Research ArticleA Ranking Procedure by Incomplete PairwiseComparisons Using Information Entropy and Dempster-ShaferEvidence Theory

Dongbo Pan Xi Lu Juan Liu and Yong Deng

Faculty of Computer and Information Science Southwest University Chongqing 400715 China

Correspondence should be addressed to Yong Deng professordeng163com

Received 8 May 2014 Accepted 30 July 2014 Published 27 August 2014

Academic Editor Baolin Sun

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

Decision-making as a way to discover the preference of ranking has been used in various fields However owing to the uncertaintyin group decision-making how to rank alternatives by incomplete pairwise comparisons has become an open issue In thispaper an improved method is proposed for ranking of alternatives by incomplete pairwise comparisons using Dempster-Shaferevidence theory and information entropy Firstly taking the probability assignment of the chosen preference into considerationthe comparison of alternatives to each group is addressed Experiments verified that the information entropy of the data itself candetermine the different weight of each grouprsquos choices objectively Numerical examples in group decision-making environmentsare used to test the effectiveness of the proposed method Moreover the divergence of ranking mechanism is analyzed briefly inconclusion section

1 Introduction

The increasing trend toward decision-making in variousfields requires computational methods for discovering thepreferences of ranking In fact methods for finding andpredicting preferences in a reasonableway are among the veryhot topics in recent science study such as information sys-tems [1] control systems [2] social choices [3 4] and so on

The term ldquopairwise comparisonsrdquo generally refers to anyprocess of comparing entities in pairs to judge which ofeach entity is preferred or has a greater amount of quanti-tative property Prominent psychometrician Thurstone firstintroduced a scientific approach to use pairwise comparisonsfor measurement in 1927 which he referred to as the lawof comparative judgment [5] Thurstone demonstrated thatthe method can be used to order items along a dimensionsuch as preference or importance using an interval-type scaleThe Bradley-Terry-Luce (BTL)model was applied to pairwisecomparison data to scale preferences [6 7] The BTL modelwas identical to Thurstonersquos model if the simple logisticfunction was used Thurstone used the normal distributionin applications of the model which the method of pairwisecomparisonswas used as an approach tomeasuring perceived

intensity of physical stimuli attitudes preferences choicesand values He also studied implications of the theory hedeveloped for opinion polls and political voting [8] If anindividual or organization expresses a preference betweentwo mutually distinct alternatives this preference can beexpressed as a pairwise comparison If pairwise comparisonsare in fact transitive then pairwise comparisons for a listalternatives (119860

1 1198602 1198603 119860

119899minus1 119860119899) can take the form

1198601⪰ 1198602⪰ 1198603⪰ sdot sdot sdot ⪰ 119860

119899minus1⪰ 119860119899 (1)

and it means that the alternative 119860119894is preferred to 119860

119895 in

which 119894 lt 119895 Or the alternatives can be expressed as

1198601≻ 1198602≻ 1198603≻ sdot sdot sdot ≻ 119860

119899minus1≻ 119860119899 (2)

and it means that the alternative119860119894is strictly preferred to119860

119895

if 119894 lt 119895Although pairwise comparison is a well-known tech-

nique in decision-making in some cases we have to be facedwith the problem of incomplete judgement to preferenceFor instance if the number of the alternatives 119899 is large theexperts may not give the full comparison one by one In order

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 904596 11 pageshttpdxdoiorg1011552014904596

2 The Scientific World Journal

to overcome such problem a decision support system (DSS)based on fuzzy information axiom (FIA) is developed in [9]However calculation procedure of information axiom is notonly incommodious but also difficult for decision makersand it is hard to deal with the incomplete pairwise problemsFor the purpose of reducing the complexity of calculationand preference eliciting process [10] points out that somecomparison between alternative can be skipped and amethodis proposed to derive the priorities of 119899 alternatives froman incomplete 119899 times 119899 pairwise comparison matrices in [11]Furthermore [12] introduces a fuzzy multiexpert multicrite-ria decision-making method in possibility measure to handlethe difficulty of conflict aggregation process Moreover thewell-known methods of eigenvector or geometric mean areused to ranking pairwise comparison also introduced in[11 13] But all of the above references only focus on thecomplexity of calculation with the complete comparisonHowever some methods have been proposed to solve theincomplete comparison problem Shiraishi et al proposed aheuristic method which is based on a property of a coefficientof the characteristic polynomial of pairwise comparisonmatrices [14] But the solving is mainly depending on thepolynomial which has infinitely many solutions and it isdifficult to get the best candidate In [15] a least squares typemethod is proposed to directly calculate the priority vector asthe solution of a constrained optimization problem insteadof calculating the missing entries of pairwise comparisonmatrices In [16] a new centroid-index ranking method offuzzy number in decision-making was proposed by Yong andQi He also proposed a method to concern with the rankingof decision alternatives based on preference judgments madeon decision alternatives over a number of criteria toMultiple-criteria decision-making (MCDM) problem in [17] And infuzzy group decision-making an optimal consensus methodin which the limit of each expertrsquos compromise was underconsideration in the process of reaching group consensuswas proposed by Liu et al in [18] Similarly in order to cal-culate the priority vector of incomplete preference a fuzzypreference relation by a goal programming approach wasproposed by Xu in [19] and then he developed a method forincomplete fuzzy preference [20] Subsequently the eigen-vector method and least square method are proposed withincomplete fuzzy preference relation in [21 22] The fuzzydecision-making can not only reduce the complexity of cal-culation but also be used in incomplete pairwise comparisonbut it is important that the little variety of preference willchange the result mainly Moreover a method for learningvalued preference structures using a natural extension of so-called pairwise classification was proposed by Hullermeierand Furnkranz which may have a potential application infuzzy classification [23 24] A weighted voting procedure isused in the Hullermeierrsquos proposed method but the problemis that the voting procedure is always lost in the dead cycle

In this paper a discount rate is derived from the informa-tion entropy which determines the certainty or uncertaintyof the preference Then an improved method to calculate theprobability assignment of preference using an index intro-duced in [25] with the discount rate is proposed Comparingto the current method the incomplete comparison results of

the proposed method are completely dependendent on theinformation itself instead of the human factor The paper isorganized as follows Some fundamental and quantifyingprinciples of Dempster-Shafer evidence theory are given inSection 2 Then the ranking procedures by comparing withsingle group of experts and with independent groups areintroduced in Section 3 In this section the Dempster-Shaferevidence theorymodel and imprecise Dirichlet model whichgive a basic ranking are concerned Next the proposedenhancedmethod on the casewhen comparisons are suppliedby independent groups of experts is presented in Section 4In this section a weight derived from the entropy of BPArsquosprobability is used in Dempster-Shafer evidence theory andimproves the existing ranking method mentioned in the pre-vious section Finally the conclusions are given in Section 5

2 Preliminaries

21 Dempster-Shafer Evidence Theory Dempster-Shafer evi-dence theory can be divided into probability distributionfunction plausibility function and Dempster evidence com-bination rule [26 27] It is rather flexible for many appliedproblems

Definition 1 Assume frame of discernment is 120579 then func-tion 119898 2

120579 rarr [0 1] satisfies 119898(120601) = 0 and sum119860cap120579

119898(119860) = 1

is called the basic probability distribution of frame of discern-ment 120579

For forall119860 sub 120579 119898(119860) is the basic probability of 119860 Themeaning of 119898(119860) is that if 119860 sub Ω and 119860 = Ω thus 119898(119860)is the accurate trust degree of 119860 if 119860 = Ω thus 119898(119860)meansthat the trust degree of 119860 can not be allocated accurately

Definition 2 As for forall119860 sub 120579 the defined function Bel 119898

2120579 rarr [0 1] by Bel(119860) = sum

119861sub119860119898(119861) is called the belief func-

tion of 120579

Definition 3 As for forall119860 sub 120579 pl is called the plausibility func-tion of Bel in pl(119860) = 1 minus Bel(119860)

The relation between belief function and plausibilityfunction is that Bel(119860) and pl(119860) are respectively referredto as the lower limit and the upper limit function of pl(119860) geBel(119860)

Even if they are the same evidences the probabilityassignmentmight be different when they came from differentsources Then the orthogonal method is used to combinethese functions by dempster-shafer evidence theory

Assume 1198981 1198982 119898

119899are the basic probability assign-

ment functions of 2Ω and their orthogonal 119898 = 1198981oplus 1198982oplus

sdot sdot sdot oplus 119898119899are

119898(120601) = 0

119898 (119860) = 119896 sdot sumcap119860119894=119860

prod1le119894le119899

119898119894(119860119894) 119860 = 120601

(3)

in which 119896minus1 = 1 minus sumcap119860119894=119860

prod1le119894le119899

119898119894(119860119894)

The Scientific World Journal 3

Several of the algorithms basic to Dempster-Shaferrsquos evi-dence theory are as follows

(1) It is known that if we assume frame of discernmentof some field is Ω = 119878

1 1198782 119878

119899and propositions

119860 119861 are the subsets of Ω the inference rule shallbe

If 119864 then 119867CF (4)

among which 119864 119867 are the logic groupings of theproposition CF is the certainty factor which is mea-sured by 119888

119894and called credibility For any proposition

119860 the certainty factor CF of credibility 119860 shall satisfy

(a) 119888119894ge 0 1 le 119894 le 119899

(b) sum1le119894le119899

119888119894le 1

(2) Evidence description assume 119898 is the defined basicprobability assignment function of 2Ω then it shallmeet the following conditions during calculation

(a) 119898(119878119894) ge 0 119878

119894isin Ω

(b) sum1le119894le119899

(119898(119878119894) ge 0) le 1

(c) 119898(Ω) = 1 minus sum1le119894le119899

119898(119878119894)

(d) 119898(119860) = 0 119860 sub Ω and |119860| gt 1 or |119860| = 0

among which |119860| means the factor numbers ofproposition 119860

(3) Inaccurate inference model

(a) suppose 119860 is one part proposition of regularcondition Under the condition of evidence 119864the matching degree of proposition 119860 and evi-dence 119864 is

MD (119860 119864) = 1 if 119864 sup 119860

0 otherwise(5)

(b) the definition of part proposition 119860 in regularcondition is

CER = MD (119860 119864) sdot 119891 (119860) (6)

22 Quantifying the Uncertainty in the Dempster-Shafer Evi-dence Theory The uncertain factor considered in evidencetheory includes both the uncertainty associated with ran-domness and the uncertainty associated with granularityThemeasure of the granular uncertainty associated with a subsetis the specificity measure introduced by [28]

Definition 4 Let 119861 be a nonempty subset of119883 A measure ofspecificity of 119861 Sp(119861) is defined using Card to denote thecardinality of a set as

Sp (119861) = 1 minusCard (119861) minus 1

Card (119883) minus 1=Card (119883) minus Card (119861)

Card (119883) minus 1 (7)

It essentially measures the degree to which 119861 has exactly oneelement The larger the specificity the lesser the uncertaintyIn [29] the measure to the case of a probability assignmentfunction119898 was extended

