MODELING INTERACTIONS BETWEEN CRITERIA IN MCDA: A COMPARATIVE ANALYSIS OF THE BIPOLAR CHOQUET INTEGRAL AND AN ELECTRE METHOD

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1 a 9 Setembro de 0 MODELING INTERATIONS BETWEEN RITERIA IN MDA: A OMPARATIVE ANALYSIS OF THE BIPOLAR HOQUET INTEGRAL AND AN ELETRE METHOD Luiz Flavio Autra Moteiro Gomes Ibmec/Rio, Av. Presidete Wilso, 8, th floor, , Rio de Jaeiro, RJ, Brazil, Phoe: 00, s: autra@ibmecrj.br Maria Augusta Soares Machado Ibmec/Rio, Av. Presidete Wilso, 8, th floor, , Rio de Jaeiro, RJ, Brazil, Phoe: 00, s: mmachado@ibmecrj.br Luis Alberto Duca Ragel EEIMVR/UFF, Av. dos Trabalhadores, 0, Vila Sata ecília,volta Redoda, RJ, Brazil, Phoe: 070, duca@metal.eeimvr.uff.br Reato Mote Araujo Lojas Reer S.A, Av. Joaquim Porto Villaova, 0, Jardim arvalho, 90-00, Porto Alegre, RS, Brazil, Phoe: -900, reatomote@uol.com.br ABSTRAT The focus of this paper is the use of a bi-capacity model for approachig the computatio of iteractios betwee criteria withi the framework of Multiple riteria Decisio Aidig.. This model takes ito cosideratio bipolar scales. The paper itroduces the hoquet itegral i the bipolar scale. The core of the paper is a comparative aalysis of the uses of the bipolar hoquet itegral ad the ELETRE IV method as decisio aidig tools i a suppliers rakig problem. The matrix of performaces is made explicit ad the bipolar hoquet itegral is used. The rakigs of alteratives produced by the bipolar hoquet itegral ad the use of the ELETRE IV method are the compared. The types of iteractio are explaied ad the compariso is carried out with a sesitivity aalysis over the orderig of criteria. The paper closes with coclusios withi the cotext of the problem uder aalysis. KEYWORDS. hoquet itegral, ELETRE IV method, Multicriteria Decisio Aid. RESUMO O foco deste artigo é o uso de um modelo bi-capacidade para abordar-se o cálculo das iterações etre critérios o Apoio Multicritério à Decisão. O modelo emprega escalas bipolares. O artigo itroduz a itegral de hoquet a escala bipolar. O cere do artigo é uma aálise comparativa etre os usos da itegral de hoquet bipolar e o método ELETRE IV como ferrametas de apoio à decisão em um problema de ordeação de forecedores. Explicita-se a matriz de desempehos e faz-se uso da itegral de hoquet bipolar. omparam-se em seguida as ordeações das alterativas produzidas por aquela itegral e pelo emprego do método ELETRE IV. Explicam-se os tipos de iterações e efetua-se a comparação com uma aálise de sesibilidade sobre as ordeações dos critérios. oclui-se o artigo com cosiderações pertietes ao cotexto da aálise realizada. PALAVRAS HAVE. Itegral de hoquet, Método ELETRE IV, Apoio Multicritério à Decisão. Área pricipal: ADM - Multicriteria Decisio Support

