Research ArticleMultiscale Probability Transformation of BasicProbability Assignment

Meizhu Li1 Xi Lu12 Qi Zhang1 and Yong Deng134

1 School of Computer and Information Science Southwest University Chongqing 400715 China2 School of Hanhong Southwest University Chongqing 400715 China3 School of Automation Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China4 School of Engineering Vanderbilt University Nashville TN 37235 USA

Correspondence should be addressed to Yong Deng ydengswueducn

Received 16 June 2014 Accepted 2 October 2014 Published 20 October 2014

Academic Editor Mohamed Abd El Aziz

Copyright copy 2014 Meizhu Li 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

Decisionmaking is still an open issue in the application of Dempster-Shafer evidence theory A lot of works have been presented forit In the transferable belief model (TBM) pignistic probabilities based on the basic probability assignments are used for decisionmaking In this paper multiscale probability transformation of basic probability assignment based on the belief function and theplausibility function is proposed which is a generalization of the pignistic probability transformation In the multiscale probabilityfunction a factor q based on the Tsallis entropy is used to make the multiscale probabilities diversified An example showing thatthe multiscale probability transformation is more reasonable in the decision making is given

1 Introduction

Since first proposed by Dempster [1] and then developedby Shafer [2] the Dempster-Shafer theory of evidencewhich is also called Dempster-Shafer theory or evidencetheory has been paid much attention for a long time andcontinually attracted growing interests [3ndash5] Even as a theoryof reasoning under the uncertain environment Dempster-Shafer theory has an advantage of directly expressing theldquouncertaintyrdquo by assigning the probability to the subsets ofthe set composed ofmultiple objects rather than to each of theindividual objects so it has been widely used in many fieldssuch as statistical learning [6 7] classification and clustering[8ndash11] decisionmaking [12ndash14] knowledge reasoning [15 16]risk assessment and evaluation [17 18] and others [19ndash21]

To improve the Dempster-Shafer theory of evidencemany studies have been devoted for combination rule ofevidence confliction problem generation of mass function[22 23] uncertain measure of evidence [24 25] and so on[26ndash28] The basic rule of combination is Dempsterrsquos ruleof combination (orthogonal sum) It needs a normalizationstep in order to preserve the basic properties of the belief

functions In [29] Zadeh has underlined that this normaliza-tion involves counterintuitive behaviours In order to solvethe problem of conflict management Yager [30] Smets [31]and Murphy [32] and more recently Liu [33] and Lefevre andElouedi [34] have proposed other combination rules

One open issue of evidence theory is the decisionmakingbased on the basic probability assignments and many workshave been done to construct a reasonable model for thedecisionmaking In the transferable beliefmodel (TBM) [35]pignistic probabilities are used for decision making [36] Thetransferable belief model is presented to represent quantifiedbeliefs based on belief functions TBM was constructed bytwo levels the credal level where beliefs are entertained andquantified by belief functions and the pignistic level wherebeliefs can be used to make decisions and are quantifiedby probability functions The main idea of the pignisticprobability transformation is to transform the multielementsubsets into singleton subsets by an average method Becausein some examples the pignistic transformation method pro-duces results that appear to be inconsistent with Demp-sterrsquos rule of combination Cobb and Shenoy [37] proposeda new plausibility transformation method for translating

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 319264 6 pageshttpdxdoiorg1011552014319264

2 Mathematical Problems in Engineering

Dempster-Shafer (D-S) belief function models to probabilitymodels Other works have been done to obtain a morereasonable transformation method [38ndash40]

Among the previous research the pignistic probabilitytransformation is the most widely used but it is not reason-able enough to describe the unknown for the multielementsubsets Hence a generalization of the pignistic probabilitytransformation called multiscale probability transformationof basic probability assignment is proposed in this paperwhich is based on the belief function and the plausibilityfunction The proposed function can be calculated with thedifference between the belief function and the plausibilityfunction we call itmultiscale probability function and denoteit as a function Mul119875 In the multiscale probability functiona factor 119902 based on the Tsallis entropy [41] is used to makethe multiscale probabilities diversified When the value of 119902equals 0 the proposed multiscale probability transformationcan be degenerated as the pignistic probability transforma-tion

The rest of this paper is organized as follows Section 2introduces some basic preliminaries about the Dempster-Shafer theory and the pignistic probability transformationIn Section 3 the multiscale probability transformation is pre-sented Section 4 uses an example to illustrate the effective-ness of themultiscale probability transformation Conclusionis given in Section 5

2 Preliminaries

21 Dempster-Shafer Theory of Evidence Dempster-Shafertheory of evidence [1 2] also called Dempster-Shafer the-ory or evidence theory is used to deal with uncertaininformation As an effective theory of evidential reason-ing Dempster-Shafer theory has an advantage of directlyexpressing various uncertainties This theory needs weakerconditions than Bayesian theory of probability so it isoften regarded as an extension of the Bayesian theory Forcompleteness of the explanation a few basic concepts areintroduced as follows

Definition 1 Let Ω be a mutually exclusive and collectivelyexhaustive set which is indicted by

Ω = 1198641 1198642 119864

119894 119864

119873 (1)

The set Ω is called frame of discernment The power set of Ωis indicated by 2Ω where

2Ω= 0 119864

1 119864

119873 1198641 1198642

1198641 1198642 119864

119894 Ω

(2)

If 119860 isin 2Ω 119860 is called a proposition

Definition 2 For a frame of discernment Ω a mass functionis a mapping119898 from 2

Ω to [0 1] formally defined by

119898 2Ω997888rarr [0 1] (3)

which satisfies the following condition

119898(0) = 0 sum

119860isin2Ω

119898(119860) = 1 (4)

In Dempster-Shafer theory a mass function is also calleda basic probability assignment (BPA) If119898(119860) gt 0119860 is calleda focal element and the union of all focal elements is calledthe core of the mass function

Definition 3 For a proposition 119860 sube Ω the belief functionBel 2Ω rarr [0 1] is defined as

Bel (119860) = sum119861sube119860

119898(119861) (5)

The plausibility function Pl 2Ω rarr [0 1] is defined as

Pl (119860) = 1 minus Bel (119860) = sum

119861cap119860 =0

119898(119861) (6)

where 119860 = Ω minus 119860Obviously Pl(119860) ge Bel(119860) and these functions Bel and

Pl are the lower limit function and upper limit function ofproposition 119860 respectively

22 Pignistic Probability Transformation In the transferablebelief model (TBM) [35] pignistic probabilities are used fordecision making The definition of the pignistic probabilitytransformation is shown as follows

Definition 4 Let 119898 be a BPA on the frame of discernmentΩ Its associated pignistic probability function Bet119875

119898 Ω rarr

[0 1] is defined as

Bet119875119898(120596) = sum

119860sube119875(Ω)120596isin119860

1

|119860|

119898 (119860)

1 minus 119898 (120601) 119898 (120601) = 1 (7)

where |119882| is the cardinality of subset119860Theprocess of pignis-tic probability transformation (PPT) is that basic probabilityassignment transferred to probability distributionThereforethe pignistic betting distance can be easily obtained by PPT

3 Multiscale Probability Transformation ofBasic Probability Assignment

In the transferable belief model (TBM) [35] pignistic proba-bilities are used for decision making The transferable beliefmodel is presented to represent quantified beliefs based onbelief functions The main idea of the pignistic probabilitytransformation is to transform the multielement subsets intosingleton subsets by an averagemethodThough the pignisticprobability transformation is widely used it is not reasonablein Example 5

Example 5 Suppose there is a frame of discernment of 119886 119887and 119888 then the BPA is given as follows

119898(119886) = 02 119898 (119887) = 07

119898 (119887 119888) = 005 119898 (119886 119887 119888) = 005

(8)

Mathematical Problems in Engineering 3

In the pignistic probability transformation for119898(119886 119887 119888) = 005 the result will be 119886 = 119887 = 119888 = 0053Actually it is not reasonable as 119898(119886 119887 119888) = 005 meansthat the sensor cannot judge to which classes the targetbelongs since it represents a meaning of ldquounknownrdquo Inother words only according to 119898(119886 119887 119888) = 005 nothingcan be obtained except ldquounknownrdquo In this situation averageis used in the pignistic probability transformation which isone of the methods to solve the problem Compared withthe average weighted average is more reasonable in manysituations

