Research Article Repairing the Inconsistent Fuzzy...

Research ArticleRepairing the Inconsistent Fuzzy Preference Matrix UsingMultiobjective PSO

Abba Suganda Girsang1 Chun-Wei Tsai2 and Chu-Sing Yang3

1Master of Information Technology at Binus Graduate Program Bina Nusantara University Jalan Kebon Jeruk Raya No 27Jakarta 11530 Indonesia2Department of Computer Science and Information Engineering National Ilan University Yilan 26041 Taiwan3Institute of Computer and Communication Engineering and Department of Electrical EngineeringNational Cheng Kung University Tainan 70101 Taiwan

Correspondence should be addressed to Abba Suganda Girsang gandagirsangyahoocom

Received 26 August 2015 Accepted 8 October 2015

Academic Editor Katsuhiro Honda

Copyright copy 2015 Abba Suganda Girsang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a method using multiobjective particle swarm optimization (PSO) approach to improve the consistency matrixin analytic hierarchy process (AHP) called PSOMOFThe purpose of this method is to optimize two objectives which conflict eachother while improving the consistency matrixThey are minimizing consistent ratio (CR) and deviation matrixThis study focuseson fuzzy preference matrix as one model comparison matrix in AHP Some inconsistent matrices are repaired successfully to beconsistent by this method This proposed method offers some alternative consistent matrices as solutions

1 Introduction

One important issue in comparison matrix of AHP is theconsistency Inmulticriteria decisionmaking (MCDM) deci-sion makers (DMs) reveal their opinion to choose somedecision alternatives by a comparison matrix [1] Howeverthe comparison matrix which is identified as inconsistentcannot be used as a judgment Meanwhile the consistencyis hard to obtain when evaluating a large number of criteria

There are twomodels of a comparisonmatrix multiplica-tive preference relations [1] and fuzzy preference relations[2 3] The element comparison matrix of multiplicative pref-erence relation is stated as 119886

119894119895which defines the dominance

of alternative 119894 over 119895 where 1 lt 119886119894119895

lt 9 and 119886119894119895

= 1119886119895119894

On fuzzy preference relations element comparison matrix isstated as 119886

119894119895 which defines the preference of alternative 119894 over

119895 where 0 lt 119886119894119895

lt 1 and 119886119894119895

+ 119886119895119894

= 1 This study focuses onfuzzy preference relations

The issues of consistency in fuzzy preference relation alsohave received attention from researchers Xu and Wang [4]proposed a revised approach by using linear programmingmodels to generate the priority weights for additive interval

fuzzy preference relations Xu and Chen [5] presented themethod to fulfill the element which is incomplete on fuzzypreference for group decision making based on additivetransitive consistency and accumulates the auxiliary valueinto a group auxiliary relationThis research was extended byXu et al [6] who deduced a function between the additivetransitivity fuzzy preference and its corresponding priorityvector Xu et al [7] proposed algorithm by eliminating thecycles of length 3 to 119899 in the digraph of the incompletereciprocal preference relation and converted it to the onewith ordinal consistency Liu et al [8] proposed a methodto solve the incompleteness of fuzzy preference matrix andalso repair the inconsistency preference matrix This methodcalculated minimal of the squared error of the incompletefuzzy preference relation and its priority weight vector tofulfill the missing values and generated the consistency fuzzypreference such that the modified one is the closest to theoriginal one Chen et al [9] presented a method for groupdecision making using incomplete fuzzy preference basedon additive consistency Chiclana et al [10] proposed afunctional equation to model the cardinal consistency inthe strength of preferences of reciprocal preference relations

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2015 Article ID 467274 10 pageshttpdxdoiorg1011552015467274

2 Advances in Fuzzy Systems

Xia et al [11] improved the consistency by using the geo-metric consistency index in complete and incomplete fuzzypreference

A research using swarm intelligence was also used tosolve the inconsistent comparison matrix such as PSO whichcombines Taguchi method [12] It improved the previousresearch using genetic algorithm [13] to solve the inconsistentcomparison matrix Both researches used the same objectivefunction to solve the problem that is summing the CR anddeviation matrix Although successful metaheuristic to solvethat problem the variations of implemented metaheuristicis rarely conducted Girsang et al [14 15] also alreadyimplemented the ant colony optimization (ACO) approach inour previous research to solve this problem with the differentobjective function that uses Yang et al [12] and Lin et al [13]In [14] besides repairing the inconsistent ratio ACO is usedto enhance the minimal deviation matrix while in [15] ACOis used to enhance the minimal consistent ratio It becomesa promising research to consider both of the two objectivefunctions using swarm intelligence Girsang et al [16] alsoimplemented PSO with multiobjective approach howeverit only focuses on repairing the multiplicative preferencematrix

2 Related Work

21 Consistent Ratio in AHP A simple illustration aboutinconsistency is described as follows The decision maker(DM) has opinion that119883 is bigger than119884 and119884 is bigger than119885 The consistent logic of this case is that 119883 should be biggerthan119885 Contrarily it would be inconsistent if DM said that119885is bigger than 119883 In AHP the opinion of decision makers isrepresented in a comparisonmatrix An element comparisonmatrix can reflect the subjective opinion that expose strengthof the preference and the feeling In a fuzzy preferencematrixthe element of comparison matrix119860 can be expressed as 119886

119894119895

with a scale value (0 sdot sdot sdot 1) where 0 lt 119886119894119895

lt 1 119886119894119895

+ 119886119895119894

= 1and 119886

119894119894= 05 Matrix 119860 as Fuzzy preference relation can be

depicted as follows

119860 = (

05 1 minus 11988621

1 minus 11988631

1 minus 11988641

11988621

05 1 minus 11988632

1 minus 11988642

11988631

11988632

05 1 minus 11988643

11988641

11988642

11988643

05

) (1)

