Multiple criteria aggregation procedure for mixed evaluations

European Journal of Operational Research 181 (2007) 1506–1515 www.elsevier.com/locate/ejor

Multiple criteria aggregation procedure for mixed evaluations Sarah Ben Amor

a,*

, Khaled Jabeur b, Jean-Marc Martel

a

a

b

De´partement des Ope´rations et Syste`mes de De´cision, F.S.A., Universite´ Laval, Que´., Canada G1K 7P4 Recherche et De´veloppement pour la de´fense Canada – Valcartier, 2459 boulevard Pie-XI Nord, Val-Be´lair, Que´., Canada G3J 1X5 Received 1 December 2004; accepted 1 November 2005 Available online 12 May 2006

Abstract Most of the existing multiple criteria decision-making methods handle one kind of the information imperfections at the same time. Stochastic methods and fuzzy methods constitute typical examples of these methods. However, several multiple criteria modelizations include simultaneously many kinds of the information imperfections. In this work, we propose a multiple criteria aggregation procedure which accepts mixed evaluations, i.e. evaluations which contain different natures of imperfections. It is based on an adaptation of the stochastic dominance results.  2006 Elsevier B.V. All rights reserved. Keywords: Decision analysis; Multiple criteria analysis; Uncertainty modeling

1. Introduction The majority of the existing discrete multiple criteria methods, which consider the information imperfections, treat only one type of imperfection at the time [1]. Let us note that most of these procedures are based on a probabilistic or a fuzzy modelization. However, many multiple criteria modelizations imply often the presence of different forms of imperfection at the same time. The multiple criteria methods accepting mixed evaluations (punctual, stochastic, fuzzy or else) are rather rare. We find in the literature at least two: NAIADE [17] and PAMSSEM [14,6].

Multiple criteria methods are based on the (A, X, E) model, where A is a set of potential alternatives, X is a finite set of attributes (or criteria) and E is a performances table or an evaluation matrix. Some methods associate to each attribute/criterion j a coefficient of relative importance (or weight) wj, P where nj¼1 wj ¼ 1. We have

*

Corresponding author. E-mail addresses: [email protected] (S.B. Amor), [email protected] (K. Jabeur), Jean-marc.martel@ fsa.ulaval.ca (J.-M. Martel). 0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.048

S.B. Amor et al. / European Journal of Operational Research 181 (2007) 1506–1515

It should be observed that the information imperfection would notably appear in the evaluations eij. The aim of the NAIADE and PAMSSEM methods is to establish binary relations between the different pairs of alternatives according to each criterion. These relations are then aggregated to produce ‘‘global’’ relations between the pairs of alternatives, which are exploited afterwards. Since the evaluations are mixed (different natures of the imperfections), the aggregation of the valued local relations for all the pairs of alternatives is sometimes difficult. In fact, the signification of the numerical values at the local level could be unclear because of the heterogeneity of the measurement scales and the natures of the evaluations. Consequently, conciliating the results of the pair comparisons according to the criteria could be difficult. This observation led us to use all available information to establish ‘‘crisp’’ relations locally between all the pairs of alternatives, then, to aggregate them. In the beginning of this paper, a general modeling framework will be presented to clarify the context in which the proposed procedure is applied (Section 2). Our multiple criteria procedure for mixed evaluations comprises two main phases. The first phase (Section 3) consists of constructing the local preference relations. This construction is based on an extension of the stochastic dominance concept to the mixed evaluations context; the stochastic dominance providing a uniform treatment in this context. The second phase (Section 4) consists of aggregating these local binary relations into global binary relations using an algorithm developed in a group decision context and adapted to the multicriteria aggregation [7]. The global preference relations are then exploited (Section 5) using original procedures [7]. Finally, in Section 6, we conclude and discuss future work.

