

(iv) if 𝜇𝐵4


𝐴4


𝐵4

(𝑥𝑖) =

𝜋𝐵4

(𝑥𝑖) = 0.

Hence 𝜇𝐴1

(𝑥𝑖) + 𝜇𝐵1

(𝑥𝑖) = 1, 𝜇

𝐴2


(𝑥𝑖) = 1,

𝜇𝐴3


(𝑥𝑖) = 1, and 𝜇

𝐴4


(𝑥𝑖) = 1.

Conversely, when𝐴 and𝐵 are nonfuzzy sets and 𝜇𝐴1

(𝑥𝑖)+

𝜇𝐵1

(𝑥𝑖) = 1, 𝜇

𝐴2


(𝑥𝑖) = 1, 𝜇

𝐴3


(𝑥𝑖) =

1, and 𝜇𝐴4


(𝑥𝑖) = 1.

If 𝜇𝐴1


𝐵1


𝐴1

(𝑥𝑖) = 𝜋𝐴1

(𝑥𝑖) = 0.

This property can be observed similarly for all otherentries. Therefore 𝐶TrFIFS(𝐴, 𝐵) = 0.

The cases 𝛾𝐴(𝑥𝑖) + 𝛾𝐵(𝑥𝑖) = 1 and 𝜋

𝐴(𝑥𝑖) + 𝜋𝐵(𝑥𝑖) = 1 can

be proved similarly.

Theorem 9. 𝐶TzFIFS(𝐴, 𝐴) = 1 ⇔ 𝐴 is a nonfuzzy set.

Proof. If 𝐴 is a nonfuzzy set, then 𝐶TzFIFS(𝐴, 𝐴) = 1 isobvious.

Conversely, it can be proved by the method of contradic-tion.

Assume 𝐴 is not a nonfuzzy set.Then 0 ≤ 𝜇

𝐴(𝑥𝑖) < 1, 0 ≤ 𝛾

𝐴(𝑥𝑖) < 1, and 0 ≤ 𝜋

𝐴(𝑥𝑖) < 1,

for some 𝑥𝑖.

Hence 𝜇2𝐴(𝑥𝑖) + 𝛾2

𝐴(𝑥𝑖) + 𝜋2

𝐴(𝑥𝑖) < 1.

That is,

𝜇2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴1

(𝑥𝑖) < 1,

𝜇2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖) < 1,

𝜇2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖) < 1,

𝜇2

𝐴4

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖) < 1.

(19)

Also,

{𝜇2

𝐴1

(𝑥𝑖) + 𝜇2

𝐴2

(𝑥𝑖) + 𝜇2

𝐴3

(𝑥𝑖) + 𝜇2

𝐴4

(𝑥𝑖)

+ 𝛾2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖) + 𝛾2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖)

+ 𝜋2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖) + 𝜋2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖)} < 1.

(20)

Then

𝐶TzFIFS (𝐴, 𝐴) =1

3

𝑛

∑

𝑖=1

{𝜇2

𝐴1

(𝑥𝑖) + 𝜇2

𝐴2

(𝑥𝑖) + 𝜇2

𝐴3

(𝑥𝑖)

+ 𝜇2

𝐴4

(𝑥𝑖) + 𝛾2

𝐴1

(𝑥𝑖) + 𝛾2

𝐴2

(𝑥𝑖)

+ 𝛾2

𝐴3

(𝑥𝑖) + 𝛾2

𝐴4

(𝑥𝑖)

+ 𝜋2

𝐴1

(𝑥𝑖) + 𝜋2

𝐴2

(𝑥𝑖)

+𝜋2

𝐴3

(𝑥𝑖) + 𝜋2

𝐴4

(𝑥𝑖)} < 1.

(21)

This is contradictory, and so 𝐴 is a nonfuzzy set.


4. WA and WG Operators for TzFIFNs

On the foundation of the definitions discussed by Wang [13],the weighted arithmetic operators for TzFIFNs are given asfollows.