To measure the multiplicative consistency in a compari-son matrix Saaty defined consistent ratio (CR) He proposedthat the threshold of CR inmultiplicative preferencematrixesis 01 The CR is defined as

119860119882 = 120582max119882 (2)

CI =

120582max minus 119899

119899 minus 1

(3)

CR =

CIRI

(4)

where 120582max and 119882 are the eigenvalue and eigenvector ofthe matrix respectively Further CI is the consistency index

Table 1 Random consistency index (RI)

Number criteria 1 2 3 4 5 6 7 8 90 0 058 09 112 124 132 141 145

119899 represents number criteria or size matrix and the RI(random consistency index) is the average index of randomlygenerated weights The value of RI on each size matrices isdescribed in Table 1 A CR less than 01 can be categorized asconsistent matrix Perfect consistency is obtained when themaximum eigenvalue equal to the number criteria (120582max =

119899)Herrera-Viedma et al [17] proposed some definitions to

reveal the consistency in a fuzzy preference matrix Theyshow that the additive consistency is more appropriate todefine the degree of consistency of fuzzy preference matrixThe relation in matrix 119860 is consistent if the element matrixcan satisfy (5) and (6)

119886119894119895

+ 119886119895119896

+ 119886119896119894

=

3

2

forall119894 119895 119896 (5)

where

119908119894=

sum119899

119895=1119886119894119895

minus 05

119899 (119899 minus 1) 2

(6)

Xu and Da [18] proposed determining the multiplicativeconsistency in the fuzzy preference matrix They used Xursquos[19] approach to determine CI in multiplicative preferencematrix Suppose 119887

119894119895is the element of multiplicative of pref-

erence matrix Xu [19] defined the CI in (7) This equation isderived from (3) By knowing theweight of each criterion andelement of matrix CI can be obtained

CI =

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[119887119894119895

sdot

119908119895

119908119894

+ 119887119895119894

sdot

119908119894

119908119895

minus 2] (7)

where 119887119894119895

is element matrix of multiplicative preferencematrix and 119908 = (119908

1 1199082 1199083 119908

119899) is the weight of each

criterion To determine the CI in a fuzzy preference matrix[4 18] used convertingwith assumption 119887

119894119895= 119886119894119895119886119895119894 where 119886

119894119895

is element matrix of the fuzzy preference matrix Thereforethey proposed determining CI as in

CI =

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[

119886119894119895

119886119895119894

sdot

119908119895

119908119894

+

119886119895119894

119886119894119895

sdot

119908119894

119908119895

minus 2] (8)

22 Deviation Matrix While the consistent ratio is repairedthe modified matrix automatically generates the deviationmatrix from the original Ideally the modified matrices arekept closer to their original matrices in order to maintainthe original judgment It means that the deviation matrix isenriched to beminimalThere are somemethods to representthe deviation such as difference index (Di) [13] and 120575 and 120590Difference index (Di) is defined as the real difference between

Advances in Fuzzy Systems 3

the same gene values in two genotypesThe other deviation isdefined as 120575 and 120590 which is denoted as

120575 = max119894119895

100381610038161003816100381610038161198861015840

119894119895minus 119886119894119895

10038161003816100381610038161003816 (9)

120590 =

radicsum119899

119894=1sum119899

119895=1(1198861015840

119894119895minus 119886119894119895

)

2

119899

(10)

where 119860 is the original matrix [119886119894119895] 119860 is the modified matrix

[1198861015840119894119895] and 119899 is the matrix sizeIn the multiplicative preference matrix the difference

index (Di) is generally used to measure the distance betweentwo matrices However in fuzzy preference matrix 120575 and 120590

are considered as more appropriate to represent the deviationmatrix Since the value preference will be 051 to 1 or 0 to049 the division each all genes in Di will not be differentsignificantly As a consequence the difference of twomatriceswill not be significant as well Therefore in this study insteadof using Di 120590 is employed to define the deviation matrix forthe preference fuzzy matrix

23 Particle Swarm Optimization PSO was firstly proposedby Kennedy and Eberhart [20] It is population-basedstochastic optimization on the social behaviors observed inanimals or insects such as bird flocking fish schooling andanimal herding In PSO each particle of swarm represents thesolution which moves to search the optimal solutions Eachparticle also broadcasts its current position to neighbourparticles The position of each particle is adjusted accordingto its velocity and the best position it has found so far Aparticle 119894 starts moving with a velocity 119881119894(119905 + 1) from itscurrent position 119883119894(119905) to the next position 119883119894(119905 + 1) as in(11) The velocity is influenced by three factors (a) previousvelocity 119881119894(119905) (b) the best previous particle position 119883119901(119905)and (c) the best previous swarmparticle position119883119892(119905) It canbe stated as (12)

119883119894 (119905 + 1) = 119883119894 (119905) + 119881119894 (119905 + 1) (11)

119881119894 (119905 + 1) = (119908 lowast 119881119894 (119905)) + (1198621lowast 1198771(119883119901 (119905) minus 119883119894 (119905))

+ (1198622lowast 1198772(119883119892 (119905) minus 119883119894 (119905)))

(12)

where 119908 is the weight to control the convergence of thevelocity 119862

1the acceleration weight cognitive element 119862

2

the weight of social parameter and 1198771and 119877

2are random

numbers in the range [0 1]