In the Dempster-Shafer theory the plausibility functionrepresents the optimistic estimate of one proposition andthe belief function represents the pessimistic estimate of oneproposition The distance between the plausibility functionand the belief function refers to the belief level of oneproposition For one proposition the bigger the belief levelthe bigger the probability In this paper the weighted averageis represented by the distance between the belief functionand the plausibility function whose definition is shown asfollows

Definition 6 Let119898 be a BPA on the frame of discernmentΩThe difference function 119889

119898is defined as

119889119898(120596) = Pl (120596) minus Bel (120596) 120596 isin Ω (9)

Definition 7 The weight is defined as

Weight119898(120596) =

119889119898(120596)

sum|119860|

120572isin119860119889119898(120572)

120596 isin 119860 119860 sube 119875 (Ω) (10)

Based on the weighted average idea a factor 119902 whichis proposed in the Tsallis entropy [41] is used to highlightthe weights Thus the definition of multiscale probabilityfunction Mul119875 is shown as follows

Definition 8 Let119898 be a BPA on the frame of discernmentΩIts associated multiscale probability function Mul119875

119898 Ω rarr

[0 1] onΩ is defined as

Mul119875119898(120596)

= sum

119860sube119875(Ω)120596isin119860

((Pl(120596) minus Bel(120596))119902

sum|119860|

120572isin119860(Pl(120572) minus Bel(120572))119902

119898(119860)

1 minus 119898 (120601))

119898 (120601) = 1

(11)

where |119882| is the cardinality of subset119860 119902 is a factor based onthe Tsallis entropy to amend the proportion of the intervalThe transformation between 119898 and Mul119875

119898is called the

multiscale probability transformationActually the part of (11) (Pl(120596) minus Bel(120596))119902sum|119882|

120572isin119882

(Pl(120572) minus Bel(120572))119902 denotes the weight of element 120596 based onnormalization which is replaced by the averaged 1|119882| inthe pignistic probability function

Theorem9 Let119898 be a BPA on the frame of discernmentΩ Itsassociatedmultiscale probabilityMul119875

119898onΩ is degenerated as

the pignistic probability Bet119875119898when 119902 equals 0

Proof When 119902 equals 0 and (Pl(120596) minus Bel(120596))119902 equals 1 themultiscale probability function will be calculated as follows

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

sum|119860|

120572isin1198601

sdot119898Ω(119882)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(12)

Then it can obtain

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

|119860|sdot

119898Ω(119860)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(13)

From (12) and (13) we can see that when the value of 119902equals 0 the proposed multiscale probability function can bedegenerated as the pignistic probability function

Theorem 10 Let119898 be a BPA on the frame of discernmentΩ Ifthe belief function equals the plausibility function its associatedmultiscale probability Mul119875

119898is degenerated as the pignistic

probability Bet119875119898

Proof Given a BPA 119898 on the frame of discernment Ω foreach 120596 isin Ω when the belief function equals the plausibilityfunction namely Bel(120596) = Pl(120596) and the bel is a probabilitydistribution 119875 [35] then Mul119875 is equal to Bet119875

For example let Ω be a frame of discernment and Ω =

119886 119887 119888 if it is satisfied with Bel(119886) = Pl(119886) Bel(119887) = Pl(119887)and Bel(119888) = Pl(119888) the BPA on the frame must be satisfiedwith119898(119886) +119898(119887) +119898(119888) = 1 In this situation the multiscaleprobability will be degenerated as the pignistic probability

Corollary 11 If bel is a probability distribution 119875 thenMul119875is equal to 119875

Theorem 12 Let 119898 be a BPA on the frame of discernmentΩ = 119886 119887 119888 If the distances between the belief function andthe plausibility function are the same themultiscale probabilitytransformation can be degenerated as the pignistic probabilitytransformation

Proof It is the same as the proof of Theorem 9

An illustrative example is given to show the calculation ofthe multiscale probability transformation step by step

Example 13 LetΩ be a frame of discernmentwith 3 elementsWe use 119886 119887 and 119888 to denote element 1 element 2 and element3 in the frame One body of BPA is given as follows

119898(119886) = 02 119898 (119887) = 03 119898 (119888) = 01

119898 (119886 119887) = 01 119898 (119886 119887 119888) = 03

(14)

4 Mathematical Problems in Engineering

Step 1 Based on (5) and (6) the values of the belief functionand the plausibility function of elements 119886 119887 and 119888 can beobtained as follows

Bel (119886) = 02 Pl (119886) = 06

Bel (119887) = 03 Pl (119887) = 07

Bel (119888) = 01 Pl (119888) = 04

(15)

Step 2 Calculate the distance between the belief function andthe plausibility function

119889119898(119886) = Pl (119886) minus Bel (119886) = 06 minus 02 = 04

119889119898(119887) = Pl (119887) minus Bel (119887) = 07 minus 03 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 04 minus 01 = 03

(16)

Step 3 Calculate the weight of each element in Ω Assumethat the value of 119902 equals 1

When 119860 = 119886 119887

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

(17)

When119882 = 119886 119887 119888

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119888) =

(Pl (119888) minus Bel (119888))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=03

(04 + 04 + 03)= 0272

(18)

Step 4 The value of the multiscale probability function canbe obtained based on the above steps

Mul119875119898(119886) = 02 + 01 lowast 05 + 03 lowast 0364 = 03592

Mul119875119898(119887) = 03 + 01 lowast 05 + 03 lowast 0364 = 04592

Mul119875119898(119888) = 01 + 03 lowast 0272 = 01816

4 Case Study

In this section an illustrative example is given to show theeffect of the multiscale probability function when the valueof 119902 changes

Example 14 LetΩ be a frame of discernmentwith 3 elementsnamely Ω = 119886 119887 119888

We are given one body of BPAs as follows

119898(119886) = 03 119898 (119887) = 01 119898 (119886 119887) = 01

119898 (119886 119888) = 02 119898 (119886 119887 119888) = 03

(19)

Based on the pignistic probability transformation theresults of the pignistic probability function are shown asfollows

Bet119875119898(119886) = 055 Bet119875

119898(119887) = 025

Bet119875119898(119888) = 020

(20)

According to the proposed function in this paper theresults of the multiscale probability function can be obtainedthrough the following steps

Firstly the values of belief function and the plausibilityfunction can be obtained as follows

Bel (119886) = 03 Pl (119886) = 09

Bel (119887) = 01 Pl (119887) = 05

Bel (119888) = 0 Pl (119888) = 05

(21)

Then the distances between the belief functions and theplausibility functions can be calculated

119889119898(119886) = Pl (119886) minus Bel (119886) = 09 minus 03 = 06

119889119898(119887) = Pl (119887) minus Bel (119887) = 05 minus 01 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 05 minus 0 = 05

(22)

Based on the definition of the multiscale probabilitytransformation the values of Mul119875

119898can be obtained There

are 20 cases where the values of 119902 are starting from Case 1with 119902 = 0 and ending with Case 11 when 119902 = 10 as shown inTable 1The values of Mul119875

119898for these 20 cases are detailed in

Table 1 and graphically illustrated in Figure 1According to Table 1 and Figure 1 on one hand when the

value of 119902 increased the probability of the element which haslarger weight is increased and the probability of the elementwhich has smaller weight is decreased For example theelement 119886 is starting with probability 05500 and ending withprobability 08250 The element 119887 is starting with probability02500 and ending with probability 01061

On the other hand according to Table 1 the optionranking of the values of Mul119875

119898can be obtained It is starting

with Mul119875119898(119886) ≻ Mul119875

119898(119887) ≻ Mul119875

119898(119888) and ending with

Mul119875119898(119886) ≻ Mul119875

119898(119888) ≻ Mul119875

119898(119887) It is mainly because

Mul119875119898is impact of the values of 119902 This principle makes the

multiscale probability function have the ability to highlightthe proportion of each element in the frame of discernment

Mathematical Problems in Engineering 5

Table 1 The values of multiscale probability function when thevalues of 119902 change

Cases Mul119875(119886) Mul119875(119887) Mul119875(119888)119902 = 0 05500 02500 02000119902 = 1 05891 02200 01909119902 = 2 06275 01931 01794119902 = 3 06638 01703 01659119902 = 4 05970 01518 01512119902 = 5 07267 01374 01360119902 = 6 07526 01266 01208119902 = 7 07751 01187 01062119902 = 8 07944 01130 00926119902 = 9 08109 01089 00801119902 = 10 08250 01061 00689