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The performance matrix for an attribute j is presented in Table 1. For each alternative ai, the evaluation eij according to the attribute j will depend on the set of the nature states Xj ¼ fxj1 ; . . . ; xjh ; . . . ; xjH g. The matrix is represented by the values xhij of the set xij ¼ fx1ij ; . . . ; xhij ; . . . ; xHij g. This modelization assumes also that we are able to indicate the consequence of choosing an alternative ai when the element xjh (h = 1, 2, . . . , H) is realized, i.e. when xhij is obtained. The a priori information available about the nature states (probability functions, belief masses, possibility measures, . . .) could be integrated in Table 2. This table is based on a modelization proposed by the theory of evidence [23], where the a priori information is represented by belief masses associated to the focal elements. These focal elements, which are subsets Bjh0  Xj ðh0 ¼ 1; . . . ; 2H Þ of the nature states set Xj ¼ fxj1 ; . . . ; xjh ; . . . ; xjH g, influence the evaluations of the alternatives according to the attribute j. According to the evidence theory, the evaluation eij of an alternative ai on the attribute 0 j is represented by a subset of values C hij  xij which depends on the subsets of the nature states (focal elements) Bjh0 ðh0 ¼ 1; . . . ; 2H Þ and Bjh  Xj . By using this modelization, we could consider that possibilistic or probabilistic modelizations are

Table 1 Performance matrix for the attribute j Alternatives

Xj xj1

a. 1 .. ..ai . am



xjh



...

.. . h ..xij .

...

xjH

2. General modeling framework The adopted framework accepts, in the uncertain decision-making situations, stochastic, possibilistic or ‘‘evidential’’ evaluations. It also includes fuzzy or ordinal evaluations. We think that among all incertitude-modeling languages (evidence theory, possibility theory and probability theory), the evidence theory presents the more general framework where possibilities and probabilities are proposed as particular cases. We exploit this property to settle our modelization.

Table 2 Performance matrix for the attribute j integrating the a priori information Alternatives

a. 1 .. a. i .. am A priori belief masses

Any focal elements Bj1

...

Bjh0

1 C .. 1j . 1 C .. ij . C 1mj mðBj1 Þ

...

h C .. 1j . h0 C .. ij . C h0 mj mðBjh0 Þ

... ... ...

0

...

Bj2H

...

2 C .. 1j . 2H C .. ij . H C 2mj mðBj2H Þ

... ... ...

H

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particular cases when the a priori information is represented by possibility or probability distributions respectively. In fact, when the attribute is a possibilistic one, the a priori information related to the evaluation eij is characterized by possibility distributions p. In such a context, the corresponding belief masses are associated with focal elements that are embedded Bj1      Bjh0      BjH 0 . The possibility measures coincide with the plausibilities of the embedded focal elements. When the attribute j is stochastic, the focal elements Bjh0 are reduced to the singletons {xjh } and the corresponding belief masses correspond to probability measures. In this case, the evaluations are characterized by random variables X hij with a priori probability distributions fijh . We have then the a priori (subjective) probabilities P hj for each state of the nature xjh ðh ¼ 1; 2; . . . ; H Þ. 3. Local preference relations The first phase of our procedure consists of establishing the local preference relations between two alternatives. Let Hj be the local relation between two alternatives ai and ak on a criterion j. We have: ai Hj ak, where Hj = {, 1, , ?} with  is the indifference relation,  is the strict preference relation, 1 is the inverse strict preference relation and ? is the incomparability relation. In order to construct these relations, we privilege the approach based on the extension of the stochastic dominance concept. The use of the stochastic dominance concept has the advantage of providing a uniform treatment for the different languages which express the information imperfections. Notice that stochastic dominance allows to conclude about the preference of an alternative ai over an alternative ak for a decision-maker whose attitude towards risk corresponds to DARA (Decreasing Absolute Risk Aversion) utility functions [15]. This type of risk aversion is observed for several economic phenomena. The link between stochastic dominance and the preference is well known for this class of utility functions. It is not always easy to make the decision-maker’s preferences explicit and we can often conclude that ai is preferred to ak if some stochastic dominance conditions are verified. In the case where the evaluation eij is a random variable, the results of stochastic dominance could be directly applied in order to establish preference relations. For instance, we could say that

