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A new method for group decision support based on ELECTRE III methodology
European Journal of Operational Research 148 (2003) 14–27 www.elsevier.com/locate/dsw
Decision Aiding
A new method for group decision support based on ELECTRE III methodology pez, Eduardo Fern Juan Carlos Leyva-Lo andez-Gonz alez
*
Facultad de Ingenierıa, Universidad Aut onoma de Sinaloa, Ciudad Universitaria, Calzada de Las Americas S/N, Culiac an, Sinaloa, Mexico, CP 80040, Mexico Received 9 November 1999; accepted 9 January 2002
Abstract Group decision is usually understood as the reduction of different individual preferences on a given set to a single collective preference. At present, there are few approaches which solve the group ranking problem with multiple criteria in a widely acceptable way. Often, they rest on a poor heuristic which makes a decision about consensus ranking difficult to support. This paper presents an extension of the ELECTRE III multicriteria outranking methodology to assist a group of decision makers with different value systems to achieve a consensus on a set of possible alternatives. Our proposal starts with N individual rankings and N corresponding valued preference functions, and uses the natural heuristic provided by ELECTRE methodology for obtaining a fuzzy binary relation representing the collective preference. A comparison of this method with PROMETHEE II for group decision is carried out. We found that, in this particular application, the proposed heuristic based on majority rules combined with concessions to significant minorities, performs relatively better than a compensatory scheme based on a net flow weighted sum function. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multicriteria analysis; Group decision; Outranking methods; ELECTRE III; Genetic algorithms
1. Introduction People make a group decision (intra-organizational or inter-organizational) when they face a common problem and they are all interested in its solution. This problem may be the purchase of a car, the acquisition of a house by a family, the design of a new product or a production planning or
* Corresponding author. Tel.: +52-667-7134281; fax: +52667-7134053. E-mail addresses: [email protected] (J.C. LeyvaL opez), [email protected] (E. Fernandez-Gonzalez).
to find the best location for the wastewater treatment plant in a city. An important characteristic of group decision is that all involved individuals belong to some organization (family, firm, government). They may differ in their perception of the problem and have different interests, but they are all responsible for the organizationÕs well-being and share responsibility for the implemented decision. When a decision situation involves multiple actors, each with different values and informational systems, the final decision will generally be the result of an interaction between this individualÕs preferences and those of others. This interaction is not free of conflict, which may be due to
0377-2217/03/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00273-4
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
any of a number of factors, e.g. different ethical or ideological beliefs, different specific objectives, or different roles within organization. Whatever the origin of the conflicting value systems, they will usually affect the evolution of the decision process in ways that were not expected at the outset (Roy, 1996; Keeney, 1992). Group decision is usually understood as the reduction of different individual preferences on a given set to a single collective preference (Jelassi et al., 1990). We will not focus on the psychological aspects of group interactions. The main goal of this work is the prescription of a final group ranking from these individual preferences once they have been established. Group decision making covers a wide range of situations. We are interested in a problem with the following characteristics: (a) Each group member participates in the process giving information about his preferences and beliefs, contributing in this way to the final decision. There usually is an overall goal which is accepted by all the members, but they differ in the ways of how this goal should be achieved. (b) Each actor considers the same set of alternatives or potential actions. (c) There are multiple criteria, which usually conflict with each other. Each actor must generate relevant criteria which may be shared (totally or partially) by some, none or all the remaining members. (d) There is a special actor (may be a single person, a small group of stakeholders) with authority for establishing consensus rules and priority information on the set of group members. Following (Keeney and Raiffa, 1976) we call this entity the Supra Decision Maker (SDM). (e) The group members accept the final decision derived from an aggregation of their opinions according to the rules and priorities defined by the SDM. Points (a), (d) and (e) define our concept of a collaborative group. The last two points are very important and they need a further discussion. Usually, the group belongs to some organization
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or works for some organization; in this case, the SDM can be seen as an entity representing the interest of that organization. Sometimes the SDM can be considered as an interpreter (may be a referee) of some system of rules describing the group constitution, the way in which the group agrees to make decisions. Sometimes the SDM is an altruistic dictator which, taking into account the different individual opinions, tries to identify the best decision. In particular cases, the SDMÕs role could be performed by the decision analyst. Up to now, there are not many approaches which solve the group ranking problem with multiple criteria (GRPMC) in an acceptable way. Often, they rest on a poor heuristic which makes a decision about consensus ranking difficult to support. Here, we present a new method which starts with N individual rankings and N corresponding valued preference relations, and uses the natural heuristic provided by ELECTRE methodology for obtaining a fuzzy outranking relation representing the SDMÕs preferences, which is also useful for deriving final collective ranking. Thus, this method is limited to group decision problems in which the individual preferences are aggregated using a fuzzy preference relation. This paper is structured as follows. In Section 2, some criticisms of the existing approaches are pointed out, followed by the presentation of our specific problem (Section 3). Section 4 is devoted to expose our proposal. Section 5 contains an example and a comparison of the proposed method with PROMETHEE II for group decision, and on this basis, we present some conclusions in Section 6.