0 1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

q

Mul

tisca

le p

roba

bilit

y

MulP(a)MulP(b)MulP(c)

Figure 1 The values of multiscale probability function when thevalues of 119902 change

Note that when the value of 119902 equals 0 the values ofpignistic probability Bet119875

119898are the same as the values of

multiscale probability Mul119875119898 which is proposed in this

paper In other words the multiscale probability function is ageneralization of the pignistic probability function

5 Conclusion

In the transferable beliefmodel (TBM) pignistic probabilitiesare used for decision making In this paper a multiscaleprobability transformation of basic probability assignmentbased on the belief function and the plausibility functionwhich is a generalization of the pignistic probability transfor-mation is proposed In the multiscale probability functiona factor 119902 is proposed to make the multiscale probabilityfunction have the ability to highlight the proportion of eachelement in the frame of discernment When the value of 119902

equals 0 the multiscale probability transformation can bedegenerated as the pignistic probability transformation Anillustrative case is provided to demonstrate the effectivenessof the multiscale probability transformation

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous reviewers for their valu-able comments and suggestions that improved this paperThe work is partially supported by the National NaturalScience Foundation of China (Grant no 61174022) Special-ized Research Fund for the Doctoral Program of HigherEducation (Grant no 20131102130002) RampD Program ofChina (2012BAH07B01) NationalHigh Technology Researchand Development Program of China (863 Program) (Grantno 2013AA013801) Open Funding Project of State KeyLaboratory of Virtual Reality Technology and Systems Bei-hang University (Grant no BUAA-VR-14KF-02) and Fun-damental Research Funds for the Central Universities noXDJK2015D009

References

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

[3] R P Srivastava and L Liu ldquoApplications of belief functions inbusiness decisions a reviewrdquo Information Systems Frontiers vol5 no 4 pp 359ndash378 2003

[4] T Denœux ldquoMaximum likelihood estimation from fuzzy datausing the EM algorithmrdquo Fuzzy Sets and Systems vol 183 no 1pp 72ndash91 2011

[5] D Wei X Deng X Zhang Y Deng and S MahadevanldquoIdentifying influential nodes in weighted networks basedon evidence theoryrdquo Physica A Statistical Mechanics and ItsApplications vol 392 no 10 pp 2564ndash2575 2013

[6] F Cuzzolin ldquoA geometric approach to the theory of evidencerdquoIEEE Transactions on Systems Man and Cybernetics Part CApplications and Reviews vol 38 no 4 pp 522ndash534 2008

[7] Y Yang D Han C Han and F Cao ldquoA novel approximationof basic probability assignment based on rank-level fusionrdquoChinese Journal of Aeronautics vol 26 no 4 pp 993ndash999 2013

[8] M-H Masson and T Denœux ldquoECM an evidential version ofthe fuzzy c-means algorithmrdquo Pattern Recognition vol 41 no4 pp 1384ndash1397 2008

[9] Z-G Liu Q Pan and J Dezert ldquoEvidential classifier forimprecise data based on belief functionsrdquo Knowledge-BasedSystems vol 52 pp 246ndash257 2013

[10] L Livi H Tahayori A Sadeghian and A Rizzi ldquoDistinguisha-bility of interval type-2 fuzzy sets data by analyzing upper andlower membership functionsrdquo Applied Soft Computing Journalvol 17 pp 79ndash89 2014

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Dempster-Shafer (D-S) belief function models to probabilitymodels Other works have been done to obtain a morereasonable transformation method [38ndash40]

Among the previous research the pignistic probabilitytransformation is the most widely used but it is not reason-able enough to describe the unknown for the multielementsubsets Hence a generalization of the pignistic probabilitytransformation called multiscale probability transformationof basic probability assignment is proposed in this paperwhich is based on the belief function and the plausibilityfunction The proposed function can be calculated with thedifference between the belief function and the plausibilityfunction we call itmultiscale probability function and denoteit as a function Mul119875 In the multiscale probability functiona factor 119902 based on the Tsallis entropy [41] is used to makethe multiscale probabilities diversified When the value of 119902equals 0 the proposed multiscale probability transformationcan be degenerated as the pignistic probability transforma-tion

The rest of this paper is organized as follows Section 2introduces some basic preliminaries about the Dempster-Shafer theory and the pignistic probability transformationIn Section 3 the multiscale probability transformation is pre-sented Section 4 uses an example to illustrate the effective-ness of themultiscale probability transformation Conclusionis given in Section 5

2 Preliminaries

21 Dempster-Shafer Theory of Evidence Dempster-Shafertheory of evidence [1 2] also called Dempster-Shafer the-ory or evidence theory is used to deal with uncertaininformation As an effective theory of evidential reason-ing Dempster-Shafer theory has an advantage of directlyexpressing various uncertainties This theory needs weakerconditions than Bayesian theory of probability so it isoften regarded as an extension of the Bayesian theory Forcompleteness of the explanation a few basic concepts areintroduced as follows

Definition 1 Let Ω be a mutually exclusive and collectivelyexhaustive set which is indicted by

Ω = 1198641 1198642 119864

119894 119864

119873 (1)

The set Ω is called frame of discernment The power set of Ωis indicated by 2Ω where

2Ω= 0 119864

1 119864

119873 1198641 1198642

1198641 1198642 119864

119894 Ω

(2)

If 119860 isin 2Ω 119860 is called a proposition

Definition 2 For a frame of discernment Ω a mass functionis a mapping119898 from 2

Ω to [0 1] formally defined by

119898 2Ω997888rarr [0 1] (3)

which satisfies the following condition

119898(0) = 0 sum

119860isin2Ω

119898(119860) = 1 (4)

In Dempster-Shafer theory a mass function is also calleda basic probability assignment (BPA) If119898(119860) gt 0119860 is calleda focal element and the union of all focal elements is calledthe core of the mass function

Definition 3 For a proposition 119860 sube Ω the belief functionBel 2Ω rarr [0 1] is defined as

Bel (119860) = sum119861sube119860

119898(119861) (5)

The plausibility function Pl 2Ω rarr [0 1] is defined as

Pl (119860) = 1 minus Bel (119860) = sum

119861cap119860 =0

119898(119861) (6)

where 119860 = Ω minus 119860Obviously Pl(119860) ge Bel(119860) and these functions Bel and

Pl are the lower limit function and upper limit function ofproposition 119860 respectively

22 Pignistic Probability Transformation In the transferablebelief model (TBM) [35] pignistic probabilities are used fordecision making The definition of the pignistic probabilitytransformation is shown as follows

Definition 4 Let 119898 be a BPA on the frame of discernmentΩ Its associated pignistic probability function Bet119875

119898 Ω rarr

[0 1] is defined as

Bet119875119898(120596) = sum

119860sube119875(Ω)120596isin119860

1

|119860|

119898 (119860)

1 minus 119898 (120601) 119898 (120601) = 1 (7)

where |119882| is the cardinality of subset119860Theprocess of pignis-tic probability transformation (PPT) is that basic probabilityassignment transferred to probability distributionThereforethe pignistic betting distance can be easily obtained by PPT

3 Multiscale Probability Transformation ofBasic Probability Assignment

In the transferable belief model (TBM) [35] pignistic proba-bilities are used for decision making The transferable beliefmodel is presented to represent quantified beliefs based onbelief functions The main idea of the pignistic probabilitytransformation is to transform the multielement subsets intosingleton subsets by an averagemethodThough the pignisticprobability transformation is widely used it is not reasonablein Example 5

Example 5 Suppose there is a frame of discernment of 119886 119887and 119888 then the BPA is given as follows

119898(119886) = 02 119898 (119887) = 07

119898 (119887 119888) = 005 119898 (119886 119887 119888) = 005

(8)

Mathematical Problems in Engineering 3

In the pignistic probability transformation for119898(119886 119887 119888) = 005 the result will be 119886 = 119887 = 119888 = 0053Actually it is not reasonable as 119898(119886 119887 119888) = 005 meansthat the sensor cannot judge to which classes the targetbelongs since it represents a meaning of ldquounknownrdquo Inother words only according to 119898(119886 119887 119888) = 005 nothingcan be obtained except ldquounknownrdquo In this situation averageis used in the pignistic probability transformation which isone of the methods to solve the problem Compared withthe average weighted average is more reasonable in manysituations