Definition 5 Assume119898 has focal element119861119895 119895 = 1 to 119902Then

Sp (119898) =119902

sum119895=1

Sp (119861119895)119898 (119861

119895) (8)

can be denoted as the expected specificity of the focal ele-ments

Noting that Sp(119898) isin [0 1] and the larger the Sp(119898) theless the uncertainty will be It is clear that the specificity issmallest when119898 is the vacuous belief function119898(119883) = 1 Inthis case Sp(119898) = 0

Klir [30] and Ayyub [31] introduced a related measurewhich he called nonspecificity

Definition 6 If 119898 is a belief function with focal element 119861119895

119895 = 1 to 119902 then its nonspecificity is defined as

119873[Sp (119898)] =119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895) (9)

where |119861119895| = Card(119861

119895)

It is clear that this takes its largest value ln(119899) where 119899 isthe cardinality of 119883 for the vacuous belief function It takesits smallest value when 119898 is Bayesian where |119861

119895| = 1 in this

case ln(|119861119895|) = 0 and119873[Sp(119898)] = 0

Yager [32] made this definition of nonspecificity cointen-sive with the preceding definition of specificity by normaliza-tion and negation hence

Sp2(119898) = 1 minus

119873 [Sp (119898)]ln (119899)

Sp2(119898) = 1 minus

1

ln (119899)

119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895)

(10)

The standard measure of uncertainty associated with a prob-ability distribution is the Shannon entropy

Definition 7 If 119883 = 1199091 119909

119899 and if 119875 is a probability

distribution on 119883 such that 119901119894is the probability of 119909

119894 then

the Shannon entropy of 119875 is

119867(119875) = minus

119902

sum119894=1

119901119894ln (119901119894) (11)

Any extension of this to belief function must be such that itreduced to the Shannon entropy when the belief structure isBayesian Yager [29] suggested an extension of the Shannonentropy to belief structures using the measure of dissonance

Definition 8 If119898 has focal element 119861119895 then the extension of

the Shannon entropy to belief structures is

119867(119898) = minus

119902

sum119895=1

ln Pl (119861119895)119898 (119861

119895) (12)

4 The Scientific World Journal

0

0806

0402

1

0

05

10

02

04

06

08

Figure 1 The relationship between the assignment of element andentropy

For a given set of weights it can be noted that the smallerthe focal elements the larger the entropy In particular thesmallest a nonempty set can be is one element And as weknew the larger the entropy the larger the uncertainty Fora given set of weights the smallest entropy for a Bayesianbelief structure shall be one element because one elementalways means very certainty information of the preferenceand the entropy is 0 and the largest entropy occurredwhen allelements have equal probability From this we can concludethat for any uncertainty and certainty information the fewerthe element the smaller the entropy and the sparser the prob-ability of element the smaller the entropy Figure 1 shows therelationship between the assignment of element and entropyin which the more balance the assignment of element thehigher the entropy

3 Ranking Procedure by IncompletePairwise Comparison

Here we suppose there is a set of alternative Λ = 1198601 1198602

119860119899 with 119899 elements Then we can get 2119899 minus 1 subset with

119899 elements as

Ω = 1198601 119860

119899 1198601 1198602 119860

119899minus1 119860119899

1198601 119860

119899

(13)

The pairwise comparison means that an expert has chosensome subset of alternatives fromΩ and compared them pair-wisely If the paired comparisons have been done for all thesubset in Ω we call it complete pairwise comparison other-wise we call it incomplete pairwise comparison For completepairwise comparison we always get the pairwise comparisonmatrix that looks likeTable 1 inwhichwe give two alternativesas an example It is supposed that experts only compare subsetof alternatives without providing preference value or weightsof preference If an expert chooses one comparison prefer-ence then the value 1 is added to the corresponding cell in thepairwise comparison matrix For example 119888

12is the number

of experts who choose the comparison preference 1198601 ⪰

Table 1 The complete pairwise comparison matrix

1198601 119860

2 119860

1 1198602

1198601 mdash 119888

1211988813

1198602 119888

21mdash 119888

23

1198601 1198602 119888

3111988832

mdash

1198602 There are many references [13 23 33] that have studied

the complete pairwise comparison matrix But in mostoccasions which have a lot of alternatives the experts cannot give the preference one by one They only choose limitedpreferences of all the alternativesThus we get the incompletepairwise comparison matrix that looks like Table 2 withthree alternatives which contain some uncertainty and (or)conflicted information In complete pairwise comparisonmatrix we can calculate the probability of each subset 119860

119894

with 119888119894119895 but in incomplete pairwise comparison we can not

get the assignment of the probability of all the subset Forinstant if an expert chooses the preference 119860

1 ⪰ 119860

2 1198603

we do not know how the probability is distributed amongthe preferences 119860

1 ⪰ 119860

2 and 119860

1 ⪰ 119860

3 without

any other supplement information In such situation we canapply the framework of Dempster-Shafer evidence theory tothe considered sets of preferences Now we suppose there arethree alternatives which need to prefer each other In orderto simplify the format we denote all the subset of the threealternatives as in Table 3 and denote119861

119894119895as the preference119861

119894⪰

119861119895 For example 119861

16means the preference 119860

1 ⪰ 119860

2 1198603

Then we define its BPA for every pairwise comparison in theextended pairwise comparison matrix as follows

119898(119861119894⪰ 119861119895) = 119898 (119861

119894119895) =

119888119894119895

119873 119873 = sum

119894119895isin12119899119894 =119895

119888119894119895

(14)

31 The Ranking Method with One Group of Preference Weassume the experts choose the following preferences fromUtkinrsquos works [25]

five experts 1198601 1198603 ⪰ 119860

2 = 11986152

two experts 1198603 ⪰ 119860

1 1198602 1198603 = 11986137

three experts 1198601 ⪰ 119860

3 = 11986113

Using the Dempster-Shafer evidence theory we describedin previous section we can get the belief function of thepreferences 119860

119894 ⪰ Λ which means 119860

119894 is the best choice

of all the subset (or alternative)Bel(119860

1 ⪰ Λ) = 119898(119861

13) = 03

Bel(1198602 ⪰ Λ) = 0

Bel(1198603 ⪰ Λ) = 119898(119861

37) = 02

and the plausibility functions of the preferencepl(119860

1 ⪰ Λ) = 119898(119861

52) + 119898(119861

13) = 08

pl(1198602 ⪰ Λ) = 0

pl(1198603 ⪰ Λ) = 119898(119861

52) + 119898(119861

37) = 07

From the belief function and the plausibility function itcan be concluded that the best ranking of the alternatives is1198601⪰ 1198603⪰ 1198602

The Scientific World Journal 5

Table 2 The incomplete pairwise comparison matrix

1198601 119860

2 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198601 mdash 119888

12mdash mdash mdash 119888

16mdash

1198602 119888

21mdash mdash mdash 119888

2511988826

mdash1198603 mdash 119888

32mdash mdash mdash mdash 119888

37

1198601 1198602 mdash mdash 119888

43mdash mdash mdash mdash

1198601 1198603 119888

5111988852

mdash mdash mdash 11988856

mdash1198602 1198603 119888

61mdash mdash mdash mdash mdash mdash

1198601 1198602 1198603 mdash mdash mdash mdash mdash mdash mdash

Table 3 The subset of three alternatives and their short notations

1198601 1198602 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198611 1198612 119861

3 119861

4 119861

5 119861

6 119861

7

32 The Ranking Method with Two Independent Groups ofPreference In fact in order to obtain more consensus resultof the preference we always choose more than one indepen-dent group experts to give their preference of alternativesThus we can use the well-established method for combiningthe independent information with the Dempster-Shafer evi-dence rule of combination

Here we suppose there are two groups of experts withoutloss of generality and denote the preference obtained fromthe first and second groups of experts by upper indices (1)and (2) respectivelyThe combined rule refers to the contentsintroduced in preliminaries Now we assume that the firstgroup (with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides thefollowing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPArsquos of all preferences

1198981(119861(1)

16) = 04 119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05 119898

2(119861(2)

34) = 02 119898

2(119861(2)

13) = 03

and the preference intersections for Dempster-Shafer combi-nation rule are in Table 4

According to the table we calculate the weight of conflictfirst

119870 = 1198981(119861(1)

16) times 119898

2(119861(2)

34) + 119898

1(119861(1)

43) times 119898

2(119861(2)

52)

+ 1198981(119861(1)

43) times 119898

2(119861(2)

34)

= 04 times 02 + 06 times 05 + 06 times 02 = 05

(15)

Then the probability of the assignment for the nonzerocombined BPArsquos preference can be calculated as follows

11989812(119861(12)

12) =

1

1 minus 119870times 1198981(119861(1)

16) times 119898

2(119861(2)

52)

= 2 times 04 times 05 = 04

11989812(119861(12)

13) =

1

1 minus 119870times (1198981(119861(1)

16) + 119898

1(119861(1)

43)) times 119898

2(119861(2)

52)

= 2 times 03 = 06

(16)

After that we can get the belief and plausibility functionsof alternatives 119860

1 1198602 1198603or preferences 119860

119894 ⪰ Λ 119894 = 1 2 3

as followsBel(119860

1 ⪰ Λ) = 119898

12(119861(12)

12) + 11989812(119861(12)

13) = 1

pl(1198601 ⪰ Λ) = 1

The belief and plausibility functions of 1198602 ⪰ Λ and 119860

3 ⪰

Λ are 0 So we can only get the result of preference from thatthe ldquobestrdquo alternative is119860

1and can not get any information of

preference about 1198602and 119860

3 From [25] the author thought

the main reason of the situation in previous example was thesmall number of expert judgements and the other reasonwas the used assumption that the sources of evidence wereabsolutely reliable So he improved the ranking method withthe imprecise Dirichlet model

33 The Ranking Method with Imprecise Dirichlet Model Aswe described at the final part of the last subsection the maindifficulty of the proposed ranking method is the possiblesmall number of experts In order to overcome this difficultyan imprecise Dirichlet model (IDM) introduced by Walley[34]was applied to extend the belief and plausibility functionssuch that a lack of sufficient statistical data could be taken intoaccount [35 36] With the method we can get the extendedbelief and plausibility functions as follows

Bel119904(119860) =

119873 sdot Bel (119860)119873 + 119904

pl119904(119860) =

119873 sdot pl (119860) + 119904

119873 + 119904

(17)

Here the hyperparameter 119904 determines how quickly upperand lower probability of events converge as statistical dataaccumulate and it should be taken to be 1 or 2 119873 is thenumber of expert judgements

6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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RoboticsJournal of

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Computational Intelligence and Neuroscience