3 Proposed Method

31 PSOMOF Algorithm In PSO each particle seeks thebest position by moving in the search space The positionin PSO can represent an element in the comparison matrixAs shown in previous section encoding position of elementmatrix can be encoded only from lower triangular matrix Ifmatrix 119860 is identified as an inconsistent matrix and needsto be repaired then the scale value of matrix should bechanged with new value To be efficient the whole elements

of comparison matrix can be represented by lower triangularmatrixTherefore the position of PSO that should be changedcan be represented only by the lower triangular matrixWhen changing the value of each node to be consistent italso changes the rate consistent ratio (CR) and deviationmatrix We use 120590 to represent the deviation matrix in thismethod Changing the value of each node means changingthe particlersquos position In PSO the position is affected by aparticles historical best position (local best) and the swarmsrsquobest position (global best) The solution (new value position)is performed to chase the consistent rate However as previ-ously mentioned there is no one solution which can achieveCR and 120590 minimal at the same time PSOMOF algorithm isproposed by constructing the nondominated solutions whichdepicts the relation between 120590 andCR Algorithm 1 shows theoutline of PSOMOFalgorithm In thismethod there are threesteps in which each step uses PSO to get the result matrices

(1) Minimize 120590 Step Firstly each particle (there are 200particles) generates its position and its velocity ran-domly The position particle means that the particlegenerates randomly the candidate for the modifiedmatrix The element matrix can be represented onlyby the lower triangularmatrix elements consecutivelyThe velocity particle means that the particle generatesthe value as addingdiminishing the position of theparticles The initial position of each particle 119883119894(119905)

is set the same as the original position The initialvelocity of each particle 119881119894(119905) is set randomly butlower than 01 The best historical particle is definedas 119883119901 and the best position for all particles isdefined as 119883119892 Initially 119883119901 is taken from the firstposition particle generated while 119883119892 is taken fromthe best position from the first position of all particlesgenerated In the next iteration based on the previousvelocity information 119883119901 119883119892 and some variables(1199081198621 1198622 1198771 1198772) the velocity of each particle is

updated as described in (12) To set the value ofvariables some experiments are conducted and thenew position will be obtained based on the updatedvelocity as described in (11) The evaluation of thefitness function is minimizing However if a particles120590 is worse than before or CR gt 01 the updatewill be cancelled The result of this fitness functionalso updates the new best historical position of eachparticle (119883119901) and the new best position of all particlesas a group (119883119892) This process is repeated until theiteration maximum is reached

(2) Minimize CR Step It is almost the same as step (1) Ifstep (1) minimizes 120590 as its fitness function then step(2) minimizes CR as its fitness function

(3) Obtain a Set of Nondominated CR-120590 Solutions Step Itis also the same as the process to minimize 120590 Yet theprocess adds some various CRs which are decreasedgradually until CRmin is reached

32 Encoding and Fractioning of Original Element MatrixThe encoding of matrix can be assembled by picking all


Initialize ()Minimize 120590 ()For each particle generates the position and velocity randomly

Xp larr the initial position Xp is the particles best historicalXg larr the initial position Xg is the best of all particlesRepeatDetermine velocity using (12)Update new position particle using (11)Determine 120590 of new position using (10) If the new position has a lower 120590 and CR lt 01 updated new position is allowedotherwise update new position is canceled and keeping the current positionChoosing the new Xp and Xg based on value 120590

Until max iterations is reachedGet minimal 120590

Minimize CR ()For each particle generates the position and velocity randomly

Xp larr the initial position Xp is the particles best historicalXg larr the initial position Xg is the best of all particlesRepeatDetermine velocity using (12)Update new position particle using (11)Determine CR of new position using (4) If the new position has a lower CR and CR lt 01 updated new position is allowedotherwise update new position is canceled and keeping the current positionChoosing the new Xp and Xg based on value CR

Until max iterations is reachedGet minimal CR

Minimize CR-120590 ()CRo larr 01Xp larr the initial position Xp is the particles best historicalXg larr the initial position Xg is the best of all particlesWhile CRmin lt CRoRepeatDetermine velocity using (12)Update new position particle using (11)Determine CR of new position using (4) If the new position has a lower CR and CR lt 01 updated new position is allowedotherwise update new position is canceled and keeping the current positionChoosing the new Xp and Xg based on value CR

Until max iterations is reachedStore the modified matrix and its CR 120590CRo larr CRo minus k k is small value in this study 119896 = 0001

EndWhileGet matrices with their CR 120590

Algorithm 1 PSOMOF algorithm

elements in matrix However because the elements of fuzzypreferencematrix (FPM) have a relation such that 119886

119894119895+119886119895119894

= 1

and 119886119894119894

= 05 encoding node can only encode the lowertriangular elements of the matrix as nodes

119860 = (

05 06 04 08

04 05 03 07

06 07 05 09

02 03 01 05

)

Encode 119860 = 04 minus 06 minus 07 minus 02 minus 03 minus 01

(13)

Equation (13) showsmatrix119860with 119899 = 4 and its encodingfor FPMby picking row by row sequentially in the elements ofthe lower triangularmatrixThe number element of encoding

119860 can be determined (1198992minus 119899)2 To obtain consistent matrix

of course the value of each elementmatrix should be changedto a new value The new values are chosen from the valuesof several candidates Candidates elements are generatedusing the original fractioned value If the original element ismore than 05 the candidates will be between 05 and 1 ifthe original element is less than 05 the candidates will bebetween 0 and 05 if the original element is 05 (neutral)the candidate is still 05 or the original data should notbe fractioned This approach makes the candidate elementnot change the judgment tendency but will only change thejudgment weightThe number of candidate elements is basedon the fraction factor (120595) For example if 120595 = 001 then thenumber candidate will be 50(= (1 minus 05)001) Suppose thatmatrix 119860 119886

