research article problems on solving matrix aggregation in...

Research ArticleProblems on Solving Matrix Aggregation in GroupDecision-Making by Glowworm Swarm Optimization

Yaping Li12

1School of Management Hefei University of Technology Hefei 230009 China2Anhui Economic Management Institute Hefei 230059 China

Correspondence should be addressed to Yaping Li yplifly126com

Received 19 May 2016 Revised 16 July 2016 Accepted 17 July 2016

Academic Editor Muhammad N Akram

Copyright copy 2016 Yaping Li This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Judgment matrix aggregation as an important part of group decision-making has been widely and deeply studied due to theuniversality and importance of group decision-making in the management field For the variety of judgment matrix in groupdecision-making the matrix aggregation result can be obtained by using the mode of glowworm swarm optimization First thispaper introduces the basic principle of the glowworm swarm optimization (GSO) algorithm and gives the improvedGSO algorithmto solve the matrix aggregation problems In this approach the consistency ratio is introduced to the objective function of theglowworm swarm optimization thus reducing the subjectivity and information loss in the aggregation processThen the improvedGSO algorithm is applied to the solution of the deterministic matrix and the fuzzymatrixThemethod optimization can provide aneffective and relatively uniform aggregation method for matrix aggregation Finally through comparative analysis it is shown thatthe method of this paper has certain advantages in terms of adaptability accuracy and stability to solving the matrix aggregationproblems

1 Introduction

In social and economic life group decision-making is widelyapplied in various management fields providing supportfor solving complicated decisions Therefore aggregationof expert opinions in group decision-making is of longstanding [1 2] At the same time Zeng et al gave anapplicable example of group decision-making under Web 20environment it was shown that group decision problem hasimportant significance in both the present and the future[3] As the application of information technology becomeswidely used expert opinion forms for group decision-makingare changing Forms of the expert judgment matrix mainlyinclude deterministic type and interval type fuzzy judgmentmatrix containing deterministic value and interval value isderived For judgment matrixes of different types the matrixaggregation method mainly includes additive aggregationmultiplicative aggregation [4 5] operators-based aggrega-tion [6] graph theory-based aggregation [7] and evidencetheory-based aggregation [8] Lu and Guo gave a matrixaggregation scheme based on an undirected connected graph

theory In the scheme an aggregation matrix with completeconsistency is rebuilt by screening more consistent expertopinions according to opinion deviations thereby obtainingthe importance sequence [7] In 2014 a matrix aggregationalgorithm based on spanning tree aggregation operatorswas proposed by Huang et al [8] which applies spanningtree aggregation operators to obtain a completely consistentaggregation matrix according to the relations between thejudgment matrix and simple undirected graph spanning treeZhai and Zhang [9] proposed a matrix aggregation methodbased on the evidence theory in group judgment The abovemethods mainly focus on the problem of aggregation in thedeterministicmatrix In addition the aggregationmethod forinterval judgment matrix is also an important research direc-tion [10 11] In 2015 L Li and J Li [12] gave an aggregationmethod for interval judgmentmatrix the three-point intervalnumber judgment matrix aggregation is transformed intooptimal two-point judgmentmatrix and optimal aggregationintervals are synthesized through plant growth simulationalgorithm For Pythagorean fuzzy multiple-criteria decision-making problems Zeng et al [13] developed a new method

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1350807 8 pageshttpdxdoiorg10115520161350807

2 Mathematical Problems in Engineering

with aggregation operators and distance measures in 2016Meanwhile Zeng et al [14] presented another aggregationoperator that uses generalized means in a unified modelbetween the probability and the OWA operator Moreovermore attention has also been paid to differential weightingresearch for expert opinions [15]

Traditional expert judgmentmatrix aggregationmethodsare mainly used for direct arithmetic and logic operationsbased on consistency regulation for the existing matrix Ithas higher dependence on the matrix type and qualitythus reduction of consistency regulation subjectivity andinformation loss in the aggregation process are still keyproblems for group decision-making in matrix aggregationThe glowworm swarm optimization (GSO) [16] proposed byKrishnanand and Ghose is a new type of swarm intelligenceoptimization algorithm it has a higher efficiency than thetraditional swarm intelligence algorithms in solving multi-modal problems [17] GSO algorithm is hereby introducedinto the matrix aggregation the expert matrix as the knownfeasible solution and the parameter of the objective functionis optimized continuously by GSO algorithm for solvingthe feasible solution with closer Euclidean distance to theexpert matrix as the aggregation result Therefore it canplay a positive role in improving the consistency of thejudgment matrix after aggregation reducing the subjectivityof the aggregation process and the information loss of theoriginal judgment matrix The researches herein mainlyinclude three parts Section 2 gives the optimization idea ofmatrix aggregation-orientedGSO algorithm Section 3 showsa matrix aggregation method based on an improved GSOalgorithm and Section 4 presents an experimental analysisfor deterministic judgment matrix aggregation and intervaljudgment matrix aggregation

2 Matrix Aggregation-OrientedGSO Algorithm

21 Basic Idea of GSO Algorithm GSO algorithm is to solvethe problem mainly through each glowworm represent-ing a feasible solution to the objective problem in spaceGlowworms will gather toward high-brightness glowwormsby mutual attraction and movement thus finding multipleextreme points in the solution space for the objective prob-lems The main idea is described as below

Step 1 (initialize the algorithm) This includes assigningrelevant parameters such as the quantity of glowwormsinitial position initial fluorescein decision-making radiusfluorescein volatilization rate and domain change rate

Step 2 (calculate the fitness according to the objective func-tion) That is calculate the fitness of each glowworm in placeaccording to the objective function for the specific issues

Step 3 (determine the moving direction and step length ofthe glowworm) Each glowworm searches for the glowwormwith higher fluorescein value within its own decision-makingradius and determines the moving direction and step lengthof the next step according to the fluorescein value and thedistance

Step 4 (update the position of the glowworm) To be specificupdate the position of each glowworm according to thedetermined moving direction and step length

Step 5 (update the decision-making radius and the fluoresceinof the glowworm) Judge whether the algorithm is termi-nated and decide whether to enter into the next iteration

