INTEGRATING IMPERFECTION OF INFORMATION INTO THE PROMETHEE MULTICRITERIA DECISION AID METHODS: A GENERAL FRAMEWORK

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1 F O U N D A T I O N S O F C O M P U T I N G A N D D E C I S I O N S C I E N C E S Vol. 7 (0) No. DOI: 0.478/v INTEGRATING IMPERFECTION OF INFORMATION INTO THE PROMETHEE MULTICRITERIA DECISION AID METHODS: A GENERAL FRAMEWORK Sara BEN AMOR*, Bertrand MARESCHAL** Abstract. Multicriteria decision aid metods are used to analyze decision problems including a series of alternative decisions evaluated on several criteria. Tey most often assume tat perfect information is available wit respect to te evaluation of te alternative decisions. However, in practice, imprecision, uncertainty or indetermination are often present at least for some criteria. Tis is a limit of most multicriteria metods. In particular te PROMETHEE metods do not allow directly for taking into account tis kind of imperfection of information. We sow ow a general framework can be adapted to PROMETHEE and can be used in order to integrate different imperfect information models suc as a.o. probabilities, fuzzy logic or possibility teory. An important caracteristic of te proposed approac is tat it makes it possible to use different models for different criteria in te same decision problem. Keywords: multiple criteria analysis, multicriteria decision aid, imperfection of information, PROMETHEE. Introduction We consider discrete multicriteria decision problems were A = {a, a,..., a n } is a set of n alternative decisions (actions) tat are evaluated on a set of k criteria denoted by f, f,.., f k. Te actions represent eiter potential decisions among wic a coice as to be made or items to evaluate. Te criteria express te different and often conflicting obectives of te decision-maker. Witout loss of generality, we may assume tat tese criteria are realvalued and ave to be maximized. Te multicriteria decision problem can tus be written as follows: * Telfer Scool of Management, University of Ottawa, Ottawa, Canada. (benamor@telfer.uottawa.ca) ** Solvay Brussels Scool of Economics and Management, Université Libre de Bruxelles, Brussels, Belgium (bmaresc@ulb.ac.be). Download Date /8/9 :5 PM

2 0 S. Ben Amor, B. Marescal max,,, k f a f a f a a A () Were we note te evaluation of action a i on criterion f as follows: i i e f a () Different approaces ave been proposed to treat multicriteria problems. Tey are based eiter on te absolute evaluation of te alternatives (suc as multi-attribute utility (MAUT) or value functions metods) or eiter on pair-wise comparisons (a.o. outranking metods). Te interested reader can for instance consult [6] for additional information. In tis paper we focus on te PROMETHEE and GAIA outranking metods ([], [], [4]). According to [] and to te online bibliograpical database available at ttp://biblio.prometee-gaia.net, tese metods are amongst te most widely used in practice. Tey ave also been implemented in several recognized software suc as PROMCALC, Decision Lab and more recently te PROMETHEE Software. Wit respect to oter outranking metods, suc as ELECTRE ([6]), PROMETHEE is based on a relatively simple principle tat makes it easier to extend for taking into account imperfection of information. It also includes te GAIA descriptive approac. PROMETHEE is a prescriptive metod tat enables to rank te actions according to te preferences of te decision-maker. Actually two rankings are produced: a partial ranking is built mostly on undisputable preferences and a complete, possibly less robust, ranking can also be obtained depending on te decision-maker s requirements. GAIA is a descriptive metod. It complements PROMETHEE by providing te decision-maker wit a syntetic visual representation of te main caracteristics of te decision problem, suc as te conflicts existing between te criteria and te specific profiles of te actions. GAIA can also play a role in te determination of te decision-maker priorities as te weigts of te criteria and teir impact on te PROMETHEE rankings are displayed. As wit most multicriteria decision aid metods PROMETHEE assume tat perfect information is available for all te actions and all te criteria. Tat means tat as well te evaluations of te actions on te criteria as te various preference parameters are known as exact numbers. In practice, tis is not always te case. Tere can be missing evaluations or imprecise information. We only consider ere te case of te evaluations. For different reasons, tis information can be difficult to obtain: tere can be uncertainty on te future outcome of decisions, tere can be some imprecision according to te evaluation procedure, or tere can be indetermination associated to te measurement scales. Some attempts ave been made to take tis into account in te PROMETHEE metods. For instance missing values are integrated into te Decision Lab software. A stocastic extension of te PROMETHEE metods as also been proposed in [8] but it was limited to stocastic modeling and as not been implemented. In section II we propose a general framework for andling different types of imperfection of information. We consider different modeling languages tat can be used ointly in tis context. Indeed several metods rely on probability teory to model te imperfection of information, some oters rely on fuzzy logic, but very few allow te use of different modeling languages in te same framework []. In fact, various models are available suc as possibility teory, fuzzy logic, interval aritmetic or te teory of evidence. Tey differ by teir underlying assumptions and by te kind of information tey Download Date /8/9 :5 PM