2. Some criticisms of the existing approaches We can distinguish two main general approaches which use a multicriteria decision aid technique for aggregating group preferences in GRPMC (in Belton and Pictet (1997) there are others approaches): (A) In the first way (e.g. Salminen et al., 1998; Hokkanen and Salminen, 1997; Rogers et al., 2000), the group is asked to agree on the
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alternatives, criteria, scores, weights, thresholds and remaining parameters before the model provides a ranking. The group discussion focuses on what actions and criteria should be considered, what weights and other necessary parameters are appropriate. Once the discussion is closed and all the individual information has been gathered, a technique is used for obtaining values of these model parameters which should represent the collective opinion. With this information, the multicriteria decision model gives us the group ranking. (B) Although members can exchange opinions and relevant information, a group consensus is needed only for defining the set of potential actions. Each member defines his own criteria, the appropriate evaluations and model parameters (weights, thresholds, etc.), and then the multicriteria decision aid method is used to get a personal ranking. Next, each actor is considered as a separate criterion, and the preferential information contained in its particular ranking is aggregated in a final collective ordering with the same (may be other) multicriteria decision approach (e.g. Macharis et al., 1998; Hwang and Lin, 1987). Some problems could arise when Approach A is used. In order to be a real collective opinion, the model parameters (e.g. weights, scores, thresholds) should represent a group consensus coming from unanimity or, at least, from a strong majority. If the values, for instance, of the weights are determined by a majority subset of the group members, the analyst should take care that other important parameters (e.g. veto thresholds) will be determined by the same subset. Otherwise, the complete set of chosen parameters could not represent anybody. In this case, it would be impossible to talk about a collective opinion. Some authors (cf. Rogers et al., 2000; Hokkanen and Salminen, 1994, 1997; Salminen et al., 1998) have reported successful results applying ELECTRE methods to GRPMC following Approach A. Average weights, upper quartile and lower quartile weights are used in the assessment, with different weighting systems reflecting the views of different groupings, used within a sensi-
tivity analysis. However, strong divergences often exist in the perceived importance of the criteria; sometimes the ratio of maximum to minimum weighting exceeds 10 (cf. Rogers et al., 2000). Fortunately, consensus was achieved about threshold levels. Considering its successful applications, Approach A performs well when the assessments of model parameters coming from the different members do not show strong divergences, or when the final ranking is sufficiently robust to handle them. If these conditions do not hold, other approaches for aggregation of group member preferences could be better. Approach B is supported by the following arguments: • There are some important similar characteristics in group decision and multicriteria decision problems (MCDP): (a) the interest in obtaining a good performance of the complete set of criteria (in MCDP), and in obtaining a ranking which satisfies to the complete set of actors (in GRPMC); (b) the concept of an ideal solution, usually unfeasible, is common; (c) the concept of dominance plays a similar role too; and (d) when the ideal solution is not feasible, it is necessary to model the decision makerÕs preferences on multiple criteria (in MCDP), or the SDMÕs preferences on multiple actors (in GRPMC), in order to find the best solution. • It seems natural considering each actor as a different point of view, that is, a different ‘‘attribute’’ of a decision problem. • Using Approach B, the SDMÕs preferences on the different ‘‘criteria’’ (his opinions and priorities concerning the different group members) can be modeled. • Some suitable ideas coming from the concept of democracy can be included in the model of SDMÕs preferences. • We do not use parameters provided by the group members. We do not need to work with poorly defined ‘‘collective parameters’’. We feel that the basis of our discussion is the meaning of the term ‘‘collective opinion’’. We think that a real ‘‘collective opinion’’ exists only in
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case of (i) unanimity or strong consensus, and (ii) when a strong majority accepts the altruistic opinion of the SDM, who performs as an interpreter of the group opinion; it happens working with collaborative groups. Approach B allows looking for the ‘‘collective opinion’’ in the sense of point (ii). The main questions in order to make operational a method based on B are the following: A1. How to build an aggregation model of SDMÕs preferences with good properties, only knowing the group member rankings and the aggregation models of the member multicriteria preferences? B1. When the member ranking is derived exploiting fuzzy preference relations (or crisp outranking relations), the effect of irrelevant alternatives produces frequent discrepancies between the ranking and the aggregation model of preferences (Vanderpooten, 1990). Then, how could this contradictory information about the preferences of a particular group member be handled? C1. How could the aggregation model of SDMÕs preferences be exploited deriving the final solution in such a way that the above discrepancies model-ranking are minimized? C1 is a very important question for improving the prescription coming from fuzzy preference modeling (Bouyssou and Vincke, 1995). We have proposed the use of a genetic algorithm for deriving final ranking from a fuzzy preference relation (cf. Leyva L opez and Fern andez Gonz alez, 1999) which has good properties and allows finding more consistent orderings than those obtained with other reported exploitation methods. In fact, this genetic search for improving the final ranking is part of our proposal (see Appendix A). However, the aim of this contribution concerns mainly with A1 and B1. PROMETHEE (Brans et al., 1984) has been applied for GRPMC using a net flow weightedsum function as a rough model of SDMÕs preferences (Brans et al., 1997; Macharis et al., 1998), giving an answer to question A1 (not to B1). Anyway, we think that a compensatory scheme is not suitable for group decision, because the natural heuristic based on majority rules combined with respect to significant minorities should be
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preserved. Therefore, a compensatory scheme does not seem to be a good model of the SDMÕs preferences. In this sense, ELECTRE is the multicriteria decision methodology closest to this heuristic based on concordance and discordance principles, which it is often used when groups perform pairwise comparisons (Lootsma, 1997). However, the use of ELECTRE for building an aggregation model of the SDMÕs preferences (following Approach B) is not straightforward, because this model requires some numeric information which the SDM can not derive directly from member particular rankings. Besides, when the group members possess different values and informational systems, Roy (1996) recommends to modify the traditional ELECTRE methods incorporating these new characteristics.
3. Problem formulation Let A be the finite set of alternatives or potential actions considered by a group and evaluated on multiple criteria. The group is composed of a set M ¼ f1; 2; . . . ; N g of members, whose work is somehow controlled by a Supra Decision Maker. Let us suppose that a multicriteria aggregation method based on fuzzy preference relation is used for deriving the group members particular rankings. Let ri : AXA ! ½0; 1 be a valued binary relation which aggregates the preferences of the ith member on the multiple criteria describing the elements of A. Let Oi be a complete ranking of A derived using some procedure for exploiting ri . The problem is: from N pairs ðri ; Oi Þ, obtaining a ranking O which performs the best compromise according to the SDMÕs preferences. Remarks: • This problem arises when the facilitator (may be the SDM) uses fuzzy outranking approaches for obtaining individual rankings, as in ELECTRE III and PROMETHEE, but is not limited to these particular methods (Fodor and Roubens, 1994). • The model of the SDM plays a fundamental role, which has been neglected or simplified by
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previous approaches. Building this model is hard and different from a typical multicriteria decision model, because: (a) it should solve the probable conflict between Oi and ri , when the last one is not transitive (point B1 of previous section); (b) the imprecise and qualitative nature of the information provided by the individual rankings and fuzzy preference relations. • A good model of the SDMÕs way of thinking should be based on a widely accepted heuristic for democracy. In the following section, we present a method strongly inspired by ELECTRE III.
4. The new method ELECTRE-GD Our approach consists of the following steps: first, it is necessary to propose a way to solve the possible conflict between the information provided by individual rankings and fuzzy preference relations. Second, we will define a fuzzy outranking relation for the SDM, which should be exploited in some rational way in order to derive final group ranking. 4.1. Solving conflicts: the preference matrix As we have previously discussed, the basic idea is considering each member k as a criterion of a new multicriteria problem. Each pair of actions ða; a0 Þ 2 A A should be compared according to the point of view of criterion k. For this comparison, the SDM considers an information composed of two significant elements: the relative position of actions a and a0 in Ok and the values rk ða; a0 Þ and rk ða0 ; aÞ. Let uk : A ! N be a function defined as uk ðai Þ ¼ cardðBÞ þ 1; where B ¼ faj 2 A : aj is ranked worse than ai in Ok g. Note that if ai is the best in Ok , then uk ðai Þ ¼ cardðAÞ. If ai is the worst ranked, uk ðai Þ ¼ 1. Now, we need to introduce two thresholds which reflect some feelings of the SDM on the fuzzy preference relation rk . Suppose that k ð0 < k < 1Þ is such that if rk ða; a0 Þ P k the SDM agrees
with the assertion ‘‘in absence of other information this is a strong reason for thinking that a is at least as good as a0 since the point of view of member k’’. Let b another threshold parameter such that if rk ða; a0 Þ 6 k b, the SDM feels that ‘‘in absence of other information, action a does not seem at least as good as a0 from the point of view of member k’’. In the interval k b 6 rk 6 k the SDM is doubtful about the outranking aSa0 . Combining these intervals for rk ða; a0 Þ and rk ða0 ; aÞ we can distinguish nine different situations (zones): I II III IV V VI VII VIII IX
rk ða; a0 Þ P k; rk ða0 ; aÞ P k, rk ða; a0 Þ P k; k b < rk ða0 ; aÞ < k, rk ða; a0 Þ P k; rk ða0 ; aÞ 6 k b, rk ða0 ; aÞ P k; k b < rk ða; a0 Þ < k, k b < rk ða; a0 Þ < k; k b < rk ða0 ; aÞ < k, rk ða0 ; aÞ 6 k b; k b < rk ða; a0 Þ < k, rk ða0 ; aÞ P k; rk ða; a0 Þ 6 k b, rk ða; a0 Þ 6 k b; k b < rk ða0 ; aÞ < k, rk ða0 ; aÞ 6 k b; rk ða; a0 Þ 6 k b.
