OntheQualityofOptimalAssignment forDataAssociationJeanDezertKaoutharBenameurAbstract. In this paper, we present a methodbased onbelief functionstoevaluatethequalityoftheoptimal assignmentsolutionofaclassicalassociation problem encountered in multiple target tracking applications.Thepurposeofthisworkisnottoprovideanewalgorithmforsolvingtheassignment problem, butasolutiontoestimatethequalityof theindividual associations(pairings)givenintheoptimal assignmentsolu-tion.Totheknowledgeofauthors,thisproblemhasnotbeenaddressedsofarintheliteratureanditssolutionmayhavepractical aspectsforimproving the performances of multisensor-multitarget tracking systems.Keywords: Dataassociation; PCR6rule;Belieffunction.1 IntroductionEcient algorithms for modern multisensor-multitarget tracking (MS-MTT) sys-tems [1, 2] require to estimate and predict the states (position, velocity, etc) of thetargets evolving in the surveillance area covered by the sensors. The estimationsandthepredictionsarebasedonsensorsmeasurementsanddynamicalmodelsassumptions.Inthemonosensorcontext,MTTrequirestosolvethedataasso-ciation(DA)problemtoassociatetheavailablemeasurementsatagiventimewiththepredictedstates of thetargets toupdatetheir tracks usinglteringtechniques(Kalman lter,Particle lter,etc).In themultisensor MTTcontext,weneedtosolvemoredicult multi-dimensional assignmentproblems underconstraints. Fortunately, ecient algorithms have been developed in operationalresearch and tracking communities for formalizing and solving these optimal as-signmentsproblems.Severalapproachesbasedondierentmodelscanbeusedtoestablishrewardsmatrix, eitherbasedontheprobabilisticframework[1, 3],or on the belief function (BF) framework [47]. In this paper, we do not focus onthe construction of the rewards matrix1, and our purpose is to provide a methodtoevaluatethequality(interpretedasacondencescore)of eachassociation(pairing)providedintheoptimal solutionbasedonitsconsistency(stability)withrespecttoallthesecondbestsolutions.1Weassumethattherewardsmatrixisknownandhasbeenobtainedbyamethodchosenbytheuser,eitherintheprobabilisticorintheBFframework.Originally published as Dezert J., Benameur K., On the quality of optimal assignment for data association, in Proc. of Belief 2014 Conf. Oxford, UK, Sept. 26-29, 2014, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4239The simple DA problem under concern can be formulated as follows. We havem>1targetsTi(i =1, . . . , m), andn>1measurements2zj(j =1, . . . , n)at agiventime k, andam nrewards (gain/payo) matrix =[(i, j)]whoseelements (i, j) 0representthepayo(usuallyhomogeneoustothelikelihood) of the association of target Tiwith measurement zj, denoted (Ti, zj).Thedataassociation problemconsistsinndingtheglobaloptimalassignmentofthetargetstosomemeasurementsbymaximizing3theoverallgaininsuchaway that no more than one target is assigned to a measurement, and reciprocally.Without loss of generality, we can assume (i, j) 0 because if some elements(i, j) of were negative, wecan always add thesamemaximalnegative valuetoallelementsof toworkwithanewpayo matrix = [(i, j)]havingallelements(i, j) 0,andwegetthesameoptimalassignmentsolutionwith andwith. Moreover,wecanalsoassume,withoutlossofgeneralitymn,becauseotherwise we can always swap theroles of targets and measurements inthe mathematical problem denition by working directly with tinstead, wherethe superscript t denotes the transposition of the matrix. The optimal assignmentproblemconsistsof ndingthem nbinaryassociationmatrixA=[a(i, j)]whichmaximizetheglobalrewardsR(, A)givenbyR(, A) m

i=1n

j=1(i, j)a(i, j) (1)Subjectto

nj=1 a(i, j) = 1 (i = 1, . . . , m)