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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2 The Scientific World Journal

to overcome such problem a decision support system (DSS)based on fuzzy information axiom (FIA) is developed in [9]However calculation procedure of information axiom is notonly incommodious but also difficult for decision makersand it is hard to deal with the incomplete pairwise problemsFor the purpose of reducing the complexity of calculationand preference eliciting process [10] points out that somecomparison between alternative can be skipped and amethodis proposed to derive the priorities of 119899 alternatives froman incomplete 119899 times 119899 pairwise comparison matrices in [11]Furthermore [12] introduces a fuzzy multiexpert multicrite-ria decision-making method in possibility measure to handlethe difficulty of conflict aggregation process Moreover thewell-known methods of eigenvector or geometric mean areused to ranking pairwise comparison also introduced in[11 13] But all of the above references only focus on thecomplexity of calculation with the complete comparisonHowever some methods have been proposed to solve theincomplete comparison problem Shiraishi et al proposed aheuristic method which is based on a property of a coefficientof the characteristic polynomial of pairwise comparisonmatrices [14] But the solving is mainly depending on thepolynomial which has infinitely many solutions and it isdifficult to get the best candidate In [15] a least squares typemethod is proposed to directly calculate the priority vector asthe solution of a constrained optimization problem insteadof calculating the missing entries of pairwise comparisonmatrices In [16] a new centroid-index ranking method offuzzy number in decision-making was proposed by Yong andQi He also proposed a method to concern with the rankingof decision alternatives based on preference judgments madeon decision alternatives over a number of criteria toMultiple-criteria decision-making (MCDM) problem in [17] And infuzzy group decision-making an optimal consensus methodin which the limit of each expertrsquos compromise was underconsideration in the process of reaching group consensuswas proposed by Liu et al in [18] Similarly in order to cal-culate the priority vector of incomplete preference a fuzzypreference relation by a goal programming approach wasproposed by Xu in [19] and then he developed a method forincomplete fuzzy preference [20] Subsequently the eigen-vector method and least square method are proposed withincomplete fuzzy preference relation in [21 22] The fuzzydecision-making can not only reduce the complexity of cal-culation but also be used in incomplete pairwise comparisonbut it is important that the little variety of preference willchange the result mainly Moreover a method for learningvalued preference structures using a natural extension of so-called pairwise classification was proposed by Hullermeierand Furnkranz which may have a potential application infuzzy classification [23 24] A weighted voting procedure isused in the Hullermeierrsquos proposed method but the problemis that the voting procedure is always lost in the dead cycle

In this paper a discount rate is derived from the informa-tion entropy which determines the certainty or uncertaintyof the preference Then an improved method to calculate theprobability assignment of preference using an index intro-duced in [25] with the discount rate is proposed Comparingto the current method the incomplete comparison results of

the proposed method are completely dependendent on theinformation itself instead of the human factor The paper isorganized as follows Some fundamental and quantifyingprinciples of Dempster-Shafer evidence theory are given inSection 2 Then the ranking procedures by comparing withsingle group of experts and with independent groups areintroduced in Section 3 In this section the Dempster-Shaferevidence theorymodel and imprecise Dirichlet model whichgive a basic ranking are concerned Next the proposedenhancedmethod on the casewhen comparisons are suppliedby independent groups of experts is presented in Section 4In this section a weight derived from the entropy of BPArsquosprobability is used in Dempster-Shafer evidence theory andimproves the existing ranking method mentioned in the pre-vious section Finally the conclusions are given in Section 5

2 Preliminaries

21 Dempster-Shafer Evidence Theory Dempster-Shafer evi-dence theory can be divided into probability distributionfunction plausibility function and Dempster evidence com-bination rule [26 27] It is rather flexible for many appliedproblems

Definition 1 Assume frame of discernment is 120579 then func-tion 119898 2

120579 rarr [0 1] satisfies 119898(120601) = 0 and sum119860cap120579

119898(119860) = 1

is called the basic probability distribution of frame of discern-ment 120579

For forall119860 sub 120579 119898(119860) is the basic probability of 119860 Themeaning of 119898(119860) is that if 119860 sub Ω and 119860 = Ω thus 119898(119860)is the accurate trust degree of 119860 if 119860 = Ω thus 119898(119860)meansthat the trust degree of 119860 can not be allocated accurately

Definition 2 As for forall119860 sub 120579 the defined function Bel 119898

2120579 rarr [0 1] by Bel(119860) = sum

119861sub119860119898(119861) is called the belief func-

tion of 120579

Definition 3 As for forall119860 sub 120579 pl is called the plausibility func-tion of Bel in pl(119860) = 1 minus Bel(119860)

The relation between belief function and plausibilityfunction is that Bel(119860) and pl(119860) are respectively referredto as the lower limit and the upper limit function of pl(119860) geBel(119860)

Even if they are the same evidences the probabilityassignmentmight be different when they came from differentsources Then the orthogonal method is used to combinethese functions by dempster-shafer evidence theory

Assume 1198981 1198982 119898

119899are the basic probability assign-

ment functions of 2Ω and their orthogonal 119898 = 1198981oplus 1198982oplus

sdot sdot sdot oplus 119898119899are

119898(120601) = 0

119898 (119860) = 119896 sdot sumcap119860119894=119860

prod1le119894le119899

119898119894(119860119894) 119860 = 120601

(3)

in which 119896minus1 = 1 minus sumcap119860119894=119860

prod1le119894le119899

119898119894(119860119894)

The Scientific World Journal 3

Several of the algorithms basic to Dempster-Shaferrsquos evi-dence theory are as follows

(1) It is known that if we assume frame of discernmentof some field is Ω = 119878

1 1198782 119878

119899and propositions

119860 119861 are the subsets of Ω the inference rule shallbe

If 119864 then 119867CF (4)

among which 119864 119867 are the logic groupings of theproposition CF is the certainty factor which is mea-sured by 119888

119894and called credibility For any proposition

119860 the certainty factor CF of credibility 119860 shall satisfy

(a) 119888119894ge 0 1 le 119894 le 119899

(b) sum1le119894le119899

119888119894le 1

(2) Evidence description assume 119898 is the defined basicprobability assignment function of 2Ω then it shallmeet the following conditions during calculation

(a) 119898(119878119894) ge 0 119878

119894isin Ω

(b) sum1le119894le119899

(119898(119878119894) ge 0) le 1

(c) 119898(Ω) = 1 minus sum1le119894le119899

119898(119878119894)

(d) 119898(119860) = 0 119860 sub Ω and |119860| gt 1 or |119860| = 0

among which |119860| means the factor numbers ofproposition 119860

(3) Inaccurate inference model

(a) suppose 119860 is one part proposition of regularcondition Under the condition of evidence 119864the matching degree of proposition 119860 and evi-dence 119864 is

MD (119860 119864) = 1 if 119864 sup 119860

0 otherwise(5)

(b) the definition of part proposition 119860 in regularcondition is

CER = MD (119860 119864) sdot 119891 (119860) (6)

22 Quantifying the Uncertainty in the Dempster-Shafer Evi-dence Theory The uncertain factor considered in evidencetheory includes both the uncertainty associated with ran-domness and the uncertainty associated with granularityThemeasure of the granular uncertainty associated with a subsetis the specificity measure introduced by [28]

Definition 4 Let 119861 be a nonempty subset of119883 A measure ofspecificity of 119861 Sp(119861) is defined using Card to denote thecardinality of a set as

Sp (119861) = 1 minusCard (119861) minus 1

Card (119883) minus 1=Card (119883) minus Card (119861)

Card (119883) minus 1 (7)

It essentially measures the degree to which 119861 has exactly oneelement The larger the specificity the lesser the uncertaintyIn [29] the measure to the case of a probability assignmentfunction119898 was extended

Definition 5 Assume119898 has focal element119861119895 119895 = 1 to 119902Then

Sp (119898) =119902

sum119895=1

Sp (119861119895)119898 (119861

119895) (8)

can be denoted as the expected specificity of the focal ele-ments

Noting that Sp(119898) isin [0 1] and the larger the Sp(119898) theless the uncertainty will be It is clear that the specificity issmallest when119898 is the vacuous belief function119898(119883) = 1 Inthis case Sp(119898) = 0

Klir [30] and Ayyub [31] introduced a related measurewhich he called nonspecificity

Definition 6 If 119898 is a belief function with focal element 119861119895

119895 = 1 to 119902 then its nonspecificity is defined as

119873[Sp (119898)] =119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895) (9)

where |119861119895| = Card(119861

119895)

It is clear that this takes its largest value ln(119899) where 119899 isthe cardinality of 119883 for the vacuous belief function It takesits smallest value when 119898 is Bayesian where |119861

119895| = 1 in this

case ln(|119861119895|) = 0 and119873[Sp(119898)] = 0

Yager [32] made this definition of nonspecificity cointen-sive with the preceding definition of specificity by normaliza-tion and negation hence

Sp2(119898) = 1 minus

119873 [Sp (119898)]ln (119899)

Sp2(119898) = 1 minus

1

ln (119899)

119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895)

(10)

The standard measure of uncertainty associated with a prob-ability distribution is the Shannon entropy

Definition 7 If 119883 = 1199091 119909

119899 and if 119875 is a probability

distribution on 119883 such that 119901119894is the probability of 119909

119894 then

the Shannon entropy of 119875 is

119867(119875) = minus

119902

sum119894=1

119901119894ln (119901119894) (11)

Any extension of this to belief function must be such that itreduced to the Shannon entropy when the belief structure isBayesian Yager [29] suggested an extension of the Shannonentropy to belief structures using the measure of dissonance

Definition 8 If119898 has focal element 119861119895 then the extension of

the Shannon entropy to belief structures is

119867(119898) = minus

119902

sum119895=1

ln Pl (119861119895)119898 (119861

119895) (12)

4 The Scientific World Journal

0

0806

0402

1

0

05

10

02

04

06

08

Figure 1 The relationship between the assignment of element andentropy

For a given set of weights it can be noted that the smallerthe focal elements the larger the entropy In particular thesmallest a nonempty set can be is one element And as weknew the larger the entropy the larger the uncertainty Fora given set of weights the smallest entropy for a Bayesianbelief structure shall be one element because one elementalways means very certainty information of the preferenceand the entropy is 0 and the largest entropy occurredwhen allelements have equal probability From this we can concludethat for any uncertainty and certainty information the fewerthe element the smaller the entropy and the sparser the prob-ability of element the smaller the entropy Figure 1 shows therelationship between the assignment of element and entropyin which the more balance the assignment of element thehigher the entropy

3 Ranking Procedure by IncompletePairwise Comparison

Here we suppose there is a set of alternative Λ = 1198601 1198602

119860119899 with 119899 elements Then we can get 2119899 minus 1 subset with

119899 elements as

Ω = 1198601 119860

119899 1198601 1198602 119860

119899minus1 119860119899

1198601 119860

119899

(13)

The pairwise comparison means that an expert has chosensome subset of alternatives fromΩ and compared them pair-wisely If the paired comparisons have been done for all thesubset in Ω we call it complete pairwise comparison other-wise we call it incomplete pairwise comparison For completepairwise comparison we always get the pairwise comparisonmatrix that looks likeTable 1 inwhichwe give two alternativesas an example It is supposed that experts only compare subsetof alternatives without providing preference value or weightsof preference If an expert chooses one comparison prefer-ence then the value 1 is added to the corresponding cell in thepairwise comparison matrix For example 119888

