Cautious OWA and Evidential Reasoning for Decision Making under Uncertainty

CautiousOWAandEvidentialReasoning forDecisionMakingunderUncertaintyJean-MarcTacnetJean DezertAbstractTomakeadecisionundercertainty, multicriteriadecisionmethodsaimstochoose, rankorsort alternativesonthebasisofquantitativeorqualitativecriteriaandpreferencesexpressedby the decision-makers. However, decisionis oftendone under uncertainty: choosing alternatives can have differentconsequences depending on the external context (or state of theword). In this paper, a new methodology called Cautious OrderedWeightedAveragingwithEvidential Reasoning(COWA-ER) isproposedfor decisionmakingunder uncertaintytotake intoaccountimperfectevaluationsofthealternativesandunknownbeliefs about groups of the possible states of the world (scenarii).COWA-ERmixes cautiouslytheprincipleof Yagers OrderedWeightedAveraging(OWA) approachwiththeefcient fusionof belief functions proposed in Dezert-Smarandache Theory(DSmT).Keywords: fusion, OrderedWeightedAveraging(OWA),DSmT, uncertainty, information imperfection, multi-criteria decision making (MCDM)I. INTRODUCTIONA. Decisions under certainty, risk or uncertaintyDecisionmakinginreal-lifesituations areoftendifcultmulti-criteria problems. In the classical Multi-Criteria De-cisionMaking(MCDM) framework, thosedecisionsconsistmainlyinchoosing,rankingorsortingalternatives,solutionsor more generally potential actions [17] on the basis ofquantitative or qualitative criteria. Existing methods differs onaggregationprinciples(total or partial), preferences weight-ing, and so on. In total aggregation multicriteria decisionmethods such as Analytic Hierarchy Process (AHP) [19], theresult for analternative is a unique value calledsynthesiscriterion. Possiblealternatives(Ai)belongingtoagivensetA = {A1, A2, . . . , Aq} are evaluated according to preferences(represented by weights wj) expressed by the decision-makerson the different criteria(Cj) (see gure 1).Decisionsareoftentakenonthebasisofimperfect infor-mation and knowledge (imprecise, uncertain, incomplete) pro-vided by several more or less reliable sources and dependingon the statesof theworld: decisionscan betaken incertain,risky or uncertain environment. In a MCDM context, decisionundercertaintymeansthat theevaluationsofthealternativeareindependentfromthestatesoftheworld.Inothercases,alternatives may be assessed differently depending on thescenarii that are considered.Figure1. Principleof amulti-criteriadecisionmethodbasedonatotalaggregation principle.In the classical framework of decision theory under uncer-tainty, ExpectedUtilityTheory(EUT)statesthat adecisionmaker chooses between risky or uncertain alternatives oractions by comparing their expected utilities [14]. Let usconsider anexampleof decisionunder uncertainty(or risk)related to natural hazards management. On the lower parts oftorrent catchment basinor anavalanchepath, riskanalysisconsists in evaluating potential damage caused due to theeffects of hazard (a phenomenon with an intensity and afrequency) onpeopleandassetsat risk. Different strategies(Ai) arepossibletoprotect theexposedareas. For eachofthem, damagewill dependonthedifferent scenarii (Sj) ofphenomenon which can be more or less uncertain. An actionAi(e.g. buildinga protectiondevice, a dam) is evaluatedthrough its potential effects rk to which are associated utilitiesu(rk) (protection level of people, cost of protection, . . . ) andprobabilities p(rk) (linked to natural events or states of natureSk). TheexpectedutilityU(a) of anactionaisestimatedthroughthesumofproductsofutilitiesandprobabilitiesofall potential consequences of the actiona:U(Ai) =u(rk) p(rk) (1)Originally published as Tacnet J.-M., Dezert J., Cautious OWA and Evidential Reasoning for Decision Making under Uncertainty, in Proc. Of Fusion 2011 Conf., Chicago, July, 2011, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4101When probabilities are known, decision is done under risk.When those probabilities becomes subjective, the prospecttheory(subjectiveexpectedutilitytheory- SEUT) [12] canapply : the objective utility (e.g. cost) u(rk) is replacedbyasubjective function (value) denotedv(u(rk)) ; the objective weightingp(rk) is replaced by a subjectivefunction(p(rk)).v() is the felt subjective value in response of the expectedcost oftheconsideredaction, and()isthefelt weightingface to the