mi=1a(i, j) 1 (j= 1, . . . , n)a(i, j){0, 1} (i = 1, . . . , mandj= 1, . . . , n)(2)Theassociationindicator value a(i, j) =1means that the correspondingtargetTiandmeasurementzjareassociated, anda(i, j)=0meansthattheyarenotassociated(i = 1, . . . , mandj= 1, . . . , n).The solutionof the optimal assignment problemstatedin(1)(2) is wellreportedintheliteratureandseveralecientmethodshavebeendevelopedintheoperational researchandtrackingcommunitiestosolveit. Themostwell-known algorithms are Kuhn-Munkres (a.k.as Hungarian) algorithm [8, 9] and itsextensiontorectangularmatricesproposedbyBourgeoisandLassallein[10],Jonker-Volgenantmethod[11], andAuction[12]. Moresophisticatedmethodsusing Murtys method [13], and some variants [3, 1419], are also able to providenot onlythe best assignment, but alsothe m-best assignments. We will notpresent indetails all theseclassical methods becausetheyhavebeenalreadywell reported in the literature [20, 21], and they are quite easily accessible on the2Inamulti-sensor context targets canbe replacedbytracks providedbyagiventracker associatedwitha type of sensor, andmeasurements canbe replacedbyanother tracks set. In dierent contexts, possible equivalents are assigning personneltojobsorassigningdeliverytruckstolocations.3Insomeproblems, =[(i, j)] representsacost matrixwhoseelementsarethenegativelog-likelihoodof associationhypotheses. Inthiscase, thedataassociationproblemsconsistsinndingthebestassignmentthatminimizestheoverallcost.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4240web. In this paper, we want to provide a condence level (i.e. a quality indicator)intheoptimal dataassociationsolution. Moreprecisely, wearesearchingananswertothequestion: howtomeasurethequalityof thepairingsa(i, j)=1providedintheoptimal assignment solutionA?Thenecessitytoestablishaqualityindicatorismotivatedbythefollowingthreemainreasons:1. Insomepractical trackingenvironmentwiththepresenceof clutter, someassociationdecisions(a(i, j)=1)aredoubtful. Fortheseunreliableassoci-ations, itisbettertowaitfornewinformation(measurements)insteadofapplyingtheharddataassociation decision,andmakingpotentiallyseriousassociation mistakes.2. Insomemultisensorsystems, itcanbealsoimportanttosaveenergycon-sumptionforpreservingahighautonomycapacitiesofthesystem.Forthisgoal, onlythe most trustful specic associations providedinthe optimalassignmenthavetobeselectedandusedinsteadofallofthem.3. Thebestoptimalassignment solutionisnotnecessarilyunique.Insuchsit-uation, the establishment of qualityindicators mayhelpinselectingoneparticularoptimalassignment solutionamongmultiplepossiblechoices.Before presenting our solution in Section 2, one must recall that the best, as wellasthe2nd-best,optimalassignment solutionsareunfortunatelynotnecessarilyunique. Therefore, wemustalsotakeintoaccountthepossiblemultiplicityofassignmentsintheanalysisof theproblem. Themultiplicityindexof thebestoptimal assignment solution is denoted 1 1, and the multiplicity index of the2nd-best optimalassignment solutionisdenoted2 1,and wewilldenotethesetsofcorrespondingassignmentmatricesbyA1={A(k1)1, k1=1 . . . , 1}andby A2= {A(k2)2, k2= 1 . . . , 2}. The next simple example illustrates a case withmultiplicityof2nd-bestassignmentsolutionsforthereward matrix 1.Example: 1= 1and 2= 4(i.e.no multiplicityof A1and multiplicityof A2)1=

1 11453017 8 382710143520

This reward matrix provides a unique best assignment A1 providing R1(1, A1) =86, and2=4second-bestassignmentsolutionsprovidingR2(1, Ak22)=82(k2= 1, 2, 3, 4) givenbyA1=