In the Dempster-Shafer theory the plausibility functionrepresents the optimistic estimate of one proposition andthe belief function represents the pessimistic estimate of oneproposition The distance between the plausibility functionand the belief function refers to the belief level of oneproposition For one proposition the bigger the belief levelthe bigger the probability In this paper the weighted averageis represented by the distance between the belief functionand the plausibility function whose definition is shown asfollows

Definition 6 Let119898 be a BPA on the frame of discernmentΩThe difference function 119889

119898is defined as

119889119898(120596) = Pl (120596) minus Bel (120596) 120596 isin Ω (9)

Definition 7 The weight is defined as

Weight119898(120596) =

119889119898(120596)

sum|119860|

120572isin119860119889119898(120572)

120596 isin 119860 119860 sube 119875 (Ω) (10)

Based on the weighted average idea a factor 119902 whichis proposed in the Tsallis entropy [41] is used to highlightthe weights Thus the definition of multiscale probabilityfunction Mul119875 is shown as follows

Definition 8 Let119898 be a BPA on the frame of discernmentΩIts associated multiscale probability function Mul119875

119898 Ω rarr

[0 1] onΩ is defined as

Mul119875119898(120596)

= sum

119860sube119875(Ω)120596isin119860

((Pl(120596) minus Bel(120596))119902

sum|119860|

120572isin119860(Pl(120572) minus Bel(120572))119902

119898(119860)

1 minus 119898 (120601))

119898 (120601) = 1

(11)

where |119882| is the cardinality of subset119860 119902 is a factor based onthe Tsallis entropy to amend the proportion of the intervalThe transformation between 119898 and Mul119875

119898is called the

multiscale probability transformationActually the part of (11) (Pl(120596) minus Bel(120596))119902sum|119882|

120572isin119882

(Pl(120572) minus Bel(120572))119902 denotes the weight of element 120596 based onnormalization which is replaced by the averaged 1|119882| inthe pignistic probability function

Theorem9 Let119898 be a BPA on the frame of discernmentΩ Itsassociatedmultiscale probabilityMul119875

119898onΩ is degenerated as

the pignistic probability Bet119875119898when 119902 equals 0

Proof When 119902 equals 0 and (Pl(120596) minus Bel(120596))119902 equals 1 themultiscale probability function will be calculated as follows

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

sum|119860|

120572isin1198601

sdot119898Ω(119882)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(12)

Then it can obtain

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

|119860|sdot

119898Ω(119860)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(13)

From (12) and (13) we can see that when the value of 119902equals 0 the proposed multiscale probability function can bedegenerated as the pignistic probability function

Theorem 10 Let119898 be a BPA on the frame of discernmentΩ Ifthe belief function equals the plausibility function its associatedmultiscale probability Mul119875

119898is degenerated as the pignistic

probability Bet119875119898

Proof Given a BPA 119898 on the frame of discernment Ω foreach 120596 isin Ω when the belief function equals the plausibilityfunction namely Bel(120596) = Pl(120596) and the bel is a probabilitydistribution 119875 [35] then Mul119875 is equal to Bet119875

For example let Ω be a frame of discernment and Ω =

119886 119887 119888 if it is satisfied with Bel(119886) = Pl(119886) Bel(119887) = Pl(119887)and Bel(119888) = Pl(119888) the BPA on the frame must be satisfiedwith119898(119886) +119898(119887) +119898(119888) = 1 In this situation the multiscaleprobability will be degenerated as the pignistic probability

Corollary 11 If bel is a probability distribution 119875 thenMul119875is equal to 119875

Theorem 12 Let 119898 be a BPA on the frame of discernmentΩ = 119886 119887 119888 If the distances between the belief function andthe plausibility function are the same themultiscale probabilitytransformation can be degenerated as the pignistic probabilitytransformation

Proof It is the same as the proof of Theorem 9

An illustrative example is given to show the calculation ofthe multiscale probability transformation step by step

Example 13 LetΩ be a frame of discernmentwith 3 elementsWe use 119886 119887 and 119888 to denote element 1 element 2 and element3 in the frame One body of BPA is given as follows

119898(119886) = 02 119898 (119887) = 03 119898 (119888) = 01

119898 (119886 119887) = 01 119898 (119886 119887 119888) = 03

(14)

4 Mathematical Problems in Engineering

Step 1 Based on (5) and (6) the values of the belief functionand the plausibility function of elements 119886 119887 and 119888 can beobtained as follows

Bel (119886) = 02 Pl (119886) = 06

Bel (119887) = 03 Pl (119887) = 07

Bel (119888) = 01 Pl (119888) = 04

(15)

Step 2 Calculate the distance between the belief function andthe plausibility function

119889119898(119886) = Pl (119886) minus Bel (119886) = 06 minus 02 = 04

119889119898(119887) = Pl (119887) minus Bel (119887) = 07 minus 03 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 04 minus 01 = 03

(16)

Step 3 Calculate the weight of each element in Ω Assumethat the value of 119902 equals 1

When 119860 = 119886 119887

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

(17)

When119882 = 119886 119887 119888

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119888) =

(Pl (119888) minus Bel (119888))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=03

(04 + 04 + 03)= 0272

(18)

Step 4 The value of the multiscale probability function canbe obtained based on the above steps

Mul119875119898(119886) = 02 + 01 lowast 05 + 03 lowast 0364 = 03592

Mul119875119898(119887) = 03 + 01 lowast 05 + 03 lowast 0364 = 04592

Mul119875119898(119888) = 01 + 03 lowast 0272 = 01816

4 Case Study

In this section an illustrative example is given to show theeffect of the multiscale probability function when the valueof 119902 changes

Example 14 LetΩ be a frame of discernmentwith 3 elementsnamely Ω = 119886 119887 119888

We are given one body of BPAs as follows

119898(119886) = 03 119898 (119887) = 01 119898 (119886 119887) = 01

119898 (119886 119888) = 02 119898 (119886 119887 119888) = 03

(19)

Based on the pignistic probability transformation theresults of the pignistic probability function are shown asfollows

Bet119875119898(119886) = 055 Bet119875

119898(119887) = 025

Bet119875119898(119888) = 020

(20)

According to the proposed function in this paper theresults of the multiscale probability function can be obtainedthrough the following steps

Firstly the values of belief function and the plausibilityfunction can be obtained as follows

Bel (119886) = 03 Pl (119886) = 09

Bel (119887) = 01 Pl (119887) = 05

Bel (119888) = 0 Pl (119888) = 05

(21)

Then the distances between the belief functions and theplausibility functions can be calculated

119889119898(119886) = Pl (119886) minus Bel (119886) = 09 minus 03 = 06

119889119898(119887) = Pl (119887) minus Bel (119887) = 05 minus 01 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 05 minus 0 = 05

(22)

Based on the definition of the multiscale probabilitytransformation the values of Mul119875

119898can be obtained There

are 20 cases where the values of 119902 are starting from Case 1with 119902 = 0 and ending with Case 11 when 119902 = 10 as shown inTable 1The values of Mul119875

119898for these 20 cases are detailed in

Table 1 and graphically illustrated in Figure 1According to Table 1 and Figure 1 on one hand when the

value of 119902 increased the probability of the element which haslarger weight is increased and the probability of the elementwhich has smaller weight is decreased For example theelement 119886 is starting with probability 05500 and ending withprobability 08250 The element 119887 is starting with probability02500 and ending with probability 01061

On the other hand according to Table 1 the optionranking of the values of Mul119875

119898can be obtained It is starting

with Mul119875119898(119886) ≻ Mul119875

119898(119887) ≻ Mul119875

119898(119888) and ending with

Mul119875119898(119886) ≻ Mul119875

119898(119888) ≻ Mul119875

119898(119887) It is mainly because

Mul119875119898is impact of the values of 119902 This principle makes the

multiscale probability function have the ability to highlightthe proportion of each element in the frame of discernment

Mathematical Problems in Engineering 5

Table 1 The values of multiscale probability function when thevalues of 119902 change

Cases Mul119875(119886) Mul119875(119887) Mul119875(119888)119902 = 0 05500 02500 02000119902 = 1 05891 02200 01909119902 = 2 06275 01931 01794119902 = 3 06638 01703 01659119902 = 4 05970 01518 01512119902 = 5 07267 01374 01360119902 = 6 07526 01266 01208119902 = 7 07751 01187 01062119902 = 8 07944 01130 00926119902 = 9 08109 01089 00801119902 = 10 08250 01061 00689

0 1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

q

Mul

tisca

le p

roba

bilit

y

MulP(a)MulP(b)MulP(c)

