Research Article Optimal Inconsistency Repairing of...

Research ArticleOptimal Inconsistency Repairing of PairwiseComparison Matrices Using Integrated LinearProgramming and Eigenvector Methods

Haiqing Zhang1 Aicha Sekhari1 Yacine Ouzrout1 and Abdelaziz Bouras12

1 DISP Laboratory Lumiere University Lyon 2 160 boulevard de lrsquoUniversite 69676 Bron Cedex France2 Computer Science Department Faculty of Engineering Qatar University amp ictQATAR PO Box 2731 Doha Qatar

Correspondence should be addressed to Haiqing Zhang haiqingzhangzhqgmailcom

Received 1 October 2013 Accepted 18 December 2013 Published 17 March 2014

Academic Editor Hao-Chun Lu

Copyright copy 2014 Haiqing Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Satisfying consistency requirements of pairwise comparison matrix (PCM) is a critical step in decision making methodologiesAn algorithm has been proposed to find a new modified consistent PCM in which it can replace the original inconsistent PCM inanalytic hierarchy process (AHP) or in fuzzy AHP This paper defines the modified consistent PCM by the original inconsistentPCM and an adjustable consistent PCM combined The algorithm adopts a segment tree to gradually approach the greatest lowerbound of the distance with the original PCM to obtain the middle value of an adjustable PCM It also proposes a theorem to obtainthe lower value and the upper value of an adjustable PCM based on two constraints The experiments for crisp elements showthat the proposed approach can preserve more of the original information than previous works of the same consistent value Theconvergence rate of our algorithm is significantly faster than previous works with respect to different parameters The experimentsfor fuzzy elements show that our method could obtain suitable modified fuzzy PCMs

1 Introduction

Analytic Hierarchy Process (AHP) is developed by Saaty[1] which is a multicriterion decision-making methodologywidely used in many real problems [2 3] The AHP method-ology expresses the relative importance of criteria by pairwisecomparisons and converts the values of pairwise comparisonsto priorities Fuzzy AHP methodology [4] is an advancedAHP methodology which is used to tackle the uncertaintyand inaccurate problems in multicriteria decision-makingprocess Fuzzy AHP derives the fuzzy priorities of criteriafrom pairwise comparisons matrix with triangular (or trape-zoidal) fuzzy elements To make sure the priorities of eachcriterion are accurate and sensible consistency of pairwisecomparison matrix (PCM) with crisp or fuzzy elements mustbe achieved

Several works [5ndash8] focus on reducing inconsistentPCMswith crisp numbers Karapetrovic and Rosenbloom [5]revised the single entry of a ratiorsquos value till the consistencyof relative matrix was at an acceptable level Xu and Wei [6]

preserved the initial ratiosrsquo value in the pairwise compari-son matrix while obtaining satisfactory consistency require-ments Cao et al [7] developed a heuristic approach whichcan preserve more of the original information compared toXu and Wei [6] However for these three works when theconsistency requirement is increased the computing timeswill be largely increased and the information on the originalmatrix cannot be well preserved Anholcerrsquos [8] work is usedto minimize the distance between inconsistent PCMs andtheir corresponding consistent PCMThe aim of this work isto find out a new modified PCM which is consistent and hasthe closest resemblance to the optimal one But the modifiednew matrix has a long distance with the original matrixaccording to the analysis of the parameters given in [6 7]

Someworks used to solve the inconsistency of PCMswithfuzzy elements are given in [9ndash12] Xu andWang [9] repairedincomplete and inconsistent fuzzy preference relations byfinding out the unusual and false element until the consis-tency ratio was at a satisfactory level Leung and Cao [10]proposed a new definition of fuzzy positive reciprocal matrix

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 989726 16 pageshttpdxdoiorg1011552014989726

2 Mathematical Problems in Engineering

by setting deviation tolerances based on an idea of allowinginconsistent information Morteza and Bafandeh [11] furtherdiscussed Leung and Caorsquos work and proposed a newmethodof fuzzy consistency tests by direct fuzzification of a QR(quick response) algorithm which is one of the methodsfor the eigenvalues calculation of an arbitrary matrix Wangand Chen [12] applied fuzzy linguistic preference relationsto construct consistent PCMs by considering reducing thenumber of pairwise comparisons However these works donot have standard parameters to verify the reliability of theirtheory so far Therefore it is very important to prove thefeasibility of amethodology that can reduce the inconsistencyof the original matrix and preserve the original matrixrsquosinformation as much as possible

In this research work we propose a modified consistentPCM as a combination of original inconsistent PCM and anadjustable consistent PCM In order to achieve the modifiedPCM this paper is structured in the following way Section 2and Section 3 give the basic concepts of the PCM withfuzzy and crisp elements consistency indices for crisp andfuzzy elements and parameters to judge the effectivenessof the modified matrix Section 4 and Section 5 propose analgorithm to obtain the middle value upper value and lowervalue of the adjustablematrixThemain idea of our algorithmis to find the optimum priority vector by solving a linearprogramming problem and use the eigenvector method toobtain the adjustable matrix based on the optimum priorityvector In Section 6 the algorithm was applied to obtain theadjustable PCM by adopting the same illustrating PCM in[6 7] the comparison results show that our algorithm canpreserve more original information than Cao et al [7] andXu and Wei [6] In Section 7 the algorithm was used tofind the sensible and the closest consistent modified PCMwith fuzzy elements The experiments show that the newmodified matrix can satisfy consistent indicesrsquo requirementsofNI [13] andCCI [14 15] and also can preservemore originalinformation under parameters of 120590fuzzy and 120575fuzzy Section 8concludeswith effective and efficient analysis to show that ouralgorithm can be performed easily

2 State of the Art

21 Notations andDefinitions The119898times119899 pairwise comparisonmatrix 119860 with triangular fuzzy elements can be described inthe following

119860 = (

(119886119871

11 119886119872

11 119886119880

11) sdot sdot sdot (119886

119871

1119899 119886119872

1119899 119886119880

1119899)

d

(119886119871

1198981 119886119872

1198981 119886119880

1198981) sdot sdot sdot (119886

119871

119898119899 119886119872

119898119899 119886119880

119898119899)

) (1)

where for each element 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) 119886119871119894119895is the lower

value 119886119872119894119895is themiddle value and 119886

119880

119894119895is the upper value In the

particular situation if 119898 = 119899 when the following conditionis satisfied then 119860 is a reciprocal matrix

119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) implies 119886

119895119894= (

1

119886119880

119894119895

1

119886119872

119894119895

1

119886119871

119894119895

)

forall119894 119895 = 1 2 119899

(2)

where 1120590 lt 119886119871

119894119895lt 119886119872

119894119895lt 119886119880

119894119895lt 120590 and 119878 = [1120590 120590] 120590 gt 1 119878 is

an interval of real numbers called fuzzy scale Crisp numbers(nonfuzzy numbers) are special cases of 119860 when 119886

119871

119894119895= 119886119872

119894119895=

119886119880

119894119895

Definition 1 Asit was proposed by Buckley [16] a fuzzypositive reciprocal matrix 119860 = [119886

119894119895]119899times119899

is consistent if andonly if 119886

119894119896⊙ 119886119896119895

asymp 119886119894119895 forall119894 119895 119896 | 1 le 119894 119895 119896 le 119899

Where the operator ⊙ is one of the operation rules oftriangular fuzzy elements this operation can be calculated bythe following equation

119886119894119896

⊙ 119886119896119895

= (119886119871

119894119896 119886119872

119894119896 119886119880

119894119896) ⊙ (119886

119871

119896119895 119886119872

119896119895 119886119880

119896119895)

= (119886119871

119894119896sdot 119886119871

119896119895 119886119872

119894119896sdot 119886119872

119896119895 119886119880

119894119896sdot 119886119880

119896119895)

asymp 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895)

(3)

When positive reciprocal matrix 119860 is crisp numbers 119860then the consistent condition is 119886

119894119896sdot 119886119896119895

= 119886119894119895 forall119894 119895 119896 | 1 le

119894 119895 119896 le 119899However this definition is too strict because it is unre-

alistic to reach perfect consistency of a PCM (crisp orfuzzy elements) Some works [10 13 17ndash23] have developedconsistency indices to accept a certain level of deviationsWe will adopt consistency indices in Section 22 to determinewhether the current PCM matrix is at an acceptable consis-tency level

22 Consistency Indices Several consistency indices havebeen proposed for crisp numbers for instance geometricconsistency index [17] singular value decompositionmethod[18] and harmonic consistency index [19] However it wasproven that all these consistency indices are linear or non-linear transformations of Sattyrsquos CR [20] Therefore we useSattyrsquos CR in this paper for measuring consistency of crispnumbers

The consistency index of CR [20] is defined as follows

CR =

120582max minus 119899

119899 minus 1

sdot

1

RI (4)

where 120582max is the principle eigenvalue of 119860 RI is randomindexwhich can be gotten by searching a defined tableWhenthe value of 0 lt CR lt 01 the consistency can be accepted

Several important works focus on the consistency ofpairwise comparison matrix with fuzzy elements The firstone is Leung and Cao [10] who proposed a notion withconsideration of a tolerance deviation However the notion

Mathematical Problems in Engineering 3

is strongly related to Sattyrsquos CR and it has shortcomings tocalculate consistency of pairwise comparison matrix withfuzzy elements [11] The second one is Ramık and Korvinyrsquos[13] work which proposes a new consistency index NI toexamine fuzzy elements based on the distance of the matrixto a special ratio matrix and compare the properties withCR This work has been further studied and has been usedby several important works [21ndash23] We have tested thisworkrsquos performance The results indicated that it can satisfyreasonable results with fuzzy elements although it has someshortcomings [23]

The consistency index of NI [13] is defined as follows

NI120590119899= 120574120590

119899sdot max119894119895

max

1003816100381610038161003816100381610038161003816100381610038161003816

120596119871

119894

120596119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119872

119894

120596119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119880

119894

120596119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(1 le 119894 119895 le 119899)

(5)

where 120574120590

119899is a normalization factor and the values of 120596119871

119894 120596119872119894

and 120596119880

119894can be obtained from 119886

119871

119894119895 119886119872119894119895 and 119886

119880

119894119895

Another successful index that can be extracted from [1415] is that they extendGCI (geometric consistency index) [24]to CCI (centric consistency index) to deal with PCM withtriangle fuzzy elementsThe consistency index CCI is definedas follows

CCI (119860) =

2

(119899 minus 1) (119899 minus 2)

times sum

119894lt119895

( log(

119886119871

119894119895+ 119886119872

119894119895+ 119886119880

119894119895

3

)

minus log(

120596119871

119894+ 120596119872

119894+ 120596119880

119894

120596119871

119895+ 120596119872

119895+ 120596119880

119895

))