The information about preferences contained in rk should be aggregated with another one that follows from ranking Ok . It is not an easy task. Many techniques for rank ordering alternatives on the basis of fuzzy binary relations have been proposed (e.g. Fodor and Roubens, 1994). No one seems to be clearly better than others (cf. Bouyssou and Vincke, 1995). In our opinion, the main criticism concerns with the fact that the relative position of alternatives a and b in the ranking is not determined by rða; bÞ and rðb; aÞ (r denotes the fuzzy preference relation). Other alternatives strongly influence on their positions, and therefore the final ranking often does not respect the Decision MakerÕs preferences that should be derived from rða; bÞ and rðb; aÞ. This drawback can lead to violations of the crisp preference that could be obtained from r (rða; bÞ much greater than rðb; aÞ and however a is ranked worse than b). The SDM could consider five different situations concerning with uk : • uk ðaÞ uk ða0 Þ. Denotes the case in which action a is ranked as one of the best action, while a0 is one of the worst (this classification is defined by
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
•
• • •
the SDM which could take into account the opinion from DMk ). uk ðaÞ > uk ða0 Þ. Denotes when action a is ranked better than a0 , but the previous situation does not hold. uk ðaÞ ¼ uk ða0 Þ. Actions a and a0 are tied in the ranking, or their difference is negligible. uk ða0 Þ > uk ðaÞ. uk ða0 Þ uk ðaÞ.
It is not possible here to assess a quantitative meaning to the statements ‘‘one of the best ranked actions’’ and ‘‘one of the worst’’. It depends on the number of ranked actions and on the SDMÕs perception about the reliability of the method used for deriving a particular ranking. But roughly, we could use the top 15% and the bottom 15%, respectively, as reference levels by default. In practice, the interaction SDM-analyst should assess the correct levels. Combining the nine possible situations for rk and uk ðaÞ uk ða0 Þ, there does exist 45 combinations. They can be arranged in a matrix with nine rows and five columns. The SDM can express preferences about these events using preference binary relations P, Q, I, R. According to Roy (1996), they mean: P (strict preference). Corresponds to the existence of clear and positive reasons that justify significant preference in favor of one (identified) of the two actions. I (indifference). Corresponds to the existence of clear and positive reasons that justify equivalence between the two actions.
Q (weak preference). Corresponds to the existence of clear and positive reasons that invalidate strict preference in favor of one (identified) of the two actions, but that are insufficient to deduce either strict preference in favor of the other action or indifference between the two actions, not allowing either of the two preceding situation to be distinguished as appropriate. R (incomparability). Corresponds to an absence of clear and positive reasons that justify any of the three preceding relations. Most of these 45 judgements are trivial ones. We can propose, by default, the ‘‘preference matrix’’ in Table 1. Some positions in the ‘‘preference matrix’’ reflect a good coincidence between rk ða; a0 Þ and uk ðaÞ uk ða0 Þ (for instance see (3,1), (1,3)). Other positions like (3,5) and (7,1) reflect strong (although possible) discrepancy. In a decision aiding process, this matrix is a result of the interaction between an analyst and the SDM. If necessary, some doubtful elements may be adjusted (for instance, in positions (3,5) and (7,1) an incomparability statement should be considered). This adjustment does not depend on a particular group member, the preference matrix is an expression of the SDMÕs beliefs, his own feeling about the reliability of methods used for deriving the fuzzy preference relations and actorÕs ranking. Note that when the preference matrix has been assessed, given a pair ða; a0 Þ we have a correspondence between actors and elements of the preference matrix. That is, for a pair ða; a0 Þ and for the
Table 1 Preference matrix Zone I II III IV V VI VII VIII IX
uðaÞ uða0 Þ 0
aPa aPa0 aPa0 aPa0 aPa0 aPa0 aQa0 aPa0 aPa0
uðaÞ > uða0 Þ 0
aQa aPa0 aPa0 aIa0 aQa0 aPa0 aIa0 aIa0 aQa0
19
uðaÞ ¼ uða0 Þ 0
aIa aQa0 aQa0 a0 Qa aIa0 aQa0 a0 Qa a0 Qa aIa0
uða0 Þ > uðaÞ 0
a Qa aIa0 aIa0 a0 Pa a0 Qa aIa0 a0 Pa a0 Pa a0 Qa
uða0 Þ uðaÞ a0 Pa a0 Pa a0 Qa a0 Pa a0 Pa a0 Pa a0 Pa a0 Pa a0 Pa
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kth actor the analyst can find the corresponding element of the preference matrix, which will be used by the algorithm exposed in the next paragraph. 4.2. A fuzzy outranking relation for aggregating SDM’s preferences Once the preference matrix has been obtained, the definition of a valued binary relation following ELECTRE III is straightforward (e.g. Roy, 1990). 4.2.1. Preliminary definitions Definition 1. We say that action a outranks action a0 from the point of view of actor k (restricted outranking relation aSk a0 ) if and only if aPk a0 , aQk a0 , or aIk a, in agreement with the corresponding element of the preference matrix. Definition 2. An actor k is in concordance with the assertion aSG a0 , (SG means group outranking), if and only if aSk a0 . In the following CðaSG a0 Þ will denote the concordant coalition, the set of actors which are in concordance with aSG a0 . Definition 3. An actor k is in discordance with the assertion aSG a0 if and only if a0 Pk a. DðaSG a0 Þ will denote the discordant coalition, joining the actors which are in discordance with aSG a0 . Definition 4. An actor k belongs to veto coalition V ðaSG a0 Þ if and only if the two following conditions hold: (i) uk ða0 Þ uk ðaÞ, (ii) rk ða0 ; aÞ rk ða; a0 Þb ((ii) could be changed by a stronger condition like rk ða0 ; aÞ P k and rk ða; a0 Þ < k b). Definition 5. An actor k belongs to the incomparability coalition CðaRG a0 Þ if aRk a0 .