119903is one of the original elements on node 119903 and 119899


Table 2 The original element and its candidate with 120595 = 001

Origin element Candidate element05 050 01 02 03 04 0 001 002 048 04906 07 08 09 1 051 052 053 099 1

is the matrix size thus the sequence of nodes traveled 119866119860

can be defined as

119866119860

= 1198861 1198862 1198863 119886

(1198992minus119899)2

(14)

Each element origin 119886119903 is fractioned into several candi-

date elements 119886119903119904 where 119904 denotes the index of the candidate

element as described in

119886119903119904

= 119886119903119904minus1

+ 120595

1198861199030

=

0 if 0 le 119886119903lt 05

05 if 05 lt 119886119903le 1

(15)

where 119903 = 1 2 3 (1198992minus 119899)2 119904 = 1 2 3 (1 minus 05)120595

Table 2 shows the original element and its candidate as aresult of being fractioned if 120595 = 001 There are 50 candidatesto substitute for the origin element

These fractioned elements can be used as candidate nodesto travel by particle in PSOMOFThe particle will move fromthe candidate in one node to the candidate in the next nodeHowever it is possible that the particle preserves the originalelement

33 Determining CI in a Fuzzy Preference Matrix Deter-mining CI on fuzzy preference matrix by using (8) as atransforming from (7) is not suitable Transformation usingoperation 119887

119894119895= 119886119894119895119886119895119894

is not appropriate because it canexceed the threshold of themultiplicativematrix element Forexample suppose 119886

119894119895= 095 and 119886

119895119894= 005 By transforming

the above formula 119887119894119895will be 19 The value 119887

119894119895exceeds 9

which is the threshold of the multiplicative element matrixTherefore in this study we use a method to transform thefuzzy preference (119886

119894119895) to the multiplicative preference (119887

119894119895) as

introduced by Herrera-Viedma et al [17] to determine themultiplicative consistency as shown in

119887119894119895

= 92lowast119886119894119895 (16)

119886119894119895

=

1

2

(1 + log9119887119894119895) (17)

Therefore if the element fuzzy preference matrix 119886119894119895

=

095 it can be transformed to be the element multiplicativematrix 119887

119894119895= 722 This transformation value is not higher

than the maximum scale of 9 By using (16) a new formulais proposed to determine the CI as shown in

CI

=

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[92sdot119886119894119895minus1

sdot

119908119895

119908119894

+ 92sdot119886119895119894minus1

sdot

119908119894

119908119895

minus 2]

(18)

Table 3 Parameter settings for the PSOMOF NSGA-2 andMOPSO

Parameter ValuePSOMOF

119882 011198621

021198622

031198771

041198772

05NSGA-2Population size 100Generation 200Rate crossover 09Rate mutation 01

MOPSONumber of particle 20Number of cycles 1000

To prove this formula the consistent ratio rate of onesample matrix as shown in (13) is determined This samplematrix is selected from Xu et al [6] According to (5)obviously matrix 119860 can be verified as a consistent matrixContrarily matrix 119860 is identified as an inconsistent matrixwith CR = 011 when it is determined by (4) and (7)However if (4) and (18) are used CR will be 005 andtherefore will be a consistent matrix as in the result of (5)Therefore in this research (18) is used to define the CI value

4 Experimental Results

41 Parameter Setting Table 3 shows the parameter settingsfor the proposed and compared methods The inconsistentmatrix can be taken from the real life application whichneeds the decision maker opinions of comparing severalcriteria to get some alternatives Once the matrix is identifiedas inconsistent PSOMOF is able to be used to repair theinconsistent matrix To see the performance of proposedmethods in repairing inconsistent matrices there are 15inconsistent fuzzy preference matrices which need to berepaired as shown in Table 4 Some matrices come from theother papers but some matrices are created randomly

42 Generating Nondominated Solutions As aforemen-tioned there are two objectives for this proposed methodthat is the best CR and deviation matrix Both objectives willconflict each other When the CR is lowest (good consistentratio) it leads to the highest (the worst) deviation and viceversa However in order to get the acceptable matrix the CRof modified matrix is limited below 01 It makes the solutionconsist of some relations (ldquoCR-deviationrdquo) which can beidentified as nondominated solutions Equations (19a) (19b)and (19c) display the performance of PSOMOF to get thebest CR-120590 for 119860

5 respectively The origin matrix 119860

5(19a)

can be transformed to the modified matrices which have thebest CR (19b) and 120590(19c) respectively


Table 4 The dataset inconsistency matrices

Matrix Elements of lower triangular matrix CRSize 4 times 4

1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015

07-08-07-03-08-06-04-07-02-02-07-03-08-07-03-01-03-01-02-08-08-03-08-01-06- 02-01-04 0457Data on a and b is picked from [8 18]

CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)

PSOMOF splits the method into three steps These areto find the optimal deviation optimal CR and the optimaldeviation with the particular value of CR Figure 1 shows

the process convergence to find the optimal deviation whileFigure 2 shows process convergence to find the optimal CRBoth of them are conducted on 119860

5

After obtaining the minimal CR and 120590 the third stepof the PSOMOF is executed to get the nondominated CR-deviation nodes By using PSOMOF for each CR theoptimal deviation can be obtained This proposed methodthus successfully generates some nodes as solutions Figure 3shows the Pareto graph which depicts the relation of CR anddeviation of matrixThe sample matrices for fuzzy preferencematrix are 119860

1 1198605 1198609 11986012 and 119860

14 It shows clearly that

they will be contradictory to each other In case of matriceswhen 120590 is minimized CR is maximized Likewise when CRis minimized 120590 is maximized