001 0000 1010 0

Ak2=12=

0 0010 0100 100

, Ak2=22=

00 1010 0000 01

, Ak2=32=

001 0000 1100 0

, Ak2=42=

0 0011 0000 010

Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 42412 QualityoftheAssociationsoftheOptimal AssignmentToestablishthequalityof thespecicassociations(pairings) (i, j) satisfyinga1(i, j) = 1belongingtotheoptimalassignmentmatrixA1,weproposetouseboth A1and 2nd-best assignment solution A2. The basic idea is to compare thevalues a1(i, j) with a2(i, j) obtained in the best and in the 2nd-best assignmentsto identify the change (if any) of the optimal pairing (i, j). Our quality indicatorwill dependonboththestabilityof thepairinganditsrelativeimpactintheglobalreward.Theproposedmethodworksalsowhenthe2nd-bestassignmentsolutionA2isnotunique(asinourexample). Theproposedmethodwill alsohelp to select thebest (most trustful) optimal assignment in case of multiplicityofA1matrices.2.1 ASimplisticMethod(MethodI)Beforepresentingoursophisticatemethodbasedonbelief functions, letsrstpresent a simplistic intuitive method (called Method I). For this, lets assume atrst that A1and A2are unique (no multiplicity occurs). The simplistic methodusesonlytheratioofglobalrewardsR2(, A2)/R1(, A1)tomeasurethelevel of uncertainty in the change (if any) of pairing (i, j) provided in A1 and A2.Moreprecisely, thequality(trustfulness)of pairingsinanoptimal assignmentsolutionA1,denoted4qI(i, j),issimplydenedasfollowsfori =1, . . . , mandj= 2, . . . , n:qI(i, j) 1, ifa1(i, j) + a2(i, j) = 01 ifa1(i, j) + a2(i, j) = 11, ifa1(i, j) + a2(i, j) = 2(3)Byadoptingsuchdenition,onecommitsthefullcondencetothecompo-nents(i, j)ofA1andA2thatperfectlymatch,andalowercondencevalue(alowerquality)of 1 tothosethat donotmatch. Totakeintoaccounttheeventualmultiplicities(when2> 1)ofthe2nd-best assignment solutionsAk22,k2= 1, 2, . . . , 2,weneedtocombinetheQI(A1, Ak22)values.Severalmethodscanbeusedforthis,inparticularwecanuseeither:A weighted averaging approach: The quality indicator component qI(i, j)isthenobtainedbyaveragingthequalitiesobtainedfromeachcomparisonofA1withAk22.Moreprecisely,onewilltake:qI (i, j) 2

k2=1w(Ak22)qk2I(i, j) (4)where qk2I(i, j) is denedas in(3) (witha2(i, j) replacedbyak22(i, j) intheformula), andwherew(Ak22) is aweightingfactor in[0, 1], suchthat

2k2=1 w(Ak22) = 1.SinceallassignmentsAk22havethesameglobalrewardvalue R2, then we suggest to take w(Ak22) = 1/2. A more elaborate method4ThesubscriptIinqI (i, j)notationreferstoMethodI.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4242wouldconsist touse the qualityindicator of Ak22basedonthe 3rd-bestsolution, which can be itself computed from the quality of the 3rd assignmentsolutionbasedonthe4th-bestsolution,andsoonbyasimilarmechanism.Wehoweverdontgivemoredetailsonthisduetospaceconstraints.A belief-based approach (see [22] for basics on belief functions): A secondmethodwouldexpressthequalitybyabelief interval [qminI(i, j), qmaxI(i, j)]in[0, 1] insteadof singlereal numberqI(i, j)in[0, 1]. Moreprecisely, onecancompute the belief andplausibilitybounds of the qualitybytakingqminI(i, j) Bel(a1(i, j)) =mink2 qk2I(i, j) andqmaxI(i, j) Pl(a1(i, j)) =maxk2 qk2I(i, j), withqk2I(i, j)givenby(3)anda2(i, j)replacedbyak22(i, j)intheformula. Henceforeachassociationa1(i, j), onecandeneabasicbelief assignment (BBA) mij(.) onthe frameof discernment {T =trustful, T= nottrustful}, which will characterize the quality of the pairing(i, j)intheoptimalassignmentsolutionA1,asfollows:mij(T) = qminI(i, j)mij(T) = 1 qmaxI(i, j)mij(T T) = qmaxI(i, j) qminI(i, j)(5)Remark: Inpractice, onlythepairings5(i, j)suchthata1(i, j) =1areuse-ful intrackingalgorithms toupdate the tracks. Therefore, wedont needtopayattention(computeandstore)thequalitiesof components(i, j)suchthata1(i, j) = 0.2.2 AMoreSophisticateandEcientMethod(MethodII)The previous method can be easily applied in practice but it does not work verywell because the quality indicator depends only on the factor, which means thatall mismatchesbetweenthebestassignmentA1andthe2nd-bestassignmentsolution A2have theirqualityimpacted inthesamemanner(theyarealltakenas1 ). Asasimpleexample, ifweconsidertherewardsmatrix1giveninourexample, wewill have=R2(1, Ak22)/R1(1, A1)=82/860.95, andwewillgetusingmethodIwiththeweightingaveragingapproach(usingsamew(Ak22) = 1/2 = 0.25fork2= 1, 2, 3, 4) thefollowingqualityindicatormatrix:QI (A1, A2) =122