22 Features of Judgment Matrix Aggregation Expert judg-ment matrix aggregation is a key element for the effec-tiveness of group decision-making The judgment matrixaggregation aims at reasonably and effectively integratingexpert opinions removing opinion deviations to the greatestextent and obtaining a synthetic judgment matrix with thehighest consistency Therefore each expert matrix hereincan be considered as a suboptimal feasible solution andthe initial distribution of glowworms is optimized Randomdistribution of glowworms in the solution space is replacedwith probability distribution by taking the initial suboptimalfeasible solution as the boundary point so as to improvethe optimizing efficiency for solving the judgment matrixaggregation The specific optimizing ideas are as follows

First narrow the initial distribution field of glow-worms each initial glowworm represents a feasible syntheticjudgment matrix The initial value of the glowworm isassumed to be within the field defined by the known expertmatrix

Assume 119868 as the value range of the element in expertjudgment matrix matrix 119860 as the expert judgment matrixandmatrix119861 as the synthetic judgmentmatrix correspondingto the glowworm Here the judgment matrix is describedaccording to 1ndash9 scale method [18]

119868 = 1

91

81

71

61

51

41

31

2 1 2 3 4 5 6 7 8 9

119860119896= [119886119896

119894119895]119899times119899

119886119896

119894119895isin 119868

119861 = [119887119894119895]119899times119899

119887119894119895 = 119909 | 119909 isin 119868 119909 isin [min (119886119894119895) max (119886119894119895)]

(1)

Second in view of the elements in the synthetic judgmentmatrix of matrix aggregation not completely restricted by 1ndash9scales (ie the value range is not 119868) the restrictions to 119887119894119895 isin 119868are canceled during initialization of glowworm distribution

119861 = [119887119894119895]119899times119899 119887119894119895 = 119909 | 119909 isin [min (119886119894119895) max (119886119894119895)] (2)

Third add probability distribution factors The aggrega-tion of expert matrix is probably closer to its mathematicalexpectation Therefore the probability for obtaining theoptimal solution is greater if the Euclidean distance to thespace point represented by the mathematical expectation iscloser Based on random glowworm distribution probabilitydistribution is introduced to improve the algorithm conver-gence speed and optimizing effect

Mathematical Problems in Engineering 3

3 Matrix Aggregation Method Based onImproved GSO Algorithm

Through the above analysis and in combination with thefeatures of group decision-making matrix aggregation thealgorithm (M-GSO) solving the judgmentmatrix aggregationby using GSO algorithm can be described as follows

Step 1 (initialize) Assume the judgment matrix is designedaccording to 1ndash9 scale method and provide transforma-tion processing for initial judgment matrixes 1198601 1198602 1198603 119860119899 to facilitate comparison for Euclidean distances Laterrefer to the construction method for the distance matrixproposed by Lu and Guo [7] and transform 19 18

17 13 12 1 2 3 7 8 9 17 scale values into 1ndash9scales such as 1 2 3 4 5 17 correspondingly that is

1198861015840

119894119895=

10 minus1

119886119894119895

119886119894119895 le 1

8 + 119886119894119895 119886119894119895 gt 1

(3)

Thereby transform 1198601 1198602 1198603 119860119899 into 119860(1) 119860(2) 119860(3)

119860(119899)

On this basis the value range of the synthetic judgmentmatrix element corresponding to each glowworm can beobtained as follows

119861 = 1198611 1198612 119861119899

119861119896 = [119887119896

119894119895]119899times119899

119887119896

119894119895isin [min119904(119886(119904)

119894119895) max119904

(119886(119904)

119894119895)]

exist119861119896 =1

119898

119898

sum

119894=1

119860(119904)

119894

(4)

In the meantime set the quantity of glowworms 119873 steplength 119904 initial fluorescein value 1198970 fluoresce in volatilizationrate 120588 domain change rate120573 decision domain initial value 1205740domain threshold 120574max and other related parameters in GSOalgorithm as well as the maximum number of iterations andthe algorithm termination condition

Step 2 (set the objective function) 119910 = max(119891(119909)) =

max(1sum119899119894=1

radic(119909 minus 1198601015840

119894)2times CI) in which the objective func-

tion is introduced into CI to express the consistency ratio ofthe 119894th expert judgment matrix

CI =120582max minus 119899

119899 minus 1 (5)

where 120582max is the maximum eigenvalue of the judgmentmatrix

120582max =119899

sum

119894=1

119860119882119894

119899119882119894

(6)

Different judgment matrixes have different consistenciesand the importance is different The greater the CI theworse the consistency leading to the smaller weight of thecorresponding judgment matrix

Meanwhile introduce corresponding average randomconsistency index RI and the consistency ratio CR [19]

CR =CIRI (7)

If CR lt 01 it is considered that the judgment matrix hasconsistency and the element weight result is acceptable Oth-erwise the judgment matrix must be subjected to necessarycorrection until CR lt 01

Step 3 (calculate the fitness) To be specific put the positionvalue of each glowworm 1198611 1198612 1198613 119861119896 into the objectivefunction to calculate the fitness of each glowworm

119891 (119861119894) =1

sum119899

119894=1radic(119861119894 minus 119860

1015840

119894)2times CI

(8)

Step 4 (update the position of the glowworm) Search forthe glowworm 119861119894 with the highest fluorescein within thefield range 120574 of the glowworm 119861

1015840

119894 in order to determine the

direction of 119861119894 moving toward 1198611015840119894

119861119894 = 119861119894 +

(1198611015840

119894minus 119861119894)

1003816100381610038161003816(1198611015840

119894minus 119861119894)

1003816100381610038161003816

times 119904 (9)

Step 5 (update the fluorescein) 119897119894(119905) represents the fluo-rescein value of the 119894th glowworm in the 119905th iteration 120588represents the fluorescein volatilization rate and 120582 representsthe extract ratio of the fitness

119897119894 (119905 + 1) = (1 minus 120588) 119897119894 (119905) + 120582119891 (119909119894 (119905)) (10)

Step 6 (update the decision domain) 120574119894(119905) represents thedecision-making radius of the 119894th glowworm in the 119905th iter-ation 120573 represents the domain change rate 119899119905 represents thequantity threshold of glowworms in the domain and 119873119894(119905)represents the quantity of glowworms within the decision-making radius of the 119894th glowworm in the 119905th iteration

120574119894 (119905 + 1) = 120574119894 (119905) + 120573 (119899119905 minus 119873119894 (119905)) (11)

Step 7 Judge the number of iterations and terminationconditions and either terminate the algorithm or turn to Step3