Definition 10. Let 𝑎𝑗= ⟨[𝜇

1𝑗, 𝜇2𝑗, 𝜇3𝑗, 𝜇4𝑗], [𝛾1𝑗, 𝛾2𝑗, 𝛾3𝑗, 𝛾4𝑗]⟩,

𝑗 = 1, 2, . . . , 𝑛, be a collection of TzFIFN values. Thetrapezoidal fuzzy intuitionistic fuzzy weighted averaging(TzFIFWA) operator, TzFIFWA : 𝑄𝑛 → 𝑄, is defined as

TzFIFWA𝜔(𝑎1, 𝑎2, . . . , 𝑎

𝑛)

=

𝑛

∑

𝑗=1

𝜔𝑗𝑎𝑗= (1 −

𝑛

∏

𝑗=1

(1 − 𝜇𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾𝑗)𝜔𝑗

)

= ([

[

1 −

𝑛

∏

𝑗=1

(1 − 𝜇1𝑗)𝜔𝑗

,

1 −

𝑛

∏

𝑗=1

(1 − 𝜇2𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝜇3𝑗)𝜔𝑗

,

1 −

𝑛

∏

𝑗=1

(1 − 𝜇4𝑗)𝜔𝑗]

]

,

[

[

𝑛

∏

𝑗=1

(𝛾1𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾2𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾3𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾4𝑗)𝜔𝑗]

]

) ,

(22)

where 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔

𝑡)𝑇 is the weight vector of 𝑎

𝑗=

⟨[𝜇1𝑗, 𝜇2𝑗, 𝜇3𝑗, 𝜇4𝑗], [𝛾1𝑗, 𝛾2𝑗, 𝛾3𝑗, 𝛾4𝑗]⟩, such that 𝜔

𝑗> 0 and

∑𝑛

𝑗=1𝜔𝑗= 1.

Definition 11. Let 𝑎𝑗= ⟨[𝜇

1𝑗, 𝜇2𝑗, 𝜇3𝑗, 𝜇4𝑗], [𝛾1𝑗, 𝛾2𝑗, 𝛾3𝑗, 𝛾4𝑗]⟩,

𝑗 = 1, 2, . . . , 𝑛, be a collection of TzFIFN values. Thetrapezoidal fuzzy intuitionistic fuzzy weighted geometric(TzFIFWG) operator, TzFIFWG : 𝑄𝑛 → 𝑄, is defined as

TzFIFWG𝜔(𝑎1, 𝑎2, . . . , 𝑎

𝑛)

=

𝑛

∑

𝑗=1

𝜔𝑗𝑎𝑗= (

𝑛

∏

𝑗=1

(𝜇𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾𝑗)𝜔𝑗

)

= ([

[

𝑛

∏

𝑗=1

(𝜇1𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇2𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇3𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇4𝑗)𝜔𝑗]

]

,

[

[

1 −

𝑛

∏

𝑗=1

(1 − 𝛾1𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾2𝑗)𝜔𝑗

,

1 −

𝑛

∏

𝑗=1

(1 − 𝛾3𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾4𝑗)𝜔𝑗]

]

) ,

(23)

where 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔

𝑡)𝑇 is the weight vector of 𝑎

𝑗=

⟨[𝜇1𝑗, 𝜇2𝑗, 𝜇3𝑗, 𝜇4𝑗], [𝛾1𝑗, 𝛾2𝑗, 𝛾3𝑗, 𝛾4𝑗]⟩, such that 𝜔

𝑗> 0 and

∑𝑛

𝑗=1𝜔𝑗= 1.

5. MADM Algorithm forTrapezoidal Fuzzy IFS

Let 𝐴 = {𝐴1, 𝐴2, . . . , 𝐴

𝑚} be a set of alternatives, and let 𝐺 =

{𝐺1, 𝐺2, . . . , 𝐺

𝑛} be the set of attributes;𝜔 = (𝜔

1, 𝜔2, . . . , 𝜔

𝑛) is

the weighting vector of the attribute𝐺𝑗, 𝑗 = 1, 2, . . . , 𝑛, where

𝜔𝑗∈ [0, 1], such that ∑𝑛

𝑗=1𝜔𝑗= 1. Suppose that

�̃�𝑘= (𝑟(𝑘)

ℎ𝑖𝑗)𝑚×𝑛

= (𝜇(𝑘)