3 Integrating imperfection of information into te PROMETHEE multicriteria... require from te decision maker. Te proposed framework makes it possible to use te best suited model for eac criterion in a given decision problem and to tus combine all te models as needed. In section III we recall te basic principles underlying te PROMETHEE & GAIA metods. A more complete description of tese metods, including a complete numerical example, can be found in [4]. In section IV we sow ow te general framework of section II can be integrated in PROMETHEE. Section V concludes te paper. It summarizes te possibilities of te proposed approac and outlines future researces.. A general framework for andling imperfection of information We consider only te imperfection of information appearing in te evaluations e i. Te general framework we use accepts, in te uncertain decision-making situations, stocastic, possibilistic or "evidential" evaluations. It also includes fuzzy or ordinal evaluations. Among all uncertainty-modeling languages (evidence teory, possibility teory and probability teory), te evidence teory presents te more general framework were possibilities and probabilities are proposed as particular cases. Tis property is used to settle te modeling framework. Let us consider imperfect information for criterion f. For eac alternative a i, te evaluation e i of a i on f will depend on te actual state of te nature witin a set of H possible nature states: We note te value of e i under nature state and x i,,,, () H H X x,, x,, x (4) i i i i Tis model assumes tat we are able to indicate te consequence of coosing an alternative a i wen te state of te nature (=,...,H) occurs, i.e. wen x i is obtained. Te a priori information available about te states of te nature (probability functions, belief masses, possibility measures, ) can be integrated as in Table. Tis table is based on a teory of evidence model settings [], were a priori information is represented by belief masses associated to focal elements. Te focal elements, wic are subsets of te states of te nature set:,,,, ', ' H B ' H (5) H determine te evaluations of te alternatives according to criterion f. Te evaluation e i of action a i on f is represented by a subset of values: Download Date /8/9 :5 PM

4 S. Ben Amor, B. Marescal C ' i X (6) i wic depends on te subsets of te states of te nature (focal elements) B. ' Table. Performance matrix for criterion f integrating te a priori information Alternatives Focal elements B B ' B H ' ' H ' a C C C ' H ' a i Ci Ci C i H ' C C a n A priori belief masses n ' n C n By using tis modeling framework, we can consider possibilistic or probabilistic models as particular cases wen te a priori information is represented by possibility or probability distributions respectively. In fact, wen te criterion is a possibilistic one, te a priori information related to te evaluation e i is caracterized by possibility distributions. In suc a context, te corresponding belief masses are associated wit focal elements tat are embedded: B B B (7) ' H ' Te possibility measures coincide wit te plausibilities of te embedded focal elements. Wen te criterion f is stocastic, te focal elements B ' are reduced to te singletons and te corresponding belief masses correspond to probability measures. In tis case, te evaluations are caracterized by random variables X i wit a priori probability distributions f i. We ten ave te a priori (subective) probabilities P for eac state of te nature (=,,H). Te next step of our procedure consists of establising te local preference relations between two actions. Let H be te local relation between two actions a i and a l on criterion f. We ave: i l ; ; ;? were is te indifference relation, is te a H a were H strict preference relation, is te inverse strict preference relation and? is te incomparability relation. In order to construct tese relations, we privilege te approac based on te extension of te stocastic dominance concept. Te use of te stocastic dominance concept as te advantage of providing a uniform treatment for te different languages wic express te information imperfections. Notice tat stocastic dominance allows to conclude about te preference of an action over anoter action for a decision-maker wose attitude towards Download Date /8/9 :5 PM