2

(6)

where 119860 = (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) are the elements for fuzzy PCM

= (119894) is a priority vector derived by logarithmic least

squares When CCI(119860) = 0 119860 is considered fully consistentThresholds remain identical with index GCI as CCI beinga fuzzy extension of GCI The thresholds are provided asfollows CCI = 03147 for 119899 = 3 CCI = 03526 for 119899 = 4and CCI = 0370 for 119899 gt 4 based on Aguaron and Moreno-Jimenez [24]

In conclusion we will adopt CR to examine the consis-tency of crisp elements and use NI and CCI to examine theconsistency of fuzzy elements

23 Parameters to Judge the Effectiveness of Modified MatrixNext we are going to describe the necessary parameterswhich can be used to measure the effectiveness of modified

PCM 119861 Xu and Wei [6] have given two parameters in thefollowing

120575 = max119894119895

10038161003816100381610038161003816119887119894119895minus 119886119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899 (7)

120590 =

radicsum119899

119894=1sum119899

119895=1(119887119894119895minus 119886119894119895)

2

119899

(8)

120590 and 120575 are used as the parameters of modificatory effec-tiveness The authors of [6 7] argue that a modified matrixthat preserves the most information of the original one mustsatisfy the following condition 0 lt 120575 lt 2 and 0 lt 120590 lt

1 Extending these two parameters to be suitable to judgemodificatory effectiveness of fuzzy elements so the range isidentical with 120590 and 120575 120590fuzzy and 120575fuzzy are

120575fuzzy = max119894119895

10038161003816100381610038161003816

119887119894119895minus 119886119894119895

10038161003816100381610038161003816

= max119894119895

10038161003816100381610038161003816119887119871

119894119895minus 119886119871

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119872

119894119895minus 119886119872

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119880

119894119895minus 119886119880

119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899

120590fuzzy = (max

(

119899

sum

119894=1

119899

sum

119895=1

(119887119871

119894119895minus 119886119871

119894119895)

2

)

(

119899

sum

119894=1

119899

sum

119895=1

(119887119872

119894119895minus 119886119872

119894119895)

2

)

119899

sum

119894=1

119899

sum

119895=1

(119887119880

119894119895minus 119886119880

119894119895)

2

)

12

119899minus1

(9)

Besides 120590 and 120575 two parameters proposed by Xu andWei[6] we propose a third parameter should be added which isCondition of Order Preservation (COP) (Table 3) [25] Forexample suppose the original matrix 119860 has alternatives (119886

1

1198862 1198863 and 119886

4) it has that the relationship 119886

1dominates 119886

2and

1198863dominates 119886

4 and the judgments indicate that the extent

to which 1198861dominates 119886

2is greater than the extent to which

1198863dominates 119886

4 then the priority vector 120596 should satisfy

120596(1198861) gt 120596(119886

2) and 120596(119886

3) gt 120596(119886

4) (preservation of order of

preference) and 120596(1198861)120596(1198862) gt 120596(119886

3)120596(1198864) (preservation of

order of intensity of preference)

3 Main Theories to Obtain Modified PCM

31 Distance Analysis between Original Matrix and ModifiedMatrix To measure the distance from original PCM 119860 (or119860) several methods can be used [26ndash30] The PCM 119860 canbe considered as the combination of three matrixes of PCM119860 If the measuring distance methodologies for crisp 119860 havebeen properly handled then they can be suitable for fuzzy119860 as well We will first discuss the distance methods forcrisp numbersThemeasurement has logarithmic least squaremethod (LSM) [26 31] eigenvectormethod [27 28] and least


squares method [29 30]The optimum eigenvector should beas close as possible to the original eigenvector (derived fromthe original matrix) The adjustable matrix could stronglyresemble the original matrix once the optimum eigenvectorhas been determined Therefore eigenvector can be used tocalculate adjustable PCM 119861

1015840(1198611015840)

According to [27 28 32] if matrix 119860 is consistent thenwe could find positive weights 120596 = 120596

1 1205962 120596

119899119879 which

can satisfy such condition 119886119894119895

= 120596119894120596119895 forall119894 119895 isin (1 2 119899)

Therefore if matrix119860 is close to consistent then it must have(120596119894120596119895) minus 119886119894119895

asymp 0 On the basis of this idea we want to obtainthe minimum value of the fastest distance between 120596

119894120596119895and

119886119894119895 The fastest distance between 120596

119894120596119895and 119886

119894119895is defined as

follows

119891 = max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

) (10)

The issue of finding the optimal modified matrix can beportrayed as resolving the minimum value of function 119891which can be expressed in the following equation

119866 (119860 120596) = min (119891) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (11)

This equation can reach the absolute minimum value(lowest point) when even the worst situation of proportionof 120596119894120596119895is closest to 119886

119894119895 The method to obtain the optimum

positive eigenvector 120596 is to find out all possible constraintsand obtain feasible solutions by selecting a suitable linearprogramming pattern

32 Constraints Analysis for Achieving Modified Matrix Thenew modified PCM 119861 (or 119861) should satisfy three conditions(1) the consistent value should be at an acceptable level(2) the farthest distance between new PCM 119861 (or 119861) andorginal PCM 119860 (or 119860) should be as small as possible (3) theobtained newmatrix should have a strong similarity with theoriginal matrix These three conditions can guarantee a newconsistent PCM 119861 (or 119861) which has the closest and maximalsimilarity with the orginal inconsistent PCM 119860 (or 119860) Tosuit these conditions we provide an adjustable PCM whichcan reduce the inconsistency of original PCM as much aspossible On the basis of the adjustable matrix we proposethe definition of the modified matrix

Definition 2 The new modified PCM 119861 is defined as acombination of the original PCM119860 and an adjustable matrixcalled 119861

1015840 which is derived from 119860 the modified PCM 119861 isdefined in

119861 = 119886120573

119894119895(1198871015840

119894119895)

1minus120573

(12)

For fuzzy elements the new modified PCM 119861 is definedas follows

119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(13)

Theproperties of the adjustable PCM119861 (or119861) and thewayto obtain them will be studied in Section 4

4 Calculation Processes of ObtainingAdjustable Matrix

41 Problem Statement The aim of this paper is to find outa consistent matrix 119861 = [

119887119894119895]119899times119899

which can preserve the mostinformation of the original matrix 119860 = [119886

119894119895]119899times119899

and be theclosest to 119860 One method to achieve this aim is that settingone element of 119861 is

119887119894119895

= (119887119871

119894119895 119887119872

119894119895 119887119880

119894119895) and one element of 119860

is 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) by making every element of 119861 closely

resemble the elements of 119860 which is 119887119871

119894119895≐ 119886119871

119894119895 119887119872119894119895

≐ 119886119872

119894119895 and

119887119880

119894119895≐ 119886119880

119894119895 Therefore we go to find 119887

119871

119894119895 119887119872

119894119895 119887119880

119894119895separately After

finding out 119887119871119894119895 119887119872

119894119895 119887119880

119894119895 combine them together to get the new

matrix 119861On the basis of this idea we build an adjustable matrix

1198611015840

= [1198871015840

119894119895]119899times119899

which can reduce the inconsistency of theoriginal matrix The new modified matrix is constructed bytwo parts (1) One is the original matrix The function ofthis matrix is to keep the original information and makesure the two matrixes are in an acceptable distance (2) Oneis the adjustable matrix The function of this matrix is tomodify the inconsistency level tomake sure the newmodifiedmatrixrsquos consistency is based on consistency indexes Themathematical expression objective is as follows

119861 = 119860 ⊙ 1198611015840 (14)

The mathematical expression objective can be developedas

119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(15)

This equation is also suitable when 119886119871

119894119895= 119886119872

119894119895= 119886119880

119894119895 which

is for crisp elements

42 Stage 1 Specify Formulas to Obtain the Middle Value 119887119872

119894119895

1015840

of the Adjustable Matrix 1198611015840 First we go to calculate 119887

119872

119894119895

1015840 ofadjustable matrix 119861

1015840 119887119872119894119895

1015840 should have a strong relationshipwith 119886

119872

119894119895 119872 is the eigenvector matrix of 119886

119872

119894119895 119887119872

119894119895

1015840 can begotten in the following equation

119887119872

119894119895

1015840

asymp

120596119872

119894

120596119872

119895

(16)

The value can be gotten when the proportion is closestto 119886119872

119894119895 Equation (17) has feasible solutions which means the

following equation reaches the minimum based value on (11)

119866(119886119872

119894119895 119872) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (17)


We can simply write the variable by introducing anadditional variable 119885 = 119866(119886

119872

119894119895 119872) Then the problem is as

follows

min 119885

st

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

le 119885 forall119894 119895 isin (1 2 119899) (18)

1

120590

le 120596119872

119894 120596119872

119895le 120590 119895 = 1 2 119899 (19)

(The calculation steps of (19) are given in the appendix)Assume the value of 119885 is given and then the constraint

(18) can be rewritten as follows

minus119885 le (119886119872

119894119895minus

120596119872

119894

120596119872

119895

) le 119885 (20)

(119886119872

119894119895minus 119885) sdot 120596

119872

119895minus 120596119872

119894le 0 (21)

120596119872

119894le (119886119872

119894119895+ 119885) sdot 120596

119872

119895 (22)

Take the reciprocal of constraint (22) then the newconstraints (23) (24) and (25) can be gotten

120596119872

119895

120596119872

119894

ge

1

(119886119872

119894119895+ 119885)

(23)

1

(119886119872

119894119895+ 119885)

sdot 120596119872

119894minus 120596119872

119895le 0 (24)

1

(119886119872

119895119894+ 119885)

sdot 120596119872

119895minus 120596119872

119894le 0 (25)

Combine constraint (21) and constraint (25) and removeone of the inequalities and then the new inequality can begotten

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895le 0 (26)

Analogously a similar inequality can be gotten

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894le 0 (27)

Next we add slack variable 119878 119879 and objective function119867 = 119878 + 119879 to change the constraints (26)-(27) to equalityconstraints

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

119867 = 119878 + 119879

0 le 119878 119879119867

(28)

Nowwe specify constraints propose formulas to calculate120596119872

119894 120596119872

119895 and add additional stopping parameter 119883 Then

constraints (28) correspond to the following linear program-ming problem

min 119883

st

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

(29)

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

(30)

119867 = 119878 + 119879 (31)

120596119872

1+ 119883 = 1 (32)