12is the number

of experts who choose the comparison preference 1198601 ⪰

Table 1 The complete pairwise comparison matrix

1198601 119860

2 119860

1 1198602

1198601 mdash 119888

1211988813

1198602 119888

21mdash 119888

23

1198601 1198602 119888

3111988832

mdash

1198602 There are many references [13 23 33] that have studied

the complete pairwise comparison matrix But in mostoccasions which have a lot of alternatives the experts cannot give the preference one by one They only choose limitedpreferences of all the alternativesThus we get the incompletepairwise comparison matrix that looks like Table 2 withthree alternatives which contain some uncertainty and (or)conflicted information In complete pairwise comparisonmatrix we can calculate the probability of each subset 119860

119894

with 119888119894119895 but in incomplete pairwise comparison we can not

get the assignment of the probability of all the subset Forinstant if an expert chooses the preference 119860

1 ⪰ 119860

2 1198603

we do not know how the probability is distributed amongthe preferences 119860

1 ⪰ 119860

2 and 119860

1 ⪰ 119860

3 without

any other supplement information In such situation we canapply the framework of Dempster-Shafer evidence theory tothe considered sets of preferences Now we suppose there arethree alternatives which need to prefer each other In orderto simplify the format we denote all the subset of the threealternatives as in Table 3 and denote119861

119894119895as the preference119861

119894⪰

119861119895 For example 119861

16means the preference 119860

1 ⪰ 119860

2 1198603

Then we define its BPA for every pairwise comparison in theextended pairwise comparison matrix as follows

119898(119861119894⪰ 119861119895) = 119898 (119861

119894119895) =

119888119894119895

119873 119873 = sum

119894119895isin12119899119894 =119895

119888119894119895

(14)

31 The Ranking Method with One Group of Preference Weassume the experts choose the following preferences fromUtkinrsquos works [25]

five experts 1198601 1198603 ⪰ 119860

2 = 11986152

two experts 1198603 ⪰ 119860

1 1198602 1198603 = 11986137

three experts 1198601 ⪰ 119860

3 = 11986113

Using the Dempster-Shafer evidence theory we describedin previous section we can get the belief function of thepreferences 119860

119894 ⪰ Λ which means 119860

119894 is the best choice

of all the subset (or alternative)Bel(119860

1 ⪰ Λ) = 119898(119861

13) = 03

Bel(1198602 ⪰ Λ) = 0

Bel(1198603 ⪰ Λ) = 119898(119861

37) = 02

and the plausibility functions of the preferencepl(119860

1 ⪰ Λ) = 119898(119861

52) + 119898(119861

13) = 08

pl(1198602 ⪰ Λ) = 0

pl(1198603 ⪰ Λ) = 119898(119861

52) + 119898(119861

37) = 07

From the belief function and the plausibility function itcan be concluded that the best ranking of the alternatives is1198601⪰ 1198603⪰ 1198602

The Scientific World Journal 5

Table 2 The incomplete pairwise comparison matrix

1198601 119860

2 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198601 mdash 119888

12mdash mdash mdash 119888

16mdash

1198602 119888

21mdash mdash mdash 119888

2511988826

mdash1198603 mdash 119888

32mdash mdash mdash mdash 119888

37

1198601 1198602 mdash mdash 119888

43mdash mdash mdash mdash

1198601 1198603 119888

5111988852

mdash mdash mdash 11988856

mdash1198602 1198603 119888

61mdash mdash mdash mdash mdash mdash

1198601 1198602 1198603 mdash mdash mdash mdash mdash mdash mdash

Table 3 The subset of three alternatives and their short notations

1198601 1198602 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198611 1198612 119861

3 119861

4 119861

5 119861

6 119861

7

32 The Ranking Method with Two Independent Groups ofPreference In fact in order to obtain more consensus resultof the preference we always choose more than one indepen-dent group experts to give their preference of alternativesThus we can use the well-established method for combiningthe independent information with the Dempster-Shafer evi-dence rule of combination

Here we suppose there are two groups of experts withoutloss of generality and denote the preference obtained fromthe first and second groups of experts by upper indices (1)and (2) respectivelyThe combined rule refers to the contentsintroduced in preliminaries Now we assume that the firstgroup (with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides thefollowing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPArsquos of all preferences

1198981(119861(1)

16) = 04 119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05 119898

2(119861(2)

34) = 02 119898

2(119861(2)

13) = 03

and the preference intersections for Dempster-Shafer combi-nation rule are in Table 4

According to the table we calculate the weight of conflictfirst

119870 = 1198981(119861(1)

16) times 119898

2(119861(2)

34) + 119898

1(119861(1)

43) times 119898

2(119861(2)

52)

+ 1198981(119861(1)

43) times 119898

2(119861(2)

34)

= 04 times 02 + 06 times 05 + 06 times 02 = 05

(15)

Then the probability of the assignment for the nonzerocombined BPArsquos preference can be calculated as follows

11989812(119861(12)

12) =

1

1 minus 119870times 1198981(119861(1)

16) times 119898

2(119861(2)

52)

= 2 times 04 times 05 = 04

11989812(119861(12)

13) =

1

1 minus 119870times (1198981(119861(1)

16) + 119898

1(119861(1)

43)) times 119898

2(119861(2)

52)

= 2 times 03 = 06

(16)

After that we can get the belief and plausibility functionsof alternatives 119860

1 1198602 1198603or preferences 119860

119894 ⪰ Λ 119894 = 1 2 3

as followsBel(119860

1 ⪰ Λ) = 119898

12(119861(12)

12) + 11989812(119861(12)

13) = 1

pl(1198601 ⪰ Λ) = 1

The belief and plausibility functions of 1198602 ⪰ Λ and 119860

3 ⪰

Λ are 0 So we can only get the result of preference from thatthe ldquobestrdquo alternative is119860

1and can not get any information of

preference about 1198602and 119860

3 From [25] the author thought

the main reason of the situation in previous example was thesmall number of expert judgements and the other reasonwas the used assumption that the sources of evidence wereabsolutely reliable So he improved the ranking method withthe imprecise Dirichlet model

33 The Ranking Method with Imprecise Dirichlet Model Aswe described at the final part of the last subsection the maindifficulty of the proposed ranking method is the possiblesmall number of experts In order to overcome this difficultyan imprecise Dirichlet model (IDM) introduced by Walley[34]was applied to extend the belief and plausibility functionssuch that a lack of sufficient statistical data could be taken intoaccount [35 36] With the method we can get the extendedbelief and plausibility functions as follows

Bel119904(119860) =

119873 sdot Bel (119860)119873 + 119904

pl119904(119860) =

119873 sdot pl (119860) + 119904

119873 + 119904

(17)

Here the hyperparameter 119904 determines how quickly upperand lower probability of events converge as statistical dataaccumulate and it should be taken to be 1 or 2 119873 is thenumber of expert judgements

6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Electrical and Computer Engineering

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 3

Several of the algorithms basic to Dempster-Shaferrsquos evi-dence theory are as follows

(1) It is known that if we assume frame of discernmentof some field is Ω = 119878

1 1198782 119878

119899and propositions

119860 119861 are the subsets of Ω the inference rule shallbe

If 119864 then 119867CF (4)

among which 119864 119867 are the logic groupings of theproposition CF is the certainty factor which is mea-sured by 119888

119894and called credibility For any proposition

119860 the certainty factor CF of credibility 119860 shall satisfy

(a) 119888119894ge 0 1 le 119894 le 119899

(b) sum1le119894le119899

119888119894le 1

(2) Evidence description assume 119898 is the defined basicprobability assignment function of 2Ω then it shallmeet the following conditions during calculation

(a) 119898(119878119894) ge 0 119878

119894isin Ω

(b) sum1le119894le119899

(119898(119878119894) ge 0) le 1

(c) 119898(Ω) = 1 minus sum1le119894le119899

119898(119878119894)

(d) 119898(119860) = 0 119860 sub Ω and |119860| gt 1 or |119860| = 0

among which |119860| means the factor numbers ofproposition 119860

(3) Inaccurate inference model

(a) suppose 119860 is one part proposition of regularcondition Under the condition of evidence 119864the matching degree of proposition 119860 and evi-dence 119864 is

MD (119860 119864) = 1 if 119864 sup 119860

0 otherwise(5)

(b) the definition of part proposition 119860 in regularcondition is

CER = MD (119860 119864) sdot 119891 (119860) (6)

22 Quantifying the Uncertainty in the Dempster-Shafer Evi-dence Theory The uncertain factor considered in evidencetheory includes both the uncertainty associated with ran-domness and the uncertainty associated with granularityThemeasure of the granular uncertainty associated with a subsetis the specificity measure introduced by [28]

Definition 4 Let 119861 be a nonempty subset of119883 A measure ofspecificity of 119861 Sp(119861) is defined using Card to denote thecardinality of a set as

Sp (119861) = 1 minusCard (119861) minus 1

Card (119883) minus 1=Card (119883) minus Card (119861)

Card (119883) minus 1 (7)

It essentially measures the degree to which 119861 has exactly oneelement The larger the specificity the lesser the uncertaintyIn [29] the measure to the case of a probability assignmentfunction119898 was extended

Definition 5 Assume119898 has focal element119861119895 119895 = 1 to 119902Then

Sp (119898) =119902

sum119895=1

Sp (119861119895)119898 (119861

119895) (8)

can be denoted as the expected specificity of the focal ele-ments

Noting that Sp(119898) isin [0 1] and the larger the Sp(119898) theless the uncertainty will be It is clear that the specificity issmallest when119898 is the vacuous belief function119898(119883) = 1 Inthis case Sp(119898) = 0

Klir [30] and Ayyub [31] introduced a related measurewhich he called nonspecificity

Definition 6 If 119898 is a belief function with focal element 119861119895

119895 = 1 to 119902 then its nonspecificity is defined as

119873[Sp (119898)] =119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895) (9)

where |119861119895| = Card(119861

119895)

It is clear that this takes its largest value ln(119899) where 119899 isthe cardinality of 119883 for the vacuous belief function It takesits smallest value when 119898 is Bayesian where |119861

119895| = 1 in this

case ln(|119861119895|) = 0 and119873[Sp(119898)] = 0

Yager [32] made this definition of nonspecificity cointen-sive with the preceding definition of specificity by normaliza-tion and negation hence

Sp2(119898) = 1 minus

119873 [Sp (119898)]ln (119899)

Sp2(119898) = 1 minus

1

ln (119899)

119902

sum119895=1

ln (1003816100381610038161003816100381611986111989510038161003816100381610038161003816)119898 (119861

119895)

(10)

The standard measure of uncertainty associated with a prob-ability distribution is the Shannon entropy

Definition 7 If 119883 = 1199091 119909

119899 and if 119875 is a probability

distribution on 119883 such that 119901119894is the probability of 119909

119894 then

the Shannon entropy of 119875 is

119867(119875) = minus

119902

sum119894=1

119901119894ln (119901119894) (11)

Any extension of this to belief function must be such that itreduced to the Shannon entropy when the belief structure isBayesian Yager [29] suggested an extension of the Shannonentropy to belief structures using the measure of dissonance

Definition 8 If119898 has focal element 119861119895 then the extension of

the Shannon entropy to belief structures is

119867(119898) = minus

119902

sum119895=1

ln Pl (119861119895)119898 (119861

119895) (12)