5 Integrating imperfection of information into te PROMETHEE multicriteria... risk corresponds to DARA (Decreasing Absolute Risk Aversion) utility functions [9]. Tis type of risk aversion is observed for several economic penomena. Te link between stocastic dominance and te preference is well known for tis class of utility functions. It is not always easy to make te decision-maker s preferences explicit and we can often conclude tat is preferred to if some stocastic dominance conditions are verified. We consider te tree first degrees of stocastic dominance. Tese are defined for continuous and discrete random variables in te following way. Let us note respectively x * * and x te lower and upper limits of te evaluation scale of criterion f. F i and F l are te cumulative probability distributions for te evaluations of alternatives a i and a l troug criterion f and x is a modality of tis criterion. Definition : First order stocastic dominance (FSD) F i FSD F l if and only if * 0 i l, H x F x F x x x x (8) * Definition : Second order stocastic dominance (SSD) F i SSD F l if and only if x * 0 *, x * H x H y dy x x x (9) Definition : Tird order stocastic dominance (TSD) F i TSD F l if and only if x * 0 *, x * H x H y dy x x x (0) In te case were te evaluation e i is a random variable, te results of stocastic dominance can be directly applied in order to establis preference relations. For instance, we could say tat: * =, e e H x F x F x s x x x () i l i l * were s > 0 is a predetermined tresold. Oterwise, for cases were H x s, ( Fi Fl ), we can say tat: e e F FSD or SSD or TSD F () i l i l e? e F non SD F () i l i l were SD means tat one of te tree dominance types is verified. Download Date /8/9 :5 PM

6 4 S. Ben Amor, B. Marescal Tese concepts can be extended to ambiguous probabilities [7]. For fuzzy, possibilistic or evidential criteria, we propose te use of some transformations wic provide to te functions caracterizing te evaluation of an alternative on a given criterion (fuzzy membersip functions, possibility distributions, belief masses, ) properties similar to tose of a probability density function. For evidential criteria we use te pignistic transformation ustified by Smets [], [4]: BetP ' ': ' ' B B B B P, m m B ' m (4) were B ' is te cardinality of in B ' and BetP( ) is te pignistic probability of. For possibilistic criteria we use te relation proposed by Dubois and al. [a]. Tis relation is equivalent to te pignistic transformation wen applied to possibility measures. Te probability p corresponding to is given by: p H H t t m (5) t t t t t were m is te belief mass m(b ), B are embedded and organized in sequence, i.e. B, B,,, B,,,,, BH suc tat B B : m B 0 and it is possible for a certain tat m B 0, and is te possibility measure defined for : H m (6) For fuzzy criteria we use te rescaling approac as proposed in Munda [0]: t f x k e i i i i i were f x dx i i i Let us note tat te construction of local preference relations for deterministic criteria can be carried out by using discrimination tresolds to distinguis between strict preference and indifference situations (quasi-criterion notion) in te context of punctual evaluations. In tis case, locally tere is no incomparability. It is one of te caracteristics of a criterion []. t (7) Download Date /8/9 :5 PM

7 Integrating imperfection of information into te PROMETHEE multicriteria Te PROMETHEE Metods.. Basic principles Te PROMETHEE metods are based on a principle of pair-wise comparisons of te actions. We recall ere te basic principles of te metods. A complete description can be found in [4] or on te PROMETHEE web site (ttp:// In a first step, a preference function as to be associated wit eac criterion in order to reflect te perception of te criterion scale by te decision-maker. Usually te preference function P (a i,a l ) is a non-decreasing function of te difference f (a i )f (a l ) between te evaluations of two actions a i and a l. Several typical sapes of preference functions are proposed in te literature ([],[]) and indications are given on te way to select appropriate functions for different types of criteria. Te value of P (a i,a l ) is a number between 0 and. It corresponds to te degree of preference tat te decision-maker expresses for a i over a l according to criterion f : 0 corresponds to no preference at all wile corresponds to a full preference. In a second step te decision-maker assesses numerical weigts to te criteria to reflect te priorities: more important criteria receive larger weigts. We note w te weigt of criterion f and we assume tat te weigts are normalized as follows: k w (8) A multicriteria pair-wise preference index is ten computed as a weigted average of te preference functions: k,, a a w P a a (9) i l i l Tree preference flows are ten computed in order to globally evaluate eac action wit respect to all te oter ones. Te leaving flow is a measure of te strengt of an action a i wit respect to te oter ones: n (0) a a, a i i l n l Te entering flow measures te weakness of action a i wit respect to te oter ones: n () a a, a i l i n l Finally te net flow is te balance between te two first ones: Download Date /8/9 :5 PM