eij j ekj () jH 1 ðxÞj ¼ jF ij ðxÞ  F kj ðxÞj 6 sj ; 8x 2 ½xj ; xj ; where xj and xj are, respectively, the inferior and superior limits of the evaluation scale of attribute/ criterion j, sj P 0 is a predetermined threshold, Fij and Fkj are the cumulated probability distributions for the evaluations of alternatives ai and ak through an attribute j and x is a modality of this attribute. Otherwise, i.e. for the cases where jH1(x)j > sj (Fij 5 Fkj), we can say that eij j ekj () eij FSDj

or

eij ?j ekj () eij non SDj

SSDj

or

TSDj ekj ;

ekj ;

where SD* means that one of the three dominance types is verified. The definitions of the dominance for first order, second order and third order (FSD, SSD and TSD) can be found in [15] and Appendix A. These concepts can be extended to ambiguous probabilities [13]. For fuzzy, possibilistic or evidential criteria, we propose the use of some transformations which provide to the functions characterizing the evaluation of an alternative on a given criterion (fuzzy membership functions, possibility distributions, belief masses, . . .) properties similar to those of a probability density function. For evidential criteria we use the pignistic transformation justified by Smets [24,25] X

BetP ðxh Þ ¼ j b

j

j h

B 0 :xh 2B 0 PðXÞ

mðBjh0 Þ jBjh0 jð1  mð;ÞÞ

;

8Bjh0 2 PðXÞ; mð;Þ 6¼ 1; where jBjh0 j is the cardinality of X in Bjh0 and BetP(xh) is the pignistic probability of xh. For possibilistic criteria we use the relation proposed by Dubois et al. [5]. This relation is equivalent to the pignistic transformation when applied to possibility measures. The probability ph corresponding to xh is given by ph ¼

H H X X 1 1  mt ¼ ðPt  Ptþ1 Þ; t t t¼h t¼h

where mh is the belief mass m(Bh) (h = 1, 2, . . . , H), Bh are embedded and organized in sequence, i.e. B1 = {x1}, B2 = {x1, x2}, . . . , Bh = {x1, x2, . . . , xh}, . . . , BH = X such that "B 5 Bh, m(B) = 0 and it is possible for a certain h that m(Bh) = 0,

S.B. Amor et al. / European Journal of Operational Research 181 (2007) 1506–1515

Ph = P({xh}) and P is the possibility P measure deH fined for {xh}: Pðfxh gÞ ¼ p‘ðfxh gÞ ¼ t¼h mt . For fuzzy criteria we use the rescaling approach as proposed in Munda [17] Z 1 fij ðxij Þ ¼ k ij lij ðeij Þ where fij ðxij Þdxij ¼ 1: 1

Let us note that the construction of local preference relations for deterministic criteria can be carried out by using discrimination thresholds to distinguish between strict preference and indifference situations (quasi-criterion notion) in the context of punctual evaluations. In this case, locally there is no incomparability. It is one of the characteristics of a criterion [20]. 4. Aggregating local preference relations Once the local preference relations are established, the second stage of our procedure aims at aggregating n binary relations Hj (ai Hj ak) in order to obtain a global binary relation H (ai H ak) by considering the relative importance of the criteria. To achieve this, the algorithm (AL3) developed by Jabeur and Martel [8] in a group decision-making context was adapted. AL3 starts by defining, for each pair of alternatives (ai, ak), an index UH(ai, ak) where H 2 {, 1, , ?}. This index measures the divergence between the global relation H and the local binary relations relating this pair of alternatives on each criterion. This divergence will be quantified by using the distance measure D between binary relations relating two pairs of alternatives [12]. The index UH(ai, ak) is determined as follows: n X wj DðH ; Rj ðai ; ak ÞÞ; UH ðai ; ak Þ ¼ j¼1