Figure 1 The values of multiscale probability function when thevalues of 119902 change

Note that when the value of 119902 equals 0 the values ofpignistic probability Bet119875

119898are the same as the values of

multiscale probability Mul119875119898 which is proposed in this

paper In other words the multiscale probability function is ageneralization of the pignistic probability function

5 Conclusion

In the transferable beliefmodel (TBM) pignistic probabilitiesare used for decision making In this paper a multiscaleprobability transformation of basic probability assignmentbased on the belief function and the plausibility functionwhich is a generalization of the pignistic probability transfor-mation is proposed In the multiscale probability functiona factor 119902 is proposed to make the multiscale probabilityfunction have the ability to highlight the proportion of eachelement in the frame of discernment When the value of 119902

equals 0 the multiscale probability transformation can bedegenerated as the pignistic probability transformation Anillustrative case is provided to demonstrate the effectivenessof the multiscale probability transformation

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous reviewers for their valu-able comments and suggestions that improved this paperThe work is partially supported by the National NaturalScience Foundation of China (Grant no 61174022) Special-ized Research Fund for the Doctoral Program of HigherEducation (Grant no 20131102130002) RampD Program ofChina (2012BAH07B01) NationalHigh Technology Researchand Development Program of China (863 Program) (Grantno 2013AA013801) Open Funding Project of State KeyLaboratory of Virtual Reality Technology and Systems Bei-hang University (Grant no BUAA-VR-14KF-02) and Fun-damental Research Funds for the Central Universities noXDJK2015D009

References

[1] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 no 2 pp 325ndash339 1967

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

[3] R P Srivastava and L Liu ldquoApplications of belief functions inbusiness decisions a reviewrdquo Information Systems Frontiers vol5 no 4 pp 359ndash378 2003

[4] T Denœux ldquoMaximum likelihood estimation from fuzzy datausing the EM algorithmrdquo Fuzzy Sets and Systems vol 183 no 1pp 72ndash91 2011

[5] D Wei X Deng X Zhang Y Deng and S MahadevanldquoIdentifying influential nodes in weighted networks basedon evidence theoryrdquo Physica A Statistical Mechanics and ItsApplications vol 392 no 10 pp 2564ndash2575 2013

[6] F Cuzzolin ldquoA geometric approach to the theory of evidencerdquoIEEE Transactions on Systems Man and Cybernetics Part CApplications and Reviews vol 38 no 4 pp 522ndash534 2008

[7] Y Yang D Han C Han and F Cao ldquoA novel approximationof basic probability assignment based on rank-level fusionrdquoChinese Journal of Aeronautics vol 26 no 4 pp 993ndash999 2013

[8] M-H Masson and T Denœux ldquoECM an evidential version ofthe fuzzy c-means algorithmrdquo Pattern Recognition vol 41 no4 pp 1384ndash1397 2008

[9] Z-G Liu Q Pan and J Dezert ldquoEvidential classifier forimprecise data based on belief functionsrdquo Knowledge-BasedSystems vol 52 pp 246ndash257 2013

[10] L Livi H Tahayori A Sadeghian and A Rizzi ldquoDistinguisha-bility of interval type-2 fuzzy sets data by analyzing upper andlower membership functionsrdquo Applied Soft Computing Journalvol 17 pp 79ndash89 2014

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

In the pignistic probability transformation for119898(119886 119887 119888) = 005 the result will be 119886 = 119887 = 119888 = 0053Actually it is not reasonable as 119898(119886 119887 119888) = 005 meansthat the sensor cannot judge to which classes the targetbelongs since it represents a meaning of ldquounknownrdquo Inother words only according to 119898(119886 119887 119888) = 005 nothingcan be obtained except ldquounknownrdquo In this situation averageis used in the pignistic probability transformation which isone of the methods to solve the problem Compared withthe average weighted average is more reasonable in manysituations

In the Dempster-Shafer theory the plausibility functionrepresents the optimistic estimate of one proposition andthe belief function represents the pessimistic estimate of oneproposition The distance between the plausibility functionand the belief function refers to the belief level of oneproposition For one proposition the bigger the belief levelthe bigger the probability In this paper the weighted averageis represented by the distance between the belief functionand the plausibility function whose definition is shown asfollows

Definition 6 Let119898 be a BPA on the frame of discernmentΩThe difference function 119889

119898is defined as

119889119898(120596) = Pl (120596) minus Bel (120596) 120596 isin Ω (9)

Definition 7 The weight is defined as

Weight119898(120596) =

119889119898(120596)

sum|119860|

120572isin119860119889119898(120572)

120596 isin 119860 119860 sube 119875 (Ω) (10)

Based on the weighted average idea a factor 119902 whichis proposed in the Tsallis entropy [41] is used to highlightthe weights Thus the definition of multiscale probabilityfunction Mul119875 is shown as follows

Definition 8 Let119898 be a BPA on the frame of discernmentΩIts associated multiscale probability function Mul119875

119898 Ω rarr

[0 1] onΩ is defined as

Mul119875119898(120596)

= sum

119860sube119875(Ω)120596isin119860

((Pl(120596) minus Bel(120596))119902

sum|119860|

120572isin119860(Pl(120572) minus Bel(120572))119902

119898(119860)

1 minus 119898 (120601))

119898 (120601) = 1

(11)

where |119882| is the cardinality of subset119860 119902 is a factor based onthe Tsallis entropy to amend the proportion of the intervalThe transformation between 119898 and Mul119875

119898is called the

multiscale probability transformationActually the part of (11) (Pl(120596) minus Bel(120596))119902sum|119882|

120572isin119882

(Pl(120572) minus Bel(120572))119902 denotes the weight of element 120596 based onnormalization which is replaced by the averaged 1|119882| inthe pignistic probability function

Theorem9 Let119898 be a BPA on the frame of discernmentΩ Itsassociatedmultiscale probabilityMul119875

119898onΩ is degenerated as

the pignistic probability Bet119875119898when 119902 equals 0

Proof When 119902 equals 0 and (Pl(120596) minus Bel(120596))119902 equals 1 themultiscale probability function will be calculated as follows

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

sum|119860|

120572isin1198601

sdot119898Ω(119882)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(12)

Then it can obtain

Mul119875119898(120596)

= sum

119860subeΩ120596isin119860

(1

|119860|sdot

119898Ω(119860)

(1 minus 119898Ω (120601))) forall120596 isin Ω

(13)

From (12) and (13) we can see that when the value of 119902equals 0 the proposed multiscale probability function can bedegenerated as the pignistic probability function

Theorem 10 Let119898 be a BPA on the frame of discernmentΩ Ifthe belief function equals the plausibility function its associatedmultiscale probability Mul119875

119898is degenerated as the pignistic

probability Bet119875119898

Proof Given a BPA 119898 on the frame of discernment Ω foreach 120596 isin Ω when the belief function equals the plausibilityfunction namely Bel(120596) = Pl(120596) and the bel is a probabilitydistribution 119875 [35] then Mul119875 is equal to Bet119875

For example let Ω be a frame of discernment and Ω =

119886 119887 119888 if it is satisfied with Bel(119886) = Pl(119886) Bel(119887) = Pl(119887)and Bel(119888) = Pl(119888) the BPA on the frame must be satisfiedwith119898(119886) +119898(119887) +119898(119888) = 1 In this situation the multiscaleprobability will be degenerated as the pignistic probability

Corollary 11 If bel is a probability distribution 119875 thenMul119875is equal to 119875

Theorem 12 Let 119898 be a BPA on the frame of discernmentΩ = 119886 119887 119888 If the distances between the belief function andthe plausibility function are the same themultiscale probabilitytransformation can be degenerated as the pignistic probabilitytransformation

Proof It is the same as the proof of Theorem 9

An illustrative example is given to show the calculation ofthe multiscale probability transformation step by step

Example 13 LetΩ be a frame of discernmentwith 3 elementsWe use 119886 119887 and 119888 to denote element 1 element 2 and element3 in the frame One body of BPA is given as follows

119898(119886) = 02 119898 (119887) = 03 119898 (119888) = 01

119898 (119886 119887) = 01 119898 (119886 119887 119888) = 03

(14)

4 Mathematical Problems in Engineering

Step 1 Based on (5) and (6) the values of the belief functionand the plausibility function of elements 119886 119887 and 119888 can beobtained as follows