4 The Scientific World Journal

0

0806

0402

1

0

05

10

02

04

06

08

Figure 1 The relationship between the assignment of element andentropy

For a given set of weights it can be noted that the smallerthe focal elements the larger the entropy In particular thesmallest a nonempty set can be is one element And as weknew the larger the entropy the larger the uncertainty Fora given set of weights the smallest entropy for a Bayesianbelief structure shall be one element because one elementalways means very certainty information of the preferenceand the entropy is 0 and the largest entropy occurredwhen allelements have equal probability From this we can concludethat for any uncertainty and certainty information the fewerthe element the smaller the entropy and the sparser the prob-ability of element the smaller the entropy Figure 1 shows therelationship between the assignment of element and entropyin which the more balance the assignment of element thehigher the entropy

3 Ranking Procedure by IncompletePairwise Comparison

Here we suppose there is a set of alternative Λ = 1198601 1198602

119860119899 with 119899 elements Then we can get 2119899 minus 1 subset with

119899 elements as

Ω = 1198601 119860

119899 1198601 1198602 119860

119899minus1 119860119899

1198601 119860

119899

(13)

The pairwise comparison means that an expert has chosensome subset of alternatives fromΩ and compared them pair-wisely If the paired comparisons have been done for all thesubset in Ω we call it complete pairwise comparison other-wise we call it incomplete pairwise comparison For completepairwise comparison we always get the pairwise comparisonmatrix that looks likeTable 1 inwhichwe give two alternativesas an example It is supposed that experts only compare subsetof alternatives without providing preference value or weightsof preference If an expert chooses one comparison prefer-ence then the value 1 is added to the corresponding cell in thepairwise comparison matrix For example 119888

12is the number

of experts who choose the comparison preference 1198601 ⪰

Table 1 The complete pairwise comparison matrix

1198601 119860

2 119860

1 1198602

1198601 mdash 119888

1211988813

1198602 119888

21mdash 119888

23

1198601 1198602 119888

3111988832

mdash

1198602 There are many references [13 23 33] that have studied

the complete pairwise comparison matrix But in mostoccasions which have a lot of alternatives the experts cannot give the preference one by one They only choose limitedpreferences of all the alternativesThus we get the incompletepairwise comparison matrix that looks like Table 2 withthree alternatives which contain some uncertainty and (or)conflicted information In complete pairwise comparisonmatrix we can calculate the probability of each subset 119860

119894

with 119888119894119895 but in incomplete pairwise comparison we can not

get the assignment of the probability of all the subset Forinstant if an expert chooses the preference 119860

1 ⪰ 119860

2 1198603

we do not know how the probability is distributed amongthe preferences 119860

1 ⪰ 119860

2 and 119860

1 ⪰ 119860

3 without

any other supplement information In such situation we canapply the framework of Dempster-Shafer evidence theory tothe considered sets of preferences Now we suppose there arethree alternatives which need to prefer each other In orderto simplify the format we denote all the subset of the threealternatives as in Table 3 and denote119861

119894119895as the preference119861

119894⪰

119861119895 For example 119861

16means the preference 119860

1 ⪰ 119860

2 1198603

Then we define its BPA for every pairwise comparison in theextended pairwise comparison matrix as follows

119898(119861119894⪰ 119861119895) = 119898 (119861

119894119895) =

119888119894119895

119873 119873 = sum

119894119895isin12119899119894 =119895

119888119894119895

(14)

31 The Ranking Method with One Group of Preference Weassume the experts choose the following preferences fromUtkinrsquos works [25]

five experts 1198601 1198603 ⪰ 119860

2 = 11986152

two experts 1198603 ⪰ 119860

1 1198602 1198603 = 11986137

three experts 1198601 ⪰ 119860

3 = 11986113

Using the Dempster-Shafer evidence theory we describedin previous section we can get the belief function of thepreferences 119860

119894 ⪰ Λ which means 119860

119894 is the best choice

of all the subset (or alternative)Bel(119860

1 ⪰ Λ) = 119898(119861

13) = 03

Bel(1198602 ⪰ Λ) = 0

Bel(1198603 ⪰ Λ) = 119898(119861

37) = 02

and the plausibility functions of the preferencepl(119860

1 ⪰ Λ) = 119898(119861

52) + 119898(119861

13) = 08

pl(1198602 ⪰ Λ) = 0

pl(1198603 ⪰ Λ) = 119898(119861

52) + 119898(119861

37) = 07

From the belief function and the plausibility function itcan be concluded that the best ranking of the alternatives is1198601⪰ 1198603⪰ 1198602

The Scientific World Journal 5

Table 2 The incomplete pairwise comparison matrix

1198601 119860

2 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198601 mdash 119888

12mdash mdash mdash 119888

16mdash

1198602 119888

21mdash mdash mdash 119888

2511988826

mdash1198603 mdash 119888

32mdash mdash mdash mdash 119888

37

1198601 1198602 mdash mdash 119888

43mdash mdash mdash mdash

1198601 1198603 119888

5111988852

mdash mdash mdash 11988856

mdash1198602 1198603 119888

61mdash mdash mdash mdash mdash mdash

1198601 1198602 1198603 mdash mdash mdash mdash mdash mdash mdash

Table 3 The subset of three alternatives and their short notations

1198601 1198602 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198611 1198612 119861

3 119861

4 119861

5 119861

6 119861

7

32 The Ranking Method with Two Independent Groups ofPreference In fact in order to obtain more consensus resultof the preference we always choose more than one indepen-dent group experts to give their preference of alternativesThus we can use the well-established method for combiningthe independent information with the Dempster-Shafer evi-dence rule of combination

Here we suppose there are two groups of experts withoutloss of generality and denote the preference obtained fromthe first and second groups of experts by upper indices (1)and (2) respectivelyThe combined rule refers to the contentsintroduced in preliminaries Now we assume that the firstgroup (with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides thefollowing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPArsquos of all preferences

1198981(119861(1)

16) = 04 119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05 119898

2(119861(2)

34) = 02 119898

2(119861(2)

13) = 03

and the preference intersections for Dempster-Shafer combi-nation rule are in Table 4

According to the table we calculate the weight of conflictfirst

119870 = 1198981(119861(1)

16) times 119898

2(119861(2)

34) + 119898

1(119861(1)

43) times 119898

2(119861(2)

52)

+ 1198981(119861(1)

43) times 119898

2(119861(2)

34)

= 04 times 02 + 06 times 05 + 06 times 02 = 05

(15)

Then the probability of the assignment for the nonzerocombined BPArsquos preference can be calculated as follows

11989812(119861(12)

12) =

1

1 minus 119870times 1198981(119861(1)

16) times 119898

2(119861(2)

52)

= 2 times 04 times 05 = 04

11989812(119861(12)

13) =

1

1 minus 119870times (1198981(119861(1)

16) + 119898

1(119861(1)

43)) times 119898

2(119861(2)

52)

= 2 times 03 = 06

(16)

After that we can get the belief and plausibility functionsof alternatives 119860

1 1198602 1198603or preferences 119860

119894 ⪰ Λ 119894 = 1 2 3

as followsBel(119860

1 ⪰ Λ) = 119898

12(119861(12)

12) + 11989812(119861(12)

13) = 1

pl(1198601 ⪰ Λ) = 1

The belief and plausibility functions of 1198602 ⪰ Λ and 119860

3 ⪰

Λ are 0 So we can only get the result of preference from thatthe ldquobestrdquo alternative is119860

1and can not get any information of

preference about 1198602and 119860

3 From [25] the author thought

the main reason of the situation in previous example was thesmall number of expert judgements and the other reasonwas the used assumption that the sources of evidence wereabsolutely reliable So he improved the ranking method withthe imprecise Dirichlet model

33 The Ranking Method with Imprecise Dirichlet Model Aswe described at the final part of the last subsection the maindifficulty of the proposed ranking method is the possiblesmall number of experts In order to overcome this difficultyan imprecise Dirichlet model (IDM) introduced by Walley[34]was applied to extend the belief and plausibility functionssuch that a lack of sufficient statistical data could be taken intoaccount [35 36] With the method we can get the extendedbelief and plausibility functions as follows

Bel119904(119860) =

119873 sdot Bel (119860)119873 + 119904

pl119904(119860) =

119873 sdot pl (119860) + 119904

119873 + 119904

(17)

Here the hyperparameter 119904 determines how quickly upperand lower probability of events converge as statistical dataaccumulate and it should be taken to be 1 or 2 119873 is thenumber of expert judgements

6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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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

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httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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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

4 The Scientific World Journal

0

0806

0402

1

0

05

10

02

04

06

08

Figure 1 The relationship between the assignment of element andentropy

For a given set of weights it can be noted that the smallerthe focal elements the larger the entropy In particular thesmallest a nonempty set can be is one element And as weknew the larger the entropy the larger the uncertainty Fora given set of weights the smallest entropy for a Bayesianbelief structure shall be one element because one elementalways means very certainty information of the preferenceand the entropy is 0 and the largest entropy occurredwhen allelements have equal probability From this we can concludethat for any uncertainty and certainty information the fewerthe element the smaller the entropy and the sparser the prob-ability of element the smaller the entropy Figure 1 shows therelationship between the assignment of element and entropyin which the more balance the assignment of element thehigher the entropy

3 Ranking Procedure by IncompletePairwise Comparison

Here we suppose there is a set of alternative Λ = 1198601 1198602

119860119899 with 119899 elements Then we can get 2119899 minus 1 subset with

119899 elements as

Ω = 1198601 119860

119899 1198601 1198602 119860

119899minus1 119860119899

1198601 119860

119899

(13)

The pairwise comparison means that an expert has chosensome subset of alternatives fromΩ and compared them pair-wisely If the paired comparisons have been done for all thesubset in Ω we call it complete pairwise comparison other-wise we call it incomplete pairwise comparison For completepairwise comparison we always get the pairwise comparisonmatrix that looks likeTable 1 inwhichwe give two alternativesas an example It is supposed that experts only compare subsetof alternatives without providing preference value or weightsof preference If an expert chooses one comparison prefer-ence then the value 1 is added to the corresponding cell in thepairwise comparison matrix For example 119888

12is the number

of experts who choose the comparison preference 1198601 ⪰

Table 1 The complete pairwise comparison matrix

1198601 119860

2 119860

1 1198602

1198601 mdash 119888

1211988813

1198602 119888

21mdash 119888

23

1198601 1198602 119888

3111988832

mdash

1198602 There are many references [13 23 33] that have studied

the complete pairwise comparison matrix But in mostoccasions which have a lot of alternatives the experts cannot give the preference one by one They only choose limitedpreferences of all the alternativesThus we get the incompletepairwise comparison matrix that looks like Table 2 withthree alternatives which contain some uncertainty and (or)conflicted information In complete pairwise comparisonmatrix we can calculate the probability of each subset 119860

119894

with 119888119894119895 but in incomplete pairwise comparison we can not

get the assignment of the probability of all the subset Forinstant if an expert chooses the preference 119860