objective probability of the realisation of the result.Prospect theoryshowsthat thefunctionv()isasymmetric:loss causes a negative reaction intensity stronger than the pos-itive reaction caused by the equivalent gain. This correspondstoanaversiontoriskychoicesintheareaofearningsandasearch of risky choices in the area of loss.Ina MCDMcontext, informationimperfectionconcernsboth the evaluation of the alternatives (in any context ofcertainty, riskor ignorance) andtheuncertaintyor lackofknowledge about the possible states of the world. Uncertaintyandimprecisioninmulti-criteria decisionmodels has beenearlyconsidered[16]. Different kindsofuncertaintycanbeconsidered: on the one hand the internal uncertainty is linkedtothestructureof themodel andthejudgmental inputsre-quired by the model, on the other hand the external uncertaintyreferstolackofknowledgeabout theconsequencesabout aparticular choice.B. Objectives and goalsSeveral decisionsupport methods exist toconsider bothinformation imperfection, sources heterogeneity, reliability,conict and the different states of the world when evaluatingthe alternatives as summarized on gure 2. A more completereviewcan be found in [28]. Here we just remind somerecent examplesofmethodsmixingMCDMapproachesandEvidential Reasoning1(ER).Figure 2. Information imperfection in the different decision support methods Dempster-Shafer-basedAHP(DS-AHP) has introducedamergingof Evidential Reasoning(ER) withAnalytic1Evidential Reasoningreferstotheuseofbelieffunctionsastheoreticalbackground, not to a specic theory of belief functions (BF) aimed forcombining, or conditioning BF. Actually, Dempster-Shafer Theory (DST) [21],Dezert-Smarandache Theory (DSmT) [22], and Smets TBM [25] are differentapproaches of Evidential Reasoning.Hierachy Process (AHP) [19] to consider the imprecisionandtheuncertaintyinevaluationofseveralalternatives.Theideaistoconsider criteriaassources[1], [3] andderive weights as discounting factors in the fusion process[5]; Dezert-Smarandache-based(DSmT-AHP) [8] takesintoaccount the partial uncertainty (disjunctions) betweenpossible alternatives and introduces newfusion rules,based on Proportional Conict Redistribution (PCR) prin-ciple, which allow to consider differences between impor-tance and reliability of sources [23]; ER-MCDA [28], [29] is based on AHP, fuzzy sets theory,possibilitytheoryandbelief functionstheorytoo. Thismethodconsiders bothimperfectionof criteriaevalua-tions, importance and reliability of sources.IntroducingignoranceanduncertaintyinaMCDMprocessconsists in considering that consequences of actions (Ai)depend of the state of nature represented by a nite setS = {S1, S2, . . . , Sn}. For eachstate, theMCDMmethodprovidesanevaluationCij. Weassumethat thisevaluationCijdonebythedecisionmaker corresponds tothechoiceof Aiwhen Sjoccurs with a given (possibly subjective)probability. Theevaluationmatrixis denedas C=[Cij]wherei = 1, . . . , qandj= 1, . . . , n._______S1 Sj SnA1C11 C11 C1n......AiCi1 Cij Cin......AqCq1 Cqj Cqn_______= C (2)Existingmethods usingevidential reasoningandMCDMhave, up to now, focused on the case of imperfect evaluationof alternatives ina context of decisionunder certainty. Inthis paper, we propose a newmethod for decision underuncertainty that mixes MCDMprinciples, decision underuncertainty principles and evidential reasoning. For thispurpose, weproposeaframeworkthat considersuncertaintyandimperfectionfor scenarii correspondingtothestateofthe world.This paper is organized as follows. In section II, webrieyrecall thebasis of DSmT. Section III presents twoexistingmethodsfor MCDMunder uncertaintyusingbelieffunctions theory: DSmT-AHP as an extension of Saatys multi-criteria decision method AHP, and Yagers Ordered WeightedAveraging(OWA)approachfordecisionmakingwithbeliefstructures. The contribution of this paper concerns the sectionIVwherewedescribeanalternativetotheclassical OWA,called cautious OWA method, where evaluations of alternativesdepend on more or less uncertain scenarii. The exibilityandadvantages of this COWAmethodare alsodiscussed.Conclusions and perspectives are given in section V.