ℎ𝑖𝑗, 𝛾(𝑘)

ℎ𝑖𝑗)𝑚×𝑛

= ([𝜇(𝑘)

1𝑖𝑗, 𝜇(𝑘)



4𝑖𝑗] ,

[𝛾(𝑘)

1𝑖𝑗, 𝛾(𝑘)



4𝑖𝑗])𝑚×𝑛

(24)

be the trapezoidal fuzzy intuitionistic fuzzy number decisionmatrix, where [𝜇(𝑘)




4𝑖𝑗] is the degree of the mem-

bership value that the alternative 𝐴𝑖satisfies the attribute 𝐺

𝑗

given by the decision maker 𝐷𝑘and [𝛾(𝑘)




4𝑖𝑗] is

the degree of nonmembership value for the alternative 𝐴𝑖,

where 𝜇(𝑘)ℎ𝑖𝑗

, 𝛾(𝑘)

ℎ𝑖𝑗⊂ [0, 1] and 𝜇(𝑘)


4𝑖𝑗≤ 1, 𝑖 = 1, 2, . . . , 𝑚,

𝑗 = 1, 2, . . . , 𝑛, and 𝑘 = 1, 2, . . . , 𝑡.The developed model of MADM is given as follows.

Step 1. Utilize the decision information given in the matrix�̃� and the TzFIFWA or TzFIFWG operator which has theassociated weighting vector 𝜔 = (𝜔

1, 𝜔2, . . . , 𝜔

𝑛)𝑇 and

aggregate the given decision matrix �̃� = (𝑟𝑖𝑗)𝑚×𝑛

into groupintegrated attribute values 𝑟

𝑖:

𝑟𝑖= TzFIFWA

𝜔(𝑎1, 𝑎2, . . . , 𝑎

𝑛)

=

𝑛

∑

𝑗=1

𝜔𝑗𝑎𝑗= (1 −

𝑛

∏

𝑗=1

(1 − 𝜇𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾𝑗)𝜔𝑗

)

= ([

[

1 −

𝑛

∏

𝑗=1

(1 − 𝜇1𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝜇2𝑗)𝜔𝑗

,


1 −

𝑛

∏

𝑗=1

(1 − 𝜇3𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝜇4𝑗)𝜔𝑗]

]

,

[

[

𝑛

∏

𝑗=1

(𝛾1𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾2𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾3𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝛾4𝑗)𝜔𝑗]

]

)

(25)

or

𝑟𝑖= TzFIFWG

𝜔(𝑎1, 𝑎2, . . . , 𝑎

𝑛)

=

𝑛

∑

𝑗=1

𝜔𝑗𝑎𝑗= (

𝑛

∏

𝑗=1

(𝜇𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾𝑗)𝜔𝑗

)

= ([

[

𝑛

∏

𝑗=1

(𝜇1𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇2𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇3𝑗)𝜔𝑗

,

𝑛

∏

𝑗=1

(𝜇4𝑗)𝜔𝑗]

]

,

[

[

1 −

𝑛

∏

𝑗=1

(1 − 𝛾1𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾2𝑗)𝜔𝑗

,

1 −

𝑛

∏

𝑗=1

(1 − 𝛾3𝑗)𝜔𝑗

, 1 −

𝑛

∏

𝑗=1

(1 − 𝛾4𝑗)𝜔𝑗]

]

) .

(26)

Step 2. Utilize the correlation coefficient (8) to (11) ofTzFIFNs to derive the closeness between the overall groupintegrated attribute values 𝑟

𝑖and the TzFIFN positive ideal

value 𝑟+, where 𝑟+ = ⟨[1, 1, 1, 1], [0, 0, 0, 0]⟩.

Step 3. Rank alternatives 𝐴𝑖, 𝑖 = 1, 2, . . . , 𝑚, and select the

best in accordance with the highest closeness obtained fromStep 2.

6. Numerical Illustration

A company intends to select one person to take the positionof Assistant Manager from four candidates. Five indicators(attributes) must be evaluated. They are shown as follows:

(i) technical skill (𝐺1),

(ii) professional ability (𝐺2),

(iii) creative ability (𝐺3),

(iv) analytical skill (𝐺4),

(v) leadership ability (𝐺5).