1 ⪰ 119860

2 1198603

we do not know how the probability is distributed amongthe preferences 119860

1 ⪰ 119860

2 and 119860

1 ⪰ 119860

3 without

any other supplement information In such situation we canapply the framework of Dempster-Shafer evidence theory tothe considered sets of preferences Now we suppose there arethree alternatives which need to prefer each other In orderto simplify the format we denote all the subset of the threealternatives as in Table 3 and denote119861

119894119895as the preference119861

119894⪰

119861119895 For example 119861

16means the preference 119860

1 ⪰ 119860

2 1198603

Then we define its BPA for every pairwise comparison in theextended pairwise comparison matrix as follows

119898(119861119894⪰ 119861119895) = 119898 (119861

119894119895) =

119888119894119895

119873 119873 = sum

119894119895isin12119899119894 =119895

119888119894119895

(14)

31 The Ranking Method with One Group of Preference Weassume the experts choose the following preferences fromUtkinrsquos works [25]

five experts 1198601 1198603 ⪰ 119860

2 = 11986152

two experts 1198603 ⪰ 119860

1 1198602 1198603 = 11986137

three experts 1198601 ⪰ 119860

3 = 11986113

Using the Dempster-Shafer evidence theory we describedin previous section we can get the belief function of thepreferences 119860

119894 ⪰ Λ which means 119860

119894 is the best choice

of all the subset (or alternative)Bel(119860

1 ⪰ Λ) = 119898(119861

13) = 03

Bel(1198602 ⪰ Λ) = 0

Bel(1198603 ⪰ Λ) = 119898(119861

37) = 02

and the plausibility functions of the preferencepl(119860

1 ⪰ Λ) = 119898(119861

52) + 119898(119861

13) = 08

pl(1198602 ⪰ Λ) = 0

pl(1198603 ⪰ Λ) = 119898(119861

52) + 119898(119861

37) = 07

From the belief function and the plausibility function itcan be concluded that the best ranking of the alternatives is1198601⪰ 1198603⪰ 1198602

The Scientific World Journal 5

Table 2 The incomplete pairwise comparison matrix

1198601 119860

2 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198601 mdash 119888

12mdash mdash mdash 119888

16mdash

1198602 119888

21mdash mdash mdash 119888

2511988826

mdash1198603 mdash 119888

32mdash mdash mdash mdash 119888

37

1198601 1198602 mdash mdash 119888

43mdash mdash mdash mdash

1198601 1198603 119888

5111988852

mdash mdash mdash 11988856

mdash1198602 1198603 119888

61mdash mdash mdash mdash mdash mdash

1198601 1198602 1198603 mdash mdash mdash mdash mdash mdash mdash

Table 3 The subset of three alternatives and their short notations

1198601 1198602 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198611 1198612 119861

3 119861

4 119861

5 119861

6 119861

7

32 The Ranking Method with Two Independent Groups ofPreference In fact in order to obtain more consensus resultof the preference we always choose more than one indepen-dent group experts to give their preference of alternativesThus we can use the well-established method for combiningthe independent information with the Dempster-Shafer evi-dence rule of combination

Here we suppose there are two groups of experts withoutloss of generality and denote the preference obtained fromthe first and second groups of experts by upper indices (1)and (2) respectivelyThe combined rule refers to the contentsintroduced in preliminaries Now we assume that the firstgroup (with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides thefollowing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPArsquos of all preferences

1198981(119861(1)

16) = 04 119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05 119898

2(119861(2)

34) = 02 119898

2(119861(2)

13) = 03

and the preference intersections for Dempster-Shafer combi-nation rule are in Table 4

According to the table we calculate the weight of conflictfirst

119870 = 1198981(119861(1)

16) times 119898

2(119861(2)

34) + 119898

1(119861(1)

43) times 119898

2(119861(2)

52)

+ 1198981(119861(1)

43) times 119898

2(119861(2)

34)

= 04 times 02 + 06 times 05 + 06 times 02 = 05

(15)

Then the probability of the assignment for the nonzerocombined BPArsquos preference can be calculated as follows

11989812(119861(12)

12) =

1

1 minus 119870times 1198981(119861(1)

16) times 119898

2(119861(2)

52)

= 2 times 04 times 05 = 04

11989812(119861(12)

13) =

1

1 minus 119870times (1198981(119861(1)

16) + 119898

1(119861(1)

43)) times 119898

2(119861(2)

52)

= 2 times 03 = 06

(16)

After that we can get the belief and plausibility functionsof alternatives 119860

1 1198602 1198603or preferences 119860

119894 ⪰ Λ 119894 = 1 2 3

as followsBel(119860

1 ⪰ Λ) = 119898

12(119861(12)

12) + 11989812(119861(12)

13) = 1

pl(1198601 ⪰ Λ) = 1

The belief and plausibility functions of 1198602 ⪰ Λ and 119860

3 ⪰

Λ are 0 So we can only get the result of preference from thatthe ldquobestrdquo alternative is119860

1and can not get any information of

preference about 1198602and 119860

3 From [25] the author thought

the main reason of the situation in previous example was thesmall number of expert judgements and the other reasonwas the used assumption that the sources of evidence wereabsolutely reliable So he improved the ranking method withthe imprecise Dirichlet model

33 The Ranking Method with Imprecise Dirichlet Model Aswe described at the final part of the last subsection the maindifficulty of the proposed ranking method is the possiblesmall number of experts In order to overcome this difficultyan imprecise Dirichlet model (IDM) introduced by Walley[34]was applied to extend the belief and plausibility functionssuch that a lack of sufficient statistical data could be taken intoaccount [35 36] With the method we can get the extendedbelief and plausibility functions as follows

Bel119904(119860) =

119873 sdot Bel (119860)119873 + 119904

pl119904(119860) =

119873 sdot pl (119860) + 119904

119873 + 119904

(17)

Here the hyperparameter 119904 determines how quickly upperand lower probability of events converge as statistical dataaccumulate and it should be taken to be 1 or 2 119873 is thenumber of expert judgements

6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

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httpwwwhindawicom Volume 2014

Advances in

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ArtificialNeural Systems

Advances in

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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

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

The Scientific World Journal 5

Table 2 The incomplete pairwise comparison matrix

1198601 119860

2 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198601 mdash 119888

12mdash mdash mdash 119888

16mdash

1198602 119888

21mdash mdash mdash 119888

2511988826

mdash1198603 mdash 119888

32mdash mdash mdash mdash 119888

37

1198601 1198602 mdash mdash 119888

43mdash mdash mdash mdash

1198601 1198603 119888

5111988852

mdash mdash mdash 11988856

mdash1198602 1198603 119888

61mdash mdash mdash mdash mdash mdash

1198601 1198602 1198603 mdash mdash mdash mdash mdash mdash mdash

Table 3 The subset of three alternatives and their short notations

1198601 1198602 119860

3 119860

1 1198602 119860

1 1198603 119860

2 1198603 119860

1 1198602 1198603

1198611 1198612 119861

3 119861

4 119861

5 119861

6 119861

7

32 The Ranking Method with Two Independent Groups ofPreference In fact in order to obtain more consensus resultof the preference we always choose more than one indepen-dent group experts to give their preference of alternativesThus we can use the well-established method for combiningthe independent information with the Dempster-Shafer evi-dence rule of combination

Here we suppose there are two groups of experts withoutloss of generality and denote the preference obtained fromthe first and second groups of experts by upper indices (1)and (2) respectivelyThe combined rule refers to the contentsintroduced in preliminaries Now we assume that the firstgroup (with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides thefollowing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPArsquos of all preferences

1198981(119861(1)

16) = 04 119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05 119898

2(119861(2)

34) = 02 119898

2(119861(2)

13) = 03

and the preference intersections for Dempster-Shafer combi-nation rule are in Table 4

According to the table we calculate the weight of conflictfirst

119870 = 1198981(119861(1)

16) times 119898

2(119861(2)

34) + 119898

1(119861(1)

43) times 119898

2(119861(2)

52)

+ 1198981(119861(1)

43) times 119898

2(119861(2)

34)

= 04 times 02 + 06 times 05 + 06 times 02 = 05

(15)

Then the probability of the assignment for the nonzerocombined BPArsquos preference can be calculated as follows

11989812(119861(12)

12) =

1

1 minus 119870times 1198981(119861(1)

16) times 119898

2(119861(2)

52)

= 2 times 04 times 05 = 04

11989812(119861(12)

13) =

1

1 minus 119870times (1198981(119861(1)

16) + 119898

1(119861(1)

43)) times 119898

2(119861(2)

52)

= 2 times 03 = 06

(16)

After that we can get the belief and plausibility functionsof alternatives 119860

1 1198602 1198603or preferences 119860

119894 ⪰ Λ 119894 = 1 2 3

as followsBel(119860

1 ⪰ Λ) = 119898

12(119861(12)

12) + 11989812(119861(12)

13) = 1

pl(1198601 ⪰ Λ) = 1

The belief and plausibility functions of 1198602 ⪰ Λ and 119860

3 ⪰

Λ are 0 So we can only get the result of preference from thatthe ldquobestrdquo alternative is119860

1and can not get any information of

preference about 1198602and 119860

3 From [25] the author thought

the main reason of the situation in previous example was thesmall number of expert judgements and the other reasonwas the used assumption that the sources of evidence wereabsolutely reliable So he improved the ranking method withthe imprecise Dirichlet model

33 The Ranking Method with Imprecise Dirichlet Model Aswe described at the final part of the last subsection the maindifficulty of the proposed ranking method is the possiblesmall number of experts In order to overcome this difficultyan imprecise Dirichlet model (IDM) introduced by Walley[34]was applied to extend the belief and plausibility functionssuch that a lack of sufficient statistical data could be taken intoaccount [35 36] With the method we can get the extendedbelief and plausibility functions as follows

Bel119904(119860) =

119873 sdot Bel (119860)119873 + 119904

pl119904(119860) =

119873 sdot pl (119860) + 119904

119873 + 119904

(17)

Here the hyperparameter 119904 determines how quickly upperand lower probability of events converge as statistical dataaccumulate and it should be taken to be 1 or 2 119873 is thenumber of expert judgements

6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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Advances in

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6 The Scientific World Journal

Table 4 The preference intersections for Dempster-Shafer combination rule

The first group1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3

119861(1)

16119861(1)

43

The second group1198601 1198603 ⪰ 119860

2 119861

(2)

521198601 ⪰ 119860

2 120601

1198603 ⪰ 119860

1 1198602 119861

(2)

34120601 120601

1198601 ⪰ 119860

3 119861

(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3

Table 5 The preference intersections for modified Dempster-Shafer combination rule

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

34120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

However the main advantage of the IDM is that itproduces the cautious inference In particular if119873 = 0 thenBel119904(119860) = 0 and pl

119904(119860) = 1 In the case 119873 rarr infin it can

be stated for any 119904 Bel119904(119860) = Bel(119860) pl

119904(119860) = pl(119860) If we

denote 120583 = 119873(119873 + 119904) then there holds119898lowast(119860119894) = 120583 sdot 119898(119860

119894)