2 a 9 Setembro de 0. Itroductio Oe importat problem i Multiple riteria Decisio Aidig (MDA) is the existece of iteractios betwee criteria. Whe usig a multiple attribute utility (or value) theoretical model there are requiremets that must be met if, for example, a liear additive fuctio is to be used (leme & Reilly, 00, pp. 7-). A importat mathematical model that has bee used for modelig iteractios betwee criteria is the hoquet itegral (hoquet, 9). I decisio theory, the hoquet itegral is a way to measure the expected utility of a ucertai evet (Gilboa & Schmeidler, 99). The hoquet itegral is ideed a geeralizatio of the weighted arithmetic mea ad has bee extesively used sice the last decade i Multiple riteria Decisio Aidig i modelig iteractios betwee criteria (Grabisch, 99; Grabisch, 00; Grabisch & Labreuche, 00, 00). Marichal (000) also modeled iteractios betwee criteria with the hoquet itegral. A critical aalysis of the use of the hoquet itegral for modelig iteractios betwee criteria was preseted by Roy (009) though. This author has poited out that the geeralizatio of the hoquet itegral kow as bipolar model (or model with bicapacities) should be utilized i order to capture some particular aspects of iteractios betwee criteria. I this paper we show a compariso of the use of the bipolar hoquet itegral agaist the use of the ELETRE IV method. This is a outrakig method that does ot require kowledge of criteria weights (Roy & Bouyssou, 99). The compariso is accomplished for a rakig problem uder multicriteria. The paper starts by reviewig the hoquet itegral i the uipolar scale as a multicriteria rakig model. The it moves to presetig the hoquet itegral i the bipolar scale as a equivalet model. Next the compariso betwee the use of the hoquet itegral i the bipolar scale is compared agaist the results obtaied from usig ELETRE IV for a suppliers rakig problem uder multiple criteria. Our approach is therefore differet from that of Figueira, Greco & Roy (009), who exteded the otio of cocordace idex that is ot preset i ELETRE IV. These three authors have show that the hoquet itegral ca be applied i order to build a trasitive ad complete pre-order relatio of type existig i the ELETRE methods. A extesio of the cocordace idex of ELETRE methods i order to deal with iteractios betwee criteria was the proposed.. The hoquet itegral i the bipolar scale Give a fiite set J = {,,...,} a fuzzy measure μ is a fuctio of the form: μ : J [ 0,] such that μ ( φ) = 0 ad (J) = (boudary coditios); () (D) if D,, D J (mootoicity coditio). Let P(J) be a set of pairs of subsets of J: P ( J ) = {(, D),, D J, D = φ}. A bi-capacity μ i J is a fuctio μ : P( J ) [0,} X[0,] such that μ (, φ) = ( c,0) ad μ ( φ, D) = ( 0, D), c, d [0,] ; μ( J, φ ) = (,0 ) ad μ ( φ, J ) = ( 0,) (boudary coditios); for each (, D),( E, F) P( J ) such that E, D F we have μ (, D) = ( c, d ) ad μ ( E, F ) = ( e, f ), c, d, e, f [0,] with c e ad d f (mootoicity coditio). Here we use the followig otatio: μ, D = (, D) = d. A bi-capacity μ, o the set J, is a fuctio μ : P( J ) [, ] ( ) c such that μ ( φ, φ) = 0 ; μ( J, φ) = ad ( φ, J ) = μ (boudary coditios); if E, D F, the μ(, D) μ( E, F) (mootoicity coditio). From each bi-polar capacity μ i J, we ca obtai a bi-capacity μ i J: μ (, D) = μ (, D) μ (, D),, D P( J ) (Greco & Figueira, 00). For each x R : x = max{x,0} is the positive part of x, for each x R ;

3 a 9 Setembro de 0 x = max{ x,0} is the egative part of x, for each positive part of x = } { x, x, x,,..., x R ; x { x, x, x,,..., x } x R, x = { x, x, x,,..., x } is the = is the egative part of x( x, x, x,,..., x ) R. Give x R we cosider a permutatio (.) of the elemets of J, such that x x... x x. For each elemet j J we have two subsets ( ) () ( j) ( ) { } { } ( j) = i J : x i x( j ) ad D( j) i J x i x( j ) = :. osiderig a bi-capacity μ i J ad a vector x R we ca defie its bipolar hoquet itegral of the positive part as follows: > h x, = x x μ, D J = j J / > 0. I the same way >( ( ) ( ) ( j ) ( j ) ( j) ( j ) μ, where { } j J we formulate the bipolar hoquet itegral of the egative part as h x = x x μ, D. Therefore the bipolar hoquet itegral is >( ( )) ( ) ( j ) ( j ) ( j ) ( j ) j J ( x) ( x, μ) h ( x, μ) h B =. Other authors have used the bipolar hoquet itegral for tacklig differet problems. For example, Greco & Ridoe (0) exteded the bipolar hoquet itegral represetatio towards bipolar umulative Prospect Theory. Greco & Ridoe (0) proposed the bipolar hoquet itegral for the case i which the uderlyig scale is bipolar ad provided a characterizatio of bipolar fuzzy itegrals. x j. omparig the use of the bipolar hoquet itegral agaist ELETRE IV To compare the hoquet itegral usig bipolar scale agaist ELETRE IV we make use of the decisio matrix take from the doctoral thesis of Alecar (00). This last author aalyzed the problem of rakig suppliers withi the cotext of group decisio makig by applyig ELETRE IV. Table displays the judgmets by experts o the relative importace of each supplyig firm with respect to every evaluatio criterio. Those criteria are the followig: = ost, = ulture, = Desig, = Quality, = Time, = Experiece. The criteria order is: > > > = =. riteria Firm # Firm # Firm # Firm # Firm # Firm # Table - Decisio matrix of riteria versus Alterative supplyig firms I order to ru a sesitive aalysis over the computatio of the bipolar hoquet itegral we adopt two hypothesis: (A) hypothesis # (H ): > > > = = with the followig iteractios betwee criteria: μ μ = μ = μ μ, μ 8μ, give that μ i = ; (B) hypothesis # (H ): > = > = = i= with the followig iteractios betwee criteria: μ μ = μ,