The weights of the indicators are = (𝜔1, 𝜔2, . . . , 𝜔

𝑛)𝑇

=

(0.18, 0.26, 0.15, 0.19, 0.22)𝑇.The individual attributes of each

candidate are to be evaluated in order to come to a gooddecision.The decision matrix �̃� = (𝑟

𝑖𝑗)𝑚×𝑛

is given as follows:

�̃�=

𝐺1

(

⟨[0.15, 0.21, 0.32, 0.42], [0.19, 0.23, 0.42, 0.51]⟩

⟨[0.21, 0.33, 0.45, 0.49], [0.12, 0.19, 0.23, 0.35]⟩

⟨[0.13, 0.19, 0.31, 0.39], [0.11, 0.23, 0.27, 0.31]⟩

⟨[0.18, 0.25, 0.32, 0.43], [0.17, 0.23, 0.34, 0.41]⟩

𝐺2

⟨[0.11, 0.19, 0.23, 0.31], [0.13, 0.24, 0.29, 0.35]⟩

⟨[0.13, 0.15, 0.25, 0.34], [0.19, 0.21, 0.26, 0.32]⟩

⟨[0.21, 0.23, 0.29, 0.42], [0.28, 0.34, 0.41, 0.46]⟩

⟨[0.22, 0.25, 0.39, 0.43], [0.33, 0.35, 0.41, 0.45]⟩

𝐺3

⟨[0.17, 0.27, 0.37, 0.43], [0.11, 0.19, 0.23, 0.30]⟩

⟨[0.21, 0.33, 0.42, 0.51], [0.25, 0.32, 0.41, 0.45]⟩

⟨[0.15, 0.20, 0.35, 0.49], [0.22, 0.26, 0.35, 0.42]⟩

⟨[0.16, 0.23, 0.36, 0.39], [0.27, 0.35, 0.41, 0.46]⟩

𝐺4

⟨[0.25, 0.35, 0.45, 0.55], [0.15, 0.20, 0.32, 0.41]⟩

⟨[0.21, 0.32, 0.43, 0.56], [0.12, 0.23, 0.33, 0.39]⟩

⟨[0.13, 0.15, 0.25, 0.29], [0.11, 0.25, 0.34, 0.41]⟩

⟨[0.15, 0.23, 0.41, 0.44], [0.21, 0.32, 0.39, 0.46]⟩

𝐺5

⟨[0.15, 0.23, 0.34, 0.47], [0.25, 0.28, 0.34, 0.45]⟩

⟨[0.18, 0.25, 0.36, 0.43], [0.26, 0.37, 0.47, 0.51]⟩

⟨[0.19, 0.23, 0.35, 0.43], [0.27, 0.31, 0.41, 0.52]⟩

⟨[0.24, 0.34, 0.45, 0.61], [0.15, 0.25, 0.28, 0.30]⟩

) .

(27)

Step 1. Utilize the decision information given in the matrix�̃�, the TzFIFWA operator, and the weighting vector of theattributes 𝜔 = (𝜔

1, 𝜔2, . . . , 𝜔

𝑛)𝑇

= (0.18, 0.26, 0.15, 0.19,

0.22)𝑇 and aggregate the decision matrix �̃�. The collective

values are as follows:

𝑟1= ⟨[0.1629, 0.2470, 0.3375, 0.4346] ,

[0.1610, 0.2297, 0.3159, 0.3986]⟩ ,

𝑟2= ⟨[0.1832, 0.2673, 0.3744, 0.4598] ,

[0.1789, 0.2532, 0.3245, 0.3937]⟩ ,

𝑟3= ⟨[0.1677, 0.2036, 0.3092, 0.4057] ,

[0.1896, 0.2813, 0.3584, 0.4248]⟩ ,

𝑟4= ⟨[0.1956, 0.2642, 0.3914, 0.4720] ,

[0.2192, 0.2963, 0.3729, 0.4078]⟩ .