One can see from the last expression for 119898lowast(119860119894) that 120583 is

the discount rate characterizing the reliability of a sourceof evidence and it depends on the number of estimates 119873Because the total probability assignment sum119898(119860

119894) = 1 we

assign the left probability to 119898lowast(11986177) that means that the

experts do not know which alternative is better than theothers and is indicated by the preference 119860

1 1198602 1198603 ⪰

1198601 1198602 1198603 By using the discount rate 120583 with 119904 = 1 we can

get 1205831= 56 ≃ 083 for the first group and 120583

2= 1011 ≃

091 Hence we can rewrite the preference intersection forDempster-Shafer combination rule in Table 5

And the modified probability assignment and conflictweight are

119898lowast

1(119861(1)

16) = 033 119898

lowast

1(119861(1)

43) = 05

119898lowast

1(119861(1)

77) = 017 119898

lowast

2(119861(2)

52) = 046

119898lowast

2(119861(2)

34) = 018 119898

lowast

2(119861(2)

13) = 027

119898lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 083 sdot 091 sdot 05 = 0378

(18)

Now the combined BPA in Table 5 are

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 161 sdot 033 sdot 046 = 0244

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 161 sdot 027 = 0435

(19)

Similarly

11989812(119861(12)

16) = 0048 119898

12(119861(12)

43) = 0072

11989812(119861(12)

52) = 0126 119898

12(119861(12)

34) = 0049

(20)

11989812(119861(12)

77) = 0026 (21)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0727

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0951

(22)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0098

Bel (1198603 ⪰ Λ) = 0049

pl (1198603 ⪰ Λ) = 0201

(23)

It can be seen from the above results that the ldquobestrdquo rank-ing is 119860

1⪰ 1198603⪰ 1198602

4 Improved Method and Numerical Analysis

The main advantage of IDM method is that it allows us todeal with comparisons of arbitrary groups of alternatives Itgives the possibility to use the framework of Dempster-Shaferevidence theory and to compute the belief and plausibilityfunctions of alternatives or ranking and provides a way to

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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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

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Electrical and Computer Engineering

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Journal of

Computer Networks and Communications

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httpwwwhindawicom Volume 2014

Advances in

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ArtificialNeural Systems

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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

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

The Scientific World Journal 7

make cautious decisionswhen the number of expert estimatesis rather small However the method depends excessively onthe number of experts rather than data itself If we changethe number of experts it may lead to a different result Forinstant we assume the number of first group of experts is 50and 20 experts choose the preference 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

and 30 experts choose the preference 1198601 1198602 ⪰ 119860

3 = 119861(1)

43

Then we get the same probability assignment for the firstgroup of experts The only change is the discount rate from083 to 120583

1= 5051 = 098 Here we recompute the probability

assignment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 098 = 0392

119898lowast

1(119861(1)

43) = 06 sdot 098 = 0588 119898

lowast

1(119861(1)

77) = 002

119898lowast

2(119861(2)

52) = 046 119898

lowast

2(119861(2)

34) = 018

119898lowast

2(119861(2)

13) = 027 119898

lowast

2(119861(2)

77) = 009

119870lowast= 12058311205832119870 = 098 sdot 091 sdot 05 = 0446

(24)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 181 sdot 0392 sdot 046 = 0326

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 181 sdot 027 = 0489

(25)

Similarly

11989812(119861(12)

16) = 0064 119898

12(119861(12)

43) = 0096

11989812(119861(12)

52) = 0017 119898

12(119861(12)

34) = 0007

(26)

11989812(119861(12)

77) = 0003 (27)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 0879

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 0995

(28)

In the same way

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 0099

Bel (1198603 ⪰ Λ) = 0007

pl (1198603 ⪰ Λ) = 0027

(29)

Thus we get that the ldquobestrdquo ranking is 1198601

⪰ 1198603

⪰ 1198602

by pessimistic decision-making and 1198601

⪰ 1198602

⪰ 1198603by

optimistic decision-making Comparing with the result inprevious subsection we will find that we only change thenumber of experts and then lead to a conflicting result Wethink the problem is that the discount rate depending on thenumber of experts is a subjective parameter It can not reflectthe objectivity of the probability assignment itself So weobtain different result with the same probability assignmentof the incomplete pairwise comparison

In order to overcome the problem we introduce animproved method to obtain the discount rate by usingentropy which we describe in the Preliminaries sectionBecause the entropy can measure the uncertainty of the datait can be applied to extend belief and plausibility functionby indicating the sparser of the probability assignment andthe fewer of the subset between different groups of pairwisecomparison In particular if the preference has the probabil-ity assignment function 119898 with focal element 119861

119894119895 we define

the entropy of the preference as

119901 (119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895) (30)

Among the definition 119902 = 2119899minus 1 and 119899 is the number of

alternatives Each group of experts can give their preferenceof alternatives with pairwise comparison and then we cancalculate every entropy of each group Because the moreuncertain the probability assignment function the greaterthe entropy and the lesser the discount rate of preferenceassigned by the group of experts we denote 119909 = 1119901(119898)

as the discount rate After that we get a sequence of thediscount rate The sequence of the discount rate can benormalized by the formula 119910 = 119886 tan(119909) sdot 2120587 119909 is the inputdata before normalization which is equal to the discountrate and 119910 is the output data after normalization Notingthat the dimension of 119901(119898) is the 1198982(119861

119894119895) we define the

weight as 120583 = radic119910 Furthermore we get the weight of eachgroup of probability assignment in the incomplete pairwisecomparison matrix Without loss of generality if 119898(119861

119894119895) =

0 we define ln(119898(119861119894119895))119898(119861

119894119895) = 0 The flowchart of the

improvedmethod using entropy to convert the conflict factoris following in Figure 2

8 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

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 The Scientific World Journal

Input

Probability assignment ofalternative

Calculate conflict factor

Calculate entropy withprobability assignment

Convert the conflictfactor with entropy

Refresh the basicprobability assignment

Rank by basic probabilityassignment

Figure 2 The flowchart of the improved method

Now we recalculate the previous example as follows

1199011(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (04 ln 04 + 06 ln 06) = 0673

1199012(119898) = minus

119902

sum119894119895=1

ln (119898 (119861119894119895))119898 (119861

119894119895)

= minus (05 ln 05 + 02 ln 02 + 03 ln 03) = 10297

(31)

Then we get the discount rate for two groups

1199091= 14859 119909

2= 09712 (32)

Next we compute the normalized discount rate

1199101= 06229 119910

2= 04907 (33)

Last we obtain the weights of each group

1205831= 079 120583

2= 07 (34)

Then the next thing is to recompute the probability assign-ment and conflict weight as follows

119898lowast

1(119861(1)

16) = 04 sdot 079 = 0316

119898lowast

1(119861(1)

43) = 06 sdot 079 = 0474 119898

lowast

1(119861(1)

77) = 021

119898lowast

2(119861(2)

52) = 05 sdot 07 = 035

119898lowast

2(119861(2)

34) = 02 sdot 07 = 014

119898lowast

2(119861(2)

13) = 03 sdot 07 = 021 119898

lowast

2(119861(2)

77) = 03

119870lowast= 12058311205832119870 = 079 sdot 07 sdot 05 = 02765

(35)

Now we recompute the combined BPA in Table 5

11989812(119861(12)

12) =

1

1 minus 119870lowastsdot 119898lowast

1(119861(1)

16) sdot 119898lowast

2(119861(2)

52)

= 13822 sdot 0316 sdot 035 = 01529

11989812(119861(12)

13) =

1

1 minus 119870lowastsdot (119898lowast

1(119861(1)

16) + 119898

lowast

1(119861(1)

43) + 119898

lowast

1(119861(1)

77))

sdot 119898lowast

2(119861(2)

13)

= 13822 sdot 021 = 00803

(36)

Similarly

11989812(119861(12)

16) = 0131 119898

12(119861(12)

43) = 01965

11989812(119861(12)

52) = 01016 119898

12(119861(12)

34) = 00406

(37)

11989812(119861(12)

77) = 00871 (38)

So the belief and plausibility functions of 119860119894 ⪰ Λ are

Bel (1198601 ⪰ Λ) = 119898

12(119861(12)

12) + 119898

12(119861(12)

13) + 119898

12(119861(12)

16)

= 04297

pl (1198601 ⪰ Λ) = Bel (119860

1 ⪰ Λ) + 119898

12(119861(12)

52)

+ 11989812(119861(12)

43) + 119898

12(119861(12)

77)

= 08149

(39)

In the same way we can get other belief and plausibilityfunctions

Bel (1198602 ⪰ Λ) = 0

pl (1198602 ⪰ Λ) = 02836

Bel (1198603 ⪰ Λ) = 00406

pl (1198603 ⪰ Λ) = 02293

(40)

According to the result of pairwise comparison we canconclude that the ldquobestrdquo ranking is 119860

1⪰ 1198603⪰ 1198602by pes-

simistic decision-making and 1198601⪰ 1198602⪰ 1198603by optimistic

decision-making Because the method is not dependent onthe amount of the experts the result will not change with theincreasing of experts

Moreover we extend the problem to lower or higherconflict with the alternative of experts Here we assume twodifferent statuses with the alternative of experts one is lowerconflict of alternative and the other is higher conflict of alter-native Then we evaluate the results respectively

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

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

The Scientific World Journal 9

Table 6 The preference intersections for lower conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 119860

2 ⪰ 119860

3 119860

2 ⪰ 119860

3

119861(2)

131198601 ⪰ 119860

3 119860

1 ⪰ 119860

3 119860

1 ⪰ 119860

3

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

Table 7 The preference intersections for higher conflict

The first group119861(1)

16119861(1)

43119861(1)77

The second group

119861(2)

521198601 ⪰ 119860

2 120601 119860

1 1198603 ⪰ 119860

2

119861(2)

23120601 120601 119860

3 ⪰ 119860

1 1198602

119861(2)

13120601 120601 119860

2 ⪰ 119860

1

119861(2)77

1198601 ⪰ 119860

2 1198603 119860

1 1198602 ⪰ 119860

3 Λ ⪰ Λ

For lower conflict condition we assume the first group(with five experts) provides the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provides the follow-ing judgements

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

2 ⪰ 119860

3 = 119861(2)

23

three experts (11988823= 3) 119860

1 ⪰ 119860

3 = 119861(2)

13

Then we get the BPA of all preference

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

23) = 02119898

2(119861(2)

13) = 03

and the preference intersections for lower conflict are inTable 6

The conflict factors are 0287 for IDMmethod and 02101for improved method respectively and the rankings are1198601

⪰ 1198602

⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

If we increase the number of experts in first group to 5020 experts choose 119860

1 ⪰ 119860

2 1198603 and the left experts choose

1198601 1198602 ⪰ 119860

3 The conflict factor shifts to 03091 for IDM

method and keeps the same as 02101 for improved methodand the ranking is the same as the prior oneThe result showsthat the IDMmethod and improvedmethod canmaintain theconsistency in lower conflict of alternative

For higher conflict condition we assume the first group(with five experts) provide the following preferences

two experts (11988811= 2) 119860

1 ⪰ 119860

2 1198603 = 119861(1)