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4102II. BELIEF FUNCTIONS AND DSMTDempster-Shafer Theory (DST) [21] offers a powerful math-ematical formalism (the belief functions) to model our beliefand uncertainty on the possible solutions of a given problem.Oneof thepillarsof DSTisDempster-Shafer rule(DS) ofcombinationof belief functions. Thepurposeof thedevel-opment of Dezert-Smarandache Theory(DSmT) [22] is toovercome the limitations of DST by proposing new underlyingmodels for theframes of discernment inorder tot betterwiththenatureofreal problems, andnewcombinationandconditioningrulesforcircumventingproblemswithDSrulespecially when the sources to combine are highly conicting.In DSmT, the elementsi,i = 1, 2, . . . , n of a given frameare not necessarily exclusive, and there is no restriction onibut their exhaustivity. Some integrity constraints (if any) canbeincludeintheunderlyingmodel oftheframe. Insteadofworking in power-set 2, we classically work on hyper-powerset D(Dedekindslattice)-see[22], Vol.1fordetailsandexamples. A (generalized) basic belief assignment (bba) givenby a source of evidence is a mappingm:D[0, 1] suchthatm() = 0 andADm(A) = 1 (3)Thegeneralizedcredibilityandplausibilityfunctionsarede-ned in almost the same manner as within DST, i.e.Bel(A) =BABDm(B) and Pl(A) =BA=BDm(B) (4)In this paper, we will work with Shafers model of the frame, i.e. all elementsiof are assumed truly exhaustive andexclusive (disjoint). ThereforeD=2and the generalizedbelief functions reduces toclassical ones. DSmTproposesa new efcient combination rules based on proportionalconict redistribution(PCR) principlefor combininghighlyconictingsourcesof evidence. Also, theclassical pignistictransformationBetP(.)[26]isreplacedbythebythemoreeffectiveDSmP(.)transformationtoestimatethesubjectiveprobabilities of hypotheses for classical decision-making. Wejust recall briey the PCR fusion rule # 5 (PCR5) and Dezert-Smarandache Probabilistic (DSmP) transformation. All details,justications with examples on PCR5 and DSmP can be foundfreelyfromthewebin[22], Vols. 2&3andwill not bereported here. TheProportionalConictRedistributionRuleno. 5:PCR5 is used generally to combine bbas in DSmT framework.PCR5 transfers the conicting mass only to the elementsinvolved in the conict and proportionally to their individualmasses, sothat thespecicityoftheinformationisentirelypreserved in this fusion process. Let m1(.) and m2(.) betwoindependent2bbas, thenthe PCR5rule is denedasfollows(see[22], Vol. 2forfulljusticationandexamples):mPCR5() = 0 and X 2\ {}2i.e. each source provides its bba independently of the other sources.mPCR5(X) =X1,X22X1X2=Xm1(X1)m2(X2)+X22X2X=[m1(X)2m2(X2)m1(X) +m2(X2)+m2(X)2m1(X2)m2(X) +m1(X2)] (5)whereall denominators in(5) aredifferent fromzero. If adenominator is zero, that fraction is discarded. AdditionalpropertiesofPCR5canbefoundin[9]. ExtensionofPCR5for combining qualitative bbas can be found in [22], Vol. 2 &3. All propositions/sets are in a canonical form. A variant ofPCR5, called PCR6 has been proposed by Martin and Osswaldin[22], Vol. 2, for combinings >2sources. Thegeneralformulas for PCR5 and PCR6 rules are given in [22], Vol. 2also. PCR6 coincides with PCR5 for the fusion of two bbas. DSmP probabilistic transformation: DSmPis a seriousalternative to the classical pignistic transformation BetPsinceit increases theprobabilisticinformationcontent (PIC), i.e.it reduces Shannon entropy of the approximate subjectiveprobabilitymeasuredrawnfromanybbasee[22], Vol. 3,Chap. 3 for details and the analytic expression ofDSmP(.).When > 0 and when the masses of all singletons arezero, DSmP(.) = BetP(.), where the well-known pignistictransformationBetP(.) is dened by Smets in [26].IntheEvidential Reasoningframework, thedecisionsareusually achieved by computing the expected utilities of the actsusing either the subjective/pignistic BetP{.} (usually adoptedinDSTframework) or DSmP(.) (as suggestedinDSmTframework) as the probabilityfunctionneededtocomputeexpectations. Usually, one uses the maximum of the pignisticprobability as decision criterion. The maximum of BetP{.