43 Comparison with Other Methods To evaluate the per-formance of PSOMOF this study uses the metric analysis[21 22] The performance is represented by the Pareto graph10 times The Pareto graph is then compared with Paretographs of two other algorithms NSGA-2 [23] and MOPSO[24] The Pareto-optimal set is generated by merging all ofthe Pareto graphs of all algorithms (PSOMOF NSGA-2 andMOPSO) into a single Pareto solution The nondominatedsolutions for each algorithm are generated by executing eachalgorithm once on a sample inconsistent matrix (119860

1 1198605

1198609 11986012 and 119860

14) There are 3 metrics to measure the

performance of nondominated solutions achieved using theproposed method Suppose a set of nondominated solutions119883 sube 119883

1198721Metric This metric measures the average distance of the

resulting nondominated set solutions to the Pareto-optimal


0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734

Figure 1 The process convergence to find the optimal deviation on 1198605

0 50 100

Iteration

A5

740025

0

002

004

006

008

CR

Figure 2 The process convergence to find the optimal CR on 1198605

set solutionsThe better value should be a lower1198721 It can be

defined as desribed in

1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)

1198722Metric This metric measures the number of distribution

nondominated solutions which are covered by a neighbour-hood parameter 119889 gt 003 A bigger 119872

2indicates better

performance 1198722can be defined as

1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)

1198723MetricThis metric measures the extent of nondominated

sets obtained Awide range of values should be covered by the

nondominated solutions The bigger 1198723is better 119872

3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)

The comparison results are shown inTable 5 It shows thatPSOMOF 119872

1metric is minimal in all of matrices compared

to MOPSO and NSGA-2 These results show that most of thePareto graphs of PSOMOF are closer to the Pareto-optimalfront than both algorithms (NSGA-2 and MOPSO) For 119872

2

metric the PSOMOF result is larger than both of the otheralgorithms except for 119860

12 This indicates that the solutions

of the proposed method are more distributed than bothalgorithms In the 119872

3metric the proposed algorithm also

outperforms as compared to the NSGA-2 and MOPSO Theproposed method returned nondominated solutions furtherthan both of the other algorithms Regarding this result theproposed method PSOMOF can be claimed as the betteralgorithm compared to the two algorithms (NSGA-2 andMOPSO)

5 Conclusions

This paper presents a study to use the multiobjective PSOto solve the inconsistent fuzzy preference matrix in AHPcalled PSOMOF There are two objectives (consistent ratioand deviation matrix) considered in rectifying the matrixin order to be consistent However they are conflicting inthat process Therefore the proposed algorithms offer somenondominated solutions which also satisfied the acceptableconsistent matrices The process in PSOMOF is split intothree parts in which each part applies the PSO process Tosee the performance 15 inconsistent comparisonmatrices arerepaired by the proposed methods Besides repairing incon-sistent comparison matrices the proposed method also can


0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14

Figure 3 The Pareto graph solutions which show relation CR-120590

generated some nondominated solution which can be classi-fied as optimal solutionsThis result shows the PSO algorithmis the potential approach to solve the inconsistent compar-ison matrix in AHP The other intelligent algorithm alsomight be used to solve this problem Further this proposedmethod might be a potential method to combine with othermethod metaheuristic (hybrid)119899 to improve the quality ofresults

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their valuable comments and suggestions on the paper


Table 5 Comparison of the metric performance of NSGA-2 MOPSO and MOBAF

Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148

This work was supported in part by the Ministry of Scienceand Technology of Taiwan under Contracts MOST103-2221-E-197-034 MOST 104-2221-E-197-005 and NSC 100-2218-E-006-028-MY3

References

[1] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resources Allocation McGraw-Hill New York NYUSA 1980

[2] S A Orlovsky ldquoDecision-making with a fuzzy preferencerelationrdquo Fuzzy Sets and Systems vol 1 no 3 pp 155ndash167 1978

[3] F Chiclana F Herrera and E Herrera-Viedma ldquoIntegratingmultiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relationsrdquo Fuzzy Setsand Systems vol 122 no 2 pp 277ndash291 2001

[4] Y Xu and H Wang ldquoEigenvector method consistency testand inconsistency repairing for an incomplete fuzzy preferencerelationrdquo Applied Mathematical Modelling vol 37 no 7 pp5171ndash5183 2013

[5] Z Xu and J Chen ldquoGroup decision-making procedure based onincomplete reciprocal relationsrdquo Soft Computing vol 12 no 6pp 515ndash521 2008

[6] Y Xu Q Da and H Wang ldquoA note on group decision-making procedure based on incomplete reciprocal relationsrdquoSoft Computing vol 15 no 7 pp 1289ndash1300 2011

[7] Y Xu J N D Gupta and H Wang ldquoThe ordinal consistencyof an incomplete reciprocal preference relationrdquo Fuzzy Sets andSystems vol 246 pp 62ndash77 2014

[8] X Liu Y Pan Y Xu and S Yu ldquoLeast square completion andinconsistency repair methods for additively consistent fuzzypreference relationsrdquo Fuzzy Sets and Systems vol 198 pp 1ndash192012

[9] S-M Chen T-E Lin and L-W Lee ldquoGroup decision makingusing incomplete fuzzy preference relations based on theadditive consistency and the order consistencyrdquo InformationSciences vol 259 pp 1ndash15 2014

[10] F Chiclana E Herrera-Viedma F Alonso and S HerreraldquoCardinal consistency of reciprocal preference relations a char-acterization of multiplicative transitivityrdquo IEEE Transactions onFuzzy Systems vol 17 no 1 pp 14ndash23 2009

[11] M Xia Z Xu and J Chen ldquoAlgorithms for improving con-sistency or consensus of reciprocal [01]-valued preferencerelationsrdquo Fuzzy Sets and Systems vol 216 pp 108ndash133 2013