where 8  if ai j ak ; > > > <  if a  a ; i j k Rj ðai ; ak Þ ¼ > ? if a ? a i j k; > > : 1  if ak j ai : wj is the normalized coefficient of the relative importance of criterion j, Rj(ai, ak) represents the local binary relation comparing ai to ak on criterion j, D is a distance measure between the binary relations relating the pairs of alternatives and H 2 {, 1, , ?} is the global relation. In the literature, some methods are proposed to establish the relative impor-

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tance of the criteria wj. Regarding this issue, one can refer to Mousseau [16] and Roy and Mousseau [21]. In order to build the D distance measure, Jabeur et al. [12] proposed first a set of five ‘‘logic’’ conditions (an axiomatic) in which they compared the distance between each pair of binary relations {, 1, , ?}. Then, they prove that D is a metric, i.e. it verifies the non-negativity, the symmetry and the triangular inequality axioms. Finally, Jabeur et al. [12] chose the centroid of the variation domain of these pairwise comparisons to associate numerical values to D (see Table 3). According to these authors, their distance measure brings some improvements to the distance measures proposed by Roy and Slowinski [22] and Ben Khe´lifa and Martel [2]. Once the divergence indexes have been computed, we identify, for each pair of alternatives (ai, ak), the set of the global relations H* that minimize the indexes of divergence UH(ai, ak), that is    H H  H ¼ H =U ¼ min U ðai ; ak Þ : H 2f;1 ;;?g

Therefore, if the set H* is reduced to a singleton, i.e. there is only one global relation H* that minimizes the indexes of divergence UH(ai, ak), then H* will be retained for the pair of alternatives (ai, ak) globally. In the opposite case, i.e. where 2, 3 or 4 distinct relations minimize the indexes of divergence UH(ai, ak), priority rules have to be applied between the binary relations {, 1, , ?}. We adopt the two priority rules proposed by Jabeur [7]. The first one points out that the preference relation priority is higher than that of the indifference relation and the second rule indicates that the indifference relation priority is higher than that of the incomparability relation. Once the priority rules are applied, we analyze the cardinality of H*. If H* contains a unique relation H*, then H* will be retained for the pair of alternatives (ai, ak) globally. In the opposite case, i.e. when H* contains two relations of the same

Table 3 Numerical values associated to the D measure of distance

ai  ak ai  ak ai ? ak ai 1 ak

ai  ak

ai  ak

ai ? ak

ai 1 ak

0 4/3 1 4/3

4/3 0 1 5/3

1 1 0 1

4/3 5/3 1 0

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Table 4 Performance matrices for X1, X2, X3 and X4 ai a1 a2 a3 a4