1

120590

le 120596119872

119894 120596119872

119895le 120590119894 119895 = 1 2 119899 (33)

0 le 119885 119878 119879119867 (34)

Note 1 (1) In constraint (29) we define 120596119872

1= 1 to normalize

vector 119872 If the stopping parameter 119883 = 0 then it means

constraint (29) has a solution and the problem (29)ndash(34) hasa feasible solution the value of 120596

119872

119894is the optimal solution

If the stopping parameter is 119883 = 1 which contradicts withconstraint (32) then the problem (29)ndash(34) is inconsistentthen the value of 120596119872

119894is not the optimal solution

(2) The equalities (29)ndash(34) can be solved by simplexalgorithm [33] The main idea of this algorithm is to walkalong edges of the polytope to find out extreme points withlower and lower objective values till the minimum value isreached or an unbounded edge is visited If the extreme pointis reached then the problem (29)ndash(34) has feasible solutions

(3) If 119885 = 1198851015840 can make problem (29)ndash(34) have a feasible

solution then it must have 119885 ge 1198851015840 that can also make the

problem (29)ndash(34) have a feasible solution In order to findthe greatest lower bound of 119885 we investigate it in Section 43


[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4

Figure 1 The structure of the segment tree

(4) If 119894 = 119895 then 119886119872

119894119894= 1 then constraint (29) can be

rewritten as follows minus120596119872

119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +

119878 = 0 This equality must always be satisfied Analogouslyconstraint (30) is always satisfied Then the equalities (29)ndash(34) can be used to find adjustable PCM 119861

1015840 for original PCM119860

(5) In this section we aim to find out the least absoluteworst distance by setting a more precise priority weightsrange and adding slack variables and an iterative way toobtain feasible solutions will be proposed in Section 43

43 Stage 2 Find Feasible Solutions to Obtain the MiddleValue 119887

119872

119894119895

1015840 of Adjustable Matrix 1198611015840 Next we focus on how

to find out the greatest lower bound of 119885 The problem canbe described as storing intervals of [0 119885max] analyzing thecorresponding 119883 value and finding the greatest lower boundof 119885 thatmakes119883 = 0This problem can be solved by segmenttree

Substep 1 (sets the initial value) Assume the accuracy level is120585 let the initial value of 120596119872

119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0

Substep 2 (builds a segment tree by using interval [0 119885]) Forexample in Figure 1 [0 119885] is the interval and P1 P2 P3 P4is the list of distinct interval endpoints We separate intervalsinto two parts in every division and terminate this processtill the value of the interval is less than the accuracy level (120585)Then obtain the 119883 value in problem (29)ndash(34) by setting thecurrent119885 value If119883 = 0 then the next119885 value is equal to thelower bound of the current node The calculation steps willend till it reaches endpoints (P1 P2 P3 P4) based on accuracylevel 120585

Substep 3 Select the greatest lower value of 119885 when 119883 = 0obtain the feasible solution of 120596119872

119894 120596119872119895to achieve 119887

119872

119894119895

1015840

Note 2 The segment tree is special for storing intervals Thebuilt time is 119874(119899 log 119899) for 119899 intervals and it uses 119874(119899 log 119899)

storage The reason we adapted to the segment tree is becausethe segments can be stored in any arbitrary manner it can

easily be adapted to counting queries and it helps us to querythe number of segments that contain a given point

44 Stage 3 Obtain the 119887119871

119894119895

1015840 and 119887119880

119894119895

1015840 of the Adjustable Matrix1198611015840 Once the optimal solution (119887119872

119894119895

1015840) has been obtained nextwe focus on obtaining the value of 119887119871

119894119895

1015840 and 119887119880

119894119895

1015840 The modifiedmatrix is a combination of the adjustable matrix and theoriginal matrix based on Definition 2 then the adjustablematrix should have the minimum fuzziness and maximumpreservation of the originalmatrixrsquos pattern If 1015840 isminimumfuzziness then fuzziness of 119861will mostly come from119860 and 119861

will be more similar with 119860 In fact the minimum fuzzinessof 1198611015840 could reduce uncertainty factors of 119861 If 119861

1015840 couldmaximally preserve the pattern of 119860 then the combinationof 1198611015840 and 119860 could reach the most potential of similarity with119860 We propose Theorem 3 to obtain the value of 119887119871

119894119895

1015840 and 119887119880

119894119895

1015840

based on the above theory

Theorem 3 The optimal solution vector is 120596119872

119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880

as arbitrary positiveconstants Define the value of120596119871

119894120596119880119894in the following formulas

120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895

1015840 is defined as follows

119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)

Proof The value of 119862119871and 119862

119880should satisfy two conditions

one is minimum fuzziness of 120596119872

119894 and the other one is

maximally maintaining similarity of original matrix 119860By the first condition we can get

120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)

By the second condition we need to consider the distancebetween 119886

119871

119894119895 119886119872119894119895 and 119886

119880

119894119895 maintain the relationship among

the original matrix and make the new matrix closest to theoriginal matrixrsquos pattern It means to find out the smallestcoefficient between 119886

119871

119894119895and 119886

119872

119894119895and the smallest coefficient


Table 1 The values of parameters for each iterative time

Iterativetime 119885max 119885min 119885 120585 119883 120596

1 100725 0 50362 100725 0 12059612 50362 25181 25181 25181 1 mdash3 37772 25181 37772 12591 0 12059624 37772 31476 31476 06296 1 mdash5 37772 34624 34624 03148 1 mdash6 37772 36198 36198 01574 1 mdash7 37772 36985 36985 00787 0 1205963

between 119886119872

119894119895and 119886119880

119894119895 We can get the following formulas based

on the second condition

119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)

Therefore we can prove formula (35) exists and provethat it is correct by (38) To be more precise for instance inFigure 2 (119887

119894119895)1015840

2is not satisfied with condition 1 and then it is

not the minimum fuzziness of associated weights If (119887119894119895)1015840

1is

not satisfied with condition 2 then it does not have the samepattern with matrix 119860 (119887

119894119895)1015840


5 An Algorithm to Obtain Modified PCM 119861

After analysis in Sections 42 43 and 44 we propose analgorithm to conclude how to obtain the modified PCM 119861

(see Algorithm 1)

6 Numerical Illustration and Comparisonwith Crisp Numbers

61 Calculation of an Illustration byUsing ProposedAlgorithmWe run the experiments by software Matlab (R2009a) on apersonal computer with Intel Core 22GHZ and 4G RAMFirst we test crisp numbers by using Algorithm 1 and thencompare them with [6 7]

The inconsistent matrix in [6 7] is the following matrix

119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)

For matrix 119860 120582max = 9669 CR = 0161 gt 01 andthe principal eigenvector is 120596 = (01730 00540 01881 0017500310 00363 01668 03332)T

The value of CR is more than 01Therefore we will adoptAlgorithm 1 to obtain the new consistency matrix

Calculation Step 1 Input of the initial value of 119860 and initialvalue of 120585

(1) The original matrix 119860 = (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) here crisp

number is a specific case of fuzzy number in ourmodel Set 119886119871

119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860

(2) The acceptable precise degree 120585 = 01 (Here we adopt01 as an example)

Calculation Step 2 Calculation process

(1) Calculate the initial value of 119885 by (17)

(2) First time119885 = 100725 substitute119885 into Algorithm 1119885max = 10075 119885min = 0 and 119885max minus 119885min gt 120585 setnew 119885 = (119885max minus 119885min)2

(3) The second time 119885 = (100725 minus 0)2 = 50362substitute new 119885 into Algorithm 1 solve the problem(29)ndash(34) the result is119883 = 0 and119908 = (10000 0346940000 05262 02902 01346 09375 14637) We putthe following calculation in Table 1

Calculation Step 3 Description parametersrsquo meaning Thevalue of 120585 is a stopping sign If current value of 120585 is less than01 stop the calculation process

The value of 119883 can determine how to change the current119885 value If 119883 = 0 it means there is a solution for problem(29)ndash(34) then save the value of vector 120596 and set 119885max = 119885else 119883 = 1 it means there is not a solution for problem (29)ndash(34) then set 119885min = 119885 119885 = 119885max


120583)

)

1120590 aLij aMij aUij 120590

ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)

Figure 2 Choosing optimal matrix 1198611015840 which is closest to matrix 119860

Algorithm Find Modified Matrix(119885max 119885min 120585 119860)Input 119885max 119885min 120585 original matrix 119860

Output 119885max 119885min optimal matrix 119861

BEGIN(1) If (119885max minus 119885min lt 120585) amp (119883 = 0)

ThenThe current 120596119872

119894is the desired optimal solution of weight vector

Adopt (16) to get 119887119872119894119895

1015840Use Theorem 3 to obtain 119887

119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840

)Obtain Modified PCM 119861 by Definition 2Stop

(2) Else Search segment treeFind the greatest lower value of 119885 when 119883 = 0

119885max = upper bound of the interval 119885min = lower bound of the intervalSolve the problem (29)ndash(34)Save the new value of 120596119872

119894

END

Algorithm 1 Find modified PCM

Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)

Now we go to discuss the results When in the 7thiterative time the gap between119885min and119885max is 00787 whichis less than 01 the process stops at this point In order tobetter understand we use the segment tree to express thechanging process in Figure 3

In Figure 3 the current 119885 value is colored with red andbold The first 119885 value is 100725 and 119883 = 0 which means

there is a solution for inequalities (29)ndash(34) the second 119885

value is 50362 and 119883 = 0 by combining the first resultit must have a solution when Z value is between 50362and 100725 Therefore the program goes to search the leftsubtree of the node the third 119885 value is 25181 and 119883 = 1which means there is not a solution for inequalities (29)ndash(34) the program needs to search the right subtree of this


node because the right child node is bigger than 25181We continue the process till the gap between two child nodesis less than 120585

The final result of positive vector is 1205963 = (10000 0909726082 02677 04345 04345 11620 33244)T

On the basis of 1205963 we construct the new consistencymatrix 119861

1015840 as in the following

1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)

Here we adopt 120573 = 08 as an e xample in Definition 2The new matrix 119861 is as follows

119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)

62 Comparison with References Xu andWei [6] defined theoriginal matrix 119860 (the elements are 119886

119894119895) that can be replaced

by the new matrix 119861 (the elements are 119887119894119895) which is showed

in the following equation

119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)

where 120572 is the positive value which is less than but approach-ing to 1

Cao et al [7] proposed an equation to obtain the newmatrix 119861 which is showed by the following

119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)

where ∘ is the symbol of Hadamard product For example119860 = 119861 ∘ 119862 means 119886