[12] I-T Yang W-C Wang and T-I Yang ldquoAutomatic repair ofinconsistent pairwise weighting matrices in analytic hierarchyprocessrdquoAutomation in Construction vol 22 pp 290ndash297 2012

[13] C-C Lin W-C Wang and W-D Yu ldquoImproving AHP forconstructionwith an adaptiveAHPapproach (A3)rdquoAutomationin Construction vol 17 no 2 pp 180ndash187 2008

[14] A S Girsang C-W Tsai and C-S Yang ldquoAnt algorithm formodifying an inconsistent pairwise weighting matrix in ananalytic hierarchy processrdquoNeural Computing and Applicationsvol 26 no 2 pp 313ndash327 2014

[15] A S Girsang C-W Tsai and C-S Yang ldquoAnt colony optimiza-tion for reducing the consistency ratio in comparison matrixrdquoin Proceedings of the International Conference on Advances inEngineering and Technology (ICAET rsquo14) pp 577ndash582 Singa-pore March 2014

[16] A S Girsang C Tsai and C Yang ldquoMulti-objective particleswarm optimization for repairing inconsistent comparisonmatricesrdquo International Journal of Computers and Applicationsvol 36 no 3 2014

[17] E Herrera-Viedma F Herrera F Chiclana and M LuqueldquoSome issues on consistency of fuzzy preference relationsrdquoEuropean Journal of Operational Research vol 154 no 1 pp 98ndash109 2004

[18] Z Xu and Q Da ldquoAn approach to improving consistencyof fuzzy preference matrixrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 3ndash12 2003

[19] Z Xu ldquoOn consistency of the weighted geometric meancomplex judgement matrix in AHPrdquo European Journal of Oper-ational Research vol 126 no 3 pp 683ndash687 2000

[20] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[21] C Garcıa-Martınez O Cordon and F Herrera ldquoA taxonomyand an empirical analysis of multiple objective ant colonyoptimization algorithms for the bi-criteria TSPrdquo EuropeanJournal of Operational Research vol 180 no 1 pp 116ndash148 2007

[22] E Zitzler K Deb and LThiele ldquoComparison of multiobjectiveevolutionary algorithms empirical resultsrdquo Evolutionary Com-putation vol 8 no 2 pp 173ndash195 2000


[23] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[24] C A Coello Coello andM S Lechuga ldquoMOPSO a proposal formultiple objective particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo02) vol 2pp 1051ndash1056 Honolulu Hawaii USA May 2002

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



Xia et al [11] improved the consistency by using the geo-metric consistency index in complete and incomplete fuzzypreference

A research using swarm intelligence was also used tosolve the inconsistent comparison matrix such as PSO whichcombines Taguchi method [12] It improved the previousresearch using genetic algorithm [13] to solve the inconsistentcomparison matrix Both researches used the same objectivefunction to solve the problem that is summing the CR anddeviation matrix Although successful metaheuristic to solvethat problem the variations of implemented metaheuristicis rarely conducted Girsang et al [14 15] also alreadyimplemented the ant colony optimization (ACO) approach inour previous research to solve this problem with the differentobjective function that uses Yang et al [12] and Lin et al [13]In [14] besides repairing the inconsistent ratio ACO is usedto enhance the minimal deviation matrix while in [15] ACOis used to enhance the minimal consistent ratio It becomesa promising research to consider both of the two objectivefunctions using swarm intelligence Girsang et al [16] alsoimplemented PSO with multiobjective approach howeverit only focuses on repairing the multiplicative preferencematrix

2 Related Work

21 Consistent Ratio in AHP A simple illustration aboutinconsistency is described as follows The decision maker(DM) has opinion that119883 is bigger than119884 and119884 is bigger than119885 The consistent logic of this case is that 119883 should be biggerthan119885 Contrarily it would be inconsistent if DM said that119885is bigger than 119883 In AHP the opinion of decision makers isrepresented in a comparisonmatrix An element comparisonmatrix can reflect the subjective opinion that expose strengthof the preference and the feeling In a fuzzy preferencematrixthe element of comparison matrix119860 can be expressed as 119886

119894119895

with a scale value (0 sdot sdot sdot 1) where 0 lt 119886119894119895

lt 1 119886119894119895

+ 119886119895119894

= 1and 119886

119894119894= 05 Matrix 119860 as Fuzzy preference relation can be

depicted as follows

119860 = (

05 1 minus 11988621

1 minus 11988631

1 minus 11988641

11988621

05 1 minus 11988632

1 minus 11988642

11988631

11988632

05 1 minus 11988643

11988641

11988642

11988643

05

) (1)

To measure the multiplicative consistency in a compari-son matrix Saaty defined consistent ratio (CR) He proposedthat the threshold of CR inmultiplicative preferencematrixesis 01 The CR is defined as

119860119882 = 120582max119882 (2)

CI =

120582max minus 119899

119899 minus 1

(3)

CR =

CIRI

(4)

where 120582max and 119882 are the eigenvalue and eigenvector ofthe matrix respectively Further CI is the consistency index

Table 1 Random consistency index (RI)

Number criteria 1 2 3 4 5 6 7 8 90 0 058 09 112 124 132 141 145

119899 represents number criteria or size matrix and the RI(random consistency index) is the average index of randomlygenerated weights The value of RI on each size matrices isdescribed in Table 1 A CR less than 01 can be categorized asconsistent matrix Perfect consistency is obtained when themaximum eigenvalue equal to the number criteria (120582max =

119899)Herrera-Viedma et al [17] proposed some definitions to

reveal the consistency in a fuzzy preference matrix Theyshow that the additive consistency is more appropriate todefine the degree of consistency of fuzzy preference matrixThe relation in matrix 119860 is consistent if the element matrixcan satisfy (5) and (6)