4 Experimental Analysis

Deterministic judgment matrix aggregation and intervaljudgmentmatrix aggregation are discussed hereinThe deter-ministic judgment matrix is mainly represented as the matrixelements which are deterministic values while the element ofthe interval judgment matrix is a numerical interval HereinM-GSO algorithm is compiled by using MATLAB Referringto the parameter design for GSO algorithm in relevantreference documents and combining actual conditions ofthe matrix aggregation relevant parameters for M-GSOalgorithm are selected as follows 119873 = 50 120588 = 04 120582 =

06 120573 = 008 119904 = 01 1198970 = 5 119899119905 = 6 with the maximumnumber of iterations as 100


41 Aggregation for the Deterministic Judgment Matrix Theaggregation method of graph theory proposed by Lu andGuo focuses on aggregation for the deterministic judgmentmatrixThebasic idea of thematrix aggregation schemebasedon undirected connected graph theory is to first select 119873 minus

1 positions in the upper triangle (or lower triangle) of thematrix 119873 times 119873 based on the basic theory of the undirectedconnected graph during aggregation for 119898119873 times 119873 judgmentmatrixes 11987211198722 119872119898 1198721(119894 119895)1198722(119894 119895) 119872119898(119894 119895)obtained correspondingly in each position (119894 119895) of 119873 minus

1 positions is required to be kept relatively consistentwith deviation as small as possible (where the methodfor deviation calculation is the sum of the absolute valuesfrom subtraction between two elements in vectors) Thencalculate the arithmetic mean (additive) or geometric mean(multiplicative) for the elements 119860 119894119895 in these positions Atlast on this basis according to the principle of proportionalrows calculate the element values in other positions therebybuilding judgment matrix with complete consistency

Four expert judgment matrixes are selected as the deter-ministic judgment matrix for M-GSO algorithm experiment(shown below)

1198601 =

[[[[[[[[[[[[

[

1 7 5 4

1

71 1

1

2

1

51 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198602 =

[[[[[[[[[[[[

[

1 6 7 5

1

61 1 1

1

71 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198603 =

[[[[[[[[[[[[

[

1 6 8 4

1

61 1

1

2

1

81 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198604 =

[[[[[[[[[[

[

1 3 2 1

1

31

1

2

1

3

1

22 1 2

1 31

21

]]]]]]]]]]

]

(12)

For the above expert matrixes some judgment matrixeswith complete consistency constructed by the aggregationmethod of graph theory can be obtained one of which is asbelow

119860lowast=

[[[[[[[[[[[[[[

[

111

2

77

16

55

24

2

111

7

8

5

12

16

77

8

71

10

21

24

55

12

5

21

101

]]]]]]]]]]]]]]

]

(13)

The weight result obtained through the aggregationmethod of graph theory is 119882 = (05476 00996 01138

02390) and the importance sequence of correspondingelements is 1199061 gt 1199064 gt 1199063 gt 1199062 in which the consistencyratio of 119860lowast is CR = 0

According to the above analysis the original judgmentmatrix is first transformed into the distance matrix by M-GSO algorithm as below

119860(1)

=

[[[[[

[

9 15 13 12

3 9 9 8

5 9 9 7

6 10 11 9

]]]]]

]

119860(2)

=

[[[[[

[

9 14 15 13

4 9 9 9

3 9 9 9

5 9 9 9

]]]]]

]

119860(3)

=

[[[[[[

[

9 14 16 12

4 9 9 8

2 9 9 9

6 10 9 9

]]]]]]

]

119860(4)

=

[[[[[

[

9 11 10 9

7 9 8 7

8 10 9 10

9 11 8 9

]]]]]

]

(14)

By running the M-GSO algorithm five times with ran-domized initial points five synthetic judgment matrixes withcomplete consistency can be obtained as below


Table 1 Aggregation for the deterministic judgment matrix

Synthetic judgment matrixes Optimal weight Importance sequence119860lowast

104455 01099 01875 02570 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

204680 01085 01804 02431 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

304620 01092 01785 02502 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

404444 01105 01891 02559 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

504449 01105 01900 02545 1199061 gt 1199064 gt 1199063 gt 1199062

Arithmetic mean 04530 01098 01851 02521 1199061gt 1199064gt 1199063gt 1199062

119860lowast

1=

[[[[[

[

1 405 238 173

025 1 059 043

042 171 1 073

058 234 137 1

]]]]]

]

119860lowast

2=

[[[[[

[

1 431 259 193

023 1 060 045

039 166 1 074

052 224 135 1

]]]]]

]

119860lowast

3=

[[[[[

[

1 423 259 185

024 1 061 044

039 163 1 071

054 229 140 1

]]]]]

]

119860lowast

4=

[[[[[

[

1 402 235 174

025 1 058 043

043 171 1 074

058 231 135 1

]]]]]

]

119860lowast

5=

[[[[[

[

1 403 234 175

025 1 058 043

043 172 1 075

057 230 134 1

]]]]]

]

(15)

where 119860lowast

1 119860lowast2 119860lowast3 119860lowast4 119860lowast5satisfy CR = 0 The se-

quence of corresponding weights can be calculated accord-ing to the matrix 119860

lowast

1 119860lowast2 119860lowast3 119860lowast4 119860lowast5 as shown in

Table 1As shown in Table 1 the arithmetic mean obtained

from five running results according to M-GSO algorithmis 119908 = (04530 01098 01851 02521) and the importancesequence is 1199061 gt 1199064 gt 1199063 gt 1199062 The sequence resultconsistent with those by the method of graph theory can bestably obtained through either single running or arithmeticmean of the operation result

However this method based on undirected connectedgraph theory does not get the correct results every time Inthis paper we change the elements 119860 119894(1 3) = (5 7 8 2) to119860 119894(1 3) = (3 2 2 13) while adjusting the value of 119860 119894(3 1)a new matrix is shown as below

1198611 =

[[[[[[[[[[[[

[

1 7 3 4

1

71 1

1

2

1

31 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198612 =

[[[[[[[[[[[[

[

1 6 2 5

1

61 1 1

1

21 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198613 =

[[[[[[[[[[[[

[

1 6 2 4

1

61 1

1

2

1

21 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198614 =

[[[[[[[[[[[[

[

1 31

31

1

31

1

2

1

3

3 2 1 2

1 31

21

]]]]]]]]]]]]

]

(16)