X1

X2

X3

x11

x12

x13

x14

x15

x21

x22

x23

x31

x32

x33

3 1 2 6

1 3 1 1

2 4 2 4

6 6 3 4

4 3 5 2

90 120 110 80

90 90 130 110

110 100 80 120

35 45 45 40

70 30 60 70

45 45 25 30

priority (necessarily the preference () and the inverse preference (1) relations), we apply the intersection rule used by Roy [19] to combine two complete orders, and therefore, the incomparability relation (H* ?) will be retained for the pair of alternatives (ai, ak) globally. The algorithm AL3 could be summarized as follows: Step 1: Let B = {(ai, ak)/(ai, ak) 2 A · A} the set of the alternatives pairs. Compute for all alternatives pairs of B and for each H 2 {,  1, , ?} the divergence index UH(ai, ak). Step 2: Identify for each alternatives pair of B the set of global relations H*. If H* contains a unique relation, then this relation would be retained for the alternatives pair (ai, ak) globally. If not, go to step 3. Step 3: Apply the priority rules. If H* contains a unique relation, then this relation would be retained for the pair of alternatives (ai, ak) globally. If not, go to step 4. Step 4: Apply the intersection rule to retain the incomparability relation for the alternatives pair (ai, ak) globally. Notice that this algorithm requires n(n  1)/2 iterations to establish a global preference relational system (p.r.s) from local preference relational systems. Moreover, this algorithm guarantees the independence towards the irrelevant alternatives. Then, adding or removing an irrelevant alternative at will not modify in any case the type of the relation between ai and ak globally. Once the global p.r.s is determined, it could be exploited according to different decisional problematics (e.g., the choice problematic, the ranking problematic, or the sorting problematic). In the literature, several exploitation procedures can be found. One can quote the works of Roy [18], Colson [4] and Jabeur and Martel [9] for the choice problematic; the works of Roy [19], Brans and Vincke [3], Colson [4] and Jabeur and

X4

Associated fuzzy numbers

Weak Fair High More or less weak

(0, .1, 0, .2) (.5, .5, .2, .2) (.9, 1, .2, 0) (.2, .2, .2, .2)

Martel [10] for the ranking problematic; and the works of Yu [26] and Jabeur and Martel [11] for the sorting problematic. 4.1. Numerical example Let us consider the following illustrative example for the proposed multiple criteria aggregation procedure: A = {a1, a2, a3, a4} and X = {X1, X2, X3, X4} where X1 is a stochastic attribute, X2 is an evidential attribute, X3 is a possibilistic attribute and X4 is a fuzzy attribute. For each attribute, the inputs consist of the performance matrix, the a priori information, and the indifference threshold. Table 4 gives the performance matrices for the four attributes. The evaluations according to X4 are linguistic variables represented by trapezoidal fuzzy numbers (a, b, c, d) (see Fig. 1). Table 5 gives the a priori information for attribute X1, while Table 6 gives the a priori information for X2 and X3. For attribute X4, the a priori information is implicitly contained in the performance matrix. We consider the same indifference threshold for all the attributes, i.e. s1 = s2 = s3 = s4 = s = 0.05.

Fig. 1. Membership function for fuzzy trapezoidal number (a, b, c, d).

Table 5 A priori information for X1 P ðx1h Þ

x11

x12

x13

x14

x15

0.28

0.23

0.10

0.28

0.11

S.B. Amor et al. / European Journal of Operational Research 181 (2007) 1506–1515

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Table 6 A priori information for X2 and X3 fx21 g

; mðB2h0 Þ mðB3h0 Þ PðB3h0 Þ

fx22 g

fx23 g

fx21 ; x22 g

0.1 0.0

0.8

0.6

a1 a2 a3 a4

Stochastic dominance results

0.2 1

0.8

fx21 ; x22 ; x23 g

1

0.3 0.6 1

The transformation relation could then be applied. For instance

H1

a1

a2

a3

a4

* IND – ?

IND * ? TSD

FSD ? * FSD

? – – *

a1

a2

a3

a4

*  1 ?

 * ? 

 ? * 

? 1 1 *

p1 ¼ P ðx33 Þ ¼

P1  P2 P2  P3 P3  P4 þ þ 1 2 3 1  :8 :8  :6 :6  0 þ þ ¼ :5: ¼ 1 2 3

B1 ¼ fx3 g and P1 ¼ Pðfx3 gÞ ¼ 1   note that PðBÞ ¼ sup pðxÞ ; x2B

B2 ¼ fx3 ; x1 g and P2 ¼ Pðfx1 gÞ ¼ 0:8; B3 ¼ fx3 ; x1 ; x2 g ¼ X and P3 ¼ Pðfx2 gÞ ¼ 0:6; P4 ¼ 0: Table 8 Pignistic probabilities for X2 and X3 fx21 g