k2=1QI (A1, Ak22) =1.0000 1.0000 0.52330.52330.5233 1.0000 0.7616 0.28490.76160.28490.7616 0.7616(6)We observethat optimal pairings (2,4) and(3,2) get the samequalityvalue0.2849withthemethodI(basedonaveraging),evenifthesepairingshavedif-ferent impacts inthe global rewardvalue, whichis abnormal. If we use themethodIwiththebelief interval measurebasedon(5), thesituationisworstbecausethethreeoptimal pairings(1,3), (2,4)and(3,2)will getexactlysamebelief interval values[0.0465,1].Totakeintoaccount, andinabetterway,the5givenintheoptimalsolutionfoundforexamplewithMurtysalgorithm.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4243rewardvaluesofeachspecicassociationgiveninthebestassignmentA1andin the2nd-best assignment Ak22, we propose to use thefollowing construction ofqualityindicatorsdependingonthetypeofmatching(calledMethodII):Whena1(i, j)=ak22(i, j)=0, onehasfull agreementonnon-association(Ti, zj)inA1and inAk22andthisnon-association (Ti, zj)hasnoimpact onthe global rewards values R1(, A1) and R2(, Ak22),and it will beuseless.Therefore,wecansetitsqualityarbitrarilytoqk2II(i, j) = 1.Whena1(i, j)=ak22(i, j)=1, onehasafull agreementontheassociation(Ti, zj) in A1 and in Ak22and this association (Ti, zj) has dierent impacts inthe global rewards values R1(, A1) and R2(, Ak22). To qualify the qualityof this matching association (Ti, zj), we dene the two BBAs on X(Ti, zj)andX X(theignorance),fors = 1, 2:

ms(X) = as(i, j) (i, j)/Rs(, As)ms(X X) = 1 ms(X)(7)Applyingtheconjunctiveruleoffusion,weget

m(X) = m1(X)m2(X) + m1(X)m2(X X) +m1(X X)m2(X)m(X X) = m1(X X)m2(X X)(8)Applying the pignistic transformation6[24], we get nally BetP(X) = m(X)+12 m(X X)andBetP(X) =12 m(X X).Therefore,wechoosethequalityindicatorasqk2II(i, j) = BetP(X).Whena1(i, j)=1andak22(i, j)=0, onehasadisagreement(conict)ontheassociation(Ti, zj)inA1andin(Ti, zj2)inAk22, wherej2isthemea-surementindexsuchthata2(i, j2)=1. Toqualifythequalityof thisnon-matching association (Ti, zj), we dene the two following basic belief assign-ments(BBAs)ofthepropositionsX(Ti, zj)andY (Ti, zj2)

m1(X) = a1(i, j) (i,j)R1(,A1)m1(X Y ) = 1 m1(X)andm2(Y ) = a2(i, j2) (i,j2)R2(,Ak22)m2(X Y ) = 1 m2(Y )(9)Applyingtheconjunctiverule,wegetm(X Y= ) = m1(X)m2(Y )andm(X) = m1(X)m2(X Y )m(Y ) = m1(X Y )m2(Y )m(X Y ) = m1(X Y )m2(X Y )(10)BecauseweneedtoworkwithanormalizedcombinedBBA,wecanchoosedierent rules of combination(Dempster-Shafers, Dubois-Prades,Yagers6WehavechosenhereBetPfor itssimplicityandbecauseit iswidelyknown, butDSmPcouldbeusedinsteadforexpectingbetterperformances[23].Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4244rule [23], etc).In thiswork, we recommend theProportional Conict Redis-tributionruleno. 6(PCR6), proposedoriginallyinDSmTframework[23],becauseithasbeen proved very ecientin practice.So,weget withPCR6:m(X) = m1(X)m2(X Y ) + m1(X) m1(X)m2(Y )m1(X)+m2(Y )m(Y ) = m1(X Y )m2(Y ) +m2(X) m1(X)m2(Y )m1(X)+m2(Y )m(X Y ) = m1(X Y )m2(X Y )(11)Applying the pignistic transformation, we get nally BetP(X) = m(X) +12m(X Y )andBetP(Y ) = m(Y ) +12 m(X Y ).Therefore,wechoosethequality indicators as follows: qk2II(i, j) = BetP(X), and qk2II(i, j2) = BetP(Y ).TheabsolutequalityfactorQabs(A1, Ak22)of theoptimal assignmentgiveninA1conditionedbyAk22,foranyk2 {1, 2, . . . , 2}isdenedasQabs(A1, Ak22) m