4.2.2. The concordance index For defining a comprehensive outranking relation SG , we take into account the fact that the role
of different actors (criteria) is not the same from the SDM point of view. As in ELECTRE methods, the importance of the jth criterion is taken into account by means of two effects: • its importance coefficient wj > 0, which is considered in the definition of a concordance degree; • its veto capacity. Following ELECTRE, in order to model the strength of the arguments in favor of the assertion aSG a0 , we define the concordance index as X wj ; j 2 CðaSG a0 Þ; ð1Þ Cða; a0 Þ ¼ 1=W where W ¼
X
wj ;
j 2 M 0;
M 0 ¼ fk 2 M : aSk a0 or a0 Sk ag: In group decision a statement of incomparability expresses the desire of no voting. Therefore, criteria belong CðaRG a0 Þ are excluded of M 0. 4.2.3. The power of veto coalition The power for determining a veto condition should be given by the number and importance of actors which belong to V ðaSG a0 Þ. In the following, we propose a model based on simple ideas taken from the natural heuristic of group pairwise comparisons. Let vj be the number of votes (for veto condition) which the SDM agrees assigning to jth member. Suppose that i denotes a middle, nonrelevant group member; then, without loss of generality, vi should be equal 1. Other values (2,3, or still more) are reserved to actors whose opinions are considered more relevant by the SDM. The number of votes for veto are close related to the number of votes for concordance wj , but they are not necessarily the same. By default, the analyst could assume identity. Let NV be the number of votes which the SDM considers necessary to become in a serious argument against aSG a0 . We define the discordance index as
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
0
dða; a Þ ¼
(P
V ðaSG a0 Þ
1
vj
. NV
P if V ðaSG a0 Þ vj 6 NV ; otherwise: ð2Þ
4.2.4. The role of incomparability coalition The credibility degree of the group outranking aSG a0 should decrease when the cardinal of CðaRG a0 Þ increase. When a pairwise comparison is performed by a group, a decision is considered valid only if an important part of the group votes effectively (50% is a usual threshold). In this sense, we propose to define the comparability index as: . P 8P P > w
w 0 0 > j j M CðaRG a Þ M 0 wj > < P P if M 0 wj P CðaRG a0 Þ wj ; rða; a0 Þ ¼ > 0 > > P P : if M 0 wj < CðaRG a0 Þ wj : ð3Þ Note that when the strength of the incomparability coalition is similar to the effective voting, r approaches zero. Its maximum value is 1, reached when CðaRG a0 Þ is empty. 4.2.5. ELECTRE-GD valued relation The precedent indexes allow us to define our fuzzy outranking relation for group decision. This is rG : A A ! ½0; 1; rG ða; a0 Þ ¼ Cða; a0 Þ ð1 dða; a0 ÞÞ rða; a0 Þ;
ð4Þ
where Cða; a0 Þ; dða; a0 Þ; rða; a0 Þ are given by (1)– (3). rG should be interpreted as a credibility value (for the SDM) of the assertion ‘‘a is at least as good as a0 ’’ for the group. When the veto and incomparability coalitions are empty, rG ða; a0 Þ ¼ Cða; a0 Þ. But this credibility value is reduced when some actors belong to above coalitions, and is reduced to zero when a veto condition holds and when the comparability index vanishes. The obvious influence of ELECTRE methodology on the present approach for group decisions permits us to call rG as ELECTRE-GD fuzzy binary relation.
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5. An example and a comparison The next example is discussed in Brans et al. (1997) and Macharis et al. (1998) using PROMETHEE method for multicriteria group decision aid. We use the same problem for testing our proposal and comparing it with PROMETHEE. An additional electricity power plant has to be build somewhere in the European Union. The problem is to identify the best possible location. The countries members of the Union introduce six possible locations: A1: Italy A2: Belgium A3: Germany
A4: Sweden A5: Austria A6: France
A comprehensive description of each proposal has been made available by the proponents. Four decision makers have been appointed to take part in the selection process. Each of them represents a specific economic interest: DM1: DM2: DM3: DM4:
(EP) Energy production (ENV) Environment (FIN) Finance (TU) Trade Unions
For this application, the 11 following criteria are formulated: C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
MP POW CC MT VIL DAN SEC CO SOC TPT FIN
Manpower (number of engineers) Power in MW Construction cost Annual maintenance cost Villages to evaluate Danger for environment Security level CO emission Social impact Transport facilities to the plant Financial return
A consensus is reached on objective data for these criteria and the following evaluation table (Table 2) is obtained. In this case, all the decision makers agree on these objective data. This is not
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Table 2 Performances of the alternatives
Italy Belgium Germany Sweden Austria France
C1 MP
C2 POW
C3 CC
C4 MT
C5 VIL
C6 DAN
C7 SEC
C8 CO
C9 SOC
C10 TPT
C11 FIN
80 55 83 40 52 94
500 580 600 450 880 960
1000 250 450 1000 900 950
5.2 3 3.8 7.5 3 3.6
8 1 4 7 3 5
0.5 4 3.5 0 4.5 3.5
9 3 7 10 2 4
0 5 65 0 10 10
2 8 6 10 5 3
300 175 125 450 150 250
4200 900 850 900 750 2000
Table 3 Min/max options and weights
DM1 EP DM2 ENV DM3 FIN DM4 TU
C1 MP
C2 POW
C3 CC
C4 MT
C5 VIL
C6 DAN
C7 SEC
C8 CO
C9 SOC
C10 TPT
C11 FIN
MIN 1
MAX 2
MIN 1
MIN 1
0
0
0
0 MIN 1
0 MIN 1
0
0 MIN 1
MAX 1 MAX 1
0
0
0 MIN 1
0
0 MIN 1 MAX 1
MIN 1 MIN 1 0
0
0
0
0
0
0
0 MAX 1
0 MIN 1
0 MAX 3
compulsory, each decision maker may have his own evaluations. The decision makers assign weights of relative importance to each criterion. Each decision maker is not necessarily interested in the 11 criteria. A weight equal to zero will be assigned to the criteria considered as non relevant. In addition, each decision maker has the freedom to maximize or to minimize each criterion. These individual choices are presented in Table 3. It is obvious that each decision maker is not interested in the complete sets of criteria. We notice that the selected criteria are completely in agreement with the concerns of the decision makers. We also notice that DM1 (EP) wants to minimize manpower (MP) in order to reduce the costs while DM4 (TU) wants to maximize it for creating as much job as possible. This is of course an extreme situation but it illustrates how opposite point of view can generate strong conflicts between the decision makers. According to the ELECTRE III methodology (e.g. Ostanello, 1984; Roy, 1990) the following choices of indifference and strict preference
0 MAX 1
0 0
0
thresholds associated to each criterion were made (Table 4). According to the additional information pointed out before, for each decision maker we applied ELECTRE III to construct a fuzzy outranking relation. After, we used the genetic algorithm presented in Leyva L opez and Fernandez Gonzalez (1999) for exploiting the outranking relation and deriving a final ranking of the alternatives in Table 4 Indifference and strict preference thresholds to each criterion Criterion
Name
Thresholds p,q
C1
MP
C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
POW CC MT VIL DAN SEC CO SOC TPT FIN
p ¼ 10 for DM4 q ¼ 10 for DM1 and DM3 p ¼ 300 q ¼ 50; p ¼ 500 q ¼ 1; p ¼ 6 – p¼5 – p¼5 q¼2 p ¼ 100 –
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27 Table 5 Credibility matrix and final ranking associated to DM1: energy production A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.86 0.94 0.91 0.86 0.71
0.45 1 0.67 0.47 0.69 0.57
0.6 0.87 1 0.47 0.73 0.49
0.69 0.71 0.77 1 0.71 0.71
0.37 0.71 0.45 0.45 1 0.71
0.55 0.69 0.71 0.47 0.87 1
Credibility matrix: ðA2 A5Þ > A3 > A6 > A4 > A1; final ranking, k0 ¼ 0:689. Table 6 Credibility matrix and final ranking associated to DM2: environment
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.32 0.50 1 0.3 0.35
0.75 1 0.50 0.75 0.43 0.5
0.75 0.77 1 0.75 0.7 0.6
0.68 0.3 0.43 1 0.28 0.32
0.75 1 0.50 0.75 1 0.75
0.75 0.93 0.75 0.75 0.85 1
Credibility matrix: ranking, k0 ¼ 0:699.
