research article repairing the inconsistent fuzzy...
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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
4 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 5
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
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
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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
4 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 5
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
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
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Distributed Sensor Networks
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
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Applied Computational Intelligence and Soft Computing
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Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
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Electrical and Computer Engineering
Journal of
Journal of
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httpwwwhindawicom Volume 2014
Advances in
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RoboticsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
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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
4 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 5
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
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
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Distributed Sensor Networks
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Volume 2014
International Journal of
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Applied Computational Intelligence and Soft Computing
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Artificial Intelligence
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Electrical and Computer Engineering
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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httpwwwhindawicom Volume 2014
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RoboticsJournal of
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Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
4 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 5
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
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Fuzzy Systems 5
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
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Fuzzy Systems 7
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
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
8 Advances in Fuzzy Systems
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
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Fuzzy Systems 9
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
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
10 Advances in Fuzzy Systems
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014