Bel (119886) = 02 Pl (119886) = 06

Bel (119887) = 03 Pl (119887) = 07

Bel (119888) = 01 Pl (119888) = 04

(15)

Step 2 Calculate the distance between the belief function andthe plausibility function

119889119898(119886) = Pl (119886) minus Bel (119886) = 06 minus 02 = 04

119889119898(119887) = Pl (119887) minus Bel (119887) = 07 minus 03 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 04 minus 01 = 03

(16)

Step 3 Calculate the weight of each element in Ω Assumethat the value of 119902 equals 1

When 119860 = 119886 119887

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

(17)

When119882 = 119886 119887 119888

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119888) =

(Pl (119888) minus Bel (119888))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=03

(04 + 04 + 03)= 0272

(18)

Step 4 The value of the multiscale probability function canbe obtained based on the above steps

Mul119875119898(119886) = 02 + 01 lowast 05 + 03 lowast 0364 = 03592

Mul119875119898(119887) = 03 + 01 lowast 05 + 03 lowast 0364 = 04592

Mul119875119898(119888) = 01 + 03 lowast 0272 = 01816

4 Case Study

In this section an illustrative example is given to show theeffect of the multiscale probability function when the valueof 119902 changes

Example 14 LetΩ be a frame of discernmentwith 3 elementsnamely Ω = 119886 119887 119888

We are given one body of BPAs as follows

119898(119886) = 03 119898 (119887) = 01 119898 (119886 119887) = 01

119898 (119886 119888) = 02 119898 (119886 119887 119888) = 03

(19)

Based on the pignistic probability transformation theresults of the pignistic probability function are shown asfollows

Bet119875119898(119886) = 055 Bet119875

119898(119887) = 025

Bet119875119898(119888) = 020

(20)

According to the proposed function in this paper theresults of the multiscale probability function can be obtainedthrough the following steps

Firstly the values of belief function and the plausibilityfunction can be obtained as follows

Bel (119886) = 03 Pl (119886) = 09

Bel (119887) = 01 Pl (119887) = 05

Bel (119888) = 0 Pl (119888) = 05

(21)

Then the distances between the belief functions and theplausibility functions can be calculated

119889119898(119886) = Pl (119886) minus Bel (119886) = 09 minus 03 = 06

119889119898(119887) = Pl (119887) minus Bel (119887) = 05 minus 01 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 05 minus 0 = 05

(22)

Based on the definition of the multiscale probabilitytransformation the values of Mul119875

119898can be obtained There

are 20 cases where the values of 119902 are starting from Case 1with 119902 = 0 and ending with Case 11 when 119902 = 10 as shown inTable 1The values of Mul119875

119898for these 20 cases are detailed in

Table 1 and graphically illustrated in Figure 1According to Table 1 and Figure 1 on one hand when the

value of 119902 increased the probability of the element which haslarger weight is increased and the probability of the elementwhich has smaller weight is decreased For example theelement 119886 is starting with probability 05500 and ending withprobability 08250 The element 119887 is starting with probability02500 and ending with probability 01061

On the other hand according to Table 1 the optionranking of the values of Mul119875

119898can be obtained It is starting

with Mul119875119898(119886) ≻ Mul119875

119898(119887) ≻ Mul119875

119898(119888) and ending with

Mul119875119898(119886) ≻ Mul119875

119898(119888) ≻ Mul119875

119898(119887) It is mainly because

Mul119875119898is impact of the values of 119902 This principle makes the

multiscale probability function have the ability to highlightthe proportion of each element in the frame of discernment

Mathematical Problems in Engineering 5

Table 1 The values of multiscale probability function when thevalues of 119902 change

Cases Mul119875(119886) Mul119875(119887) Mul119875(119888)119902 = 0 05500 02500 02000119902 = 1 05891 02200 01909119902 = 2 06275 01931 01794119902 = 3 06638 01703 01659119902 = 4 05970 01518 01512119902 = 5 07267 01374 01360119902 = 6 07526 01266 01208119902 = 7 07751 01187 01062119902 = 8 07944 01130 00926119902 = 9 08109 01089 00801119902 = 10 08250 01061 00689

0 1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

q

Mul

tisca

le p

roba

bilit

y

MulP(a)MulP(b)MulP(c)

Figure 1 The values of multiscale probability function when thevalues of 119902 change

Note that when the value of 119902 equals 0 the values ofpignistic probability Bet119875

119898are the same as the values of

multiscale probability Mul119875119898 which is proposed in this

paper In other words the multiscale probability function is ageneralization of the pignistic probability function

5 Conclusion

In the transferable beliefmodel (TBM) pignistic probabilitiesare used for decision making In this paper a multiscaleprobability transformation of basic probability assignmentbased on the belief function and the plausibility functionwhich is a generalization of the pignistic probability transfor-mation is proposed In the multiscale probability functiona factor 119902 is proposed to make the multiscale probabilityfunction have the ability to highlight the proportion of eachelement in the frame of discernment When the value of 119902

equals 0 the multiscale probability transformation can bedegenerated as the pignistic probability transformation Anillustrative case is provided to demonstrate the effectivenessof the multiscale probability transformation

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous reviewers for their valu-able comments and suggestions that improved this paperThe work is partially supported by the National NaturalScience Foundation of China (Grant no 61174022) Special-ized Research Fund for the Doctoral Program of HigherEducation (Grant no 20131102130002) RampD Program ofChina (2012BAH07B01) NationalHigh Technology Researchand Development Program of China (863 Program) (Grantno 2013AA013801) Open Funding Project of State KeyLaboratory of Virtual Reality Technology and Systems Bei-hang University (Grant no BUAA-VR-14KF-02) and Fun-damental Research Funds for the Central Universities noXDJK2015D009

References

[1] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 no 2 pp 325ndash339 1967

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

[3] R P Srivastava and L Liu ldquoApplications of belief functions inbusiness decisions a reviewrdquo Information Systems Frontiers vol5 no 4 pp 359ndash378 2003

[4] T Denœux ldquoMaximum likelihood estimation from fuzzy datausing the EM algorithmrdquo Fuzzy Sets and Systems vol 183 no 1pp 72ndash91 2011

[5] D Wei X Deng X Zhang Y Deng and S MahadevanldquoIdentifying influential nodes in weighted networks basedon evidence theoryrdquo Physica A Statistical Mechanics and ItsApplications vol 392 no 10 pp 2564ndash2575 2013

[6] F Cuzzolin ldquoA geometric approach to the theory of evidencerdquoIEEE Transactions on Systems Man and Cybernetics Part CApplications and Reviews vol 38 no 4 pp 522ndash534 2008

[7] Y Yang D Han C Han and F Cao ldquoA novel approximationof basic probability assignment based on rank-level fusionrdquoChinese Journal of Aeronautics vol 26 no 4 pp 993ndash999 2013

[8] M-H Masson and T Denœux ldquoECM an evidential version ofthe fuzzy c-means algorithmrdquo Pattern Recognition vol 41 no4 pp 1384ndash1397 2008

[9] Z-G Liu Q Pan and J Dezert ldquoEvidential classifier forimprecise data based on belief functionsrdquo Knowledge-BasedSystems vol 52 pp 246ndash257 2013

[10] L Livi H Tahayori A Sadeghian and A Rizzi ldquoDistinguisha-bility of interval type-2 fuzzy sets data by analyzing upper andlower membership functionsrdquo Applied Soft Computing Journalvol 17 pp 79ndash89 2014

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Step 1 Based on (5) and (6) the values of the belief functionand the plausibility function of elements 119886 119887 and 119888 can beobtained as follows

Bel (119886) = 02 Pl (119886) = 06

Bel (119887) = 03 Pl (119887) = 07

Bel (119888) = 01 Pl (119888) = 04

(15)

Step 2 Calculate the distance between the belief function andthe plausibility function

119889119898(119886) = Pl (119886) minus Bel (119886) = 06 minus 02 = 04

119889119898(119887) = Pl (119887) minus Bel (119887) = 07 minus 03 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 04 minus 01 = 03

(16)

Step 3 Calculate the weight of each element in Ω Assumethat the value of 119902 equals 1

When 119860 = 119886 119887

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04)= 05

(17)

When119882 = 119886 119887 119888

Weight119898(119886) =

(Pl (119886) minus Bel (119886))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119887) =

(Pl (119887) minus Bel (119887))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=04

(04 + 04 + 03)= 0364

Weight119898(119888) =

(Pl (119888) minus Bel (119888))sum|119882|

120572isin119882(Pl (120572) minus Bel (120572))

