Sigmoidal Model for Belief Function-Based Electre Tri MethodJean Dezert Jean-Marc TacnetAbstract. Main decision-making problems can be described into choice, ranking orsorting of a set of alternatives or solutions. The principle of Electre TRI (ET) methodis to sort alternatives ai according to criteria gj into categories Ch whose lower andupper limitsarerespectivelybhandbh+1.Thesortingprocedure isbasedontheevaluations ofoutranking relationsbasedrstlyoncalculationofpartialconcor-dance and discordance indexes and secondly on global concordance and credibilityindexes. In this paper, we propose to replace the calculation of the original concor-dance and discordance indexes of ET method by a more effective sigmoidal model.Such model is part of a newBelief Function ET (BF-ET) method under developmentand allows a comprehensive, elegant and continuous mathematical representation ofdegree of concordance, discordance and the uncertainty level which is not directlytaken into account explicitly in the classical Electre Tri.1 IntroductionThe Electre Tri (ET) method, developed by Yu [13], remains one of the most suc-cessful and applied methods for multiple criteria decision aiding (MCDA) sortingproblems [5]. ET method assigns a set of given alternatives ai A, i =1, 2, . . . , n ac-cording to criteria gj, j = 1, 2, . . . , m to a pre-dened (and ordered) set of categoriesChC, h =1, 2, . . . , p+1 whose lower and upper limits are respectively bh and bh+1for all h = 1, . . . , p), with b0 b1 b2 . . . bh1 bh . . . bp. The assign-ment of an alternative ai to a categoryCh (limited by proles bh and bh+1 ) consistsin four steps involving at rst the computation of global concordance c(ai, bh) anddiscordance d(ai, bh) indexes1(steps 1 & 2), secondly their fusion into a credibility1Themselves computed from partial concordance and discordance indexes based on a givenset criteria gj(.), j J.Originally published as Dezert J., Tacnet J.-M., Sigmoidal Model for Belief Function-based Electre Tri method, in Belief 2012, Compigne, May 2012, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4109index (ai, bh) (step 3), and nally the decision and choice of the category basedon the evaluations of outranking relations [13, 6] (step 4). The partial concordanceindex cj(ai, bh) measures the concordance of ai and bh in the assertion ai is at leastas good as bh. The partial discordance index dj(ai, bh) measures the opposition ofaiand bhin the assertion aiis at least as good as bh. The global concordanceindex c(ai, bh) measures the concordance of aiand bhon all criteria in the asser-tion ai outranks bh. The degree of credibility of the outranking relation denotedas (ai, bh) expresses to which extent aioutranks bh according to c(ai, bh) anddj(ai, bh) for all criteria. The main steps of ET method are described below:1. Concordance Index: The concordance index c(ai, bh) [0, 1] between the al-ternative ai and the category Chis computed as the weighted average of partialconcordance indexes cj(ai, bh), that isc(ai, bh) =jJwjcj(ai, bh) (1)wheretheweightswi [0, 1] representtherelativeimportanceofeachcrite-riongj(.)intheevaluationoftheglobal concordanceindex. Theymust sat-isfyjJwj= 1. Thepartial concordanceindexcj(ai, bh) [0, 1] basedonagivencriteriongj(.) is computedfromthedifferenceof thecriteriaeval-uatedfor theprol bh, andthecriterionevaluatedfor thealternativeai. Ifthedifferencegj(bh) gj(ai)isless(orequal)toagivenpreferencethresh-oldqj(gj(bh))thenaiand Chareconsideredasdifferentbasedonthecrite-riongj(.)sothat apreferenceofaiwithrespect toChcanbeclearlydone.Ifthedifferencegj(bh) gj(ai)isstrictlygreatertoanothergiventhresholdpj(gj(bh)) then ai and Ch are considered as indifferent (similar) based on gj(.)).When