119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863

1015840 is themodified deviation matrix which is showed in the followingequation

1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)

where 119863 is the deviation matrix and DI is a zero deviationmatrix when [119889

119894119895] = 1 The value of 120574 is between 0 and 1

120572 and 120574 has differentmeanings for two papers but the twoparameters should be as close to 1 as possible In two paperstheymentioned that 120572 = 120574 = 098 is themost suitable value toget the optimal new matrix We will discuss 120573rsquos meaning andthe relationship with 120572 and 120574 in Section 8 We compare withtwo references in two situations one is the required criticalratio (CR) less than 01 (Table 2) the other one is the requiredcritical ratio (CR) close to 0 (Table 4)

From Table 2 we reach the outcome that the CR value islower than [6 7] at the same time the method could achievelower values of 120575 and 120590 in short iterative times It meansthat our method can preserve more original information andobtain a more consistent new matrix in short iterative timesWhen we rank the priority weight 120596 derived from 119860 theranking results are the same except when 120585 = 01 in twosimilar weights which means the priority weight 120596 which isderived from our method is acceptable

On the basis of Table 2 we go to discover the differenceof COP parameter in our method and two references Thepreference value between every two alternatives is gottenfrom priority weight 120596 of the column ldquopriority weightrdquo inTable 2 For example the value of 119886

1 1198862 is obtained in

120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855

The modified new matrix 119861XU 119861Cao 119861Our(005)119861Our(01) and 119861Our(02) is in the following


119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)

Thegap of elements in every twonewmatrixes (119861XU119861Cao119861Our(005) 119861Our(01) and 119861Our(02)) is less than 15 lt 2which proves these five matrixes are considerably close witheach other but in Table 2 the CR value of our method isless than the other two methods which means 119861Our(005)

119861Our(01) and 119861Our(02) are more consistent than 119861XU and119861Cao Meanwhile the value of 120575 and 120590 for 119861Our(005) isthe lowest which means 119861Our(005) is the best solution fororiginal matrix119860 Next we go to discuss the second situationwhen CR is close to 0 The analysis results are in Table 4


The iterative times for [6 7] 820 and 1619 are quite largewhichwill cost the running timeThe value of 120575 and120590 in thesetwo references is far away from the acceptable value whichmeans the priority weight of 120596XU and 120596Cao is not acceptableat all but the iterative times of our method are less than 10times the value of 120575 is less than 4 and the value of 120590 isless than 2 which means our result is acceptable to someextent

7 Numerical Illustration and Comparisonwith Fuzzy Numbers

To judge the effectiveness of Algorithm 1 for fuzzy PCMs weanalyze several parameters including the inconsistency indexNI [13] consistency index CCI [14 15] 120575fuzzy 120590fuzzy COPiteration times and running time

We adopt the same inconsistent matrix with Ramık andKorviny [13] which is

119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)

For matrix 119860 NI93(119860) = 0219 It is an inconsistent PCM

with fuzzy elements We adopt Algorithm 1 to obtain themodified new matrix under different parameters The resultsare shown in Table 5

For this matrix the modified matrices 119861 are quite similarwhen the accurate degrees are 120585 = 01 and 120585 = 001 andtherefore we only give the modified matrix 119861 when 120585 = 01

and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)

In Table 5 the inconsistency index NI is less than 01 andconsistency index CCI is less than 03147 which indicates themodified matrix can reach a consistent requirement 120575fuzzysatisfies 0 lt 120575fuzzy lt 2 and 120590fuzzy satisfies 0 lt 120590fuzzy lt 1 whichshows the obtained new matrix is in an acceptable distancewith119860 the value of COP which is gotten from priority vector of119861 can preserve order of preference and order of intensitypreference which presents 119861 can maintain the pattern of 119860(similarity) Meanwhile Algorithm 1 has high convergencespeed based on the less iteration times and running timeThe gap among these four matrixes is quite narrow 119861 (120585 =

01 120573 = 05) is the best choice when we select the smallestvalue of the consistency index (NI CCI) 120575fuzzy and 120590fuzzy

8 Discussion

Tables 2 and 4 conclude the effectiveness of Algorithm 1 bycomparing with Cao et al [7] and Xu and Wei [6] basedon PCMs with crisp elements Algorithm 1 can retain moreoriginal information and achieve lower value of 120575 and 120590whenboth 120572 [6] and 120574 [7] approach 1 Table 5 summarizes the

effectiveness of Algorithm 1 for PCMs with fuzzy elementsThe modified PCMs with fuzzy elements can maintainoriginal information and reach acceptable consistency levelas well

In Algorithm 1 120573 has different meaning with 120572 and 120574 InCao et al [7] and Xu andWei [6] the results gotten by using 120572

and 120574 equal to 098 are significantly better than those by using120572 and 120574 less than 098 Yet it is not always the case for 120573 Thetrue meaning of 120573 is the proportion of the original matrix toadjustablematrixThe approximate prefect range of120573 is [02508] based on the experimentsrsquo results and there is no cluewhich value is the best one for 120573 as the inconsistency level ofPCMs is different As a matter of fact changing 120573rsquos value willnot add running time because this operation runs in constanttime As a future research it would be interesting to figureout the exact prefect range of 120573 In this section we focus onefficiency analysis of Algorithm 1 as effective analysis is donein Sections 6 and 7

81 Efficiency Analysis Algorithm Running Time To demon-strate the overall efficiency of Algorithm 1 we measure


[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0

Figure 3 An example of iterative times in segment tree structure

3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)

Figure 4 Average running time with fuzzy and crisp elements (in seconds)

the average running time and iteration times for crispand fuzzy elements The elements of crisp matrix 119860 andfuzzy 119860 are gotten from randomly generated numbers byprogramming The size (119899) of matrix 119860 and 119860 varies from 3to 10 (in real life problems the size is usually no more than10) We adopt twenty matrixes for each size

The assumed accuracy level 120585 is the following threenumbers 01 001 and 0001 The value of 120573 in the final step(119861 = 119886

120573

119894119895(1198871015840

119894119895)1minus120573) will not affect the running time The average

running time of crisp elements is shown in Figure 4(a)The average running time of fuzzy elements is shown inFigure 4(b) In all cases the running times are less than half asecond which is acceptable in real experiments As expectedthe higher the accuracy degree the longer the running timein the same matrix size The figure shows the growth rateof fuzzy matrixesrsquo running time is more irregular than crispmatrixes

82 Efficiency Analysis Convergence Rate The speed ofconvergence is one important factor of the efficiency for aniterative method In Algorithm 1 the only parameter thatcan influence iteration times is the accuracy degree (120585) 120572

and 120574 are the parameters which can influence the number ofiterations for [6 7] respectively According to the numericalillustration in Section 6 we study the convergence rate withrespect to the values of 120585 120572 and 120574 by using Algorithm 1 and[6 7] respectively Meanwhile we examine the convergencerate for fuzzy elements based on the data in Section 7Figure 5 shows the results

The convergence rate of Cao et al [7] is slower than Xuand Wei [6] when 120574 = 120572 For these two methods the lowerthe parameters value the higher the rate of convergenceAlgorithm 1 can approach its limits faster than Cao et al[7] and Xu and Wei [6] The maximum iteration time forAlgorithm 1 is less than twenty when 120585 = 00001 while themaximum iteration times forCao et al [7] andXu andWei [6]


0 10 20 30 40 50 600

005

01

015

02

Number of iterations

CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)

Figure 5 Convergence rate of Cao et al [7] Xu and Wei [6] and Algorithm 1

are 1619 and 820 respectively For fuzzy elements the conver-gence rate of Algorithm 1 has low growth rate as well

9 Conclusions

In this work an algorithm has been proposed to derivea consistent PCM with crisp or fuzzy elements from aninconsistent one The presented approach used the samenumerical example used by [6 7 13] The experimentsreveal that the proposed approach could retain more originalinformation and the convergence rate is faster than Cao et al[7] andXu andWei [6] in different values of CR Two effectivecriteria 120575 and 120590 show that the distance between modifiedPCM and original PCM is acceptable for both crisp and fuzzyelements A new effectiveness criterion COP reflects that themodified PCM resembles the original PCM (crisp and fuzzyelements) In conclusion this approach could enhance thequality of vague and inaccuracy data for decision makers and

also could better handle the inconsistency problem of AHPand fuzzy AHP

In the futurework wewill apply this approach to differentapplications Meanwhile efforts should be made to explorethe prefect range of 120573 and study the issue of selecting thesituation where we can supply a consistency PCM to aninconsistency one based on real-life meaning

Appendix

The Calculation Steps of (19)Before we go to find these elements define the range ofevery value of 119887

119871

119894119895 119887119872119894119895 119887119880119894119895 Given fuzzy reciprocal matrix 119860

= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899

it has the support SUPP(119886119894119895) sube

119878 = [1120590 120590] where 120590 gt 1 forall119894 119895 isin 1 2 119899 This means1120590 le 119886

119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590


Table 2 120572 = 120574 = 098 Iteration times CR le 01 120575 120590 120582max and priority weight results

Methods Iteration neededfor CR le 01 CR 120575 120590 120582max Priority weight

Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)

Table 3 COP (condition of order preservation) parameter valueOriginalmatrix 120596

120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

ConclusionCao et al [7] are closer to the original matrix 120596 but the gap among Cao et al [7] Xu and Wei [6] and our method is very

narrow (less than 1) the three methods almost in the same extends to close to original 120596 Thus our method is acceptable inthis parameter

Given a vector of fuzzy weights = (1 2

119899)

which is the corresponding eigenvector matrix closest tooriginal matrix 119860 the value of eigenvector matrix shouldsatisfy the following formula

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899

le (120590 sdot 120590 sdot sdot sdot 120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(120590 sdot 120590 sdot sdot sdot 120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)

The eigenvector of modified matrix 119861 and the adjustablematrix 119861

1015840 should also be in this range to be clearer the rangeof eigenvector is

1

120590

1

120590

1

120590

le 120596119894le 120590 120590 120590 forall119894119894 isin (1 2 sdot sdot sdot 119899) 119899 isin R

1

120590

1

120590

1

120590

le 120596119894

1015840le 120590 120590 120590 forall119894119894 isin (1 2 sdot sdot sdot 119899) 119899 isin R

(A2)


Table 4 120572 = 120574 = 098 Iteration times CR = 0 120575 120590 120582max and priority weight results

Methods Iteration neededfor CR = 0 CR 120575 120590 120582max Priority weight

Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao

Our method (120585 = 005 120573 = 02) 8 0 32554 14458 8 120596Our(005) 119877Our(005)



120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)