} isoften considered as a balanced strategy between the two otherstrategiesfordecisionmaking:themaxofplausibility(opti-misticstrategy)orthemax. ofcredibility(pessimisticstrat-egy). Themaxof DSmP(.)isconsideredasmoreefcientfor practical applications since DSmP(.) is more informative(it has a higher PIC value) thanBetP(.) transformation. ThejusticationofDSmPasafairandusefultransformationfordecision-making support can also be found in [10]. Note thatin the binary frame case, all the aforementioned decisionstrategies yields same nal decision.III. BELIEF FUNCTIONS AND MCDMTwo simple methods for MCDMunder uncertainty arebriey presented: DSmT-AHPapproach and Yagers OWAapproach. The new Cautious OWA approach that we proposewill be developed in the next section.A. DSmT-AHP approachDSmT-AHPaimedtoperformasimilar purposeasAHP[18], [19], SMART [30] or DS/AHP [1], [3], etc. that is to ndthe preferences rankings of the decision alternatives (DA), orgroups of DA. DSmT-AHP approach consists in three steps: Step 1: we extend the construction of the matrix for takinginto account the partial uncertainty (disjunctions) betweenAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4103possiblealternatives. If nocomparisonisavailablebe-tweenelements, thenthecorrespondingelementsinthematrixis zero.Eachbbarelatedtoeach (sub-)criterionis the normalized eigenvector associated with the largesteigenvalue of the uncertain knowledge matrix (as donein standard AHP approach). Step 2: we use the DSmT fusion rules, typically the PCR5rule, tocombinebbasdrawnfromstep1togetanalMCDM priority ranking. This fusion step must take intoaccount the different importances (if any) of criteria as itwill be explained in the sequel. Step 3: decision-making can be based either on themaximum of belief, or on the maximum of the plausibilityofDA, aswellasonthemaximumoftheapproximatesubjective probability of DA obtained by different prob-abilistic transformations.The MCDMproblemdeals with several criteria havingdifferent importances and the classical fusion rules cannot beapplied directly as in step 2. In AHP, the fusion is done fromtheproductofthebbasmatrixwiththeweightingvectorofcriteria. Such AHP fusion is nothing but a simple component-wise weighted average of bbas and it doesnt actually processefciently the conicting information between the sources. Itdoesnt preservetheneutralityof afull ignorant sourceinthefusion. Topalliatetheseproblems, wehaveproposedanewsolutionforcombiningsourcesofdifferentimportancesin [23]. Briey, the reliability of a source is usually taken intoaccount with Shafers discounting method [21] dened by:{m(X) = m(X), forX = m() = m() + (1 )(6)where [0; 1] isthereliabilitydiscountingfactor. =1whenthesourceisfullyreliableand=0ifthesourceistotally unreliable. We characterize the importance of a sourcebyanimportancefactor in[0, 1]. factor isusuallynotrelatedwiththereliabilityof thesourceandcanbechosento any value in[0, 1] by the designer for his/her own reason.By convention, =1means the maximalimportanceof thesource and = 0 means no importance granted to this source.From thisfactor, we dene the importance discounting by{m(X) = m(X), forX = m() = m() + (1 )(7)Here, we allow to deal with non-normal bba since m() 0as suggested by Smets in [24]. This new discounting preservesthe specicity of the primary information since all focal ele-ments are discounted with same importance factor. Based onthisimportancediscounting,onecanadaptPCR5(orPCR6)rulefor N 2discountedbbasm,i(.), i =1, 2, . . . Ntoget withPCR5fusionrule(seedetailsin[23]) aresultingbbawhichisthennormalizedbecauseintheAHPcontext,theimportancefactorscorrespondtothecomponentsofthenormalizedeigenvector w. It isimportant tonotethat suchimportance discounting method cannot be used in DST whenusing Dempster-Shafers rule of combination because this ruleisnot respondingtothediscountingof sourcestowardstheemptyset (seeTheorem1in[23]forproof). Thereliabilityandimportanceof sourcescanbetakenintoaccount easilyinthefusionprocessandseparately. Thepossibilitytotakethem into account jointly is more difcult, because in generalthereliabilityandimportancediscountingapproachesdonotcommute, but wheni=i=1. In order to deal both withreliabilitiesandimportancesfactorsandbecauseof thenoncommutativityofthesediscountings, twomethodshavealsobeen proposed in [23] and not reported here.B. Yagers OWA approachLets introduceYagers OWAapproach[33] for decisionmaking with belief structures. One considers a collection ofqalternatives belongingtoaset A={A1, A2, . . . , Aq}anda nite set S ={S1, S2, . . . , Sn}of states of the nature.Weassumethat thepayoff/gainCij of thedecisionmakerin choosing Ai when Sjoccurs are given by positive (or null)numbers. Thepayoffsq nmatrixisdenedbyC=[Cij]where i =1, . . . , q and j =1, . . . , nas ineq. (2). Thedecision-making problem consists in choosing the alternativeA Awhichmaximizesthepayofftothedecisionmakergiven