gj(bh) gj(ai) [qj(gj(bh)), pj(gj(bh))], the partial concordance indexcj(ai, bh)iscomputedfromalinearinterpolation.Mathematically,thepartialconcordance index is obtained by:cj(ai, bh) 1 if gj(bh) gj(ai) qj(gj(bh))0 if gj(bh) gj(ai) > pj(gj(bh))gj(ai)+pj(gj(bh))gj(bh)pj(gj(bh))qj(gj(bh))otherwise(2)2. Discordance Index: The discordance index between the alternative aiand thecategory Chdepends on a possible veto condition expressed by the choice of aveto threshold vj(gj(bh)) imposed on some criterion gj(.). The (global) discor-dance index d(ai, bh) is computed from the partial discordance indexes:dj(ai, bh) 1 if gj(bh) gj(ai) > vj(gj(bh))0 if gj(bh) gj(ai) pj(gj(bh))gj(bh)gj(ai)pj(gj(bh))vj(gj(bh))pj(gj(bh))otherwise(3)Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4110OnedenesbyVthesetofindexes j Jwherethevetoapplies(wherethepartial discordance index is greater than the global concordance index), that isV { j J|dj(ai, bh) > c(ai, bh)} (4)Then a global discordance index can be dened [12] asd(ai, bh)

1 if V = / 0jV1dj(ai,bh)1cj(ai,bh)if V = / 0(5)3. Global Credibility Index: In ET method, the (global) credibility index (ai, bh)is computed by the simple discounting of the concordance index c(ai, bh) givenby (1) by the discordance index (discounting factor) d(ai, bh) given in (5). Math-ematically, this is given by(ai, bh) = c(ai, bh)d(ai, bh) (6)4. Assignment Procedure: The assignment of a given action ai to a certain cate-gory Chresults from the comparison of aito the prole dening the lower andupper limits of the categories. For a given category limit bh, this comparison re-lies on the credibility of the assertions ai outranks bh. Once all credibility indexes(ai, bh) for i = 1, 2, . . . , m and h = 1, 2, . . . , k have been computed, the assign-ment matrix M[(ai, bh)] is available for helping in the nal decision-makingprocess. In ELECTRE TRI method, a simple -cutting level strategy (for a givenchoice of [0.5, 1]) is used in order to transform the fuzzy outranking relationinto a crisp one to determine if each alternative outranks (or not) each category.This is done by testing if (ai, bh) .If the inequality is satised,it meansthat indeed ai outranks the category Ch. Based on outranking relations betweenall pairs of alternatives and proles of categories, two approches are proposedin ELECTRETRItonallyassign thealternatives into categories, see[5]fordetails: Pessimistic (conjunctive) approach: aiis compared with bk, bk1, bk2, . . . ,untilaioutranks bhwhere h k.Thealternative aiisthenassigned tothehighest category Ch if (ai, bh) for a given threshold . Optimistic (disjunctive) approach: ai is compared with b1, b2, . . . bh, . . . untilbhoutranks ai.Thealternative aiisassignedtothelowestcategory Chforwhich the upper prole bh is preferred to ai.The objective and motivation of this paper is to develop a new Belief Function basedET method taking into account the potential of BF to model uncertainties. The wholeBF-ET method is under development and will be presented and evaluated on a de-tailed practical example in a forthcoming publication. Due to space limitation con-straints, we just present here what we propose to compute the new concordance anddiscordance indexes useful in our BF-ET.