8 6 S. Ben Amor, B. Marescal n ai ai, al al, ai n l () a a i Eac preference flow induces a ranking on te set of actions. Obviously te best actions sould ave a ig + value (close to ) and a low value (close to 0), and tus a ig positive value. Te PROMETHEE rankings are based on te preference flows. Tey are discussed in te next section. Te GAIA descriptive metod is based on unicriterion net preference flows. Tese are computed in a similar way to (6) but for eac criterion separately. For criterion f, te unicriterion net flow is defined as follows: a P a, a P a, a i i l l i n l k ai w ai n i ().. PROMETHEE rankings Te PROMETHEE I ranking is obtained by looking at te leaving (0) and entering () flows and keeping te preferences tat are confirmed by bot flows. It is a partial ranking as te two flows usually give a different ranking of te actions because tey syntesize te pair-wise comparisons of te actions in two different ways. Teir common part can tus be considered as more reliable. Te PROMETHEE II complete ranking is obtained by te net flow values (). All actions are compared but te differences between te leaving and entering flows are lost, leading to a possibly less robust ranking... GAIA Te PROMETHEE analysis is prescriptive. It relies on te preference parameters determined by te decision-maker. Canges in tese parameters, especially te weigts of te criteria, can ave an important impact on te PROMETHEE rankings. Te GAIA analysis is based on te unicriterion net flows (). Eac action is ten represented by a point in te k-dimensional space defined by tese flows. A principal components analysis is ten applied to tese points to obtain a two-dimensional representation of te decision problem ([], [6]). Unit axes for te criteria are also proected on te GAIA plane in order to sow te conflicts between te criteria. Download Date /8/9 :5 PM

9 Integrating imperfection of information into te PROMETHEE multicriteria Imperfection of information in te PROMETHEE metods Introducing imperfection of information in te PROMETHEE metods can be done at different levels. Indeed te PROMETHEE metods rely not only on te evaluations of te actions on te criteria but also on preference modeling parameters suc as te preference function tresolds and te weigts of te criteria. For te latter, sensitivity analyses are one possible way to take into account te imperfection of information. Suc analyses are made possible in te PROMETHEE software ([4]). We terefore focus on te evaluations. As it is sown in section III, te PROMETHEE metods rely on pair-wise comparisons of actions. In case of perfect information, te evaluations are real numbers wose values are known precisely. Te values of te preference functions P (a i,a l ) are ten computed directly from te deviations between te evaluations. Let us note: p P a, a (4) il i l From te definition of te PROMETHEE preference functions, we ave te following properties: 0 p and p p (5) il il li Moreover bot coefficients cannot be greater tan 0 at te same time as in te deterministic case, preference is only possible in one way. As PROMETHEE uses valued preferences, we are limited to te situations in Table. In particular, no incomparability is possible in tis case. Table. PROMETHEE local preference relations in te deterministic case p il pli Local preference relation 0 0 Indifference 0 - strict preference 0 < < 0 weaker preference 0 - inverse strict preference 0 0 < < inverse weaker preference In case of imperfect information, an alternative way of computation of te P (a i,a l ) values must be defined. We propose to use te local preference relations as tey are defined in Section III based on te general framework and on stocastic dominance. Tis approac corresponds to Table were te PROMETHEE preference coefficients are determined according to te local preference relation. Table. PROMETHEE preference coefficients in imperfect information case p il p li Local preference relation Indifference strict preference 0 - inverse strict preference 0? - incomparability Download Date /8/9 :5 PM