For these matrixes we can get some conflicting resultsthrough the aggregation method of graph theory The con-flicting importance sequence of corresponding elements is1199061 gt 1199064 gt 1199063 gt 1199062 and 1199064 gt 1199061 gt 1199063 gt 1199062 Thecalculation process is described in detail in another articleby the author [20] At the same time we can obtain theimportance sequence 1199061 gt 1199064 gt 1199063 gt 1199062 stably using theM-GSO algorithm

42 Aggregation for the Interval Judgment Matrix For theindeterminacy and inconsistency in the judgment matrix


the relations between the basic consistency and consensus ofgroup decision-making are discussed A scheme for solvingthe importance sequence based on integrated arithmeticmean is given byZhai andZhang [9]Three interval judgmentmatrixes are shown below

119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)

According to the aggregationmethod based on integratedarithmetic mean the combined judgment matrix 119860lowast is thearithmeticmean of the original judgmentmatrixMeanwhile119860lowast is a reciprocal matrix

119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)

The weight result obtained from aggregation in theabove interval judgment matrix by the method based onintegrated arithmetic mean is 1199081 = [0397 0637] 1199082 =

[0083 0116] 1199083 = [0328 0487] and the importancesequence of corresponding elements is (1199061 gt 1199063 gt 1199062)

Now the algorithm accuracy (120579) will be defined by thearithmetic mean of the length of the weight interval

120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)

The difference between the aggregation for interval judg-ment matrix and the above deterministic judgment matrix isnot just in the transformation of matrix elements from thenumerical values into intervals at the same time the valuerange is not according to 17 grades corresponding to 1ndash9 scalesanymore but expanded to the continuous interval of 0ndash9The consistency index for the deterministic matrix cannot beused Thus the upper and lower limits of the modal interval

judgment matrix are split into two deterministic judgmentmatrixes by M-GSO algorithm that is

119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)

The judgment matrix is used as the suboptimal feasiblesolution for optimization by M-GSO algorithm and theconsistency of the judgment matrix is introduced into theobjective function as the parameter so the influence of theconsistency of the initial judgment matrix on the aggregationresult is relatively small Moreover more optimal feasiblesolutions can also be obtained by running M-GSO algorithmfor the deterministic judgment matrix after the transforma-tion of interval judgment matrix Also by running M-GSOalgorithm for 119860(1)

1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2

in the abovedeterministic judgment matrix five times we can also obtainthe two sets of synthetic judgment matrixes with completeconsistency 119860lowast

1 119860lowast2 119860lowast3 119860lowast4 119860lowast5and 119861lowast

1 119861lowast2 119861lowast3 119861lowast4 119861lowast5 The

results as shown in Table 2 can be obtainedThe two sets of weight results obtained from the operation

results of M-GSO algorithm and the arithmetic mean areconsidered as feasible solutions of weights for the elements1199061 1199062 1199063 thereby obtaining the weight range of elements 11990611199062 1199063 with the weight intervals as1199081 = [0592 0677] 1199082 =

[0151 0169] 1199083 = [0154 0256] thus obtaining theimportance sequence of 1199061 gt 1199063 gt 1199062 It is consistent withthe weight sequence obtained through the method based onintegrated arithmetic mean

According to the definition of algorithm accuracy theaccuracy (1205791015840) of the above weight range can be expressed asfollows

1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085


Table 2 Aggregation for interval judgment matrix

Synthetic judgmentmatrixes Optimal weight of the first group of judgment matrixes Optimal weight of the second group of judgment

matrixes119860lowast

1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)

The following conclusions can be obtained 120579119894 gt 1205791015840

119894and

120579 gt 1205791015840

Compared with1199081 = [0397 0637] 1199082 = [0083 0116]1199081 = [0328 0487] the weight intervals 1199081 = [0592 0677]1199082 = [0151 0169] 1199083 = [0154 0256] obtained from M-GSO algorithm are remarkably narrowed on interval size forexample with higher calculation accuracy

43 Comparative Analyses The above experimental resultsshow that M-GSO algorithm provides a relative uniformsolution for the deterministic judgment matrix aggregationand interval judgment matrix aggregation The traditionalaggregation methods often focus on special judgment matrixtype thus in practical application consistent judging meth-ods cannot be provided for different expert judgment waysof the same problem A uniform solution can be providedby M-GSO algorithm because the expert judgment matrixis considered as the feasible solution in M-GSO algorithmand more optimal feasible weight solutions can be obtainedthrough optimization while traditional matrix aggregationmethods are often to operate the judgment matrix hencethere are more requirements on the expert matrix

431 Algorithm Robustness The expert judgment matrix issubjected to differentiation by M-GSO algorithm Differentexpert matrixes have different subjective recognition and theactual qualities are different for example expert judgmentmatrixes with poor quality have negative impact on the finalweight result Thus M-GSO algorithm is to differentiateexpert matrixes by adding the consistency index of thejudgment matrix to the objective function consequently toweaken the negative impact of the expert matrix with poorconsistency on the final result and meanwhile to furtherreduce the dependency of matrix aggregation scheme on theconsistency of the initial matrix

432 Algorithm Stability Compared with the method ofgraph theory the operation results of M-GSO algorithm are

more stabilized According to the specific case analysis forthe method of graph theory the weight sequence resultsare instable when multiple element values are equal in thedistance deviationmatrix but the weight sequence results areobtained through constant optimizing in M-GSO algorithmso those results are more stable which is the same as provedabove

433 Algorithm Accuracy By defining the algorithm accu-racy the advantages and disadvantages of the weight rangecan be compared more intuitively Due to 120579119894 gt 120579

1015840

119894and 120579 gt 120579

1015840

in Section 42 the accuracy of weight interval of M-GSOalgorithm is higher Therefore compared with the arithmeticmean it is more optimal for selecting optimal solutions frommultiple poles For weight results in practical application thesmaller the weight interval the more accurate the provideddecision-making support for example the larger the weightinterval the smaller the actual significance Final weightresults are obtained through searching for feasible solutionsand using feasible intervals among feasible solutions by M-GSO algorithm thereby obtaining more effective weightintervals

5 Summary

The result comparison of different decision alternatives ordecision factors is usually given in the form of a judgmentmatrix in group decision-making The aggregation of thematrix is equivalent to the synthesis of various expertopinions In this paper the weight or importance sequenceobtained by the M-GSO algorithm can be understood as theimportance of decision scheme or decision factors in thespecific application This is the application value of the M-GSO algorithm in group decision-making