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 41112 Limitations of the Classical Electre TriET method remains rather based on heuristic approach than on a theoretical one foreach of its steps. Belief functions can improve ET method because of their abilityto model and manage conicting as well as uncertainty information in a theoreticalframework. We only focus here on steps 1 and 2 and we propose a solution to over-come their limitations in the next section.Example 1: Lets consider gj(ai) [0, 100], and lets take gj(bh) = 50 and the fol-lowing thresholds: qj(gj(bh)) = 20 (indifference threshold), pj(gj(bh)) = 25 (pref-erence threshold) and vj(gj(bh) = 40 (veto threshold). Then the local concordanceand discordances indexes obtained in steps 1 and 2 of ET are shown on the Fig. 1.0 10 20 30 40 50 60 70 80 90 10010.500.511.52gj(ai)cj(ai,bh) and dj(ai,bh) ELECTRE TRI modeling cj(ai,bh)dj(ai,bh)Fig. 1Example of partial concordance and discordance indexes.Fromthis very simple example, one sees that ET modeling of partial concordanceanddiscordanceindexesisnotverysatisfactorysincethereisnoclear(explicitand consistent) modeling of the uncertainty area where the action aiis not totallydiscordant, nor totally concordant with the prole bh. In such simplistic modeling,there exist points gj(ai) (lying on the slope of the blue or red curves) that can benot totally concordant while being totally not discordant (and vice-versa), which iscounter-intuitive and rather abnormal. This drawback will be solved using our newsigmoidal basic belief assignment (bba) modeling presented in the next section.3 Sigmoidal Model for Concordance and Discordance IndexesIn fact, there areseveral ways tocompute partial concordances and discordancesindexesandtocombinetheminordertoprovidetheglobal credibilityindexes(ai, bh). Electre Tri proposes a simple and basic approach based on hard threshold-ing techniques for doing this. It can fail to work efciently in practice in some cases,or may require a lot of experience to calibrate/tune all setting parameters in order toapply it to get pertinent results for decision-making support. Usually, a sensitivityanalysis must be done very carefully before applying ET in real applications. Here,Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4112we propose a more exible approach based on sigmoidal modeling where no hardthresholding technique is required.In ET approach, we are mainly concerned in the evaluation of the credibility in-dexes (ai, bh) [0, 1] for i = 1, 2, . . . , m and h = 1, 2, . . . , k (step 3) from which thenal decision(assignment) willbedrawn instep4. Step3isconditioned by theresults of steps 1 and 2 which can be improved using belief functions. For such pur-pose, we consider, a binary frame of discernment2{c, c} where c means thatthe alternative aiis concordant with the assertion aiis at least as good as prolebh, and c means that the alternative ai is opposed (discordant) to this assertion. Thismust obviously be done with all the assertions to check in the ET framework. Thebasic idea is for each pair (ai, bh) to evaluate its bba mih(.) dened on the power-setof , denoted 2. Such bbas have of course to be dened from the combination(fusion) of the local bbas mjih(.) evaluated from each possible criteria gj(.) (as insteps 1 and 2). The main issue is to derive the local bbas mjih(.) dened in 2fromthe knowledge of the criteria gj(.) and preference, indifference and veto thresholdspj(gj(bh)), qj(gj(bh)) and vj(gj(bh)) respectively. It turns out that this can be easilyobtained from the new method of construction of bba presented in [4] and adaptedhere in the ET context as follows: Letgj(ai) be the evaluation of thecriterion gj(.)for thealternative ai, follow-ing ET approach when gj(ai) gj(bh) qj(gj(bh)) then the belief in concordancecmust behigh (close toone), whereas itmust below (closetozero) as soonasgj(ai) < gj(bh) pj(gj(bh)).Similarly, thebelief indiscordance cmust behigh(closetoone) ifgj(ai) < gj(bh) vj(gj(bh)),andit must below(close tozero)when gj(ai) gj(bh) pj(gj(bh)).Such behavior can bemodeled directly fromthe sigmoid functions dened byfs,t(g)1/(1 +es(gt)) where g is the criterionmagnitudeofthealternativeunderconsideration;t istheabscissaoftheinec-tion