the knowledge on the state of the nature and the payoffsmatrix C. A Ais called the best alternative or thesolution (if any) of the decision-making problem. Dependingtheknowledgethedecision-maker has onthestates of thenature, he/she is face on different decision-making problems:1 Decision-making under certainty: onlyone state ofthenatureisknownandcertaintooccur, saySj. Thenthedecision-making solution consists in choosingA= Aiwithiarg maxi{Cij}.2 Decision-making under risk: the true state of the natureisunknownbutoneknowsalltheprobabilitiespj=P(Sj),j =1, . . . , nof the possible states of the nature. Inthiscase, weusethemaximumofexpectedvaluesfordecision-making. For eachalternative Ai, we compute its expectedpayoffE[Ci]=j pj Cij, then we chooseA=Ai withiarg maxi{E[Ci]}.3 Decision-making under ignorance: one assumes noknowledge about the true state of the nature but that it belongstoS. Inthiscase, YagerproposestousetheOWAoperatorassuming a given decision attitude taken by the decision-maker. Given a set of values/payoffs c1,c2, ...,cn, OWA con-sists in choosing a normalized set of weighting factorsW=[w1, w2, . . . wn] where wj [0, 1] andj wj= 1 and for anyset of valuesc1, c2, ..., cncompute OWA(c1, c2, . . . , cn) asOWA(c1, c2, . . . , cn) =jwj bj(8)where bjis thejth largest element in the collection c1, c2, ...,cn. As seen in (8), the OWA operator is nothing but a simpleweighted average of ordered values of a variable.BasedonsuchOWAoperators, the idea consists for eachalternative Ai, i =1, . . . , q tochoose a weightingvectorWi=[wi1, wi2, . . . win] andcomputeitsOWAvalueVi

OWA(Ci1, Ci2, . . . , Cin) =j wij bijwhere bijis theAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4104jthlargestelementinthecollectionofpayoffsCi1, Ci2,. . . ,Cin. Then, as for decision-making under risk, we chooseA= Aiwithiarg maxi{Vi}. The determination ofWidepends on the decision attitude taken by the decision-maker.The pessimistic attitude considers for all i = 1, 2, . . . , q,Wi=[0, 0, . . . , 0, 1]. Inthiscase, weassigntoAitheleastpayoff and we choose the best worst (the max of least payoffs).It is a Max-Minstrategysince i=arg maxi(minj Cij).The optimistic attitude considers for all i = 1, 2, . . . , q,Wi=[1, 0, . . . , 0, 0]. Wecommit toAiitsbest payoff andwe select the best best. It is a Max-Max strategy sincei=arg maxi(maxj Cij). Betweenthesetwoextremeatti-tudes, we can dene an innity of intermediate attitudes likethenormative/neutral attitude(whenor all i =1, 2, . . . , q,Wi=[1/n, 1/n, . . . , 1/n, 1/n]) which corresponds to thesimplearithmeticmean, orHurwiczattitude(i.e. aweightedaverage of pessimistic and optimistic attitudes), etc. To justifythe choice of OWA method, Yager denes an optimistic index [0, 1] fromthe components of Wiand proposes tocompute(bymathematical programming)thebest weightingvector Wicorrespondingtoapriori chosenoptimisticindexand having the maximal entropy (dispersion). If = 1(optimisticattitude)thenofcourseWi=[1, 0, . . . , 0, 0] andif =0(pessimisticattitude) thenWi=[0, 0, . . . , 0, 1]. Itheory, Yagers method doesnt exclude the possibility to adoptan hybrid attitude depending on the alternative we consider. Inother words, we are not forced to consider the same weightingvectors for all alternatives.Example 1: Lets take statesS= {S1, S2, S3, S4}, alterna-tivesA = {A1, A2, A3} and the payoffs matrix:__S1S2S3S4A110 0 20 30A21 10 20 30A330 10 2 5__(9)If oneadopts thepessimisticattitudeinchoosingW1=W2=W3=[0, 0, 0, 1], then one gets for each alterna-tive Ai, i =1, 2, 3thefollowingvalues of OWAs: V1=OWA(10, 0, 20, 30) =0, V2=OWA(1, 10, 20, 30) =1andV3=OWA(30, 10, 2, 5)=2. Thenal decisionwill bethealternativeV3since it offers the best expected payoff.If one adopts the optimistic attitude inchoosing W1=W2=W3=[1, 0, 0, 0], then one gets for each alterna-tive Ai, i =1, 2, 3thefollowingvalues of OWAs: V1=OWA(10, 0, 20, 30) = 30, V2= OWA(1, 10, 20, 30) = 30 andV3= OWA(30, 10, 2, 5) = 30. All alternatives offer the sameexpectedpayoff andthusthenal decisionmust bechosenrandomly or purely ad-hoc since there is no best alternative.If one adopts the normative attitude inchoosing W1=W2= W3= [1/4, 1/4, 1/4, 1/4] (i.e. one assumes thatall states of natureareequiprobable), thenonegets: V1=OWA(10, 0, 20, 30) = 60/4, V2=OWA(1, 10, 20, 30) =61/4 andV3= OWA(30, 10, 2, 5) = 47/4. The nal decisionwill