4 a 9 Setembro de 0 μ μ μ = μ bipolar hoquet itegral. =, give that μ i =. Next, we show the steps for computig the i= Step Determiatio of fuzzy measures I Table we preset the fuzzy measures for each hypothesis ad for every criterio. Idividual fuzzy measures Fuzzy measures H /H riteria μ 0./ 0. μ 0.09/ 0.8 μ 0.09/ 0.8 μ 0.09/ 0. μ 0./ 0. μ 0.8/ 0. Table - Fuzzy measures for each hypothesis ad for every criterio Step Determiatio of the fuzzified decisio matrix The fuzzified decisio matrix is obtaied by multiplyig each elemet of decisio matrix by its fuzzy measure. I Table we preset the results for hypothesis H. A equivalet table has bee determied for hypothesis H. riteria Firm # Firm # Firm # Firm # Firm # Firm # Table - Fuzzified decisio matrix for hypothesis H Step : omputatio of the bipolar hoquet itegral I Table we display all the computatioal results for hypothesis H. Equivalet results are obtaied for hypothesis H. riteria Firm # Firm # Firm # Firm # Firm # Firm #

5 a 9 Setembro de 0 Mi operator.9 E E E E E E-0 Max operator Values of the bipolar hoquet itegral Rakig of alteratives Table - omputatio of the bipolar hoquet itegral I table we show the comparisos betwee the ELETRE IV ad the bipolar hoquet rakigs. Rakig from ELETRE IV assumig that Rakig from bipolar hoquet for H, Rakig from bipolar hoquet for H, > > > = = > > > = = > = > = = Firm # Firm # Firm # Firm # Firm # Firm # Firms # ad Firm # Firm # Firm # Firm # Firm # Firm # Firm # Firm # Firm # Firm # Table - compariso of results from ELETRE IV agaist bipolar hoquet. Results It ca be see that by usig bipolar hoquet itegral a complete pre-order is produced ad rakig is differet from that obtaied by ELETRE IV. I both hypotheses, whe > > > = = (H ) ad whe > = > = = (H ), bipolar hoquet itegral leads to the same rakig. Accordig to both approaches, Firm # raks as secod alterative. Firm # has low cost ad low delay as compared agaist the other alteratives. Usig ELETRE IV leads to the coclusio that Firm # raks as first alterative. Firm # has lowest cost ad cost ( ) is the most importat. osiderig time, the secod most importat criterio ( ), Firm # has the lowest value. Firm # raks i the fifth positio by usig bipolar hoquet. By usig bipolar hoquet Firm # is the best alterative, although its cost is three times higher tha Firm #. O the other had, Firm # has four times more experiece ( ) tha Firm #, ad this is ideed the third most importat criterio. Firm # is also oe ad a half times lower i time ( ), that is the secod more importat criterio; it also has more quality ( ), that is the less importat criterio. Firm # has lower culture ( ), that is the fourth importat criterio, ad desig ( ), that has relatively low importace as compared agaist other criteria. I the third positio we have, from usig ELETRE IV, a tie: Firms # ad. From usig bipolar hoquet we obtai Firm # i that third positio. It would be fair to say that bipolar hoquet leads to hoquet bipolar is more realistic whe we cosider that Firm # has lower price ad higher experiece. ELETRE IV classifies Firm # i the fourth positio ad hoquet bipolar classifies Firms # i that same positio. This firm has higher cost but has more experiece, better time, desig ad culture. Fially, Firm # is classified i the fifth positio accordig to ELETRE IV ad bipolar hoquet classifies Firm # i the first positio as commeted before.. oclusios We have show i this paper that applicatios of the bipolar hoquet itegral ad ELETRE IV are comparable, with a advatage to that first model whe a bipolar scale is used.