=03

(04 + 04 + 03)= 0272

(18)

Step 4 The value of the multiscale probability function canbe obtained based on the above steps

Mul119875119898(119886) = 02 + 01 lowast 05 + 03 lowast 0364 = 03592

Mul119875119898(119887) = 03 + 01 lowast 05 + 03 lowast 0364 = 04592

Mul119875119898(119888) = 01 + 03 lowast 0272 = 01816

4 Case Study

In this section an illustrative example is given to show theeffect of the multiscale probability function when the valueof 119902 changes

Example 14 LetΩ be a frame of discernmentwith 3 elementsnamely Ω = 119886 119887 119888

We are given one body of BPAs as follows

119898(119886) = 03 119898 (119887) = 01 119898 (119886 119887) = 01

119898 (119886 119888) = 02 119898 (119886 119887 119888) = 03

(19)

Based on the pignistic probability transformation theresults of the pignistic probability function are shown asfollows

Bet119875119898(119886) = 055 Bet119875

119898(119887) = 025

Bet119875119898(119888) = 020

(20)

According to the proposed function in this paper theresults of the multiscale probability function can be obtainedthrough the following steps

Firstly the values of belief function and the plausibilityfunction can be obtained as follows

Bel (119886) = 03 Pl (119886) = 09

Bel (119887) = 01 Pl (119887) = 05

Bel (119888) = 0 Pl (119888) = 05

(21)

Then the distances between the belief functions and theplausibility functions can be calculated

119889119898(119886) = Pl (119886) minus Bel (119886) = 09 minus 03 = 06

119889119898(119887) = Pl (119887) minus Bel (119887) = 05 minus 01 = 04

119889119898(119888) = Pl (119888) minus Bel (119888) = 05 minus 0 = 05

(22)

Based on the definition of the multiscale probabilitytransformation the values of Mul119875

119898can be obtained There

are 20 cases where the values of 119902 are starting from Case 1with 119902 = 0 and ending with Case 11 when 119902 = 10 as shown inTable 1The values of Mul119875

119898for these 20 cases are detailed in

Table 1 and graphically illustrated in Figure 1According to Table 1 and Figure 1 on one hand when the

value of 119902 increased the probability of the element which haslarger weight is increased and the probability of the elementwhich has smaller weight is decreased For example theelement 119886 is starting with probability 05500 and ending withprobability 08250 The element 119887 is starting with probability02500 and ending with probability 01061

On the other hand according to Table 1 the optionranking of the values of Mul119875

119898can be obtained It is starting

with Mul119875119898(119886) ≻ Mul119875

119898(119887) ≻ Mul119875

119898(119888) and ending with

Mul119875119898(119886) ≻ Mul119875

119898(119888) ≻ Mul119875

119898(119887) It is mainly because

Mul119875119898is impact of the values of 119902 This principle makes the

multiscale probability function have the ability to highlightthe proportion of each element in the frame of discernment

Mathematical Problems in Engineering 5

Table 1 The values of multiscale probability function when thevalues of 119902 change

Cases Mul119875(119886) Mul119875(119887) Mul119875(119888)119902 = 0 05500 02500 02000119902 = 1 05891 02200 01909119902 = 2 06275 01931 01794119902 = 3 06638 01703 01659119902 = 4 05970 01518 01512119902 = 5 07267 01374 01360119902 = 6 07526 01266 01208119902 = 7 07751 01187 01062119902 = 8 07944 01130 00926119902 = 9 08109 01089 00801119902 = 10 08250 01061 00689

0 1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

q

Mul

tisca

le p

roba

bilit

y

MulP(a)MulP(b)MulP(c)

Figure 1 The values of multiscale probability function when thevalues of 119902 change

Note that when the value of 119902 equals 0 the values ofpignistic probability Bet119875

119898are the same as the values of

multiscale probability Mul119875119898 which is proposed in this

paper In other words the multiscale probability function is ageneralization of the pignistic probability function

5 Conclusion

In the transferable beliefmodel (TBM) pignistic probabilitiesare used for decision making In this paper a multiscaleprobability transformation of basic probability assignmentbased on the belief function and the plausibility functionwhich is a generalization of the pignistic probability transfor-mation is proposed In the multiscale probability functiona factor 119902 is proposed to make the multiscale probabilityfunction have the ability to highlight the proportion of eachelement in the frame of discernment When the value of 119902

equals 0 the multiscale probability transformation can bedegenerated as the pignistic probability transformation Anillustrative case is provided to demonstrate the effectivenessof the multiscale probability transformation

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous reviewers for their valu-able comments and suggestions that improved this paperThe work is partially supported by the National NaturalScience Foundation of China (Grant no 61174022) Special-ized Research Fund for the Doctoral Program of HigherEducation (Grant no 20131102130002) RampD Program ofChina (2012BAH07B01) NationalHigh Technology Researchand Development Program of China (863 Program) (Grantno 2013AA013801) Open Funding Project of State KeyLaboratory of Virtual Reality Technology and Systems Bei-hang University (Grant no BUAA-VR-14KF-02) and Fun-damental Research Funds for the Central Universities noXDJK2015D009

References

[1] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 no 2 pp 325ndash339 1967

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

[3] R P Srivastava and L Liu ldquoApplications of belief functions inbusiness decisions a reviewrdquo Information Systems Frontiers vol5 no 4 pp 359ndash378 2003

[4] T Denœux ldquoMaximum likelihood estimation from fuzzy datausing the EM algorithmrdquo Fuzzy Sets and Systems vol 183 no 1pp 72ndash91 2011

[5] D Wei X Deng X Zhang Y Deng and S MahadevanldquoIdentifying influential nodes in weighted networks basedon evidence theoryrdquo Physica A Statistical Mechanics and ItsApplications vol 392 no 10 pp 2564ndash2575 2013

[6] F Cuzzolin ldquoA geometric approach to the theory of evidencerdquoIEEE Transactions on Systems Man and Cybernetics Part CApplications and Reviews vol 38 no 4 pp 522ndash534 2008

[7] Y Yang D Han C Han and F Cao ldquoA novel approximationof basic probability assignment based on rank-level fusionrdquoChinese Journal of Aeronautics vol 26 no 4 pp 993ndash999 2013

[8] M-H Masson and T Denœux ldquoECM an evidential version ofthe fuzzy c-means algorithmrdquo Pattern Recognition vol 41 no4 pp 1384ndash1397 2008

[9] Z-G Liu Q Pan and J Dezert ldquoEvidential classifier forimprecise data based on belief functionsrdquo Knowledge-BasedSystems vol 52 pp 246ndash257 2013

[10] L Livi H Tahayori A Sadeghian and A Rizzi ldquoDistinguisha-bility of interval type-2 fuzzy sets data by analyzing upper andlower membership functionsrdquo Applied Soft Computing Journalvol 17 pp 79ndash89 2014

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 The values of multiscale probability function when thevalues of 119902 change

Cases Mul119875(119886) Mul119875(119887) Mul119875(119888)119902 = 0 05500 02500 02000119902 = 1 05891 02200 01909119902 = 2 06275 01931 01794119902 = 3 06638 01703 01659119902 = 4 05970 01518 01512119902 = 5 07267 01374 01360119902 = 6 07526 01266 01208119902 = 7 07751 01187 01062119902 = 8 07944 01130 00926119902 = 9 08109 01089 00801119902 = 10 08250 01061 00689

0 1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

q

Mul

tisca

le p

roba

bilit

y

MulP(a)MulP(b)MulP(c)

Figure 1 The values of multiscale probability function when thevalues of 119902 change

Note that when the value of 119902 equals 0 the values ofpignistic probability Bet119875

119898are the same as the values of

multiscale probability Mul119875119898 which is proposed in this

paper In other words the multiscale probability function is ageneralization of the pignistic probability function

5 Conclusion

In the transferable beliefmodel (TBM) pignistic probabilitiesare used for decision making In this paper a multiscaleprobability transformation of basic probability assignmentbased on the belief function and the plausibility functionwhich is a generalization of the pignistic probability transfor-mation is proposed In the multiscale probability functiona factor 119902 is proposed to make the multiscale probabilityfunction have the ability to highlight the proportion of eachelement in the frame of discernment When the value of 119902

equals 0 the multiscale probability transformation can bedegenerated as the pignistic probability transformation Anillustrative case is provided to demonstrate the effectivenessof the multiscale probability transformation