A4 > A1 > A2 > A5 > A3 > A6;
final
Table 7 Credibility matrix and final ranking associated to DM3: Finance
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.5 0.5 0.45 0.5 0.33
0.62 1 0.28 0.69 0.33 0.67
0.82 1 1 0.72 0.35 0.67
0.83 0.83 0.33 1 0.33 0.83
0.76 1 0.83 0.84 1 0.83
0.98 0.5 0.5 0.38 0.5 1
Credibility matrix: ranking, k0 ¼ 0:619.
A1 > A6 > A2 > A4 > A3 > A5;
final
decreasing order of preferences. Tables 5–8 show the obtained results. As in Macharis et al. (1998), the best choice is not the same for each decision maker. DM1 DM2 DM3 DM4
(EP): Belgium (ENV): Sweden (FIN): Italy (TU): Germany
The classification of the alternatives for each decision maker can be seen better in Table 9. The next step is to solve a new multicriteria decision problem, with the same set of actions, but now
23
Table 8 Credibility matrix and final ranking associated to DM4: trade unions
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.5 0.9 0.5 0.5 0.75
0.5 1 1 0.5 0.62 0.56
0.43 0.43 1 0.5 0.44 0.35
0.7 0.75 0.6 1 0.5 0.5
0.5 0.94 1 0.5 1 0.75
0.62 0.7 0.75 0.5 0.65 1
Credibility matrix: ranking, k0 ¼ 0:699.
final
A3 > A2 > A6 > A1 > A4 > A5;
Table 9 Group performances. Classification of sites according to each new criterion Italy Belgium Germany Sweden Austria France
DM1:EP
DM2:ENV DM3:FIN
DM4:TU
6 1 3 5 2 4
2 3 5 1 4 6
4 2 1 5 6 3
1 3 5 4 6 2
working with the preference matrix of Section 4.1. Usually, the values of parameters k and b needed for defining the nine zones of rk should be determined by the interaction analyst – SDM – DMk ; because in this test problem we cannot access to this information, we consider for b the value 0.15 and for k the credibility threshold k0 obtained when the genetic algorithm of Leyva and Fernandez (see Appendix A) derives the ranking of DMk . On this basis, knowing the ranking and the fuzzy outranking relation previously obtained for each Decision Maker, assuming by default the preference matrix of Table 1, and using Definition 1 of point 4.2, we can derive the restricted outranking relations Sk , which are shown in Tables 10–13. Table 10 Restricted outranking relation associated to DM1
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 1 1 1 1 1
0 1 0 0 0 0
0 1 1 0 1 0
0 1 1 1 1 1
0 1 0 0 1 0
0 1 1 0 1 1
Note: ðAi; AjÞ ¼ 1 means ‘‘Ai outranks Aj’’ ðAi; AjÞ ¼ 0 means ‘‘Ai no outranks Aj’’.
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Table 11 Restricted outranking relation associated to DM2
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0 0 1 0 0
1 1 0 1 0 0
1 1 1 1 1 0
0 0 0 1 0 0
1 1 0 1 1 0
1 1 1 1 1 1
Table 12 Restricted outranking relation associated to DM3 A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0 0 0 0 0
1 1 0 0 0 1
1 1 1 1 0 1
1 1 0 1 0 1
1 1 1 1 1 1
1 0 0 0 0 1
Table 13 Restricted outranking relation associated to DM4 A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 1 1 0 0 1
0 1 1 0 0 0
0 0 1 0 0 0
1 1 1 1 0 1
1 1 1 1 1 1
0 1 1 0 0 1
For example, the position (2,3) of Table 10 associated to DM1 is calculated of the following way: From Table 5 we have r1 ðA2; A3Þ ¼ 0:87 and r1 ðA3; A2Þ ¼ 0:67 with k ¼ 0:689. Then the comparison between A2 and A3 lies in the zone II of Section 4.1. Additionally, we can see from Table 9, first column, that u1 ðA2Þ > u1 ðA3Þ. On these basis and accepting the preference matrix of Table 1 we concluded that
With this information, the global analysis of ELECTRE-GD can be performed. Tables 14 and 15 contain the information about concordance and discordance applying expressions (1) and (2) of Section 4.2. Following (Macharis et al., 1998), criteria (actors) have the same weight (0.25 each one). In Table 16, the ELECTRE-GD valued relation given by expression (4) is pointed out. In this example, the comparability index is 1, and therefore it does not affect rG . For example, the position (2,3) in the concordance matrix is Cð2; 3Þ ¼ 0:75 which follows from expression (1) because DM1, DM2 and DM3 are in favor of the assertion A2SG A3.
Table 14 Concordance matrix A1 A2 A3 A4 A5 A6
A2S1A3: Hence the cell (2,3) in Table 10 is cellð2; 3Þ ¼ 1:
A2
A3
A4
A5
A6
0.50 1 0.25 0.25 0.0 0.25
0.50 0.75 1 0.50 0.50 0.25
0.50 0.75 0.50 1 0.25 0.75
0.75 1 0.50 0.75 1 0.50
0.50 0.75 0.75 0.25 0.50 1
Table 15 Discordance matrix
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
0 0.0 0.50 0.0 0.50 0.50
0.50 0.0 0.0 0.50 0.0 0.0
0.0 0.0 0.0 0.0 0.50 0.0
0.0 0.0 0.50 0.0 0.0 0.50
0.50 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.50 0.0
A1
A2
A3
A4
A5
A6
1 0.50 0.25 0.50 0.12 0.25
0.25 1 0.25 0.12 0.0 0.25
0.50 0.75 1 0.50 0.25 0.25
0.50 0.75 0.25 1 0.25 0.37
0.37 1 0.50 0.75 1 0.50
0.50 0.75 0.75 0.25 0.25 1
Note: NV ¼ 2.
Table 16 Credibility matrix
A2PA3 and by Definition 1 it follows that
A1 1 0.50 0.50 0.50 0.25 0.50
A1 A2 A3 A4 A5 A6
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
Note that DM4 does not belong to the concordant coalition CðA2SG A3Þ but also he does not belong to veto coalition V ðA2SG A3Þ because u4 ðA3Þ is not much greater than u4 ðA2Þ (see column 4 in Table 9). Hence dðA2; A3Þ ¼ 0:0 (position (2,3) in Table 15). With this information and applying expression (4) we fill the cell (2,3) of Table 16 with: rG ðA2; A3Þ ¼ 0:75:
Thus, we have three actors with important objections when the ranking provided by PROMETHEE is analyzed. However, in the ranking derived by ELECTRE-GD, we only observe the last discrepancy. This is an obvious improvement, because the SDM can hardly support an ordering in which 75% of total number of members have strong disagreement.