i=1n

j=1a1(i, j)qk2II(i, j) (12)Example(continued): IfweapplytheMethodII(usingPCR6fusionrule)tothe rewards matrix 1, thenwewill get the followingqualitymatrix(usingweightedaveraging approach)QII(A1, A2) =122

k2=1QII(A1, Ak22) =

1.0000 1.0000 0.74400.70220.7200 1.0000 0.8972 0.57530.86950.49570.9119 0.8861

withtheabsolutequalityfactorsQabs(A1, Ak2=12)1.66, Qabs(A1, Ak2=22)1.91,Qabs(A1, Ak2=32) 2.19,Qabs(A1, Ak2=42) 1.51.Naturally,wegetQabs(A1, Ak2=32) > Qabs(A1, Ak2=22) > Qabs(A1, Ak2=12) > Qabs(A1, Ak2=42)becauseA1hasmorematchingpairingswithAk2=32thanwithother2nd-bestassignmentAk22(k2= 3),andthosepairingshavealsothestrongestimpactsintheglobal reward value.Oneseesthat thequalitymatrixQIIdierentiatesthequalitiesof eachpairingintheoptimal assignmentA1asexpected(contrari-wisetoMethodI).Clearly, withMethodIweobtain thesamequalityindicatorvalue0.2849 forthespecicassociations (2,4)and(3,2) whichseemsintuitivelynot veryreasonablebecausethespecicrewardsof theseassociationsimpactdierentlytheglobal rewardsresult. If themethodII basedonthebelief in-tervalmeasurecomputedfrom(5)ispreferred7,wewillgetrespectivelyforthethree optimal pairings (1,3), (2,4) and(3,2) the three distinct belief interval[0.5956,0.8924],[0.4113,0.7699]and[0.3524,0.6529].Thesebelief intervalsshowthat the ordering of quality of optimal pairings (based either on the lower bound,or on theupperbound of beliefinterval) isconsistent with theordering of qual-ityofoptimalpairingsinQII(A1, A2)computedwiththeaveragingapproach.Method II provides a better eective and comprehensive solution to estimate thequalityofeachspecicassociationprovidedintheoptimalassignmentsolutionA1.7justincaseofmultiplicityofsecondbestassignments.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 42453 ConclusionInthispaperwehaveproposedamethodbasedonbelief functionsforestab-lishingthequalityofpairingsbelongingtotheoptimaldataassociation(oras-signment)solution provided byachosen algorithm.Ourmethodisindependentofthechoiceofthealgorithmusedinndingtheoptimalassignmentsolution,and, incaseof multipleoptimal solutions, itprovidesalsoawaytoselectthebestoptimal assignmentsolution(theonehavingthehighestabsolutequalityfactor). The method developed in this paper is general in the sense that it can beapplied to dierent types of association problems corresponding to dierent setsof constraints.ThismethodcanbeextendedtoSD-assignmentproblems. Theapplication ofthisapproach inarealistic multi-target tracking contextisunderinvestigationsandwillbereportedinaforthcomingpublicationifpossible.References1. Bar-Shalom, Y., Willet, P.K., Tian, X.: TrackingandDataFusion: AHandbookofAlgorithms.YBSPublishing,Storrs,CT,USA(2011)2. Hall, D.L., Chong C.Y., Llinas, Liggins II, M.: Distributed Data Fusion forNetwork-CentricOperations.CRCPress(2013)3. He, X., Tharmarasa, R., Pelletier, M., Kirubarajan, T.: AccurateMurtysAlgo-rithmfor Multitarget TopHypothesis. In: Proc. of Fusion2011, Chicago, USA(2011)4. Dezert, J., Smarandache, F.,Tchamova, A.: On the BlackmansAssociationProb-lem.In:Proc.ofFusion2003, Cairns,Australia(2003)5. Tchamova, A., Dezert, J., Semerdjiev, Tz., Konstantinova, P.: Target Tracking withGeneralizedDataAssociationBasedontheGeneral DSmRuleof Combination.In:Proc.ofFusion2004, Stockholm,Sweden(2004)6. Dezert,J.,Tchamova,A.,Semerdjiev,T.,Konstantinova,P.:PerformanceEvalu-ationof FusionRulesForMultitargetTrackinginClutterBasedonGeneralizedDataAssociation.In:Proc.ofFusion2005,Philadelphia,USA(2005)7. DenuxT.,ElZoghbyN.,CherfaouiV.,JougletA.:OptimalObjectAssociationinthe Dempster- ShaferFramework.To appearinIEEE Trans.onCybern. (2014)8. Kuhn, H.W.: The Hungarian Method for the Assignment Problem. Naval ResearchLogisticQuarterly,vol.2,8397(1955)9. Munkres, J.: Algorithms for the Assignment and Transportation Problems. JournaloftheSocietyofIndustrialandAppliedMathematics, vol.5(1),3238(1957)10. Bourgeois, F.,Lassalle, J.C.: AnExtensionoftheMunkresAlgorithmfortheAs-signment ProblemtoRectangular Matrices. Comm. of the ACM, vol. 14 (12),802804, Dec.(1971)11. Jonker, R., Volgenant, A.: Ashortestaugmentingpathalgorithmfordenseandsparselinearassignmentproblems.J.ofComp.,vol.38(4),325340(1987)12. Bertsekas, D.: Theauctionalgorithm: ADistributedRelaxationMethodfortheAssignmentProblem.AnnalsofOperationsResearch,vol.14(1),105123(1988)13. Murty, K.G.: An Algorithm for Ranking all the Assignments in Order of IncreasingCost.OperationsResearch,vol.16(3),682687(1968)14. Chegireddy, C.R., Hamacher, H.W.: Algorithms for Finding K-best Perfect Match-ing.DiscreteAppliedMathematics, vol.18,155165(1987)Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 424615. Danchick, R., Newnam, G.E.: AFastMethodforFindingtheExactN-bestHy-potheses for Multitarget Tracking. IEEE Trans. on AES, vol. 29 (2), 555560 (1993)16. Miller, M.L., Stone, H.S., Cox, I.J.: Optimizing Murtys Ranked AssignmentMethod.IEEETrans.onAES,vol.33(3),851862, July(1997)17. Pascoal, M., Captivo, M.E., Climaco, J.: ANote onaNewVariant of MurtysRankingAssignmentsAlgorithm.4ORJournal,vol.1,243255(2003)18. Ding, Z., Vandervies, D.: AModiedMurtysAlgorithmforMultipleHypothesisTracking.In:SPIESign.andDataProc.ofSmallTargets,vol.6236,(2006)19. Fortunato, E.,et al.:GeneralizedMurtys AlgorithmwithApplicationto MultipleHypothesisTracking.In:Proc.ofFusion2007,18, Quebec,July912(2007)20. Hamacher, H.W., Queyranne, M.: K-best Solutions to Combinatorial OptimizationProblems.AnnalsofOperationsResearch,no.4,123143(1985/6)21. DellAmico,M.,Toth,P.:AlgorithmsandCodesforDenseAssignmentProblems:theStateoftheArt.DiscreteAppliedMathematics,vol.100,1748(2000)22. Shafer, G.: AMathematical Theory of Evidence. Princeton University Press,Princeton,NewJersey,USA(1976)23. Smarandache,F.,Dezert,J.:Advances andApplicationsofDSmTforInformationFusion,Volumes1,2&3.ARP(20042009)http://fs.gallup.unm.edu/DSmT.htm24. Smets, P., Kennes, R.: TheTransferableBelief Model. Articial Intelligence, vol.66(2),191234(1994)Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4247