This paper presents the M-GSO algorithm to solve thematrix aggregation problems Unlike the traditional matrixaggregation methods this method not only can be used indeterministic matrix but also can be used for the intervalmatrix The aggregation mode of M-GSO algorithm is nolonger to conduct operation for the judgment matrix itselfbut to take the judgment matrix as the restrictions foroptimization In the meantime consideration is given to thequality difference of judgmentmatrixes inM-GSOalgorithmIt is unnecessary to manually regulate the judgment matrixeswith relatively poor consistency thus this optimization ismore adaptive Through comparative analysis the M-GSO


algorithm has good robustness stability and computationalaccuracy

In addition the effects of random distribution of glow-worm position can also be noticed The M-GSO algorithmshould be improved in convergence and accuracy so thatit can be better applied to solve group decision-makingproblems

Competing Interests

The author declares no competing interests

Acknowledgments

The work was supported by the fund of the ProvincialExcellent YoungTalents of Colleges andUniversities of AnhuiProvince (no 2013SQRW115ZD) and the fund of the NaturalScience of Colleges and Universities of Anhui Province (noKJ2016A162)

References

[1] C-L Hwang and K P Yoon Multiple Attribute DecisionMaking Methods and Applications vol 186 of Lecture Notes inEconomics and Mathematical Systems 1981

[2] G Stasser and W Titus ldquoPooling of unshared information ingroup decision making Biased information sampling duringdiscussionrdquo Journal of Personality and Social Psychology vol 48no 6 pp 1467ndash1478 1985

[3] S Zeng J Gonzalez andC Lobato ldquoThe effect of organizationallearning and Web 20 on innovationrdquo Management Decisionvol 53 no 9 pp 2060ndash2072 2015

[4] T L Saaty Decision Making for Leasers Wadsworth IncBelmont Calif USA 1982

[5] E U Choo and W C Wedley ldquoComparing fundamentals ofadditive and multiplicative aggregation in ratio scale multi-criteria decision makingrdquo Open Operational Research Journalvol 2 pp 1ndash7 2008

[6] I Beg and T Rashid ldquoA geometric aggregation operator fordecision makingrdquo Vietnam Journal of Computer Science vol 2no 4 pp 243ndash255 2015

[7] Y-J Lu and X-R Guo ldquoEffective aggregation method for theAHP judgement matrix in group decision-makingrdquo SystemEngineeringTheory and Practice vol 27 no 7 pp 132ndash136 2007

[8] C Huang B Jiao F Huang et al ldquoAn aggregation method forgroup AHP judgment matrices based on the evidence theoryrdquoScience Technology and Engineering vol 14 no 15 pp 1ndash4 2014

[9] X Zhai and X Zhang ldquoThemethods on aggregation of intervalnumber judgment matrixes and calculation of its priorities inthe group decision-makingrdquo Systems Engineering vol 23 no 9pp 103ndash107 2005

[10] E Han P Guo and J Zhao ldquoMethod for multi-attribute groupdecision making based on interval grey uncertain linguisticassessmentrdquo Journal of Frontiers of Computer Science amp Tech-nology vol 10 no 1 pp 93ndash102 2016

[11] E K Zavadskas J Antucheviciene S H Razavi Hajiagha andS SHashemi ldquoThe interval-valued intuitionistic fuzzyMULTI-MOORA method for group decision making in engineeringrdquoMathematical Problems in Engineering vol 2015 Article ID560690 13 pages 2015

[12] L Li and J Li ldquoStudy of three point interval number judgmentmatrix decomposition and aggregation based on fermat andalgorithmrdquoMathematics in Practice andTheory no 17 pp 214ndash221 2015

[13] S Zeng J Chen and X Li ldquoA hybrid method for Pythagoreanfuzzy multi-criteria decision makingrdquo International Journal ofInformation Technology amp Decision Making vol 15 no 2 pp403ndash422 2016

[14] S Zeng W Su and C Zhang ldquoIntuitionistic fuzzy generalizedprobabilistic ordered weighted averaging operator and its appli-cation to group decision makingrdquo Technological and EconomicDevelopment of Economy vol 22 no 2 pp 177ndash193 2016

[15] JMMerigo D Palacios-Marques and S Zeng ldquoSubjective andobjective information in linguisticmulti-criteria group decisionmakingrdquo European Journal of Operational Research vol 248 no2 pp 522ndash531 2016

[16] K N Krishnanand and D Ghose ldquoGlowworm swarm opti-mization for simultaneous capture of multiple local optima ofmultimodal functionsrdquo Swarm Intelligence vol 3 no 2 pp 87ndash124 2009

[17] N Zainal A M Zain N H M Radzi et al ldquoGlowwormswarm optimization (GSO) for optimization of machiningparametersrdquo Journal of Intelligent Manufacturing vol 27 no 4pp 797ndash804 2016

[18] N Kotani and Y Kodono ldquoA study on the pairwise comparisonscale in AHPrdquo Journal of International Studies vol 19 pp 1ndash132006

[19] M T Escobar J Aguaron and J M Moreno-Jimenez ldquoSomeextensions of the precise consistency consensus matrixrdquo Deci-sion Support Systems vol 74 pp 67ndash77 2015

[20] Y Li ldquoOptimization of matrix aggregation method based onundirected connection graph theoryrdquo Journal of ChongqingTechnology and Business (Natural Sciences Edition) vol 31 no5 pp 10ndash13 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with aggregation operators and distance measures in 2016Meanwhile Zeng et al [14] presented another aggregationoperator that uses generalized means in a unified modelbetween the probability and the OWA operator Moreovermore attention has also been paid to differential weightingresearch for expert opinions [15]

Traditional expert judgmentmatrix aggregationmethodsare mainly used for direct arithmetic and logic operationsbased on consistency regulation for the existing matrix Ithas higher dependence on the matrix type and qualitythus reduction of consistency regulation subjectivity andinformation loss in the aggregation process are still keyproblems for group decision-making in matrix aggregationThe glowworm swarm optimization (GSO) [16] proposed byKrishnanand and Ghose is a new type of swarm intelligenceoptimization algorithm it has a higher efficiency than thetraditional swarm intelligence algorithms in solving multi-modal problems [17] GSO algorithm is hereby introducedinto the matrix aggregation the expert matrix as the knownfeasible solution and the parameter of the objective functionis optimized continuously by GSO algorithm for solvingthe feasible solution with closer Euclidean distance to theexpert matrix as the aggregation result Therefore it canplay a positive role in improving the consistency of thejudgment matrix after aggregation reducing the subjectivityof the aggregation process and the information loss of theoriginal judgment matrix The researches herein mainlyinclude three parts Section 2 gives the optimization idea ofmatrix aggregation-orientedGSO algorithm Section 3 showsa matrix aggregation method based on an improved GSOalgorithm and Section 4 presents an experimental analysisfor deterministic judgment matrix aggregation and intervaljudgment matrix aggregation