6. Conclusions
Finally we obtained the group ranking using the genetic algorithm proposed by the authors in Leyva L opez and Fern andez Gonz alez (1999), exploiting the fuzzy relation presented in Table 16. A2 > A1 > A4 > A3 > A6 > A5;
25
k0 ¼ 0:49:
ð5Þ Our result partially agrees with PROMETHEE, which gives the ranking: A2 > A3 > A1 > A4 > A6 > A5: ð6Þ We can observe a significant difference in the relative positions of A3 and (A1, A4). It is produced by the veto effect included in our proposal which is not considered by PROMETHEE. Note that: (a) CðA3; A1Þ ¼ CðA1; A3Þ ¼ 0:5, (b) CðA3; A4Þ ¼ CðA4; A3Þ ¼ 0:5, (c) dðA3; A1Þ ¼ 0:5 because DM3 belongs to V ðA3SG A1Þ, (d) dðA3; A4Þ ¼ 0:5 because DM2 belongs to V ðA3SG A4Þ, however, (c1) dðA1; A3Þ ¼ 0, (d1) dðA4; A3Þ ¼ 0 because any actor does not belong to the respective veto coalition. From points (c) and (d) follow that the values of rG ðA3; A1Þ and rG ðA3; A4Þ are affected, and therefore rG ðA1; A3Þ > rG ðA3; A1Þ þ b and rG ðA4; A3Þ > rG ðA3; A4Þ þ b conditions which explain why, in the ordering derived from rG using the genetic algorithm of Leyva and Fernandez, A1 and A4 are ranked better than A3. Other arguments in favor of our proposal could be the following: Consider the number of actors that have significant disagreement with some positions of a particular ranking. We can see that DM3 strongly disagrees with A3 > A1; DM2 strongly disagrees with A3 > A4; DM1 strongly disagrees with A1 > A5.
ELECTRE-GD is a natural extension of the ELECTRE III approach to collaborative group decision, using a genetic algorithm with very good properties for exploiting the fuzzy outranking relation, which is derived from ELECTRE ideas of concordance, discordance, veto and incomparability. A crucial point of our proposal is the existence of a Supra Decision Maker, a special actor (may be a single person, a small group of stakeholders) with authority for establishing consensus rules and priority information on the set of group members. ELECTRE-GD is in fact a way of modeling the SDMÕs preferences. The proposed method does not have structural properties which limit its application. The analyst can use ELECTRE III, PROMETHEE, or other method based on build up a fuzzy preference relation, in order to obtain the group members particular rankings. After, these pieces of information are used in the process of modeling the SDMÕs preferences. In this sense, ELECTRE-GD can be seen as a complement of these methods for group decisions. ELECTRE-GD performed very well in some examples in the sense of quality of solution as well in the sense of the effort needed for modeling. Once the individual rankings have been obtained this effort is probably greater than one from other simpler methods, but not more than one from a typical application of ELECTRE III to a multicriteria decision problem. Our proposal performs better than PROMETHEE in some test problems, probably because a compensatory scheme is not always well suited for group decisions, where veto effects are often very important. ELECTRE-GD works with the natural heuristic used by collaborative groups for making reasonable or consensus agreements, based on universally accepted majority rules combined
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with the necessary observance of significant minorities.
Appendix A In this appendix, we explain some elements of the genetic algorithm which allows us to exploit a known fuzzy outranking relation with the purpose of constructing a prescription for the multicriteria ranking problem. A potential solution of a ranking problem is represented as an ordinal representation. In general, a potential solution is a ranking of the set of actions by decreasing order of preference. These actions (known as genes in GAs) are joined together forming a string of values (known as chromosome). Any symbol in this string is referred to as an allele (Goldberg, 1989; Michalewicz, 1996). The chromosome is represented as the string of n-ary alphabet where n is the number of actions into the decision problem. In such a representation, each action is coded into n-ary form. Actions are then linked together to produce one long n-ary string or chromosome. An action coded with aki value in the ith entry of the string means that the action coded with aki value is ranking in the ith place of the ordering and aki is preferred to akj if i < j, where aki 2 A ¼ fa1 ; a2 ; . . . ; an g, i ¼ 1; 2; . . . ; n, and ½k1 ; k2 ; . . . ; kn is a permutation of ½1; 2; . . . ; n. Each individual is associated with a number k ð0 6 k 6 1Þ which will be connected with the credibility level of a crisp outranking defined on the set of genes. The fitness of an individual with credibility level k is calculated according to a given fitness function. The approach for defining individualÕs fitness involves separating the single fitness measure into two, one is called fitness and the other is called unfitness. We chose the fitness function f of an individual p with credibility level k as follows: Let p ¼ ak1 ak2 ; . . . ; akn be the schematic representation of an individualÕs chromosome and suppose that given aki and akj , two actions such that rðaki ; akj Þ P k and rðakj ; aki Þ 6 k b ðb > 0, representing a threshold level), we accept that ‘‘aki outranks akj ’’ ðaki S k akj Þ and ‘‘akj does not outrank aki ’’ ðakj nS k aki Þ. In this case into the
crisp outranking relation generated by k; SAk , a presumed preference favoring aki holds. Then: f ðpÞ ¼ jððaki ; akj Þ : aki nS akj and akj nS aki i ¼ 1; 2; . . . ; n 1; j ¼ 2; 3; . . . ; n; i < jÞj where ½k1 ; k2 ; . . . ; kn is a permutation of ½1; 2; . . . ; n. f(p) is the number of incomparabilities between pairs of actions ðaki ; akj Þ into the individual p ¼ ak1 ak2 ; . . . ; akn in the sense of the crisp relation SAk . Note that the quality of solution increases with decreasing fitness score. The unfitness u of an individual p measures the amount of unfeasibility (in relative terms) and we chose to define it as: uðpÞ ¼ jððaki ; akj Þ : aki S akj and akj nS aki i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n; i > jÞj. u(p) is the number of preferences between actions into the individual p which are not ‘‘wellordered’’ in the sense of SAk . An individual P is feasible if uðpÞ ¼ 0 and infeasible if uðpÞ > 0. Defining the unfitness taking the zero minimum value if and only if the solution is feasible seems a natural approach. Each individual P can then be represented by a triad of values f ; u and k. We are interested in: (i) Individuals whose unfitness function value is equal to zero. This assures us that the ordering represented by the individual is transitive; this is one of two characteristics that should be exhibited by all prescription (solution) of ranking problems (Vanderpooten, 1990). (ii) Individuals whose fitness function value is equal (or near) to zero. This objective improves the comparability of S on A. (iii) Individuals whose credibility level k is near to 1. This indicates us that the ordering represented by the individual with credibility level k is more trusty whenever the fitness and unfitness function values are zero or near to zero. In practice, the requirement connected to fitness function does not permit that k values approach to 1 because in this case, we could have many incomparable genes. Then, we use a genetic search for solving the multiobjective problem
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Min u; f Max k; Rs ; k 2 ½0; 1 k P k0 ; where Rs is a strict total order of A.