(28)

Step 2. Utilize the correlation coefficient of TzFIFNs to derivethe closeness between the overall group integrated attributevalues 𝑟

𝑖and the TzFIFN positive ideal value 𝑟+, where


𝑟+= ⟨[1, 1, 1, 1], [0, 0, 0, 0]⟩. Using (8) to (11), the correlation

coefficients are calculated using the following formula:

𝐾TzFIFS (𝐴, 𝐵) =𝐶TzFIFS (𝐴, 𝐵)

√𝐸TzFIFS (𝐴) ⋅ 𝐸TzFIFS (𝐵),

0 ≤ 𝐾TzFIFS (𝐴, 𝐵) ≤ 1.

(29)

Hence the calculated values are given as follows:

𝐶TzFIFS (𝑟1, 𝑟+) = 0.2955, 𝐾TzFIFS (𝑟1, 𝑟

+) = 0.4663,


+) = 0.5140,


+) = 0.4332,


+) = 0.5347.

(30)

Step 3. Rank alternatives𝐴𝑖, 𝑖 = 1, 2, . . . , 𝑚, from the highest

closeness value (correlation coefficient) obtained from Step 2;the result is obtained as follows:

𝐴4> 𝐴2> 𝐴1> 𝐴3. (31)

Hence the best alternative is 𝐴4.

Following the MADMmodel using the TzFIFWA opera-tor, we have the following numerical results for the MADMmodel using TzFIFWG operator.

Step 1. Utilize the decision information given in the matrix�̃�𝑘, the TzFIFWG operator, and the weighting vector of the

attributes 𝜔 = (𝜔1, 𝜔2, . . . , 𝜔

𝑛)𝑇

= (0.18, 0.26, 0.15, 0.19,

0.22)𝑇 and aggregate the same decision matrix �̃�. The collec-

tive values are as follows:𝑟1= ⟨[0.1553, 0.2388, 0.3245, 0.4202] ,

[0.1695, 0.2325, 0.3236, 0.4088]⟩

𝑟2= ⟨[0.1792, 0.2514, 0.3608, 0.4467] ,

[0.1906, 0.2654, 0.3431, 0.4045]⟩

𝑟3= ⟨[0.1635, 0.2006, 0.3059, 0.3973] ,

[0.2094, 0.2857, 0.3385, 0.4347]⟩

𝑟4= ⟨[0.1917, 0.2600, 0.3874, 0.4596] ,

[0.2331, 0.3024, 0.3670, 0.4163]⟩ .

(32)

Step 2. Utilize the correlation coefficient of TzFIFNs to derivethe closeness between the overall group integrated attributevalues 𝑟

𝑖and the TzFIFN positive ideal value 𝑟+, where

𝑟+= ⟨[1, 1, 1, 1], [0, 0, 0, 0]⟩. Using (8) to (11), the correlation

coefficients are calculated as follows:𝐶TzFIFS (𝑟1, 𝑟

+) = 0.2847, 𝐾TzFIFS (𝑟1, 𝑟

+) = 0.4493,


+) = 0.4966,


+) = 0.4281,


+) = 0.5270.

(33)

Step 3. Rank alternatives𝐴𝑖, 𝑖 = 1, 2, . . . , 𝑚, from the highest

closeness (correlation coefficient) obtained from Step 2; theresult obtained is

𝐴4> 𝐴2> 𝐴1> 𝐴3. (34)

Hence the best alternative is 𝐴4.

7. Conclusion

In this paper an MADM model was proposed based on thecorrelation coefficient of TzFIFS for ranking the alternativestogether with WA and WG operators. The trapezoidal fuzzyintuitionistic fuzzy weighted averaging (TzFIFWA) opera-tor and the trapezoidal fuzzy intuitionistic fuzzy weightedgeometric (TzFIFWG) operator were used to aggregate thetrapezoidal fuzzy intuitionistic fuzzy information given ina decision matrix. A numerical illustration was given toshow the effectiveness of the proposed approach in usingcorrelation coefficient of TzFIFSs because it preserved thelinear relationship between the variables.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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Research Article MADM Problems with Correlation Coefficient of … · 2019. 5. 9. · Research...

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Transcript of Research Article MADM Problems with Correlation Coefficient of … · 2019. 5. 9. · Research...