6 a 9 Setembro de 0 The tradeoffs betwee criteria were explaied i a realist way ad measures of iteractios were associated to the fuzzy measures related to two differet hypotheses. For this particular umerical example we saw that iteractios betwee criteria were ideed quite relevat. Aother importat dimesio of the advatage of usig bipolar hoquet istead of ELETRE IV was that this method has distictly grouped all firms. The key coclusios from this case study are listed below: (i) the use of the hoquet itegral miimizes the calculatios by TODIM sice it is uecessary to ormalize the raw data; (ii) ot oly precise values ca be used but also iterval data; this secod situatio would lead to usig a fuzzy triagular umber; (iii) by usig the hoquet itegral more complex additive models ca be used that allow for takig depedecies betwee criteria ito cosideratio. lies: More research alog the lie pursued i this paper will likely proceed alog two major (i) (ii) carryig out similar comparisos for larger problems; ad focusig the compariso o multiple criteria decisio aidig models that rely o bipolar umulative Prospect Theory (Greco & Ridoe, 0). I particular, ew cocepts such as geeralizatios of hoquet itegral as well as the bipolar umulative Prospect Theory should be cosidered. Ackowledgemet Research leadig to this article was partially supported by the Natioal oucil for Scietific ad Techological Developmet (NPq) of Brazil through projects 07/0-9 ad N.07/009-. Refereces Alecar, L.H. (00), Multicriteria group decisio model for the selectio of suppliers i project maagemet. Doctoral thesis i Productio Egieerig, Federal Uiversity of Perambuco, Recife: UFPE. hoquet, G. (9), Theory of apacities. Aales de l Istitut Fourier, -9. leme, R.T. & Reilly, T. (00), Makig Hard Decisios with Decisio Tools. d rev. ed., Pacific Grove: Duxbury. Figueira, J.R.; Greco, S. & Roy, B. (009), ELETRE methods with iteractio betwee criteria: A extesio of the cocordace idex. Europea Joural of Operatioal Research, 99, Gilboa, I. & Schmeidler, D. (99), Additive represetatios of o-additive measures ad the hoquet itegral. eter for Mathematical Studies i Ecoomics ad Maagemet, Evasto: Northwester Uiversity. Grabisch, M. (99), The applicatio of fuzzy itegrals i multicriteria decisio makig. Europea Joural of Operatioal Research, 89, -. Grabisch, M. (00), L utilisatio de l itegrale de hoquet e aide multicritère à la décisio. Europea Workig group Multiple riteria Decisio Aidig, Series, o, Fall, - 0. Grabisch, M. & Labreuche,. (00), Fuzzy measures ad itegrals i MDA. I: Multiple riteria Decisio Aalysis State of the Art Surveys, Figueira, J.; Greco, S. & Ehrgott, M. (eds.), -08, Series: Iteratioal Series i Operatios Research & Maagemet Sciece, New York: Spriger ScieceBusiess Media, LL. 7

7 a 9 Setembro de 0 Grabisch, M. & Labreuche,. (00), A decade of applicatio of the hoquet ad Sugeo itegrals i multi-criteria decisio aid. Aals of Operatios Research, 7,, S. Greco, S., Matarazzo, B. & Giove, S. (0) The hoquet itegral with respect to a level depedet capacity. Fuzzy Sets ad Systems 7, -. Olie versio, doi: 0.0/j.fss Greco, S. & Ridoe, F. (0), Bipolar fuzzy itegrals, i: <arxiv:0.9v [math.fa] Apr 0>, captured i April 0, 0. Greco, S. & Ridoe, F. (0), The bipolar hoquet itegral represetatio, i: < captured i April 0, 0. Marichal, J.-L. (000), Behavioral aalysis of aggregatio i multicriteria decisio aid, i: Prefereces ad Decisios uder Icomplete Kowledge, J. Fodor, J.; B. De Baets, B. & Pery, P. (eds.), -78. Series: Studies i Fuzziess ad Soft omputig,, Heidelberg: Physica Verlag. Roy, B. (009), À propos de la sigificatio des dépedeces etre critères: quelle place et quels modes de prise em compte pour l aide à la décisio? RAIRO,, -7. Roy, B. & Bouyssou, D. (99), Aide Multicritère à la Décisio : méthodes et as, Paris: Ecoomica. 8