2 Matrix Aggregation-OrientedGSO Algorithm

21 Basic Idea of GSO Algorithm GSO algorithm is to solvethe problem mainly through each glowworm represent-ing a feasible solution to the objective problem in spaceGlowworms will gather toward high-brightness glowwormsby mutual attraction and movement thus finding multipleextreme points in the solution space for the objective prob-lems The main idea is described as below

Step 1 (initialize the algorithm) This includes assigningrelevant parameters such as the quantity of glowwormsinitial position initial fluorescein decision-making radiusfluorescein volatilization rate and domain change rate

Step 2 (calculate the fitness according to the objective func-tion) That is calculate the fitness of each glowworm in placeaccording to the objective function for the specific issues

Step 3 (determine the moving direction and step length ofthe glowworm) Each glowworm searches for the glowwormwith higher fluorescein value within its own decision-makingradius and determines the moving direction and step lengthof the next step according to the fluorescein value and thedistance

Step 4 (update the position of the glowworm) To be specificupdate the position of each glowworm according to thedetermined moving direction and step length

Step 5 (update the decision-making radius and the fluoresceinof the glowworm) Judge whether the algorithm is termi-nated and decide whether to enter into the next iteration

22 Features of Judgment Matrix Aggregation Expert judg-ment matrix aggregation is a key element for the effec-tiveness of group decision-making The judgment matrixaggregation aims at reasonably and effectively integratingexpert opinions removing opinion deviations to the greatestextent and obtaining a synthetic judgment matrix with thehighest consistency Therefore each expert matrix hereincan be considered as a suboptimal feasible solution andthe initial distribution of glowworms is optimized Randomdistribution of glowworms in the solution space is replacedwith probability distribution by taking the initial suboptimalfeasible solution as the boundary point so as to improvethe optimizing efficiency for solving the judgment matrixaggregation The specific optimizing ideas are as follows

First narrow the initial distribution field of glow-worms each initial glowworm represents a feasible syntheticjudgment matrix The initial value of the glowworm isassumed to be within the field defined by the known expertmatrix

Assume 119868 as the value range of the element in expertjudgment matrix matrix 119860 as the expert judgment matrixandmatrix119861 as the synthetic judgmentmatrix correspondingto the glowworm Here the judgment matrix is describedaccording to 1ndash9 scale method [18]

119868 = 1

91

81

71

61

51

41

31

2 1 2 3 4 5 6 7 8 9

119860119896= [119886119896

119894119895]119899times119899

119886119896

119894119895isin 119868

119861 = [119887119894119895]119899times119899

119887119894119895 = 119909 | 119909 isin 119868 119909 isin [min (119886119894119895) max (119886119894119895)]

(1)

Second in view of the elements in the synthetic judgmentmatrix of matrix aggregation not completely restricted by 1ndash9scales (ie the value range is not 119868) the restrictions to 119887119894119895 isin 119868are canceled during initialization of glowworm distribution

119861 = [119887119894119895]119899times119899 119887119894119895 = 119909 | 119909 isin [min (119886119894119895) max (119886119894119895)] (2)

Third add probability distribution factors The aggrega-tion of expert matrix is probably closer to its mathematicalexpectation Therefore the probability for obtaining theoptimal solution is greater if the Euclidean distance to thespace point represented by the mathematical expectation iscloser Based on random glowworm distribution probabilitydistribution is introduced to improve the algorithm conver-gence speed and optimizing effect






1198861015840

119894119895=

10 minus1

119886119894119895

119886119894119895 le 1

8 + 119886119894119895 119886119894119895 gt 1

(3)


119860(119899)


119861 = 1198611 1198612 119861119899

119861119896 = [119887119896

119894119895]119899times119899

119887119896

119894119895isin [min119904(119886(119904)

119894119895) max119904

(119886(119904)

119894119895)]

exist119861119896 =1

119898

119898

sum

119894=1

119860(119904)

119894

(4)



max(1sum119899119894=1

radic(119909 minus 1198601015840



CI =120582max minus 119899

119899 minus 1 (5)


120582max =119899

sum

119894=1

119860119882119894

119899119882119894

(6)



CR =CIRI (7)



119891 (119861119894) =1

sum119899

119894=1radic(119861119894 minus 119860

1015840

119894)2times CI

(8)


1015840



119861119894 = 119861119894 +

(1198611015840

119894minus 119861119894)

1003816100381610038161003816(1198611015840

119894minus 119861119894)

1003816100381610038161003816

times 119904 (9)


119897119894 (119905 + 1) = (1 minus 120588) 119897119894 (119905) + 120582119891 (119909119894 (119905)) (10)


120574119894 (119905 + 1) = 120574119894 (119905) + 120573 (119899119905 minus 119873119894 (119905)) (11)










1198601 =

[[[[[[[[[[[[

[

1 7 5 4

1

71 1

1

2

1

51 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198602 =

[[[[[[[[[[[[

[

1 6 7 5

1

61 1 1

1

71 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198603 =

[[[[[[[[[[[[

[

1 6 8 4

1

61 1

1

2

1

81 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198604 =

[[[[[[[[[[

[

1 3 2 1

1

31

1

2

1

3

1

22 1 2

1 31

21

]]]]]]]]]]

]

(12)


119860lowast=

[[[[[[[[[[[[[[

[

111

2

77

16

55

24

2

111

7

8

5

12

16

77

8

71

10

21

24

55

12

5

21

101

]]]]]]]]]]]]]]

]

(13)




119860(1)

=

[[[[[

[

9 15 13 12

3 9 9 8

5 9 9 7

6 10 11 9

]]]]]

]

119860(2)

=

[[[[[

[

9 14 15 13

4 9 9 9

3 9 9 9

5 9 9 9

]]]]]

]

119860(3)

=

[[[[[[

[

9 14 16 12

4 9 9 8

2 9 9 9

6 10 9 9

]]]]]]

]

119860(4)

=

[[[[[

[

9 11 10 9

7 9 8 7

8 10 9 10

9 11 8 9

]]]]]