Table 5 Effective analysis for Algorithm 1 based on nine parameters

120573 = 05 NI [13] CCI [14 15] 120575fuzzy 120590fuzzy COP Iteration times Running time (seconds)120585 = 01 00887 01378 12841 05688 Keep 5 0033120585 = 001 00887 01378 12841 05688 Keep 8 0031120585 = 0001 00886 01378 12815 05691 Keep 11 0042120573 = 06 NI CCI 120575fuzzy 120590fuzzy COP Iteration times Running time (seconds)120585 = 01 01007 01991 09979 04692 Keep 5 0021120585 = 001 01007 01991 09979 04692 Keep 8 0034120585 = 0001 01007 01991 09960 04695 Keep 11 0046

Conflict of Interests

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

References

[1] T L Saaty The Analytic Hierarchy Process McGraw-Hill NewYork NY USA 1980

[2] H Q Zhang Y Ouzrout A Bouras V Della Selva and MSavino ldquoSelection of optimal product lifecycle managementcomponents based on AHP methodologiesrdquo in Proceedingsof the International Conference on Advanced Logistics andTransport (ICALT rsquo13) pp 523ndash528 Sousse Tunisia May 2013

[3] H Q Zhang Y Ouzrout A Bouras A Mazza and M SavinoldquoPLM components selection based on a maturity assessmentand AHP methodologyrdquo in Proceedings of the InternationalConference on Product Lifecycle Management Conference (PLMrsquo13) Nantes France July 2013

[4] F Torfi R Z Farahani and S Rezapour ldquoFuzzy AHP todetermine the relative weights of evaluation criteria and FuzzyTOPSIS to rank the alternativesrdquo Applied Soft ComputingJournal vol 10 no 2 pp 520ndash528 2010

[5] S Karapetrovic and E S Rosenbloom ldquoA quality controlapproach to consistency paradoxes in AHPrdquo European Journalof Operational Research vol 119 no 3 pp 704ndash718 1999

[6] Z S Xu and C P Wei ldquoA consistency improving method inthe analytic hierarchy processrdquo European Journal of OperationalResearch vol 116 no 2 pp 443ndash449 1999

[7] D Cao L C Leung and J S Law ldquoModifying inconsistentcomparison matrix in analytic hierarchy process a heuristicapproachrdquoDecision Support Systems vol 44 no 4 pp 944ndash9532008

[8] M Anholcer Algorithm for Deriving Priorities from InconsistentPairwise Comparison Matrices 2013

[9] Y J Xu and HMWang ldquoEigenvector method consistency testand inconsistency repairing for an incomplete fuzzy preferencerelationrdquoAppliedMathematicalModelling vol 37 pp 5171ndash51832013

[10] L C Leung and D Cao ldquoOn consistency and ranking ofalternatives in fuzzy AHPrdquo European Journal of OperationalResearch vol 124 no 1 pp 102ndash113 2000

[11] MMorteza andA R Bafandeh ldquoA newmethod for consistencytest in fuzzy AHPrdquo Journal of Intelligent and Fuzzy Systems vol25 no 2 pp 457ndash461 2013


[12] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[13] J Ramık and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[14] E Bulut O Duru T Kececi and S Yoshida ldquoUse of consistencyindex expert prioritization and direct numerical inputs forgeneric fuzzy-AHP modeling a process model for shippingasset managementrdquo Expert Systems with Applications vol 39no 2 pp 1911ndash1923 2012

[15] E Dopazo K Lui S Chouinard and J Guisse ldquoA parametricmodel for determining consensus priority vectors from fuzzycomparison matricesrdquo Fuzzy Sets and Systems 2013

[16] J J Buckley ldquoFuzzy hierarchical analysisrdquo Fuzzy Sets andSystems vol 17 no 3 pp 233ndash247 1985

[17] J Aguaron and J M Moreno-Jimenez ldquoThe geometric con-sistency index approximated thresholdsrdquo European Journal ofOperational Research vol 147 no 1 pp 137ndash145 2003

[18] S I Gass and T Rapcsak ldquoSingular value decomposition inAHPrdquo European Journal of Operational Research vol 154 no3 pp 573ndash584 2004

[19] W E Stein and P J Mizzi ldquoThe harmonic consistency index forthe analytic hierarchy processrdquo European Journal of OperationalResearch vol 177 no 1 pp 488ndash497 2007

[20] T L SaatyMulti-Criteria Decision Making-the Analytical Hier-archy Process vol 1 RWS Publications Pittsburgh Pa USA1991

[21] D Dubois ldquoThe role of fuzzy sets in decision sciences oldtechniques and newdirectionsrdquo Fuzzy Sets and Systems vol 184no 1 pp 3ndash28 2011

[22] M Yang F I Khan and R Sadiq ldquoPrioritization of envi-ronmental issues in offshore oil and gas operations a hybridapproach using fuzzy inference system and fuzzy analytichierarchy processrdquo Process Safety and Environmental Protectionvol 89 no 1 pp 22ndash34 2011

[23] M Brunelli ldquoA note on the article ldquoinconsistency of pair-wisecomparison matrix with fuzzy elements based on geometricmeanrdquo Fuzzy Sets and Systems vol 161 1604ndash1613 2010rdquo FuzzySets and Systems vol 176 no 1 pp 76ndash78 2011

[24] J Aguaron and J M Moreno-Jimenez ldquoThe geometric con-sistency index approximated thresholdsrdquo European Journal ofOperational Research vol 147 pp 137ndash145 2003

[25] C A Bana e Costa and J-C Vansnick ldquoA critical analysis of theeigenvalue method used to derive priorities in AHPrdquo EuropeanJournal of Operational Research vol 187 no 3 pp 1422ndash14282008

[26] P de Jong ldquoA statistical approach to Saatyrsquos scaling method forprioritiesrdquo Journal ofMathematical Psychology vol 28 no 4 pp467ndash478 1984

[27] T L Saaty and G Hu ldquoRanking by eigenvector versus othermethods in the analytic hierarchy processrdquo Applied Mathemat-ics Letters vol 11 no 4 pp 121ndash125 1998

[28] T L Saaty ldquoDecision-making with the AHP why is the prin-cipal eigenvector necessaryrdquo European Journal of OperationalResearch vol 145 no 1 pp 85ndash91 2003

[29] T S Almulhim L Mikhailov and D-L Xu ldquoDeriving weightsfrom group fuzzy pairwise comparison judgement matricesrdquo inAdvances in Information Systems and Technologies pp 545ndash555Springer 2013

[30] W Mogi N Izumi-cho and M Shinohara ldquoOptimum priorityweight estimation method for pairwise comparison matrixrdquo inProceedings of the 10th International Symposium on the AnalyticHierarchyNetwork Process 2009

[31] G Crawford and C Williams ldquoA note on the analysis of sub-jective judgment matricesrdquo Journal of Mathematical Psychologyvol 29 no 4 pp 387ndash405 1985

[32] G Kou D Ergu Y Peng and Y Shi ldquoA new consistency testindex for the data in the AHPANPrdquo in Data Processing for theAHPANP pp 11ndash27 Springer 2013

[33] W L Winston M Venkataramanan and J B GoldbergIntroduction to Mathematical Programming vol 1 Thom-sonBrooksCole 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


by setting deviation tolerances based on an idea of allowinginconsistent information Morteza and Bafandeh [11] furtherdiscussed Leung and Caorsquos work and proposed a newmethodof fuzzy consistency tests by direct fuzzification of a QR(quick response) algorithm which is one of the methodsfor the eigenvalues calculation of an arbitrary matrix Wangand Chen [12] applied fuzzy linguistic preference relationsto construct consistent PCMs by considering reducing thenumber of pairwise comparisons However these works donot have standard parameters to verify the reliability of theirtheory so far Therefore it is very important to prove thefeasibility of amethodology that can reduce the inconsistencyof the original matrix and preserve the original matrixrsquosinformation as much as possible

In this research work we propose a modified consistentPCM as a combination of original inconsistent PCM and anadjustable consistent PCM In order to achieve the modifiedPCM this paper is structured in the following way Section 2and Section 3 give the basic concepts of the PCM withfuzzy and crisp elements consistency indices for crisp andfuzzy elements and parameters to judge the effectivenessof the modified matrix Section 4 and Section 5 propose analgorithm to obtain the middle value upper value and lowervalue of the adjustablematrixThemain idea of our algorithmis to find the optimum priority vector by solving a linearprogramming problem and use the eigenvector method toobtain the adjustable matrix based on the optimum priorityvector In Section 6 the algorithm was applied to obtain theadjustable PCM by adopting the same illustrating PCM in[6 7] the comparison results show that our algorithm canpreserve more original information than Cao et al [7] andXu and Wei [6] In Section 7 the algorithm was used tofind the sensible and the closest consistent modified PCMwith fuzzy elements The experiments show that the newmodified matrix can satisfy consistent indicesrsquo requirementsofNI [13] andCCI [14 15] and also can preservemore originalinformation under parameters of 120590fuzzy and 120575fuzzy Section 8concludeswith effective and efficient analysis to show that ouralgorithm can be performed easily

2 State of the Art

21 Notations andDefinitions The119898times119899 pairwise comparisonmatrix 119860 with triangular fuzzy elements can be described inthe following

119860 = (

(119886119871

11 119886119872

11 119886119880

11) sdot sdot sdot (119886

119871

1119899 119886119872

1119899 119886119880

1119899)

d

(119886119871

1198981 119886119872

1198981 119886119880

1198981) sdot sdot sdot (119886

119871

119898119899 119886119872

119898119899 119886119880

119898119899)

) (1)

where for each element 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) 119886119871119894119895is the lower

value 119886119872119894119895is themiddle value and 119886

119880

119894119895is the upper value In the

particular situation if 119898 = 119899 when the following conditionis satisfied then 119860 is a reciprocal matrix

119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895) implies 119886

119895119894= (

1

119886119880

119894119895

1

119886119872

119894119895

1

119886119871

119894119895

)

forall119894 119895 = 1 2 119899

(2)

where 1120590 lt 119886119871

119894119895lt 119886119872

119894119895lt 119886119880

119894119895lt 120590 and 119878 = [1120590 120590] 120590 gt 1 119878 is

an interval of real numbers called fuzzy scale Crisp numbers(nonfuzzy numbers) are special cases of 119860 when 119886

119871

119894119895= 119886119872

119894119895=

119886119880

119894119895

Definition 1 Asit was proposed by Buckley [16] a fuzzypositive reciprocal matrix 119860 = [119886

119894119895]119899times119899

is consistent if andonly if 119886

119894119896⊙ 119886119896119895

asymp 119886119894119895 forall119894 119895 119896 | 1 le 119894 119895 119896 le 119899