10 8 S. Ben Amor, B. Marescal In tis way, Table is completely compatible wit te use of te Usual preference function ([4]) in te deterministic case. Tis is particularly well suited for qualitative criteria. Imperfect information introduces te notion of incomparability as te local level as it becomes possible to esitate as to wic action sould be preferred to te oter wen teir evaluations are not known wit certainty. 5. NUMERICAL EXAMPLE Let s recall ere te example used in [] wit four alternatives a, a, a, a 4 and four criteria of different nature: f, f, f, f 4 were f is a stocastic criterion, f is an evidential criterion, f is a possibilistic criterion and f 4 is a fuzzy criterion. For eac criterion, we ave te corresponding performance matrix and te a priori information. Te stocastic dominance indifference tresolds for all te criteria are s = s = s = s 4 = 0.05, and te weigt distribution is: W = [0.4, 0., 0., 0.]. Table 4 gives te performance matrices for te four criteria. Te evaluations according to f 4 are linguistic variables represented by trapezoidal fuzzy numbers (a, b, c, d) (see figure ). Table 4. Performance matrices for f, f, f, f 4 a i 4 5 f 4 Associated fuzzy numbers a weak (0,., 0,.) a fair (.5,.5,.,.) a ig (.9,,., 0) a more or less weak (.,.,.,.) Figure. Membersip function for fuzzy trapezoidal number (a, b, c, d). Download Date /8/9 :5 PM

11 Integrating imperfection of information into te PROMETHEE multicriteria... 9 Table 5 gives te a priori information for criterion f, wile table 6 gives te a priori information for criteria f and f. For criterion f 4, te a priori information is implicitly contained in te performance matrix. Table 5. A priori information for f 4 5 P ' Table 6. A priori information for f (=) and f (=) J J J J J, J J, J J J J J,,, m B m B ' B ' Applying stocastic dominance results as in [] as led to te local preference relations H, =,,, 4 listed in Table 7. Table 7. Local preference relations H, =,,, 4. H H H H 4 a a a a 4 a a a a 4 a a a a 4 a a a a 4 a *? *?? * * a *? *? * * a? *?? * * * a 4? *? * * * Using te PROMETHEE preference coefficients suggested in Table, one can translate te obtained local preference relations H, =,,, 4 into numerical values as sown in Table 8. a, a are computed along wit Multicriteria pair-wise preference index i l PROMETHEE leaving flow a i, entering flow a i and net flow a i Table 9. as sown in Download Date /8/9 :5 PM

12 0 S. Ben Amor, B. Marescal Table 8. PROMETHEE preference coefficients for te numerical example H H H H 4 a a a a 4 a a a a 4 a a a a 4 a a a a 4 a * * * * a 0 * * * * 0 a * * 0 0 * 0 * a * * 0 0 * 0 0 * Table 9. PROMETHEE multicriteria indexes, leaving, entering and net flows Actions a a a a 4 a i a * a 0. * a * a *.5 a i a i Depending on te decisional problematic, leaving, entering and net flows can be exploited according to PROMETHEE I or PROMETHEE II. In fact, using te net flow we obtain te PROMETHEE II complete ranking: a, a, a, a 4. In tis case, te PROMETHEE I partial ranking induced by te leaving and entering flows is te same as te PROMETHEE II complete ranking. Anoter approac to deal wit imperfect information consists of replacing imperfect data by a punctual value tat summarizes imperfect information. Te use of expected values is one possible way of obtaining suc summary values. Wen applied to te example at and, tis approac leads to te following PROMETHEE I partial ranking and PROMETHEE II complete ranking: a, a, a, a 4. One can see tat tis ranking is different from te one obtained above using te proposed approac. It is wort noticing te very different position of a 4 in bot rankings. Tis can be explained by te larger uncertainty tat affects te evaluations of a 4 on te most important criteria f, f, f. Te proposed approac provides a lower, and tus, a more prudent ranking of a Conclusion Te proposed approac makes it possible to introduce ointly different modeling languages in order to take into account imperfection of information in te evaluation of te actions. It Download Date /8/9 :5 PM