]

(14)





104455 01099 01875 02570 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

204680 01085 01804 02431 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

304620 01092 01785 02502 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

404444 01105 01891 02559 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

504449 01105 01900 02545 1199061 gt 1199064 gt 1199063 gt 1199062


119860lowast

1=

[[[[[

[

1 405 238 173

025 1 059 043

042 171 1 073

058 234 137 1

]]]]]

]

119860lowast

2=

[[[[[

[

1 431 259 193

023 1 060 045

039 166 1 074

052 224 135 1

]]]]]

]

119860lowast

3=

[[[[[

[

1 423 259 185

024 1 061 044

039 163 1 071

054 229 140 1

]]]]]

]

119860lowast

4=

[[[[[

[

1 402 235 174

025 1 058 043

043 171 1 074

058 231 135 1

]]]]]

]

119860lowast

5=

[[[[[

[

1 403 234 175

025 1 058 043

043 172 1 075

057 230 134 1

]]]]]

]

(15)

where 119860lowast



lowast





1198611 =

[[[[[[[[[[[[

[

1 7 3 4

1

71 1

1

2

1

31 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198612 =

[[[[[[[[[[[[

[

1 6 2 5

1

61 1 1

1

21 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198613 =

[[[[[[[[[[[[

[

1 6 2 4

1

61 1

1

2

1

21 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198614 =

[[[[[[[[[[[[

[

1 31

31

1

31

1

2

1

3

3 2 1 2

1 31

21

]]]]]]]]]]]]

]

(16)





119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)


119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)




120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)



119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)


1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2








1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085





1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198861015840

119894119895=

10 minus1

119886119894119895

119886119894119895 le 1

8 + 119886119894119895 119886119894119895 gt 1

(3)


119860(119899)


119861 = 1198611 1198612 119861119899

119861119896 = [119887119896

119894119895]119899times119899

119887119896

119894119895isin [min119904(119886(119904)

119894119895) max119904

(119886(119904)

119894119895)]

exist119861119896 =1

119898

119898

sum

119894=1

119860(119904)

119894

(4)



max(1sum119899119894=1

radic(119909 minus 1198601015840



CI =120582max minus 119899

119899 minus 1 (5)


120582max =119899

sum

119894=1

119860119882119894

119899119882119894

(6)



CR =CIRI (7)



119891 (119861119894) =1

sum119899

119894=1radic(119861119894 minus 119860

1015840

119894)2times CI

(8)


1015840



119861119894 = 119861119894 +

(1198611015840

119894minus 119861119894)

1003816100381610038161003816(1198611015840

119894minus 119861119894)

1003816100381610038161003816

times 119904 (9)


119897119894 (119905 + 1) = (1 minus 120588) 119897119894 (119905) + 120582119891 (119909119894 (119905)) (10)


120574119894 (119905 + 1) = 120574119894 (119905) + 120573 (119899119905 minus 119873119894 (119905)) (11)










1198601 =

[[[[[[[[[[[[

[

1 7 5 4

1

71 1

1

2

1

51 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198602 =

[[[[[[[[[[[[

[

1 6 7 5

1

61 1 1

1

71 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198603 =

[[[[[[[[[[[[

[

1 6 8 4

1

61 1

1

2

1

81 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198604 =

[[[[[[[[[[

[

1 3 2 1

1

31

1

2

1

3

1

22 1 2

1 31

21

]]]]]]]]]]

]

(12)


119860lowast=

[[[[[[[[[[[[[[

[

111

2

77

16

55

24

2

111

7

8

5

12

16

77

8

71

10

21

24

55

12

5

21

101

]]]]]]]]]]]]]]

]

(13)




119860(1)

=

[[[[[

[

9 15 13 12

3 9 9 8

5 9 9 7

6 10 11 9

]]]]]

]

119860(2)

=

[[[[[

[

9 14 15 13

4 9 9 9

3 9 9 9

5 9 9 9

]]]]]

]

119860(3)

=

[[[[[[

[

9 14 16 12

4 9 9 8

2 9 9 9

6 10 9 9

]]]]]]

]

119860(4)

=

[[[[[

[

9 11 10 9

7 9 8 7

8 10 9 10

9 11 8 9

]]]]]

]

(14)





104455 01099 01875 02570 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

204680 01085 01804 02431 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

304620 01092 01785 02502 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

404444 01105 01891 02559 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

504449 01105 01900 02545 1199061 gt 1199064 gt 1199063 gt 1199062


119860lowast

1=

[[[[[

[

1 405 238 173

025 1 059 043

042 171 1 073

058 234 137 1

]]]]]

]

119860lowast

2=

[[[[[

[

1 431 259 193

023 1 060 045

039 166 1 074

052 224 135 1

]]]]]

]

119860lowast

3=

[[[[[

[

1 423 259 185

024 1 061 044

039 163 1 071

054 229 140 1

]]]]]

]

119860lowast

4=

[[[[[

[

1 402 235 174

025 1 058 043

043 171 1 074

058 231 135 1

]]]]]

]

119860lowast

5=

[[[[[

[

1 403 234 175

025 1 058 043

043 172 1 075

057 230 134 1

]]]]]

]

(15)

where 119860lowast



lowast





1198611 =

[[[[[[[[[[[[

[

1 7 3 4

1

71 1

1

2

1

31 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198612 =

[[[[[[[[[[[[

[

1 6 2 5

1

61 1 1

1

21 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198613 =

[[[[[[[[[[[[

[

1 6 2 4

1

61 1

1

2

1

21 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198614 =

[[[[[[[[[[[[

[

1 31

31

1

31

1

2

1

3

3 2 1 2

1 31

21

]]]]]]]]]]]]

]

(16)





119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)


119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)




120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)



119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)


1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2








1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085





1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198601 =

[[[[[[[[[[[[

[

1 7 5 4

1

71 1

1

2

1

51 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198602 =

[[[[[[[[[[[[

[

1 6 7 5

1

61 1 1

1

71 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198603 =

[[[[[[[[[[[[

[

1 6 8 4

1

61 1

1

2

1

81 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198604 =

[[[[[[[[[[

[

1 3 2 1

1

31

1

2

1

3

1

22 1 2

1 31

21

]]]]]]]]]]

]

(12)


119860lowast=

[[[[[[[[[[[[[[

[

111

2

77

16

55

24

2

111

7

8

5

12

16

77

8

71

10

21

24

55

12

5

21

101

]]]]]]]]]]]]]]