119886119894119895

+ 119886119895119896

+ 119886119896119894

=

3

2

forall119894 119895 119896 (5)

where

119908119894=

sum119899

119895=1119886119894119895

minus 05

119899 (119899 minus 1) 2

(6)

Xu and Da [18] proposed determining the multiplicativeconsistency in the fuzzy preference matrix They used Xursquos[19] approach to determine CI in multiplicative preferencematrix Suppose 119887

119894119895is the element of multiplicative of pref-

erence matrix Xu [19] defined the CI in (7) This equation isderived from (3) By knowing theweight of each criterion andelement of matrix CI can be obtained

CI =

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[119887119894119895

sdot

119908119895

119908119894

+ 119887119895119894

sdot

119908119894

119908119895

minus 2] (7)

where 119887119894119895

is element matrix of multiplicative preferencematrix and 119908 = (119908

1 1199082 1199083 119908

119899) is the weight of each

criterion To determine the CI in a fuzzy preference matrix[4 18] used convertingwith assumption 119887

119894119895= 119886119894119895119886119895119894 where 119886

119894119895

is element matrix of the fuzzy preference matrix Thereforethey proposed determining CI as in

CI =

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[

119886119894119895

119886119895119894

sdot

119908119895

119908119894

+

119886119895119894

119886119894119895

sdot

119908119894

119908119895

minus 2] (8)

22 Deviation Matrix While the consistent ratio is repairedthe modified matrix automatically generates the deviationmatrix from the original Ideally the modified matrices arekept closer to their original matrices in order to maintainthe original judgment It means that the deviation matrix isenriched to beminimalThere are somemethods to representthe deviation such as difference index (Di) [13] and 120575 and 120590Difference index (Di) is defined as the real difference between



120575 = max119894119895

100381610038161003816100381610038161198861015840

119894119895minus 119886119894119895

10038161003816100381610038161003816 (9)

120590 =

radicsum119899

119894=1sum119899

119895=1(1198861015840

119894119895minus 119886119894119895

)

2

119899

(10)






119883119894 (119905 + 1) = 119883119894 (119905) + 119881119894 (119905 + 1) (11)


+ (1198622lowast 1198772(119883119892 (119905) minus 119883119894 (119905)))

(12)



2


2are random


3 Proposed Method





















119894119895+119886119895119894

= 1

and 119886119894119894


119860 = (

05 06 04 08

04 05 03 07

06 07 05 09

02 03 01 05

)


(13)









can be defined as

119866119860

= 1198861 1198862 1198863 119886

(1198992minus119899)2

(14)




119886119903119904

= 119886119903119904minus1

+ 120595

1198861199030

=

0 if 0 le 119886119903lt 05

05 if 05 lt 119886119903le 1

(15)





119894119895= 119886119894119895119886119895119894


119894119895= 095 and 119886



119894119895exceeds 9



119894119895) as


119887119894119895

= 92lowast119886119894119895 (16)

119886119894119895

=

1

2

(1 + log9119887119894119895) (17)


=




CI

=

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[92sdot119886119894119895minus1

sdot

119908119895

119908119894

+ 92sdot119886119895119894minus1

sdot

119908119894

119908119895

minus 2]

(18)



119882 011198621

021198622

031198771

041198772








5(19a)





1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015


CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)



5


1 1198605 1198609 11986012 and 119860




1 1198605

1198609 11986012 and 119860






0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120575 = max119894119895

100381610038161003816100381610038161198861015840

119894119895minus 119886119894119895

10038161003816100381610038161003816 (9)

120590 =

radicsum119899

119894=1sum119899

119895=1(1198861015840

119894119895minus 119886119894119895

)

2

119899

(10)






119883119894 (119905 + 1) = 119883119894 (119905) + 119881119894 (119905 + 1) (11)


+ (1198622lowast 1198772(119883119892 (119905) minus 119883119894 (119905)))

(12)



2


2are random


3 Proposed Method





















119894119895+119886119895119894

= 1

and 119886119894119894


119860 = (

05 06 04 08

04 05 03 07

06 07 05 09

02 03 01 05

)


(13)









can be defined as

119866119860

= 1198861 1198862 1198863 119886

(1198992minus119899)2

(14)




119886119903119904

= 119886119903119904minus1

+ 120595

1198861199030

=

0 if 0 le 119886119903lt 05

05 if 05 lt 119886119903le 1

(15)





119894119895= 119886119894119895119886119895119894


119894119895= 095 and 119886



119894119895exceeds 9



119894119895) as


119887119894119895

= 92lowast119886119894119895 (16)

119886119894119895

=

1

2

(1 + log9119887119894119895) (17)


=




CI

=

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[92sdot119886119894119895minus1

sdot

119908119895

119908119894

+ 92sdot119886119895119894minus1

sdot

119908119894

119908119895

minus 2]

(18)



119882 011198621

021198622

031198771

041198772








5(19a)





1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015


CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)



5


1 1198605 1198609 11986012 and 119860




1 1198605

1198609 11986012 and 119860






0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















119894119895+119886119895119894

= 1

and 119886119894119894


119860 = (

05 06 04 08

04 05 03 07

06 07 05 09

02 03 01 05

)


(13)









can be defined as

119866119860

= 1198861 1198862 1198863 119886

(1198992minus119899)2

(14)




119886119903119904

= 119886119903119904minus1

+ 120595

1198861199030

=

0 if 0 le 119886119903lt 05

05 if 05 lt 119886119903le 1

(15)





119894119895= 119886119894119895119886119895119894


119894119895= 095 and 119886



119894119895exceeds 9



119894119895) as


119887119894119895

= 92lowast119886119894119895 (16)