16

three experts (11988812= 3) 119860

1 1198602 ⪰ 119860

3 = 119861(1)

43

And the second group (with ten experts) provide the follow-ing option

five experts (11988821= 5) 119860

1 1198603 ⪰ 119860

2 = 119861(2)

52

two experts (11988822= 2) 119860

3 ⪰ 119860

1 1198602 = 119861(2)

34

three experts (11988823= 3) 119860

2 ⪰ 119860

1 = 119861(2)

21

Then we get the BPA of all preferences

1198981(119861(1)

16) = 04119898

1(119861(1)

43) = 06

1198982(119861(2)

52) = 05119898

2(119861(2)

34) = 02119898

2(119861(2)

21) = 03

and the preference intersections for higher conflict are inTable 7

The conflict factors are 07134 for IDM method and04424 for improved method respectively and the rankingsare 119860

1⪰ 1198602⪰ 1198603both by pessimistic decision-making

and optimistic decision-making either in IDM method or inimproved method

In the sameway we increase the number of experts in firstgroup to 50 and 20 experts choose 119860

1 ⪰ 119860

2 1198603 and 30

experts choose 1198601 1198602 ⪰ 119860

3 The conflict factor shifts to

06042 for IDMmethod and the same as 04424 for improvedmethod The ranking is 119860

1⪰ 1198602

⪰ 1198603by pessimistic

decision-making and 1198601⪰ 1198603⪰ 1198602by optimistic decision-

making in IDM method and the same result in improvedmethod as prior one The result indicates that the IDMmethod can not keep the availability in higher conflict situ-ation and the improved method maintains the effectivenessand reliability both in lower and higher conflict situationsThe results of comparison are in Table 8

Analyzing the result of improved method previous sec-tion there is no doubt that the ldquobestrdquo alternative is 119860

1 and

the problem is the ranking between 1198602 and 119860

3 Moreover

we find that 60 of experts of the first group choose thepreferences 119860

1 1198602 ⪰ 119860

3 and 70of experts of the second

group choose the preferences 1198601 1198603 ⪰ 119860

2 and 119860

3 ⪰

1198601 1198602which include the information about 119860

2 and 119860

3

So the ranking between 1198602 and 119860

3 depends on the weight

10 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

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 The Scientific World Journal

Table 8 Results of comparison between Utkinrsquos method and proposed method

ResultsUtkinrsquos method Proposed method

5 experts Low conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

50 expertsLow conflict factor 119860

1 ⪰ 119860

2 ⪰ 119860

3 119860

1 ⪰ 119860

2 ⪰ 119860

3

High conflict factor 1198601 ⪰ 119860

2 ⪰ 119860

3 (pessimistic)

1198601 ⪰ 119860

2 ⪰ 119860

3

1198601 ⪰ 119860

3 ⪰ 119860

2 (optimistic)

of the ranking chosen by the first group and the second groupof experts Noticing that the weight of imprecise Dirichletmethod relies heavily on the number of experts the weightof the second group with ten experts is greater than the firstgroup with five experts Thus the result is propitious to thepreferences 119860

1 1198603 ⪰ 119860

2 and 119860

3 ⪰ 119860

1 1198602

5 Conclusion

Different from the imprecise Dirichlet method we proposean improved method to deal with the incomplete pair-wise comparison by any groups of alternatives and expertsThe proposed method assigns the probability to the beliefand (or) plausibility of alternatives or ranking using theweightedDEMPSTER-SHAFER evidencemethodMoreoverin order to objectively consider the fairness of decision-making between different groups of experts the proposedmethod introduces the entropy which indicates the valueof information to calculate the weight The weight in theproposed method is decided by the initial data or probabilityassignment itself rather than the number of experts It takesinto account the factor of the number of the elements and thesparse degree of the probability assignment and pays moreattention on the establishmentmechanism of the data At lasta numerical analysis illustrates the method and shows thedifference from the imprecise Dirichlet method

Conflict of Interests

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

Acknowledgments

The Ph D candidates of the corresponding author in Shang-hai Jiao Tong University Xiaoyan Su and Peida Xu thePh D candidate DaijunWei and graduate students XiaogeZhang Xinyang Deng and Bingyi Kang provided valuablediscussions on the algorithm in this paper The work is par-tially supported by the Chongqing Natural Science Founda-tion (for Distinguished Young Scholars) Grant no CSCT2010BA2003 the National Natural Science Foundation ofChina Grant no 61174022 the National High TechnologyResearch andDevelopment Program of China (863 Program)(nos 2013AA013801 and 2012AA041101) and the Funda-mental Research Founds for the Central Universities (noXDJK2013C029)

References

[1] S Alonso E Herrera-Viedma F Chiclana and F Herrera ldquoAweb based consensus support system for group decisionmakingproblems and incomplete preferencesrdquo Information Sciencesvol 180 no 23 pp 4477ndash4495 2010

[2] S Bozooki ldquoAn inconsistency control system based on incom-plete pairwise comparisonmatricesrdquo in Proceedings of the Inter-national Symposium on the Analytic Hierarch Process (ISAHPrsquo11) Naples Italy 2011

[3] T L Saaty and L G Vargas ldquoThe possibility of group choicepairwise comparisons and merging functionsrdquo Social Choiceand Welfare vol 38 no 3 pp 481ndash496 2012

[4] Y Deng and F T S Chan ldquoA new fuzzy dempster MCDMmethod and its application in supplier selectionrdquoExpert Systemswith Applications vol 38 no 8 pp 9854ndash9861 2011

[5] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[6] R A Bradley and M E Terry ldquoRank analysis of incompleteblock designs I The method of paired comparisonsrdquo Biomet-rika vol 39 pp 324ndash345 1952

[7] R D Luce Individual Choice Behavior A Theoretical AnalysisCourier Dover 2012

[8] L L Thurstone The Measurement of Values University ofChicago Press 1959

[9] S Cebi and C Kahraman ldquoDeveloping a group decisionsupport system based on fuzzy information axiomrdquoKnowledge-Based Systems vol 23 no 1 pp 3ndash16 2010

[10] P T Harker ldquoAlternative modes of questioning in the analytichierarchy processrdquoMathematical Modelling vol 9 no 3ndash5 pp353ndash360 1987

[11] P T Harker ldquoIncomplete pairwise comparisons in the analytichierarchy processrdquo Mathematical Modelling vol 9 no 11 pp837ndash848 1987

[12] M Noor-E-Alam T F Lipi M Ahsan Akhtar Hasin and A MM S Ullah ldquoAlgorithms for fuzzy multi expert multi criteriadecisionmaking (ME-MCDM)rdquoKnowledge-Based Systems vol24 no 3 pp 367ndash377 2011

[13] M W Herman and W W Koczkodaj ldquoA Monte Carlo study ofpairwise comparisonrdquo Information Processing Letters vol 57 no1 pp 25ndash29 1996

[14] S Shiraishi T Obata and M Daigo ldquoProperties of a positivereciprocal matrix and their application to AHPrdquo Journal of theOperations Research Society of Japan vol 41 no 3 pp 404ndash4141998

[15] Q Chen and E Triantaphyllou ldquoEstimating data for multi-criteria decision making problems optimization techniquesrdquoEncyclopedia of Optimization vol 2 pp 27ndash36 2001

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

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

The Scientific World Journal 11

[16] D Yong and L Qi ldquoA topsis-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[17] Y Deng F T S Chan Y Wu and D Wang ldquoA new linguis-tic MCDM method based on multiple-criterion data fusionrdquoExpert Systems with Applications vol 38 no 6 pp 6985ndash69932011

[18] J Liu F T S Chan Y Li Y Zhang and Y Deng ldquoA new optimalconsensusmethodwithminimumcost in fuzzy groupdecisionrdquoKnowledge-Based Systems vol 35 pp 357ndash360 2012

[19] Z S Xu ldquoGoal programming models for obtaining the priorityvector of incomplete fuzzy preference relationrdquo InternationalJournal of Approximate Reasoning vol 36 no 3 pp 261ndash2702004

[20] Z Xu ldquoA procedure for decision making based on incompletefuzzy preference relationrdquo Fuzzy Optimization and DecisionMaking vol 4 no 3 pp 175ndash189 2005

[21] 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

[22] Y Xu R Patnayakuni andHWang ldquoLogarithmic least squaresmethod to priority for group decision making with incompletefuzzy preference relationsrdquo Applied Mathematical Modellingvol 37 no 4 pp 2139ndash2152 2013

[23] E Hullermeier and J Furnkranz ldquoComparison of rankingprocedures in pairwise preference learningrdquo in Proceedings ofthe 10th International Conference on Information Processing andManagement of Uncertainty in Knowledge-Based Systems (IPMUrsquo04) Perugia Italy 2004

[24] E Hullermeier and J Furnkranz ldquoRanking by pairwise compar-ison a note on risk minimizationrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems vol 1 pp 97ndash102IEEE Budapest Hungary July 2004

[25] L V Utkin ldquoA new ranking procedure by incomplete pairwisecomparisons using preference subsetsrdquo Intelligent Data Analy-sis vol 13 no 2 pp 229ndash241 2009

[26] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 pp 325ndash339 1967

[27] G Shafer A Mathematical Theory of Evidence vol 1 PrincetonUniversity Press Princeton NJ USA 1976

[28] O Kaynak ldquoComputational intelligence soft computing andfuzzy-neuro integration with applicationsrdquo in Proceedings of theNATO Advanced Study Institute on Computational Intelligence(Fuzzy-Neural Integration) Springer Antalya Turkey August1998

[29] R R Yager ldquoEntropy and specificity in a mathematical theoryof evidencerdquo International Journal of General Systems vol 9 no4 pp 249ndash260 1983

[30] G J Klir Uncertainty and Information Foundations of Gener-alized Information Theory Wiley-Interscience New York NYUSA 2005

[31] B M Ayyub Uncertainty Modeling and Analysis in Engineeringand the Sciences CRC Press 2006

[32] R R Yager ldquoOn the fusion of non-independent belief struc-turesrdquo International Journal of General Systems vol 38 no 5pp 505ndash531 2009

[33] K G Jamieson and R Nowak ldquoActive ranking using pairwisecomparisonsrdquo in Advances in Neural Information ProcessingSystems pp 2240ndash2248 2011

[34] P Walley ldquoInferences from multinomial data learning about abag of marblesrdquo Journal of the Royal Statistical Society Series BMethodological vol 58 no 1 pp 3ndash57 1996

[35] L V Utkin ldquoProbabilities of judgments provided by unknownexperts by using the imprecise dirichlet modelrdquo Risk Decisionand Policy vol 9 no 4 pp 371ndash389 2004

[36] L V Utkin ldquoExtensions of belief functions and possibilitydistributions by using the impreciseDirichletmodelrdquoFuzzy Setsand Systems vol 154 no 3 pp 413ndash431 2005

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

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Artificial Intelligence

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Electrical and Computer Engineering

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Computer Networks and Communications

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Biomedical Imaging

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ArtificialNeural Systems

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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

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014