bethealternative V2sinceit offers thebest expectedpayoff.4Decision-makingunderuncertainty: thiscorrespondstothe general case where the knowledge onthe states ofthenatureischaracterizedbyabeliefstructure.Clearly,oneassumes that a priori knowledge on the frame S of the differentstates of the nature is given by a bbam(.) : 2S [0, 1]. Thiscaseincludesall previouscasesdependingonthechoiceofm(.). Decision under certainty is characterized by m(Sj) = 1;Decisionunderriskischaracterizedbym(s)>0forsomestatess S; Decisionunderfull ignoranceischaracterizedby m(S1S2. . . Sn) = 1, etc. Yagers OWA for decision-making under uncertainty combines the schemes used fordecision making under risk and ignorance. It is based on thederivationof ageneralizedexpectedvalueCiof payoff foreach alternativeAias follows:Ci=rk=1m(Xk)Vik(10)where r is the number of focal elements of the belief structure(S, m(.)). m(Xk)isthemassofbeliefofthefocalelementXk 2S, and Vikis the payoff we get when we selectAiandthe state of the nature lies in Xk. The derivationof Vikis done similarlyas for thedecisionmakingunderignorance when restricting the states of the nature to the subsetof states belonging toXkonly. Therefore forAiand a focalelement Xk, insteadof usingall payoffs Cij, weconsideronly the payoffs in the set Mik={Cij|Sj Xk} andVik= OWA(Mik)forsomedecision-makingattitudechosena priori. Once generalized expected valuesCi,i = 1, 2, . . . , qare computed, we select the alternative which has its highestCi as the best alternative (i.e. the nal decision). The principleof this methodis very simple,but its implementationcan bequite greedy in computational resources specially if one wantstoadopt aparticular attitudefor agivenlevel of optimism,specially if the dimension of the frame S is large: one needs tocompute by mathematical programming the weighting vectorsgenerating the optimism level having the maximum of entropy.Asillustrativeexample, wetakeYagersexample3[33]witha pessimistic, optimistic and normative attitudes.Example 2: Lets take statesS= {S1, S2, S3, S4, S5} withassociatedbbam(S1 S3 S4)=0.6, m(S2 S5)=0.3andm(S1 S2 S3 S4 S5)=0.1. LetsalsoconsideralternativesA = {A1, A2, A3, A4} and the payoffs matrix:C=__7 5 12 13 612 10 5 11 29 13 3 10 96 9 11 15 4__(11)Ther= 3 focal elements ofm(.) areX1= S1 S3 S4,X2=S2 S5andX3=S1 S2 S3 S4 S5. X1andX2arepartialignorancesandX3isthefullignorance. Oneconsiders the following submatrix (called bags by Yager) for3There is a mistake/typo error in original Yagers example [33].Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4105the derivation ofVik, fori = 1, 2, 3, 4 andk = 1, 2, 3.M(X1) =__M11M21M31M41__=__7 12 1312 5 119 3 106 11 15__M(X2) =__M12M22M32M42__=__5 610 213 99 4__M(X3) =__M13M23M33M43__=__7 5 12 13 612 10 5 11 29 13 3 10 96 9 11 15 4__= CUsingpessimisticattitude, andapplyingtheOWAop-erator on each row of M(Xk) for k = 1 to r, onegets nally4: V (X1) =[V11, V21, V31, V41]t=[7, 5, 3, 6]t,V (X2) =[V12, V22, V32, V42]t=[5, 2, 9, 4]tandV (X3). =[V13, V23, V33, V43]t= [5, 2, 3, 4]t. Applying formula (10)for i =1, 2, 3, 4onegetsnallythefollowinggeneralizedexpected values using vectorial notation:[C1, C2, C3, C4]t=r=3k=1m(Xk) V (Xk) = [6.2, 3.8, 4.8, 5.2]tAccordingtothesevalues, thebest alternativetotakeisA1since it has the highest generalized expected payoff. Using optimistic attitude, one takes the max value of eachrow, and applying OWA on each row ofM(Xk) fork = 1 tor, one gets:V (X1) = [V11, V21, V31, V41]t= [13, 12, 10, 15]t,V (X2) = [V12, V22, V32, V42]t= [6, 10, 13, 9]t, andV (X3) =[V13, V23, V33, V43]t= [13, 12, 13, 15]t. One nally gets[C1, C2, C3, C4]t=[10.9, 11.4, 11.2, 13.2]tandthebest al-ternative to take with optimistic attitude isA4 since it has thehighest generalized expected payoff. Using normative attitude, one takes W1= W2=W3=W4=[1/|Xk|, 1/|Xk|, . . . , 1/|Xk|] where |Xk| is thecardinality of the focal element Xkunder consideration. ThenumberofelementsinWiisequalto |Xk|.Thegeneralizedexpected values are [C1, C2, C3, C4]t=[9.1, 8.3, 8.4, 9.4]tand the best alternative with the normative attitude is A4 (sameas with optimistic attitude) since it has the highest generalizedexpected payoff.C. Using expected utility theoryIn this section, we propose to use a much simpler ap-proach than OWA Yagers approach for decision making underuncertainty. Theideais toapproximatethebba