fx22 g

fx23 g

0.4 0.3

0.5 0.2

0.1 0.5

3 X Pt  Ptþ1 t t¼1

¼

4.1.1. Local preference relations Stochastic dominance results could be directly applied in the case of X1. The obtained results and the corresponding local preference relations are given in Table 7. For X2 and X3 we must transform the a priori information (see Table 8). The pignistic transformation is then applied for X2 to obtain BetP ðx2h Þ. For instance, BetP ðx22 Þ= 0.1 + 0.6/2 + 0.3/3 = 0.5. For X3, the Dubois et al. [5] relation is applied to obtain P ðx3h Þ. The subsets B3h0 are embedded and organized in sequence. Thus, we have

BetP ðx2h Þ P ðx3h Þ

fx22 ; x23 g

0.6 0.2 1.0

Table 7 Stochastic dominance results and local preference relations for X1 Alternatives

fx21 ; x23 g

The application of the dominance stochastic results for Xj leads to the local preference relations Hj, j = 2, 3, 4, listed in Table 9. 4.1.2. Aggregating local preference relations This step consists in applying the (AL3) algorithm. Thus, we start by computing the divergence index UH(ai, ak) for each relation H 2 {,  1, , ?} and for each alternatives pair. This operation is performed by using Tables 7 and 9 and the distance measure given in Table 3. By considering the relative importance vector Wj = [.4, .2, .3, .1], the obtained divergence indexes are given in Table 10. For instance, we have Table 10 Indexes of divergence 1

Alternative pairs

U

U

U?

U

(a1, a2) (a1, a3) (a1, a4) (a2, a3) (a2, a4) (a3, a4)

0.8 1.267 1.133 1.133 1.333 1.333

1.033 0.367 0.767 0.767 0.667 1.167

1 0.8 0.4 0.4 1 1

1.033 1.367 1.1 1.1 1 0.5

a3

a4

Table 9 Local preference relations Hj for attributes Xj, j = 2, 3, 4 Relations alternatives

H2 a1

a1 a2 a3 a4

*  ? ?

H3 a2 1

 * ? 1

a3 ? ? * 1

a4 ?   *

a1 * 1 1 1

H4 a2  * 1  1

a3   * 

a4    *

a1

1

*   

a2 1

 *  1

1

 1 * 1

1   *

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Table 11 Global preference relations Alternatives a1 a2 a3 a4

a1

a2

a3

a4

*  1 ?

 * ? 1

 ? * 

?  1 *

U ða1 ; a2 Þ ¼ 0:4Dð; Þ þ 0:2Dð; 1 Þ þ 0:3Dð; Þ þ 0:1Dð; 1 Þ ¼ ð0:4 0Þ þ ð0:2 4=3Þ þ ð0:3 4=3Þ þ ð0:1 4=3Þ ¼ 0:8: For each alternatives pair, the set of global relations H* minimizing the indexes of divergence is identified. In this case, the minimum is unique for each alternatives pair. Therefore, H* is reduced to a singleton. The global preference relations are given in Table 11. For instance, we have for a1 and a2 H

MinH U ða1 ; a2 Þ ¼ Minf0:8; 1:033; 1; 1:033g ¼ 0:8;

thus H ¼ fg:

Let us note that the previous matrix can be exploited according to different decisional problematic as it is proposed by Jabeur [7]. 5. Exploitation of the global preference relational system The exploitation of the graph associated to the global p.r.s is carried out by using Jabeur [7] procedures. These authors propose a detailed review of original procedures to exploit a p.r.s according to three decisional problematic: the choice problematic, the ranking problematic and the sorting problematic. For example, in the case of choice problematic, Jabeur and Martel [9] suggest mainly two exploitation procedures: (i) a procedure based on purified kernel concept and (ii) a procedure based on the alternative performance concept. The first procedure is composed of three operations: graph-prs shrinkage, kernel identification and kernel purification. In graph theory, when the graph is without circuits then its kernel N exists and is unique. However, the way used by algorithm AL3 to aggregate local preference relations does not guarantee that the graph associated to the global p.r.s, called graph-prs, will be without circuits. Thus, in order to identify the kernel, all circuits included in a graph-prs must be identified and