point of the sigmoid. s/4 is the slope3of the tangent at the inection point. Itcan beeasilyveried that the bba mjih(.)satisfying the expected behavior can beobtained bythefusion4ofthetwofollowing simplebbas dened by: where theabscisses of inection points are given by tc = gj(bh) 12(pj(gj(bh)) +qj(gj(bh)))and t c =gj(bh) 12(pj(gj(bh)) +vj(gj(bh))) and the parameters sc and s c are givenby5sc = 4/(pj(gj(bh)) qj(gj(bh))) and s c = 4/(vj(gj(bh)) pj(gj(bh))).Table 1Construction of m1(.) and m2(.).focal element m1(.) m2(.)c fsc,tc(g) 0 c 0 fs c,t c(g)c c 1 fsc,tc(g) 1 fs c,t c(g)2Here we assume that Shafers model holds, that is c c = / 0.3i.e. the ratio of the vertical and horizontal distances between two points on a line; zero ifthe line is horizontal, undened if it is vertical.4With averaging rule, PCR5 rule, or Dempster-Shafer rule [8].5The coefcient 4 appearing in sc and s c expressions comes from the fact that for a sigmoidof parameter s, the tangent at its inection point is s/4.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4113 From the setting of threshold parameterspj(gj(bh)), qj(gj(bh)) and vj(gj(bh)),it is easy to compute the parameters of the sigmoids (tc, sc) and (t c, t c), and thus togetthevaluesofbbasm1(.)andm2(.). Oncethishasbeendonethelocalbbamjih(.)iscomputedbythefusion(denoted )ofbbasm1(.)andm2(.), that ismjih(.) = [m1m2](.). As shown in [4], the choice of a particular rule of combination(Dempster, PCR5, or hybrid rule) has only a little impact on the result of the com-bined bba mjih(.). But since PCR5 proposes a better management of conicting bbasyielding to more specic results than with other rules [1], we use it to combine m1(.)with m2(.) to compute mjih(.) associated with the criterion gj(.) and the pair (ai, bh).Inadopting suchsigmoidal modeling, wegetnowfrommjih(.)afullyconsistentand elegant representation of local concordance cj(ai, bh) (step 1 of ET), local dis-cordance dj(ai, bh) (step 2 of ET), as well as of the local uncertainty uj(ai, bh) byconsidering: cj(ai, bh) mjih(c) [0, 1], dj(ai, bh) mjih( c) [0, 1] and uj(ai, bh) mjih(c c) [0, 1]. Of course, one has also cj(ai, bh) +dj(ai, bh) +uj(ai, bh) = 1.4 Example of a Sigmoidal ModelIf onetakesbacktheexample1, theinectionpointsofthesigmoids f1(g) fsc,tc(g) andf2(g)fs c,t c(g) have the following abscisses tc =50(25+20)/2=27.5 and t c =50(25+40)/2 =17.5 and parameters sc =4/(2520) =4/5 =0.8and s c =4/(4025) =4/15 0.2666. The two sigmoidsf1(gj(ai)) andf2(gj(ai))are shown on the Fig. 2.0 10 20 30 40 50 60 70 80 90 10010.500.511.52gj(ai)Sigmoids used in BFELECTRE TRI modeling f1(.)f2(.)Fig. 2 f1(gj(ai)) andf2(gj(ai)) sigmoids.It is interesting to note the resemblance of Fig. 2 with Fig. 1. From these sig-moids, the bbas m1(.) and m2(.) are computed according to Table 1 and shown onthe Figure 3.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 41140 10 20 30 40 50 60 70 80 90 10010.500.511.52gj(ai)bba m1(.) m1(c)m1 (not c)m1 (c U notc)0 10 20 30 40 50 60 70 80 90 10010.500.511.52gj(ai)bba m2(.) m2(c)m2 (not c)m2 (c U notc)Fig. 3Bbas m1(.) and m2(.) to combine.The construction of the consistent bba mjih(.) is obtained by the PCR5 fusion ofthe bbas m1(.) and m2(.). The result is shown on Fig. 4.0 10 20 30 40 50 60 70 80 90 10010.500.511.52gj(ai)bba mPCR5(.) = PCR5 fusion of m1(.) with m2(.) mPCR5(c)mPCR5 (not c)mPCR5 (c U notc)Fig. 4mjih(.) obtained from the PCR5 fusion of m1(.) with m2(.).From this new sigmoidal modeling, we can compute the local bbas mjih(.) de-rived from the knowledge of criterion gj(.) and setting parameters. This is a smoothappealing and elegant technique to build all the local bbas: no hard thresholding isnecessary because of the continuity of sigmoid functions.One