References Belton, V., Pictet, J., 1997. A framework for group decision using a MCDA model. Sharing, aggregating or comparing individual information? Journal of decision systems 6 (3), 283–303. Bouyssou, D., Vincke, Ph., 1995. Ranking alternatives on the basis of preference relations: a progress report with special emphasis on outranking relations, Serie Mathematiques de la Gestion, Universite Libre de Bruxelles. Brans, J.P., Mareschal, B., Vincke, Ph., 1984. PROMETHEE: A new family of outranking methods in MCDM. In: Brans, J.P. (Ed.), Operational Research. North Holland, Amsterdam, pp. 477–490 (1100pp.). Brans, J.P., Macharis, C., Mareschal, B., 1997. The GDSS PROMETHEE. Report STOOTW/277, Service de Mathematiques de la Gestion, Universite Libre de Bruxelles. Fodor, J., Roubens, M., 1994. Fuzzy Preference Modeling and Multicriteria Decision Support. Kluwer Academic Press, Dordrecht. Goldberg, D., 1989. Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading, MA. Michalewicz, Z., 1996. Genetic Algorithms + Data Structures ¼ Evolution Programs. Springer-Verlag, Berlin. Hwang, C.L., Lin, M.J., 1987. Group Decision Making under Multiple Criteria. In: Lecture Notes in Economics and Mathematical Systems, 281. Springer-Verlag, Berlin. Hokkanen, J., Salminen, P., 1994. Choice of a solid waste management system by using the ELECTRE III method. In: Paruccini, M. (Ed.), Applying MCDA for Decision to Environmental Management. Kluwer Academic Publishers, Dordrecht, Holland.
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Hokkanen, J., Salminen, P., 1997. Choosing a solid waste management system using multicriteria decision analysis. European Journal of Operational Research 98, 19–36. Jelassi, T., Kersten, G., Ziont, S., 1990. An introduction to group decision and negotiation support. In: Bana e Costa, (Ed.), Readings in Multiple Criteria Decision Aid. SpringerVerlag, Berlin, pp. 537–568. Keeney, R.L., 1992. Value Focused Thinking: A Path to Creative Decision Making. Harvard University Press, Cambridge, MA. Keeney, R., Raiffa, H., 1976. Decision with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, New York. Leyva L opez, J.C., Fernandez Gonzalez, E., 1999. A Genetic algorithm for deriving final ranking from a fuzzy outranking relation. Foundations of Computing and Decision Sciences 24 (1), 33–47. Lootsma, F.A., 1997. Fuzzy Logic for Planning and Decision Making. Kluwer Academic Press, Dordrecht. Macharis, C., Brans, J.P., Mareschal, B., 1998. The GDSS PROMETHEE Procedure. Journal of Decision Systems, 7SI. Ostanello, A., 1984. Outranking methods. In: Proceeding of the first summer school on MCDM, Sicilia, pp. 41–60. Rogers, M., Bruen, M., Maystre, L., 2000. ELECTRE and Decision Support. Kluwer Academic Press, Boston, Dordrecht, London. Roy, B., 1990. The outranking approach and the foundations of ELECTRE methods. In: Bana e Costa, C.A. (Ed.), Reading in multiple criteria decision aid. Springer-Verlag, Berlin, pp. 155–183. Roy, B., 1996. Multicriteria Methodology for Decision Aiding. Kluwer Academic Press, Dordrecht. Salminen, P., Hokkanen, J., Lahdelma, R., 1998. Comparing multicriteria methods in the context of environmental problems. European Journal of Operational Research 104, 485–496. Vanderpooten, D., 1990. The construction of prescriptions in outranking methods. In: Bana e Costa, C.A. (Ed.), Reading in Multiple Criteria Decision Aid. Springer-Verlag, Berlin, pp. 184–215.
Decision Aiding
A new method for group decision support based on ELECTRE III methodology pez, Eduardo Fern Juan Carlos Leyva-Lo andez-Gonz alez
*
Facultad de Ingenierıa, Universidad Aut onoma de Sinaloa, Ciudad Universitaria, Calzada de Las Americas S/N, Culiac an, Sinaloa, Mexico, CP 80040, Mexico Received 9 November 1999; accepted 9 January 2002
Abstract Group decision is usually understood as the reduction of different individual preferences on a given set to a single collective preference. At present, there are few approaches which solve the group ranking problem with multiple criteria in a widely acceptable way. Often, they rest on a poor heuristic which makes a decision about consensus ranking difficult to support. This paper presents an extension of the ELECTRE III multicriteria outranking methodology to assist a group of decision makers with different value systems to achieve a consensus on a set of possible alternatives. Our proposal starts with N individual rankings and N corresponding valued preference functions, and uses the natural heuristic provided by ELECTRE methodology for obtaining a fuzzy binary relation representing the collective preference. A comparison of this method with PROMETHEE II for group decision is carried out. We found that, in this particular application, the proposed heuristic based on majority rules combined with concessions to significant minorities, performs relatively better than a compensatory scheme based on a net flow weighted sum function. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multicriteria analysis; Group decision; Outranking methods; ELECTRE III; Genetic algorithms
1. Introduction People make a group decision (intra-organizational or inter-organizational) when they face a common problem and they are all interested in its solution. This problem may be the purchase of a car, the acquisition of a house by a family, the design of a new product or a production planning or
* Corresponding author. Tel.: +52-667-7134281; fax: +52667-7134053. E-mail addresses: [email protected] (J.C. LeyvaL opez), [email protected] (E. Fernandez-Gonzalez).