13 Integrating imperfection of information into te PROMETHEE multicriteria... is tus possible to coose te best suited modeling language for eac criterion in a same decision problem. Te approac as also several limits. It corresponds to te use of te Usual preference function in te deterministic case and tus doesn t explicitly take into account te deviations between te evaluations in te pairwise comparisons. Future researc will investigate ow to integrate tis aspect in te proposed approac. Te approac could also be extended to take into account imperfection of information in oter multicriteria metods suc as for instance te ELECTRE metods ([6]). Imperfect information related to preference parameters (preference functions tresolds and criteria weigts) is also an interesting topic tat could be furter investigated. ACKNOWLEDGMENT We would like to tank Prof. Wotek Micalowski. Witout im, tis paper would probably ave never been written. REFERENCES [] M. Bezadian, R.B. Kazemzad, A. Albadvi D. and M. Agdasi, PROMETHEE: A compreensive literature review on metodologies and applications, European Journal of Operational Researc, in press. [] S. Ben Amor, K. Jabeur and J-M. Martel, Multiple criteria aggregation procedure for mixed evaluations, European Journal of Operational Researc, 8(), pp , 007. [] J.P. Brans and B. Marescal PROMCALC & GAIA: A new decision support system for multicriteria decision aid, Decision Support Systems,, pp.97-0, 994. [4] J.P. Brans and B. Marescal PROMETHEE Metods in Multiple Criteria Decision Analysis: State of te Art, edited by J. Figueira, S. Greco and M. Ergott, pp.6-96, Kluwer, 005. [5] D. Dubois, H. Prade and S. Sandri, On possibility/probability transformations, Lowen R, Roubens M (eds), Fuzzy Logic: State of te Art, Kluwer Academic Publ., Dordrect, pp.0-, 99. [6] J. Figueira, S. Greco and M. Ergott, Multiple Criteria Decision Analysis: State of te Art., Kluwer, 005. [7] A. Langewis and F. Coobine, Stocastic dominance tests for ranking alternatives under ambiguity, European Journal of Operational Researc, 95, pp.9-54, 996. [8] B. Marescal, Stocastic multicriteria decision-making under uncertainty. European Journal of Operational Researc 6 (), 58 64, 986. [9] B. Marescal and J.P. Brans Geometrical representations for MCDA, European Journal of Operational Researc, 9, pp. 84-9, 989. [0]J-M. Martel and K. Zaras, Stocastic dominance in multicriterion analysis under risk, Teory and Decision, 9, -49, 995. []G. Munda, Multicriteria evaluation in a fuzzy environment, Pysica-Verlag, Heidelberg, 995. Download Date /8/9 :5 PM

14 S. Ben Amor, B. Marescal []B. Roy, Métodologie Multicritère d Aide à la Décision, Economica, Paris, 985. []G. Safer, A Matematical Teory of Evidence, Princeton University Press, Princeton N.J., 976. [4]P. Smets, Decision making in te TBM: te necessity of te pignistic transformation, International Journal o Approximate Reasoning, 8, pp.-47, 005. [5]P. Smets, Constructing te pignistic probability function in a context of uncertainty, in: Henrion M, Sacter RD, Kanal LN and Lemmer JF (eds), Uncertainty in Artificial Intelligence 5, Nort Holland, Amsterdam, pp.9-40, 990. Received July, 0 APPENDIX Table A.. Performance matrix and expected values for f 4 E 5 a i a a a 5.8 a P Table A.. Performance matrix, pignistic probabilities and expected values for f E( a i ) a a a a ( ) BetP Download Date /8/9 :5 PM

15 Integrating imperfection of information into te PROMETHEE multicriteria... Table A.. Performance matrix, pignistic probabilities and expected values for f E( a i ) a a a a BetP( ) Table A.4. Performance matrix and expected values for f 4 a b c d E( a, b, c. d) a a a a Table A.5. Performance matrix using expected values E( a i ) f f f f 4 a a a a Download Date /8/9 :5 PM