Where the operator ⊙ is one of the operation rules oftriangular fuzzy elements this operation can be calculated bythe following equation

119886119894119896

⊙ 119886119896119895

= (119886119871

119894119896 119886119872

119894119896 119886119880

119894119896) ⊙ (119886

119871

119896119895 119886119872

119896119895 119886119880

119896119895)

= (119886119871

119894119896sdot 119886119871

119896119895 119886119872

119894119896sdot 119886119872

119896119895 119886119880

119894119896sdot 119886119880

119896119895)

asymp 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880

119894119895)

(3)

When positive reciprocal matrix 119860 is crisp numbers 119860then the consistent condition is 119886

119894119896sdot 119886119896119895

= 119886119894119895 forall119894 119895 119896 | 1 le

119894 119895 119896 le 119899However this definition is too strict because it is unre-

alistic to reach perfect consistency of a PCM (crisp orfuzzy elements) Some works [10 13 17ndash23] have developedconsistency indices to accept a certain level of deviationsWe will adopt consistency indices in Section 22 to determinewhether the current PCM matrix is at an acceptable consis-tency level

22 Consistency Indices Several consistency indices havebeen proposed for crisp numbers for instance geometricconsistency index [17] singular value decompositionmethod[18] and harmonic consistency index [19] However it wasproven that all these consistency indices are linear or non-linear transformations of Sattyrsquos CR [20] Therefore we useSattyrsquos CR in this paper for measuring consistency of crispnumbers

The consistency index of CR [20] is defined as follows

CR =

120582max minus 119899

119899 minus 1

sdot

1

RI (4)

where 120582max is the principle eigenvalue of 119860 RI is randomindexwhich can be gotten by searching a defined tableWhenthe value of 0 lt CR lt 01 the consistency can be accepted

Several important works focus on the consistency ofpairwise comparison matrix with fuzzy elements The firstone is Leung and Cao [10] who proposed a notion withconsideration of a tolerance deviation However the notion




NI120590119899= 120574120590

119899sdot max119894119895

max

1003816100381610038161003816100381610038161003816100381610038161003816

120596119871

119894

120596119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119872

119894

120596119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119880

119894

120596119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(1 le 119894 119895 le 119899)

(5)

where 120574120590


119894 120596119872119894

and 120596119880


119871

119894119895 119886119872119894119895 and 119886

119880

119894119895


CCI (119860) =

2

(119899 minus 1) (119899 minus 2)

times sum

119894lt119895

( log(

119886119871

119894119895+ 119886119872

119894119895+ 119886119880

119894119895

3

)

minus log(

120596119871

119894+ 120596119872

119894+ 120596119880

119894

120596119871

119895+ 120596119872

119895+ 120596119880

119895

))

2

(6)

where 119860 = (119886119871

119894119895 119886119872

119894119895 119886119880







120575 = max119894119895

10038161003816100381610038161003816119887119894119895minus 119886119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899 (7)

120590 =

radicsum119899

119894=1sum119899

119895=1(119887119894119895minus 119886119894119895)

2

119899

(8)



120575fuzzy = max119894119895

10038161003816100381610038161003816

119887119894119895minus 119886119894119895

10038161003816100381610038161003816

= max119894119895

10038161003816100381610038161003816119887119871

119894119895minus 119886119871

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119872

119894119895minus 119886119872

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119880

119894119895minus 119886119880

119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899

120590fuzzy = (max

(

119899

sum

119894=1

119899

sum

119895=1

(119887119871

119894119895minus 119886119871

119894119895)

2

)

(

119899

sum

119894=1

119899

sum

119895=1

(119887119872

119894119895minus 119886119872

119894119895)

2

)

119899

sum

119894=1

119899

sum

119895=1

(119887119880

119894119895minus 119886119880

119894119895)

2

)

12

119899minus1

(9)


1

1198862 1198863 and 119886


1dominates 119886

2and

1198863dominates 119886




1198863dominates 119886


120596(1198861) gt 120596(119886

2) and 120596(119886

3) gt 120596(119886









1015840(1198611015840)


1 1205962 120596

119899119879 which


= 120596119894120596119895 forall119894 119895 isin (1 2 119899)



119894120596119895and


119894120596119895and 119886


follows

119891 = max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

) (10)


119866 (119860 120596) = min (119891) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (11)







119861 = 119886120573

119894119895(1198871015840

119894119895)

1minus120573

(12)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(13)




119887119894119895]119899times119899


119894119895]119899times119899


119887119894119895

= (119887119871

119894119895 119887119872

119894119895 119887119880


is 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880



119894119895≐ 119886119871

119894119895 119887119872119894119895

≐ 119886119872

119894119895 and

119887119880

119894119895≐ 119886119880


119871

119894119895 119887119872

119894119895 119887119880


finding out 119887119871119894119895 119887119872

119894119895 119887119880



1198611015840

= [1198871015840

119894119895]119899times119899


119861 = 119860 ⊙ 1198611015840 (14)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(15)


119894119895= 119886119872

119894119895= 119886119880

119894119895 which



119894119895

1015840


119872

119894119895


1015840 119887119872119894119895


119872


119872

119894119895 119887119872

119894119895


119887119872

119894119895

1015840

asymp

120596119872

119894

120596119872

119895

(16)




119866(119886119872

119894119895 119872) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (17)



119872


follows

min 119885

st

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

le 119885 forall119894 119895 isin (1 2 119899) (18)

1

120590

le 120596119872

119894 120596119872

119895le 120590 119895 = 1 2 119899 (19)



minus119885 le (119886119872

119894119895minus

120596119872

119894

120596119872

119895

) le 119885 (20)

(119886119872

119894119895minus 119885) sdot 120596

119872

119895minus 120596119872

119894le 0 (21)

120596119872

119894le (119886119872

119894119895+ 119885) sdot 120596

119872

119895 (22)


120596119872

119895

120596119872

119894

ge

1

(119886119872

119894119895+ 119885)

(23)

1

(119886119872

119894119895+ 119885)

sdot 120596119872

119894minus 120596119872

119895le 0 (24)

1

(119886119872

119895119894+ 119885)

sdot 120596119872

119895minus 120596119872

119894le 0 (25)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895le 0 (26)


minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894le 0 (27)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

119867 = 119878 + 119879

0 le 119878 119879119867

(28)


119894 120596119872



min 119883

st

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

(29)

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

(30)

119867 = 119878 + 119879 (31)

120596119872

1+ 119883 = 1 (32)

1

120590

le 120596119872

119894 120596119872

119895le 120590119894 119895 = 1 2 119899 (33)

0 le 119885 119878 119879119867 (34)


1= 1 to normalize



119872









[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4


(4) If 119894 = 119895 then 119886119872



119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +





119872

119894119895




119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0



119894 120596119872119895to achieve 119887

119872

119894119895

1015840





119894119895

1015840 and 119887119880

119894119895


119894119895


119894119895

1015840 and 119887119880

119894119895




119894119895

1015840 and 119887119880

119894119895

1015840



119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880



120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895


119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)






120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)


119871

119894119895 119886119872119894119895 and 119886

119880



119871

119894119895and 119886

119872






between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












NI120590119899= 120574120590

119899sdot max119894119895

max

1003816100381610038161003816100381610038161003816100381610038161003816

120596119871

119894

120596119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119872

119894

120596119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

120596119880

119894

120596119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(1 le 119894 119895 le 119899)

(5)

where 120574120590


119894 120596119872119894

and 120596119880


119871

119894119895 119886119872119894119895 and 119886

119880

119894119895


CCI (119860) =

2

(119899 minus 1) (119899 minus 2)

times sum

119894lt119895

( log(

119886119871

119894119895+ 119886119872

119894119895+ 119886119880

119894119895

3

)

minus log(

120596119871

119894+ 120596119872

119894+ 120596119880

119894

120596119871

119895+ 120596119872

119895+ 120596119880

119895

))

2

(6)

where 119860 = (119886119871

119894119895 119886119872

119894119895 119886119880







120575 = max119894119895

10038161003816100381610038161003816119887119894119895minus 119886119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899 (7)

120590 =

radicsum119899

119894=1sum119899

119895=1(119887119894119895minus 119886119894119895)

2

119899

(8)



120575fuzzy = max119894119895

10038161003816100381610038161003816

119887119894119895minus 119886119894119895

10038161003816100381610038161003816

= max119894119895

10038161003816100381610038161003816119887119871

119894119895minus 119886119871

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119872

119894119895minus 119886119872

119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119887119880

119894119895minus 119886119880

119894119895

10038161003816100381610038161003816 119894 119895 = 1 2 119899

120590fuzzy = (max

(

119899

sum

119894=1

119899

sum

119895=1

(119887119871

119894119895minus 119886119871

119894119895)

2

)

(

119899

sum

119894=1

119899

sum

119895=1

(119887119872

119894119895minus 119886119872

119894119895)

2

)

119899

sum

119894=1

119899

sum

119895=1

(119887119880

119894119895minus 119886119880

119894119895)

2

)

12

119899minus1

(9)


1

1198862 1198863 and 119886


1dominates 119886

2and

1198863dominates 119886




1198863dominates 119886


120596(1198861) gt 120596(119886

2) and 120596(119886

3) gt 120596(119886









1015840(1198611015840)


1 1205962 120596

119899119879 which


= 120596119894120596119895 forall119894 119895 isin (1 2 119899)



119894120596119895and


119894120596119895and 119886


follows

119891 = max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

) (10)


119866 (119860 120596) = min (119891) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (11)







119861 = 119886120573

119894119895(1198871015840

119894119895)

1minus120573

(12)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(13)




119887119894119895]119899times119899


119894119895]119899times119899


119887119894119895

= (119887119871

119894119895 119887119872

119894119895 119887119880


is 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880



119894119895≐ 119886119871

119894119895 119887119872119894119895

≐ 119886119872

119894119895 and

119887119880

119894119895≐ 119886119880


119871

119894119895 119887119872

119894119895 119887119880


finding out 119887119871119894119895 119887119872

119894119895 119887119880



1198611015840

= [1198871015840

119894119895]119899times119899


119861 = 119860 ⊙ 1198611015840 (14)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(15)


119894119895= 119886119872

119894119895= 119886119880

119894119895 which



119894119895

1015840


119872

119894119895


1015840 119887119872119894119895


119872


119872

119894119895 119887119872

119894119895


119887119872

119894119895

1015840

asymp

120596119872

119894

120596119872

119895

(16)




119866(119886119872

119894119895 119872) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (17)