m(.) byasubjectiveprobabilitymeasurethroughagivenprobabilistictransformation. We suggest touse either BetP or (better)DSmPtransformationsfor doingthisasexplainedin[22](Vol.3, Chap. 3). Letstakebackthepreviousexampleandcompute the BetP(.) and DSmP(.) values fromm(.).4whereXtdenotes the transpose ofX.Onegetsthesamevaluesinthisparticularexampleforany>0 because we dont have singletons as focal elements ofm(.), whichis normal. Here BetP(S1) =DSmP(S1) =0.22, BetP(S2) = DSmP(S2) = 0.17, BetP(S3) =DSmP(S3) = 0.22, BetP(S4) = DSmP(S4) = 0.22and BetP(S5) = DSmP(S2) = 0.17. Based on theseprobabilities, wecancomputetheexpectedpayoffsforeachalternative as for decision making under risk (e.g. forC1, weget7 0.22 +5 0.17 +12 0.22 +13 0.22 +6 0.17 = 8.91).For the4 alternatives, we nally get:EBetP[C] = EDSmP[C] = [8.91, 8.20, 8.58, 9.25]tAccordingtothesevalues, oneseesthat thebest alternativewith this pignistic or DSmattitude is A4(same as withYagersoptimisticornormativeattitudes)sinceit offersthehighest pignistic or DSm expected payoff. This much simplerapproach must be used with care however because there is aloss of information through the approximation of the bba m(.)into any subjective probability measure. Therefore, we do notrecommend to use it in general.IV. THE NEW COWA-ER APPROACHYagers OWAapproachis basedonthechoiceof givenattitudemeasuredbyanoptimisticindexin[0, 1] toget theweighting vectorW. How is chosen such an index/attitude ?This choice is ad-hoc and very disputable for users. What todoifwedont knowwhichattitudetoadopt ?Therationalanswertothisquestionistoconsidertheresultsofthetwoextreme attitudes (pessimistic and optimistic ones) jointly andtrytodevelopanewmethodfordecisionunderuncertaintybasedontheimprecisevaluationofalternatives. Thisistheapproach developed in this paper and we call it Cautious OWAwith Evidential Reasoning (COWA-ER) because it adopts thecautious attitude (based on the possible extreme attitudes) andER, as explained in the sequel.Lets take backthe previous example andtake the pes-simistic andoptimistic valuations of the expectedpayoffs.TheexpectedpayoffsE[Ci]areimprecisesincetheybelongtointerval [Cmini, Cmaxi] where bounds are computedwithextreme pessimistic and optimistic attitudes, and one hasE[C] =__E[C1]E[C2]E[C3]E[C4]____[6.2; 10.9][3.8; 11.4][4.8; 11.2][5.2; 13.2]__Therefore, one has 4 sources of information about theparameter associated with the best alternative to choose.For decisionmakingunder imprecision, weproposetousehereagainthebelief functionsframeworkandtoadopt thefollowing very simple COWA-ER methodology based on thefollowing four steps: Step 1: normalization of imprecise values in[0, 1]; Step2: conversionof eachnormalizedimprecisevalueinto elementary bbami(.); Step 3: fusion of bba mi(.) with an efcient combinationrule (typically PCR5);Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4106 Step 4: choice of the nal decision based on the resultingcombined bba.Lets describe in details each step of COWA-ER. In step 1,we divide each bound of intervals by the max of the boundstoget a newnormalizedimprecise expectedpayoff vectorEImp[C]. In our example, one gets:EImp[C] =__[6.2/13.2; 10.9/13.2][3.8/13.2; 11.4/13.2][4.8/13.2; 11.2/13.2][5.2/13.2; 13.2/13.2]____[0.47; 0.82][0.29; 0.86][0.36; 0.85][0.39; 1.00]__In step 2, we convert each imprecise value into its bbaaccordingtoa verynatural andsimple transformation[7].Here, we need to consider as frame of discernment, the niteset of alternatives ={A1, A2, A3, A4}andthe sourcesofbeliefassociatedwiththemobtainedfromthenormalizedimprecise expected payoff vectorEImp[C]. The modeling forcomputingabbaassociatedtothehypothesis Aifromanyimprecisevalue[a; b] [0; 1] isverysimpleandisdoneasfollows:___mi(Ai) = a,mi( Ai) = 1 bmi(Ai Ai) = mi() = b a(12)whereAiisthecomplement of Aiin. Withsuchsimpleconversion, onesees that Bel(Ai) =a, Pl(Ai) =b. Theuncertaintyisrepresentedbythelengthoftheinterval [a; b]and it corresponds to the imprecision of the variable (here theexpectedpayoff)onwhichisdenedthebelieffunctionforAi. In the example, one gets:AlternativesAimi(Ai) mi( Ai) mi(Ai Ai)A10.47 0.18 0.35A20.29 0.14 0.57A30.36 0.15 0.49A40.39 0 0.61Table IBASIC BELIEF ASSIGNMENTS OF THE ALTERNATIVESInstep3,weneedtocombinebbasmi(.)byanefcientruleofcombination. Here, wesuggesttousethePCR5ruleproposedinDSmTframeworksinceithasbeenprovedveryefcient todeal