119886119894119895

=

1

2

(1 + log9119887119894119895) (17)


=




CI

=

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[92sdot119886119894119895minus1

sdot

119908119895

119908119894

+ 92sdot119886119895119894minus1

sdot

119908119894

119908119895

minus 2]

(18)



119882 011198621

021198622

031198771

041198772








5(19a)





1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015


CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)



5


1 1198605 1198609 11986012 and 119860




1 1198605

1198609 11986012 and 119860






0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







can be defined as

119866119860

= 1198861 1198862 1198863 119886

(1198992minus119899)2

(14)




119886119903119904

= 119886119903119904minus1

+ 120595

1198861199030

=

0 if 0 le 119886119903lt 05

05 if 05 lt 119886119903le 1

(15)





119894119895= 119886119894119895119886119895119894


119894119895= 095 and 119886



119894119895exceeds 9



119894119895) as


119887119894119895

= 92lowast119886119894119895 (16)

119886119894119895

=

1

2

(1 + log9119887119894119895) (17)


=




CI

=

1

119899 (119899 minus 1)

sum

1le119894le119895le119899

[92sdot119886119894119895minus1

sdot

119908119895

119908119894

+ 92sdot119886119895119894minus1

sdot

119908119894

119908119895

minus 2]

(18)



119882 011198621

021198622

031198771

041198772








5(19a)





1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015


CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)



5


1 1198605 1198609 11986012 and 119860




1 1198605

1198609 11986012 and 119860






0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198601

09-04-02-03-06-01a 06871198602

08-04-01-01-03-07 03641198603

04-06-04-07-04-03b 01831198604

04-03-04-03-01-09 0427Size 5 times 5

1198605

04-03-04-07-08-02-04-04-06-02 03191198606

01-02-03-09-06-08-07-04-06-03 03431198607

07-02-01-03-08-08-07-01-06-04 03591198608

01-03-01-08-08-04-06-08-06-07 0479Size 6 times 6

1198609

08-02-01-04-08-09-04-02-04-07-09-08-07-04-03 044011986010

03-01-08-08-03-07-02-04-04-07-07-06-04-08-01 053111986011

02-08-01-07-08-04-04-06-07-06-01-04-06-03-07 0437Size 7 times 7

11986012

07-02-04-07-03-06-04-03-09-02-07-04-06-08-08-08-03-09-02-07-09 031511986013

07-08-03-04-06-07-02-07-02-03-08-03-03-02-06-04-07-03-02-01-07 0353Size 8 times 8

11986014

08-08-08-03-06-08-07-07-04-07-07-09-07-04-04-04-03-03-04-07-02-06-02-02-08-07-0207 031311986015


CR = 0319

(

(

(

05 06 07 03 06

04 05 06 02 06

03 04 05 08 04

07 08 02 05 08

04 04 06 02 05

)

)

)

(19a)

CR = 0003 and 120590 = 0161

(

(

(

05 05146 05072 04923 05190

04854 05 05088 04504 05097

04928 04912 05 05177 04882

05077 05496 04823 05 05312

04810 04903 05112 04688 05

)

)

)

(19b)

CR = 0099 and 120590 = 0073

(

(

(

05 05717 06188 03855 06477

04283 05 052 026 05982

03812 04473 05 06630 04645

06145 06554 03370 05 07341

03523 04018 05355 02659 05

)

)

)

(19c)



5


1 1198605 1198609 11986012 and 119860




1 1198605

1198609 11986012 and 119860






0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

004

008

012

016

02

120590

0 20 40 60 80 100

Iteration

A5

81 00734


0 50 100

Iteration

A5

740025

0

002

004

006

008

CR




1198721(1198831015840) =

1

1198831015840

sum

1198861015840isin1199091015840

min

100381710038171003817100381710038171198861015840minus 119886

10038171003817100381710038171003817

119886 isin 119883 (20)



2indicates better


1198722(1198831015840) =

1

10038161003816100381610038161198831015840minus 1

1003816100381610038161003816

sum

1198861015840isin1199091015840

100381610038161003816100381610038161198871015840isin 1199091015840

100381710038171003817100381710038171198861015840minus 119887101584010038171003817100381710038171003817119886 gt 119889

10038161003816100381610038161003816 (21)




3can be

defined as

1198723(1198831015840) = radic

119898

sum

119894=1

max 10038171003817100381710038171198861015840

119894minus 1198871015840

119894

1003817100381710038171003817 1198861015840 1198871015840isin 1198831015840 (22)




2






5 Conclusions



0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 005 01

CR

0 005 01

CR0 005 01

CR

0 005 01

CR

0 005 01

CR

0

006

012

018

024

03

120590

120590

120590

120590

120590

004

007

01

013

016

019

01

014

018

022

026

008

011

014

017

02

01

012

014

016

018

02

00012 02333

0099 01315

00025 01608

0098 00734

00067

02229

0099 01244

00069 02098

0099 01182

00067 01882

0099 01169

A1 A5

A9A12

A14





Acknowledgments




Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





Method 1198601

1198605

1198609

11986012

11986014

NSGA-21198721

0000896 0000913 000127 000146 0001331198722

240 262 159 204 1411198723

176 150 134 121 111MOPSO

1198721

0000830 0000701 000110 0000930 0001221198722

287 279 196 261 1771198723

189 164 160 139 129PSOMOF

1198721

0000728 0000688 0000957 0000926 00009981198722

325 279 206 257 1901198723

198 192 165 158 148


References

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in













Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Repairing the Inconsistent Fuzzy...

Documents

Transcript of Research Article Repairing the Inconsistent Fuzzy...