119872


follows

min 119885

st

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

le 119885 forall119894 119895 isin (1 2 119899) (18)

1

120590

le 120596119872

119894 120596119872

119895le 120590 119895 = 1 2 119899 (19)



minus119885 le (119886119872

119894119895minus

120596119872

119894

120596119872

119895

) le 119885 (20)

(119886119872

119894119895minus 119885) sdot 120596

119872

119895minus 120596119872

119894le 0 (21)

120596119872

119894le (119886119872

119894119895+ 119885) sdot 120596

119872

119895 (22)


120596119872

119895

120596119872

119894

ge

1

(119886119872

119894119895+ 119885)

(23)

1

(119886119872

119894119895+ 119885)

sdot 120596119872

119894minus 120596119872

119895le 0 (24)

1

(119886119872

119895119894+ 119885)

sdot 120596119872

119895minus 120596119872

119894le 0 (25)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895le 0 (26)


minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894le 0 (27)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

119867 = 119878 + 119879

0 le 119878 119879119867

(28)


119894 120596119872



min 119883

st

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

(29)

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

(30)

119867 = 119878 + 119879 (31)

120596119872

1+ 119883 = 1 (32)

1

120590

le 120596119872

119894 120596119872

119895le 120590119894 119895 = 1 2 119899 (33)

0 le 119885 119878 119879119867 (34)


1= 1 to normalize



119872









[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4


(4) If 119894 = 119895 then 119886119872



119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +





119872

119894119895




119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0



119894 120596119872119895to achieve 119887

119872

119894119895

1015840





119894119895

1015840 and 119887119880

119894119895


119894119895


119894119895

1015840 and 119887119880

119894119895




119894119895

1015840 and 119887119880

119894119895

1015840



119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880



120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895


119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)






120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)


119871

119894119895 119886119872119894119895 and 119886

119880



119871

119894119895and 119886

119872






between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1015840(1198611015840)


1 1205962 120596

119899119879 which


= 120596119894120596119895 forall119894 119895 isin (1 2 119899)



119894120596119895and


119894120596119895and 119886


follows

119891 = max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

) (10)


119866 (119860 120596) = min (119891) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895minus

120596119894

120596119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (11)







119861 = 119886120573

119894119895(1198871015840

119894119895)

1minus120573

(12)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(13)




119887119894119895]119899times119899


119894119895]119899times119899


119887119894119895

= (119887119871

119894119895 119887119872

119894119895 119887119880


is 119886119894119895

= (119886119871

119894119895 119886119872

119894119895 119886119880



119894119895≐ 119886119871

119894119895 119887119872119894119895

≐ 119886119872

119894119895 and

119887119880

119894119895≐ 119886119880


119871

119894119895 119887119872

119894119895 119887119880


finding out 119887119871119894119895 119887119872

119894119895 119887119880



1198611015840

= [1198871015840

119894119895]119899times119899


119861 = 119860 ⊙ 1198611015840 (14)


119861 = ((119886119871

119894119895)

120573

sdot (119887119871

119894119895

1015840

)

1minus120573

) ((119886119872

119894119895)

120573

sdot (119887119872

119894119895

1015840

)

1minus120573

)

((119886119880

119894119895)

120573

sdot (119887119880

119894119895

1015840

)

1minus120573

)

(15)


119894119895= 119886119872

119894119895= 119886119880

119894119895 which



119894119895

1015840


119872

119894119895


1015840 119887119872119894119895


119872


119872

119894119895 119887119872

119894119895


119887119872

119894119895

1015840

asymp

120596119872

119894

120596119872

119895

(16)




119866(119886119872

119894119895 119872) = min(max(

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

)) (17)



119872


follows

min 119885

st

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

le 119885 forall119894 119895 isin (1 2 119899) (18)

1

120590

le 120596119872

119894 120596119872

119895le 120590 119895 = 1 2 119899 (19)



minus119885 le (119886119872

119894119895minus

120596119872

119894

120596119872

119895

) le 119885 (20)

(119886119872

119894119895minus 119885) sdot 120596

119872

119895minus 120596119872

119894le 0 (21)

120596119872

119894le (119886119872

119894119895+ 119885) sdot 120596

119872

119895 (22)


120596119872

119895

120596119872

119894

ge

1

(119886119872

119894119895+ 119885)

(23)

1

(119886119872

119894119895+ 119885)

sdot 120596119872

119894minus 120596119872

119895le 0 (24)

1

(119886119872

119895119894+ 119885)

sdot 120596119872

119895minus 120596119872

119894le 0 (25)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895le 0 (26)


minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894le 0 (27)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

119867 = 119878 + 119879

0 le 119878 119879119867

(28)


119894 120596119872



min 119883

st

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

(29)

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

(30)

119867 = 119878 + 119879 (31)

120596119872

1+ 119883 = 1 (32)

1

120590

le 120596119872

119894 120596119872

119895le 120590119894 119895 = 1 2 119899 (33)

0 le 119885 119878 119879119867 (34)


1= 1 to normalize



119872









[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4


(4) If 119894 = 119895 then 119886119872



119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +





119872

119894119895




119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0



119894 120596119872119895to achieve 119887

119872

119894119895

1015840





119894119895

1015840 and 119887119880

119894119895


119894119895


119894119895

1015840 and 119887119880

119894119895




119894119895

1015840 and 119887119880

119894119895

1015840



119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880



120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895


119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)






120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)


119871

119894119895 119886119872119894119895 and 119886

119880



119871

119894119895and 119886

119872






between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872


follows

min 119885

st

1003816100381610038161003816100381610038161003816100381610038161003816

119886119872

119894119895minus

120596119872

119894

120596119872

119895

1003816100381610038161003816100381610038161003816100381610038161003816

le 119885 forall119894 119895 isin (1 2 119899) (18)

1

120590

le 120596119872

119894 120596119872

119895le 120590 119895 = 1 2 119899 (19)



minus119885 le (119886119872

119894119895minus

120596119872

119894

120596119872

119895

) le 119885 (20)

(119886119872

119894119895minus 119885) sdot 120596

119872

119895minus 120596119872

119894le 0 (21)

120596119872

119894le (119886119872

119894119895+ 119885) sdot 120596

119872

119895 (22)


120596119872

119895

120596119872

119894

ge

1

(119886119872

119894119895+ 119885)

(23)

1

(119886119872

119894119895+ 119885)

sdot 120596119872

119894minus 120596119872

119895le 0 (24)

1

(119886119872

119895119894+ 119885)

sdot 120596119872

119895minus 120596119872

119894le 0 (25)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895le 0 (26)


minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894le 0 (27)


minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

119867 = 119878 + 119879

0 le 119878 119879119867

(28)


119894 120596119872



min 119883

st

minus120596119872

119894+ (max

(119886119872

119894119895minus 119885)

1

(119886119872

119895119894+ 119885)

)120596119872

119895+ 119878 = 0

(29)

minus120596119872

119895+ (max

(119886119872

119895119894minus 119885)

1

(119886119872

119894119895+ 119885)

)120596119872

119894+ 119879 = 0

(30)

119867 = 119878 + 119879 (31)

120596119872

1+ 119883 = 1 (32)

1

120590

le 120596119872

119894 120596119872

119895le 120590119894 119895 = 1 2 119899 (33)

0 le 119885 119878 119879119867 (34)


1= 1 to normalize



119872









[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4


(4) If 119894 = 119895 then 119886119872



119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +





119872

119894119895




119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0



119894 120596119872119895to achieve 119887

119872

119894119895

1015840





119894119895

1015840 and 119887119880

119894119895


119894119895


119894119895

1015840 and 119887119880

119894119895




119894119895

1015840 and 119887119880

119894119895

1015840



119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880



120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895


119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)






120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)


119871

119894119895 119886119872119894119895 and 119886

119880



119871

119894119895and 119886

119872






between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[0 Z2] X

[0 Z] X

[0 Z4] X

[Z2 Z] X

[Z4 Z2] X [Z2 Z4] X [3Z4Z] X

P1P2 P3 P4


(4) If 119894 = 119895 then 119886119872



119894+ (max(119886119872

119894119894minus 119885) 1(119886

119872

119894119894+ 119885)) +





119872

119894119895




119894be (prod

119899

119895=1119886119872

119894119895)

1119899 and 120596119872

1= 1 let

119885max = 119885 = 119866(119886119872

119894119895 119872) and 119885min = 119885 = 0



119894 120596119872119895to achieve 119887

119872

119894119895

1015840





119894119895

1015840 and 119887119880

119894119895


119894119895


119894119895

1015840 and 119887119880

119894119895




119894119895

1015840 and 119887119880

119894119895

1015840



119894 then 119861

119872

119894119895

1015840

=

120596119872

119894120596119872

119895 119894 = 1 2 119899 Set 119862

119871 119862119880



120596119871

119894= 119862119871sdot 120596119872

119894

119908ℎ119890119903119890 119862119871= min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

120596119880

119894= 119862119880

sdot 120596119872

119894

119908ℎ119890119903119890 119862119880

= max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(35)

Then the value of 119887119871119894119895

1015840 and 119887119880

119894119895


119887119871

119894119895

1015840

=

120596119871

119894

120596119871

119895

119887119880

119894119895

1015840

=

120596119880

119894

120596119880

119895

(36)






120596119871

119894le 120596119872

119894le 120596119880

119894

120596119871

119894= 119862min120596

119872

119894

120596119880

119894= 119862max120596

119872

119894

(37)


119871

119894119895 119886119872119894119895 and 119886

119880



119871

119894119895and 119886

119872






between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













between 119886119872

119894119895and 119886119880



119862min = min

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119871

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119871

119894119895)

1119899

119862max = max

sum119899

119895=1119886119872

119894119895

sum119899

119895=1119886119880

119894119895

(prod119899

119895=1119886119872

119894119895)

1119899

(prod119899

119895=1119886119880

119894119895)

1119899

(38)


119894119895)1015840



1is


119894119895)1015840




(see Algorithm 1)




119860 =

(((((((((((

(

10000 50000 30000 70000 60000 60000 03333 02500

02000 10000 03333 50000 30000 30000 02000 01429

03333 30000 10000 60000 30000 40000 60000 02000

01429 02000 01667 10000 03333 02500 01429 01250

01667 03333 03333 30000 10000 05000 02000 01667

01667 03333 02500 40000 20000 10000 02000 01667

30000 50000 01667 70000 50000 50000 10000 05000

40000 70000 50000 80000 60000 60000 20000 10000

)))))))))))

)

(39)





119894119895 119886119872

119894119895 119886119880

119894119895) here crisp


119894119895= 119886119872

119894119895= 119886119880

119894119895= 119860









120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120583)