]

(13)




119860(1)

=

[[[[[

[

9 15 13 12

3 9 9 8

5 9 9 7

6 10 11 9

]]]]]

]

119860(2)

=

[[[[[

[

9 14 15 13

4 9 9 9

3 9 9 9

5 9 9 9

]]]]]

]

119860(3)

=

[[[[[[

[

9 14 16 12

4 9 9 8

2 9 9 9

6 10 9 9

]]]]]]

]

119860(4)

=

[[[[[

[

9 11 10 9

7 9 8 7

8 10 9 10

9 11 8 9

]]]]]

]

(14)





104455 01099 01875 02570 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

204680 01085 01804 02431 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

304620 01092 01785 02502 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

404444 01105 01891 02559 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

504449 01105 01900 02545 1199061 gt 1199064 gt 1199063 gt 1199062


119860lowast

1=

[[[[[

[

1 405 238 173

025 1 059 043

042 171 1 073

058 234 137 1

]]]]]

]

119860lowast

2=

[[[[[

[

1 431 259 193

023 1 060 045

039 166 1 074

052 224 135 1

]]]]]

]

119860lowast

3=

[[[[[

[

1 423 259 185

024 1 061 044

039 163 1 071

054 229 140 1

]]]]]

]

119860lowast

4=

[[[[[

[

1 402 235 174

025 1 058 043

043 171 1 074

058 231 135 1

]]]]]

]

119860lowast

5=

[[[[[

[

1 403 234 175

025 1 058 043

043 172 1 075

057 230 134 1

]]]]]

]

(15)

where 119860lowast



lowast





1198611 =

[[[[[[[[[[[[

[

1 7 3 4

1

71 1

1

2

1

31 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198612 =

[[[[[[[[[[[[

[

1 6 2 5

1

61 1 1

1

21 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198613 =

[[[[[[[[[[[[

[

1 6 2 4

1

61 1

1

2

1

21 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198614 =

[[[[[[[[[[[[

[

1 31

31

1

31

1

2

1

3

3 2 1 2

1 31

21

]]]]]]]]]]]]

]

(16)





119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)


119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)




120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)



119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)


1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2








1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085





1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












104455 01099 01875 02570 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

204680 01085 01804 02431 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

304620 01092 01785 02502 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

404444 01105 01891 02559 1199061 gt 1199064 gt 1199063 gt 1199062

119860lowast

504449 01105 01900 02545 1199061 gt 1199064 gt 1199063 gt 1199062


119860lowast

1=

[[[[[

[

1 405 238 173

025 1 059 043

042 171 1 073

058 234 137 1

]]]]]

]

119860lowast

2=

[[[[[

[

1 431 259 193

023 1 060 045

039 166 1 074

052 224 135 1

]]]]]

]

119860lowast

3=

[[[[[

[

1 423 259 185

024 1 061 044

039 163 1 071

054 229 140 1

]]]]]

]

119860lowast

4=

[[[[[

[

1 402 235 174

025 1 058 043

043 171 1 074

058 231 135 1

]]]]]

]

119860lowast

5=

[[[[[

[

1 403 234 175

025 1 058 043

043 172 1 075

057 230 134 1

]]]]]

]

(15)

where 119860lowast



lowast





1198611 =

[[[[[[[[[[[[

[

1 7 3 4

1

71 1

1

2

1

31 1

1

3

1

42 3 1

]]]]]]]]]]]]

]

1198612 =

[[[[[[[[[[[[

[

1 6 2 5

1

61 1 1

1

21 1 1

1

51 1 1

]]]]]]]]]]]]

]

1198613 =

[[[[[[[[[[[[

[

1 6 2 4

1

61 1

1

2

1

21 1 1

1

42 1 1

]]]]]]]]]]]]

]

1198614 =

[[[[[[[[[[[[

[

1 31

31

1

31

1

2

1

3

3 2 1 2

1 31

21

]]]]]]]]]]]]

]

(16)





119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)


119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)




120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)



119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)


1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2








1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085





1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860(1)

=[[

[

1 [20 30] [40 50]

1 [015 02]

1

]]

]

119860(2)

=[[

[

1 [15 25] [40 55]

1 [02 025]

1

]]

]

119860(3)

=[[

[

1 [26 35] [45 55]

1 [015 018]

1

]]

]

(17)


119860lowast=1

3

3

sum

119896=1

119860(119896)

=[[

[

1 [203 30] [417 513]

[033 049] 1 [017 021]

[019 024] [476 588] 1

]]

]

(18)




120579119894 = max (119908119894) minusmin (119908119894)

1205791 = 0637 minus 0397 = 024

1205792 = 0116 minus 0083 = 0033

1205793 = 0487 minus 0328 = 0159

120579 =1

3

3

sum

119894=1

120579119894 =1

3(024 + 0033 + 0159) = 0144

(19)



119860(1)

1=[[

[

1 20 40

1 015

1

]]

]

119860(1)

2=[[

[

1 30 50

1 02

1

]]

]

119860(2)

1=[[

[

1 15 40

1 02

1

]]

]

119860(2)

2=[[

[

1 25 55

1 025

1

]]

]

119860(3)

1=[[

[

1 26 45

1 015

1

]]

]

119860(3)

2=[[

[

1 35 55

1 018

1

]]

]

(20)


1 119860(2)1 119860(3)1

and 119860(1)2 119860(2)2 119860(3)2








1205791015840

119894= max (119908119894) minusmin (119908119894)

1205791015840

1= 0677 minus 0592 = 0085





1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 119861lowast1

054004 014557 031439 067608 016878 015514119860lowast

2 119861lowast2

060355 016131 023514 067654 016903 015443119860lowast

3 119861lowast3

058768 015270 025962 068154 016772 015074119860lowast

4 119861lowast4

059078 015350 025572 067602 016907 015491119860lowast

5 119861lowast5

063984 014281 021734 067606 016903 015491Arithmetic mean 059238 015118 025644 067725 016872 015402

1205791015840

2= 0151 minus 0169 = 0018

1205791015840

3= 0154 minus 0256 = 0102

1205791015840=1

3

3

sum

119894=1

1205791015840

119894=1

3(0085 + 0018 + 0102) = 0068

(21)


119894and

120579 gt 1205791015840







1015840

119894and 120579 gt 120579

1015840


5 Summary






Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article problems on solving matrix aggregation in...

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