removed: its the shrinkage operation (operation 1). Indeed, this operation consists in grouping all alternatives constituting each maximal circuit in a single ‘‘macro-alternative’’. A maximal circuit is a circuit which is not included in any other circuit. Moreover, a graph-prs can include several independent maximal circuits, i.e. which do not have common arcs. At the end of the shrinkage operation, we obtain a graph-prs without circuits in which it is possible to identify a unique kernel (operation 2). According to the kernel definition, it is supposed including the ‘‘best’’ alternatives and those that are incomparable. However, the kernel can also contain bad alternatives (see example given by Jabeur and Martel [9]). In order to rectify this situation, Jabeur and Martel [9] propose in the third operation to purify, when it is possible, the kernel. In order to carry out this operation, he defines first the two following sets: Wþ ðai Þ ¼ fak =ai  ak ; i 6¼ kg and W ðai Þ ¼ fak =ak  ai ; i 6¼ kg: Then, he removes from the kernel any alternative ai which verifies the following condition: card(W(ai)) > card(W+(ai)). In fact, this condition eliminates from the kernel any alternative ai for which the number of alternatives that are preferred to ai is greater than the number of alternatives that ai is preferred to. The main idea of the second exploitation procedure is to build, without erasing any information contained in the graph-prs, a subset of the ‘‘best’’ alternatives, called H, by using their performance (this concept will be defined later). This procedure determines, first, for each alternative ai its successors’ set W+(ai), its predecessors’ set W(ai) as mentioned above and the two following sets: • W(ai) = {ak/ai  ak, i 5 k}, the set of all alternatives with which ai is indifferent; • Nisol = {ai/W+(ai) = W(ai) = W(ai) = ;}, the set of alternatives which are incomparable. Second, we calculate for each alternative ai 2 AnNisol an index, called P(ai), which measures its performance in the graph-prs. This performance is defined by the difference between the number of alternatives which ai is preferred to and the number of alternatives which are preferred to ai, let Pðai Þ ¼ cardðWþ ðai ÞÞ  cardðW ðai ÞÞ:

S.B. Amor et al. / European Journal of Operational Research 181 (2007) 1506–1515

Note that the index P(ai) considers only the preference relations for estimating the performance of an alternative ai. Thus, in order to take into account the effect of the indifference relations in the performance of an alternative ai, Jabeur and Martel [9] add to P (ai) the average of the performances of all the alternatives that are indifferent to ai in the graph-prs, let 8 b i Þ ¼ Pðai Þ if cardðW ðai ÞÞ ¼ 0; < Pða P 1 b i Þ ¼ Pðai Þ þ Pðak Þ otherwise:  : Pða cardðW ðai ÞÞ

ak 2W ðai Þ

It is noteworthy to mention that the additional P ak 2W ðai Þ Pðak Þ component allows to correct upward or downward the initial performance of the alternative ai expressed by the index P(ai). Finally, Jabeur and Martel [9] build the H set in b where N b ¼ the following manner: H ¼ N isol [ N b i Þg. In other words, H will contain fai =Maxi Pða the alternatives which have obtained the highest perb formance PðÞ and those which are incomparable. The last step of the second exploitation procedure consists in purifying H. In this step Jabeur and Martel [9] remove from the graph-prs engendered by H, i.e. the subgraph-prs composed of binary relations which link only the alternatives belonging to H, any alternative ai such as $ak 2 H and ak 2 W(ai). If the purified H is nonempty, then it will contain the ‘‘best’’ alternatives. In the opposite case, i.e. the purified H is empty, we say that before this operation, H contained the alternatives which obtained the best score. According to Jabeur and Martel [9], the second exploitation procedure is more appropriate than the first one when the graphs-prs contains circuits. This is due to the fact that the first procedure regroups, in its shrinkage operation, all alternatives that constitute each circuit in one macro-alternative and consequently it erases all information contained in each circuit. However, when the graph-prs is without circuits, both of these procedures give the same subset of the ‘‘best’’ alternatives, i.e. H purified N purified. 1 cardðW ðai ÞÞ