)


ij

A

a

bij

(a)

120583)

)

1120590 aMij 120590

)9984002

(b )9984001

)9984003

ij

b998400Mij

a

ij

(bij

(bij

B

(b)







Adopt (16) to get 119887119872119894119895


119871

119894119895

1015840 and 119887119880

119894119895

1015840

Adjustable PCM 1198611015840= (119887119871

119894119895

1015840

119887119872

119894119895

1015840

119887119880

119894119895

1015840




119894

END


Note 3 Consider

1205961 = (10000 03469 40000 05262 02902 01346 09375 14637)119879

1205962 = (10000 04499 12228 01259 04499 04499 05501 14953)119879

1205963 = (10000 09097 26082 02677 04345 04345 11620 33244)119879

(40)










1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198611015840=

((((

(

10000 10993 03834 37354 23015 23015 08606 03008

09097 10000 03488 33979 20936 20936 07828 02736

26082 28672 10000 97424 60027 60027 22445 07845

02677 02943 01026 10000 06161 06161 02304 00805

04345 04777 01666 16230 10000 10000 03739 01307

04345 04777 01666 16230 10000 10000 03739 01307

11620 12775 04455 43406 26745 26745 10000 03495

33244 36546 12746 124179 76512 76512 28609 10000

))))

)

(41)


119861 =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

(42)





119887119894119895

= 119886120572

119894119895(

120596119894

120596119895

)

1minus120572

(43)



119887119894119895

= (

120596119894

120596119895

) ∘ 1198631015840 (44)


119894119895= 119887119894119895times 119888119894119895 forall119894 119895 = 1 119899 and 119863


1198631015840= [1198891015840

119894119895] = 120574119863 oplus (1 minus 120574)DI (45)







120596XU = (10000 03241 10329 01034 01843 02172 0980419353) which is 103241 = 30855



119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119861Xu =

((((((

(

10000 29597 10501 94467 52285 44066 10207 05116

03379 10000 03548 31918 17666 14889 03449 01729

09523 28185 10000 89961 49791 41964 09720 04872

01059 03133 01112 10000 05535 04665 01080 00542

01913 05661 02008 18068 10000 08428 01952 00978

02269 06717 02383 21438 11865 10000 02316 01161

09797 28997 10288 92553 51226 43173 10000 05012

19546 57851 20525 184648 102197 86133 19950 10000

))))))

)

119861Cao =

((((

(

10000 32055 09198 98822 55733 47634 10371 05193

03120 10000 02869 30829 17387 14860 03235 01620

10872 34851 10000 107443 60595 51789 11275 05646

01012 03244 00931 10000 05640 04820 01049 00525

01794 05751 01650 17731 10000 08547 01861 00932

02099 06729 01931 20746 11700 10000 02177 01090

09643 30909 08869 95290 53741 45932 10000 05007

19258 61729 17712 190308 107329 91731 19971 10000

))))

)

119861Our (005) =

((((

(

10000 42959 22710 67491 49705 49705 04642 02934

02328 10000 03303 43498 24082 24082 02602 01582

04403 30273 10000 63266 30273 38107 49705 02602

01482 02299 01581 10000 03459 02748 01656 01184

02012 04152 03303 28906 10000 05743 02602 01789

02012 04152 02624 36387 17411 10000 02602 01789

21543 38428 02012 60374 38428 38428 10000 04569

34088 63228 38428 84449 55892 55892 21886 10000

))))

)

119861Our (01) =

((((

(

10000 36932 19881 61737 49536 49536 04030 02594

02708 10000 03364 46283 27917 27917 02628 01627

05030 29730 10000 66108 34464 43383 49288 02629

01620 02161 01513 10000 03769 02994 01572 01145

02019 03582 02902 26531 10000 05743 02267 01588

02019 03582 02305 33397 17411 10000 02267 01588

24817 38058 02029 63619 44119 44119 10000 04655

38547 61468 38041 87354 62989 62989 21484 10000

))))

)

119861Our (02) =

((((

(

10000 42516 23133 71790 49193 49193 04680 03044

02352 10000 03400 46750 24082 24082 02651 01658

04323 29414 10000 66066 29414 37026 49193 02651

01393 02139 01514 10000 03219 02557 01570 01155

02033 04152 03400 31067 10000 05743 02651 01876

02033 04152 02701 39107 17411 10000 02651 01876

21369 37727 02033 63702 37727 37727 10000 04702

32854 60312 37727 86577 53315 53315 21266 10000

))))

)

(46)








119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119860 = (

(1 1 1) (2 3 4) (4 5 6)

(

1

4

1

3

1

3

) (1 1 1) (3 4 5)

(

1

6

1

5

1

4

) (

1

5

1

4

1

3

) (1 1 1)

) (47)




and 120585 = 0001 under 120573 = 05 and 120573 = 06

119861 (120585 = 01 120573 = 05) = (

[1 1 1] [2021 2453 2718] [5284 5480 5800]

[0350 0407 0424] [1 1 1] [3202 3460 3895]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 05) = (

[1 1 1] [2017 2449 2714] [5281 5477 5797]

[0350 0408 0425] [1 1 1] [3205 3463 3899]

[0154 0182 0211] [0241 0288 0331] [1 1 1]

)

119861 (120585 = 01 120573 = 06) = (

[1 1 1] [2016 2553 2936] [4997 5380 5839]

[0327 0391 0404] [1 1 1] [31608 3562 4094]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

119861 (120585 = 0001 120573 = 06) = (

[1 1 1] [2014 2550 2933] [4995 5378 5837]

[0327 0392 0405] [1 1 1] [3163 3565 4098]

[0156 0185 0218] [0232 0280 0331] [1 1 1]

)

(48)



8 Discussion







[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[0 100725] 0

[0 50362] 0 [50362 100725] 0

[0 25181] 1 [25181 50362] 0 [50362 75543] 0 [75543 100725] 0

[25181 37772] 0 [37772 50362] 0

[25181 31476] 1 [31476 37772] 1

[31476 34624] 1 [34624 37772] 0

[34624 36198] 1 [36198 37772] 0

[36198 36985] 0 [36985 37772] 0


3 4 5 6 7 8 9 100

00501

01502

Size (n) of matrix

Crisp

runn

ing

time (

s)

120585 = 01

120585 = 001

120585 = 0001

(a)

3 4 5 6 7 8 9 100

00501

01502

Fuzz

y ru

nnin

g tim

e (s)

120585 = 01

120585 = 001

120585 = 0001

Size (n) of matrix

(b)




120573

119894119895(1198871015840







0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 600

005

01

015

02


CR

120574 = 098

120574 = 09

120574 = 075120574 = 05

(a)

0 10 20 30 40 50 60 700

005

01

015

02


CR

120572 = 098

120572 = 09

120572 = 075

120572 = 05

(b)

0 1 2 3 4 50

5

10

15

20

Num

ber o

f ite

ratio

ns

Accuracy degree 120576 = 01 120576 = 001 120576 = 0001 120576 = 00001

(c)

2 3 4 5 6 7 8 90

5

10

15

20

Num

ber o

f ite

ratio

ns

120576 = 01120576 = 001

120576 = 0001120576 = 00001

Size (n) matrix

(d)



9 Conclusions




Appendix


119871


= 119886119894119895119899times119899

= 119886119871

119894119895 119886119872

119894119895 119886119880

119894119895119899times119899



119871

119894119895le 119886119872

119894119895le 119886119871

119894119895le 120590




Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Xu and Wei [6] 12 00972 1845 0589 8955 120596XU 119877XU

Cao et al [7] 18 00997 1713 0448 89844 120596Cao 119877Cao

Our method (120585 = 005 120573 = 08) 8 00964 11572 04710 89017 120596Our(005)

119877Our(005)

Our method (120585 = 01 120573 = 08) 7 00964 07354 04689 89017 120596Our(01)

119877Our(01)

Our method (120585 = 02 120573 = 08) 6 00964 12273 04993 89017 120596Our(02)

119877Our(02)

120596XU 119877XU120596XU = (10000 03241 10329 01034 01843 02172 09804 19353)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03191 10695 01019 01824 02142 09699 19340)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 03430 10856 01143 02182 02493 08808 19064)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 03990 12401 01249 02189 02501 10147 21558)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 03466 10657 01074 02205 02518 08737 18375)119877Our(02) = (8 3 1 7 2 6 5 4)


120596XU 120596Cao 120596Our(005) 120596Our(01) 120596Our(02)

1198861 1198862 32055 30855 32037 29152 25062 28852

1198861 1198863 09198 09681 09197 09212 08064 09383

1198861 1198864 98822 96712 98857 87519 80060 93096

1198861 1198865 55733 54259 55806 45830 45674 45358

1198861 1198866 47634 46041 47658 40119 39984 39706





119899)


120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119871

119894119895)

1119899


119899

)

1119899


119899

)

1119899


119899

)

1119899

= 120590 120590 120590

120596119894asymp

(

119899

prod

119895=1

119886119871

119894119895)

1119899

(

119899

prod

119895=1

119886119872

119894119895)

1119899

(

119899

prod

119895=1

119886119880

119894119895)

1119899

ge

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

(

1

120590

sdot

1

120590

sdot sdot sdot

1

120590⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

)

1119899

=

1

120590

1

120590

1

120590

(A1)



1

120590

1

120590

1

120590


1

120590

1

120590

1

120590

le 120596119894


(A2)




Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Xu and Wei [6] 820 0 104648 19069 8 120596XU 119877XU

Cao et al [7] 1619 0 110308 20714 8 120596Cao 119877Cao




120596XU 119877XU120596XU = (10000 03120 10873 01012 01794 02100 09644 19259)

119877XU = (8 3 1 7 2 6 5 4)

120596Cao 119877Cao120596Cao = (10000 03121 10873 01012 01792 02098 09642 19260)

119877Cao = (8 3 1 7 2 6 5 4)120596Our(005)

119877Our(005)

120596Our(005) = (10000 04109 12324 01565 03665 03796 06374 18322)119877Our(005) = (8 3 1 7 2 6 5 4)

120596Our(01)

119877Our(01)

120596Our(01) = (10000 07522 20985 02235 03715 03848 11224 29959)119877Our(01) = (8 3 7 1 2 6 5 4)

120596Our(02)

119877Our(02)

120596Our(02) = (10000 04499 12228 01259 04499 04499 05501 14953)119877Our(02) = (8 3 1 7 2 6 5 4)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Optimal Inconsistency Repairing of...

Documents

Transcript of Research Article Optimal Inconsistency Repairing of...