withpossiblyhighlyconictingsourcesofevidence. PCR5hasbeenalreadyappliedsuccessfullyinallapplications where it has been used so far [22]. We callthis COWA-ERmethod based on PCR5 as COWA-PCR5.Obviously, we could replace PCR5 rule by any other rule (DSrule, Dubois& Prade, Yagers rule, etc and thus dene easilyCOWA-DS, COWA-DP, COWA-Y, etc variants of COWA-ER. This is not thepurposeof this paper andthis has nofundamental interest inthis presentation. The result of thecombinationof bbas withPCR5for our exampleis givenin of Table II.The last step 4 is the decision-making from the resulting bbaofthefusionstep3. Thisproblemisrecurrentinthetheoryofbelieffunctionsandseveral attitudesarealsopossibleasFocal Element mPCR5(.)A10.2488A20.1142A30.1600A40.1865A1 A40.0045A2 A40.0094A1 A2 A40.0236A3 A40.0075A1 A3 A40.0198A2 A3 A40.0374A1 A2 A3 A40.1883Table IIFUSION OF THE FOUR ELEMENTARY BBAS WITH PCR5explainedat theendofsectionII. TableIIIshowswhat arethe values of credibilities, plausibilities, BetPand DSmP=0for each alternative in our example.AiBel(Ai) BetP(Ai) DSmP(Ai) Pl(Ai)A10.2488 0.3126 0.3364 0.4850A20.1142 0.1863 0.1623 0.3729A30.1600 0.2299 0.2242 0.4130A40.1865 0.2712 0.2771 0.4521Table IIICREDIBITITY AND PLAUSIBILITY OF AiBasedontheresultsofTableIII, it isinterestingtonotethat, inthisexample, thereisnoambiguityinthedecisionmakingwhatevertheattitudeistakenbythedecision-maker(the max of Bel, the max of Pl, the max of BetP or the max ofDSmP), the decision to take will always be A1. Such behaviorisprobablynotgeneralinallproblems,butatleastitshowsthat in some cases like in Yagers example, the ambiguity indecision can be removed when using COWA-PCR5 instead ofOWAwhichisanadvantageofourapproach. Itisworthtonote that Shannon entropy of BetP isHBetP= 1.9742 bits isbiggerthanShannonentropy ofDSmPisHDSmP=1.9512bits whichis normal since DSmPhas beendevelopedforincreasing the PIC value.Advantages and extension of COWA-ER: COWA-PCR5allows also to take easily a decision, not only on a single alter-native, but also if one wants on a group/subset of alternativessatisfying a min of credibility (or plausibility level) selected bythe decision-maker. Using such approach, it is of course veryeasy to discount each bba mi(.) entering in the fusion processusingreliabilityorimportancediscountingtechniqueswhichmakes this approach more appealing and exible for the userthanclassical OWA. COWA-PCR5issimpler toimplementbecause it doesnt require the evaluation of all weightingvectors for the bags by mathematical programming. Onlyextremeandverysimpleweightingvectors [1, 0, . . . , 0] and[0, . . . , 0, 1] are used in COWA-ER. Of course, COWA-ER canalso be extended directly for the fusion of several sources ofinformations when each source can provide a payoffs matrix. Itsufces to apply COWA-ER on each matrix to get the bbas ofstep 3, then combine them with PCR5 (or any other rule) andthenapplystep4ofCOWA-ER. WecanalsodiscounteachAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4107sourceeasilyifneeded.AlltheseadvantagesmakesCOWA-ERapproachveryexibleandappealingfor MCDMunderuncertainty. In summary, the original OWA approach considersseveral alternatives Aievaluatedinthecontext of differentuncertain scenarii and includes several ways (pessimistic,optimistic, hurwicz, normative) to interpret and aggregate theevaluations with respect to a given scenario. COWA-ER usessimultaneously the two extreme pessimistic and optimisticdecisionattitudescombinedwithanefcient fusionruleasshown on Figure 3. In order to save computational resources(if required), we alsohave proposeda less efcient OWAapproach using the classical concept of expected utility basedon DSmP or BetP.Figure 3. COWA-ER: Two evolutions of Yagers OWA method.V. CONCLUSIONIn this work, Yagers Ordered Weighted Averaging (OWA)operators are extended and simplied with evidential reasoning(ER) for MCDM under uncertainty. The new Cautious OWA-ERmethodisveryexibleandrequireslesscomputationalloadthanclassical OWA. COWA-ERimproves theexistingframeworkfor MCDMsince it candeal alsowithseveralmoreorlessreliablesources. Furtherdevelopmentsarenowplanned to combine uncertainty about states of the world withthe imperfectionanduncertaintyof alternatives evaluationsaspreviouslyintroducedintheER-MCDAandDSmT-AHPmethods in order to connect them with COWA-ER.REFERENCES[1] M. Beynon, B. Curry, P.H. 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