can then compute the global concordance and discordance indexes of steps1 and 2 from the computation of the combined bba mih(.) resulting of the fusion oflocal bbas mjih(.) taking eventually into account their importance and reliability6(ifone wants). This can be done using the recent fusion techniques proposed in [9],orbyasimpleweighted averaging. Frommih(.)wecanusethesamecredibilityindex as in step 3 of ET, or just skip this third step and dene a decision-makingbased directly on the bba mih(.) using classical approaches used in belief functionframework (say the max of belief, plausibility, or pignistic probability, etc).6In classical ET, the reliability of criteria is not taken into account.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 41155 ConclusionsAfter a brief presentation of the classical ET method, we have proposed a new ap-proach to model and compute the concordance and discordance indexes based onbelief functions in order to overcome the limitations of steps 1 and 2 of the ET ap-proach. The advantages of our modeling is to provide an elegant and simple way notonly to compute the concordance and discordance indexes, but also the uncertaintylevel that may occur when information appears partially concordant and discordant.The Improvements of other steps of ET method are under development. In futurereaserch works, we will evaluate and compare on real MCDA problem our BF-ETwith the original ET method and with other belief functions based methods alreadyavailable in MCDA frameworks [10, 11].References1. Dezert, J., Smarandache, F.: Proportional Conict Redistribution Rules for InformationFusion. In: [8], vol. 2, pp. 368 (2006)2. Dezert, J., Smarandache, F.: An introduction to DSmT. In: [8], vol. 3, pp. 373 (2009)3. Dezert, J.,Smarandache, F.,Tacnet, J.-M., Batton-Hubert, M.:Multi-criteriadecisionmakingbasedonDSmT/AHP. In:InternationalWorkshoponBeliefFunctions,Brest,France (April 2010)4. Dezert, J., Liu, Z., Mercier,G.: Edge Detection in Color Images Based on DSmT. In:Proceedings of Fusion 2011, Chicago, USA (July 2011)5. Figueira, J., Mousseau, V., Roy, B.: ELECTRE methods. In: Multiple Criteria DecisionAnalysis: State of Art Surveys, ch. 4. Springer Science+Business Media Inc. (2005)6. Mousseau, V., Slowinski, R., Zielniewicz, P.: Electre tri 2.0a - methological guide andusers manual - document no 111. In: Cahier et Documents du Lamsade, Lamsade, Uni-versit e Paris-Dauphine, Paris (1999)7. Shafer, G.: A mathematical theory of evidence. Princeton University Press (1976)8. Smarandache, F., Dezert, J.: Advances and applications of DSmT for informa-tionfusion(Collectedworks),vol.1-3.AmericanResearchPress(2004-2009), http://fs.gallup.unm.edu/DSmT.htm9. Smarandache, F., Dezert, J., Tacnet, J.-M.: Fusion of sources of evidence with differentimportances and reliabilities. In: Proc. of Fusion 2010 Conf., Edinburgh, UK (July 2010)10. Tacnet, J.-M., Dezert, J.: Cautious OWA and Evidential Reasoning for Decision Makingunder Uncertainty. In: Proceedings of Fusion 2011, Chicago, USA (July 2011)11. Tacnet, J.-M., Batton-Hubert, M., Dezert, J.: A two-step fusion process for multi-criteriadecision applied to natural hazards in mountains. In: Proceedings of International Work-shop on the Theory of Belief Functions, Belief 2010, Brest (France), April 1-2 (2010)12. Tervonen, T., Figueira, J.R., Lahdelma, R., Dias, J.A., Salminen, P.: Astochasticmethodforrobustnessanalysisinsortingproblems. EuropeanJournal ofOperationalResearch 192, 236242 (2009)13. Yu, W.: Aide multicrit` ere ` a la d ecision dans le cadre de la probl ematique du tri: Concepts,m ethodes et applications. Ph.D Thesis, University Paris-Dauphine, France (1992)Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4116