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous reviewers for their valu-able comments and suggestions that improved this paperThe work is partially supported by the National NaturalScience Foundation of China (Grant no 61174022) Special-ized Research Fund for the Doctoral Program of HigherEducation (Grant no 20131102130002) RampD Program ofChina (2012BAH07B01) NationalHigh Technology Researchand Development Program of China (863 Program) (Grantno 2013AA013801) Open Funding Project of State KeyLaboratory of Virtual Reality Technology and Systems Bei-hang University (Grant no BUAA-VR-14KF-02) and Fun-damental Research Funds for the Central Universities noXDJK2015D009

References

[1] A P Dempster ldquoUpper and lower probabilities induced by amultivalued mappingrdquo The Annals of Mathematical Statisticsvol 38 no 2 pp 325ndash339 1967

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

[3] R P Srivastava and L Liu ldquoApplications of belief functions inbusiness decisions a reviewrdquo Information Systems Frontiers vol5 no 4 pp 359ndash378 2003

[4] T Denœux ldquoMaximum likelihood estimation from fuzzy datausing the EM algorithmrdquo Fuzzy Sets and Systems vol 183 no 1pp 72ndash91 2011

[5] D Wei X Deng X Zhang Y Deng and S MahadevanldquoIdentifying influential nodes in weighted networks basedon evidence theoryrdquo Physica A Statistical Mechanics and ItsApplications vol 392 no 10 pp 2564ndash2575 2013

[6] F Cuzzolin ldquoA geometric approach to the theory of evidencerdquoIEEE Transactions on Systems Man and Cybernetics Part CApplications and Reviews vol 38 no 4 pp 522ndash534 2008

[7] Y Yang D Han C Han and F Cao ldquoA novel approximationof basic probability assignment based on rank-level fusionrdquoChinese Journal of Aeronautics vol 26 no 4 pp 993ndash999 2013

[8] M-H Masson and T Denœux ldquoECM an evidential version ofthe fuzzy c-means algorithmrdquo Pattern Recognition vol 41 no4 pp 1384ndash1397 2008

[9] Z-G Liu Q Pan and J Dezert ldquoEvidential classifier forimprecise data based on belief functionsrdquo Knowledge-BasedSystems vol 52 pp 246ndash257 2013

[10] L Livi H Tahayori A Sadeghian and A Rizzi ldquoDistinguisha-bility of interval type-2 fuzzy sets data by analyzing upper andlower membership functionsrdquo Applied Soft Computing Journalvol 17 pp 79ndash89 2014

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

[11] S Agrawal R Panda and L Dora ldquoA study on fuzzy clusteringfor magnetic resonance brain image segmentation using softcomputing approachesrdquo Applied Soft Computing vol 24 pp522ndash533 2014

[12] J Liu Y Li R Sadiq and Y Deng ldquoQuantifying influenceof weather indices on pm based on relation maprdquo StochasticEnvironmental Research and Risk Assessment vol 28 no 6 pp1323ndash1331 2014

[13] S Huang X Su Y Hu S Mahadevan and Y Deng ldquoA newdecision-making method by incomplete preferences based onevidence distancerdquo Knowledge-Based Systems vol 56 pp 264ndash272 2014

[14] C S Lin C T Chen F S Chen and W Z Hung ldquoA novelmultiperson game approach for linguistic multicriteria decisionmaking problemsrdquo Mathematical Problems in Engineering vol2014 Article ID 592326 20 pages 2014

[15] B Kang Y Deng R Sadiq and S Mahadevan ldquoEvidentialcognitive mapsrdquo Knowledge-Based Systems vol 35 pp 77ndash862012

[16] T Denoeux ldquoMaximum likelihood estimation from uncertaindata in the belief function frameworkrdquo IEEE Transactions onKnowledge andData Engineering vol 25 no 1 pp 119ndash130 2013

[17] X Zhang Y Deng F T S Chan P Xu S Mahadevan and YHu ldquoIFSJSP a novel methodology for the Job-Shop SchedulingProblem based on intuitionistic fuzzy setsrdquo International Jour-nal of Production Research vol 51 no 17 pp 5100ndash5119 2013

[18] X Deng Y Hu Y Deng and S Mahadevan ldquoSupplier selectionusing AHP methodology extended by D numbersrdquo ExpertSystems with Applications vol 41 no 1 pp 156ndash167 2014

[19] S Chen Y Deng and J Wu ldquoFuzzy sensor fusion based on evi-dence theory and its applicationrdquo Applied Artificial Intelligencevol 27 no 3 pp 235ndash248 2013

[20] X Deng Y Hu Y Deng and S Mahadevan ldquoEnvironmentalimpact assessment based on D numbersrdquo Expert Systems withApplications vol 41 no 2 pp 635ndash643 2014

[21] C Zhang Y Hu F T Chan R Sadiq and Y Deng ldquoA newmethod to determine basic probability assignment using coresamplesrdquo KnowledgemdashBased Systems vol 69 pp 140ndash149 2014

[22] T Burger and S Destercke ldquoHow to randomly generate massfunctionsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 21 no 5 pp 645ndash673 2013

[23] Z-G Liu Q Pan and J Dezert ldquoA belief classification rule forimprecise datardquo Applied Intelligence vol 40 no 2 pp 214ndash2282014

[24] M E Basiri A R Naghsh-Nilchi and N Ghasem-AghaeeldquoSentiment prediction based on dempster-shafer theory ofevidencerdquo Mathematical Problems in Engineering vol 2014Article ID 361201 13 pages 2014

[25] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems 2014

[26] F Karahan and S Ozkan ldquoOn the persistence of income shocksover the life cycle evidence theory and implicationsrdquo Reviewof Economic Dynamics vol 16 no 3 pp 452ndash476 2013

[27] Z Zhang C Jiang XHanDHu and S Yu ldquoA response surfaceapproach for structural reliability analysis using evidence the-oryrdquo Advances in Engineering Software vol 69 pp 37ndash45 2014

[28] X Zhang Y Deng F T Chan A Adamatzky and S Mahade-van ldquoSupplier selection based on evidence theory and analyticnetwork processrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture 2015

[29] L A Zadeh ldquoA simple view of the dempster-shafer theory ofevidence and its implication for the rule of combinationrdquo TheAI Magazine vol 7 no 2 pp 85ndash90 1986

[30] R R Yager ldquoOn the dempster-shafer framework and newcombination rulesrdquo Information Sciences vol 41 no 2 pp 93ndash137 1987

[31] P Smets ldquoCombination of evidence in the transferable beliefmodelrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 5 pp 447ndash458 1990

[32] C K Murphy ldquoCombining belief functions when evidenceconflictsrdquo Decision Support Systems vol 29 no 1 pp 1ndash9 2000

[33] W Liu ldquoAnalyzing the degree of conflict among belief func-tionsrdquo Artificial Intelligence vol 170 no 11 pp 909ndash924 2006

[34] E Lefevre and Z Elouedi ldquoHow to preserve the conflict as analarm in the combination of belief functionsrdquoDecision SupportSystems vol 56 no 1 pp 326ndash333 2013

[35] P Smets and R Kennes ldquoThe transferable belief modelrdquoArtificial Intelligence vol 66 no 2 pp 191ndash234 1994

[36] P Smets ldquoDecision making in the TBM the necessity of thepignistic transformationrdquo International Journal of ApproximateReasoning vol 38 no 2 pp 133ndash147 2005

[37] B R Cobb and P P Shenoy ldquoOn the plausibility transformationmethod for translating belief function models to probabilitymodelsrdquo International Journal of Approximate Reasoning vol41 no 3 pp 314ndash330 2006

[38] M Daniel ldquoOn transformations of belief functions to probabil-itiesrdquo International Journal of Intelligent Systems vol 21 no 3pp 261ndash282 2006

[39] J M Merigo and M Casanovas ldquoDecision making withDempster-Shafer theory using fuzzy induced aggregation oper-atorsrdquo in Recent Developments in the Ordered Weighted Aver-aging Operators Theory and Practice vol 265 of Studies inFuzziness and Soft Computing pp 209ndash228 Springer 2011

[40] E Nusrat and K Yamada ldquoA descriptive decision-makingmodel under uncertainty combination of dempster-sha fer the-ory and prospect theoryrdquo International Journal of UncertaintyFuzziness and Knowlege-Based Systems vol 21 no 1 pp 79ndash1022013

[41] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of