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sists in removing all circuits included in the graphprs. Thus, only one circuit is identified in the above graph-prs, let: {(a1, a2)}. The two alternatives forming this circuit are grouped in a single ‘‘macro-alternative’’ called a 0 . Once that all circuits are eliminated, it is possible to identify the kernel: {a 0 } (see Fig. 3). It is important to note that in this numerical example the purification operation is not carried out since the kernel contains only one alternative a 0 . In order to identify H, i.e. the ‘‘best’’ alternatives set, without erasing the slightest information contained in the graph-prs, the exploitation procedure based on the alternative performance determines, for each alternative ai, the sets W+(ai), W(ai), W(ai) and the performance indexes P(ai) and b i Þ. All these results are presented in Table 12. Pða According to this table, H contains the alternatives a1 and a2 since they have obtained the best performance. In order to carry out the purification operation, we will be interested only in the subgraph-prs generated by the alternatives a1 and a2. Since in this subgraph-prs these two alternatives are indifferent, none of them will be eliminated

a1

a2

a3

a4

Preference

Indifference

Incomparability

Fig. 2. Graph-prs associated to the global p.r.s.

Kernel ≡ Purified kernel

a'

5.1. Numerical example (continuation) On the basis of Table 11, we can draw the graphprs associated to the global preference relations (Fig. 2). The procedure based on the purified kernel concept starts by the shrinkage operation which con-

a3

a4

Fig. 3. Kernel and purified kernel of the graph-prs.

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Table 12 The results of the exploitation procedure based on the alternatives performance ai

+

W (ai)

W (ai)

W (ai)

P(ai)

b iÞ Pða

a1 a2 a3 a4

{a3} {a4} – {a3}

– – {a1, a4} {a2}

{a2} {a1} – –

1 1 2 0

1 1 2 0





from H. Finally, the ‘‘best’’ alternative set will contain both a1 and a2, i.e. H = {a1, a2}. 6. Conclusion The proposed multiple criteria aggregation procedure is designed to consider mixed evaluations. It leads to non-valued global preference relations that could be exploited according to different decisional problematics. This procedure could be extended to include the weak preference and the inverse weak preference relations. The stochastic dominance results would then be adapted and an extended measure of distance has to be built. The stochastic dominance could also be extended to conclude about the preference for a decision-maker whose attitude towards risk corresponds to INARA (INcreasing Absolute Risk Aversion) utility function [27]. The algorithm AL3 could be applied afterwards to lead to a richer global preference relational system. Acknowledgement This study was made possible with the financial support of Defence R&D Valcartier Canada. Appendix A We will consider the stochastic dominance set as the following: FSD (first-degree stochastic dominance), SSD (second-degree stochastic dominance) and TSD (third-degree stochastic dominance). These stochastic dominances are defined for continuous and discrete random variables in the following way: Definition 1. Fij FSD Fkj if and only if Fij 5 Fkj H 1 ðxj Þ ¼ F j ðxij Þ  F j ðxkj Þ 6 0

for all xj 2 ½xj ; xj :

Definition 2. Fij SSD Fkj if and only if Fij 5 Fkj

H 2 ðxj Þ ¼

Z

xj

H 1 ðyÞ dy 6 0

for all xj 2 ½xj ; xj :

xj

Definition 3. Fij TSD Fkj if and only if Fij 5 Fkj H 3 ðxj Þ ¼

Z

xj

H 2 ðyÞdy 6 0

for all xj 2 ½xj ; xj :

xj

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