to find the best location for the wastewater treatment plant in a city. An important characteristic of group decision is that all involved individuals belong to some organization (family, firm, government). They may differ in their perception of the problem and have different interests, but they are all responsible for the organizationÕs well-being and share responsibility for the implemented decision. When a decision situation involves multiple actors, each with different values and informational systems, the final decision will generally be the result of an interaction between this individualÕs preferences and those of others. This interaction is not free of conflict, which may be due to
0377-2217/03/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-2217(02)00273-4
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any of a number of factors, e.g. different ethical or ideological beliefs, different specific objectives, or different roles within organization. Whatever the origin of the conflicting value systems, they will usually affect the evolution of the decision process in ways that were not expected at the outset (Roy, 1996; Keeney, 1992). Group decision is usually understood as the reduction of different individual preferences on a given set to a single collective preference (Jelassi et al., 1990). We will not focus on the psychological aspects of group interactions. The main goal of this work is the prescription of a final group ranking from these individual preferences once they have been established. Group decision making covers a wide range of situations. We are interested in a problem with the following characteristics: (a) Each group member participates in the process giving information about his preferences and beliefs, contributing in this way to the final decision. There usually is an overall goal which is accepted by all the members, but they differ in the ways of how this goal should be achieved. (b) Each actor considers the same set of alternatives or potential actions. (c) There are multiple criteria, which usually conflict with each other. Each actor must generate relevant criteria which may be shared (totally or partially) by some, none or all the remaining members. (d) There is a special actor (may be a single person, a small group of stakeholders) with authority for establishing consensus rules and priority information on the set of group members. Following (Keeney and Raiffa, 1976) we call this entity the Supra Decision Maker (SDM). (e) The group members accept the final decision derived from an aggregation of their opinions according to the rules and priorities defined by the SDM. Points (a), (d) and (e) define our concept of a collaborative group. The last two points are very important and they need a further discussion. Usually, the group belongs to some organization
15
or works for some organization; in this case, the SDM can be seen as an entity representing the interest of that organization. Sometimes the SDM can be considered as an interpreter (may be a referee) of some system of rules describing the group constitution, the way in which the group agrees to make decisions. Sometimes the SDM is an altruistic dictator which, taking into account the different individual opinions, tries to identify the best decision. In particular cases, the SDMÕs role could be performed by the decision analyst. Up to now, there are not many approaches which solve the group ranking problem with multiple criteria (GRPMC) in an acceptable way. Often, they rest on a poor heuristic which makes a decision about consensus ranking difficult to support. Here, we present a new method which starts with N individual rankings and N corresponding valued preference relations, and uses the natural heuristic provided by ELECTRE methodology for obtaining a fuzzy outranking relation representing the SDMÕs preferences, which is also useful for deriving final collective ranking. Thus, this method is limited to group decision problems in which the individual preferences are aggregated using a fuzzy preference relation. This paper is structured as follows. In Section 2, some criticisms of the existing approaches are pointed out, followed by the presentation of our specific problem (Section 3). Section 4 is devoted to expose our proposal. Section 5 contains an example and a comparison of the proposed method with PROMETHEE II for group decision, and on this basis, we present some conclusions in Section 6.
2. Some criticisms of the existing approaches We can distinguish two main general approaches which use a multicriteria decision aid technique for aggregating group preferences in GRPMC (in Belton and Pictet (1997) there are others approaches): (A) In the first way (e.g. Salminen et al., 1998; Hokkanen and Salminen, 1997; Rogers et al., 2000), the group is asked to agree on the
16
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alternatives, criteria, scores, weights, thresholds and remaining parameters before the model provides a ranking. The group discussion focuses on what actions and criteria should be considered, what weights and other necessary parameters are appropriate. Once the discussion is closed and all the individual information has been gathered, a technique is used for obtaining values of these model parameters which should represent the collective opinion. With this information, the multicriteria decision model gives us the group ranking. (B) Although members can exchange opinions and relevant information, a group consensus is needed only for defining the set of potential actions. Each member defines his own criteria, the appropriate evaluations and model parameters (weights, thresholds, etc.), and then the multicriteria decision aid method is used to get a personal ranking. Next, each actor is considered as a separate criterion, and the preferential information contained in its particular ranking is aggregated in a final collective ordering with the same (may be other) multicriteria decision approach (e.g. Macharis et al., 1998; Hwang and Lin, 1987). Some problems could arise when Approach A is used. In order to be a real collective opinion, the model parameters (e.g. weights, scores, thresholds) should represent a group consensus coming from unanimity or, at least, from a strong majority. If the values, for instance, of the weights are determined by a majority subset of the group members, the analyst should take care that other important parameters (e.g. veto thresholds) will be determined by the same subset. Otherwise, the complete set of chosen parameters could not represent anybody. In this case, it would be impossible to talk about a collective opinion. Some authors (cf. Rogers et al., 2000; Hokkanen and Salminen, 1994, 1997; Salminen et al., 1998) have reported successful results applying ELECTRE methods to GRPMC following Approach A. Average weights, upper quartile and lower quartile weights are used in the assessment, with different weighting systems reflecting the views of different groupings, used within a sensi-
tivity analysis. However, strong divergences often exist in the perceived importance of the criteria; sometimes the ratio of maximum to minimum weighting exceeds 10 (cf. Rogers et al., 2000). Fortunately, consensus was achieved about threshold levels. Considering its successful applications, Approach A performs well when the assessments of model parameters coming from the different members do not show strong divergences, or when the final ranking is sufficiently robust to handle them. If these conditions do not hold, other approaches for aggregation of group member preferences could be better. Approach B is supported by the following arguments: • There are some important similar characteristics in group decision and multicriteria decision problems (MCDP): (a) the interest in obtaining a good performance of the complete set of criteria (in MCDP), and in obtaining a ranking which satisfies to the complete set of actors (in GRPMC); (b) the concept of an ideal solution, usually unfeasible, is common; (c) the concept of dominance plays a similar role too; and (d) when the ideal solution is not feasible, it is necessary to model the decision makerÕs preferences on multiple criteria (in MCDP), or the SDMÕs preferences on multiple actors (in GRPMC), in order to find the best solution. • It seems natural considering each actor as a different point of view, that is, a different ‘‘attribute’’ of a decision problem. • Using Approach B, the SDMÕs preferences on the different ‘‘criteria’’ (his opinions and priorities concerning the different group members) can be modeled. • Some suitable ideas coming from the concept of democracy can be included in the model of SDMÕs preferences. • We do not use parameters provided by the group members. We do not need to work with poorly defined ‘‘collective parameters’’. We feel that the basis of our discussion is the meaning of the term ‘‘collective opinion’’. We think that a real ‘‘collective opinion’’ exists only in
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
case of (i) unanimity or strong consensus, and (ii) when a strong majority accepts the altruistic opinion of the SDM, who performs as an interpreter of the group opinion; it happens working with collaborative groups. Approach B allows looking for the ‘‘collective opinion’’ in the sense of point (ii). The main questions in order to make operational a method based on B are the following: A1. How to build an aggregation model of SDMÕs preferences with good properties, only knowing the group member rankings and the aggregation models of the member multicriteria preferences? B1. When the member ranking is derived exploiting fuzzy preference relations (or crisp outranking relations), the effect of irrelevant alternatives produces frequent discrepancies between the ranking and the aggregation model of preferences (Vanderpooten, 1990). Then, how could this contradictory information about the preferences of a particular group member be handled? C1. How could the aggregation model of SDMÕs preferences be exploited deriving the final solution in such a way that the above discrepancies model-ranking are minimized? C1 is a very important question for improving the prescription coming from fuzzy preference modeling (Bouyssou and Vincke, 1995). We have proposed the use of a genetic algorithm for deriving final ranking from a fuzzy preference relation (cf. Leyva L opez and Fern andez Gonz alez, 1999) which has good properties and allows finding more consistent orderings than those obtained with other reported exploitation methods. In fact, this genetic search for improving the final ranking is part of our proposal (see Appendix A). However, the aim of this contribution concerns mainly with A1 and B1. PROMETHEE (Brans et al., 1984) has been applied for GRPMC using a net flow weightedsum function as a rough model of SDMÕs preferences (Brans et al., 1997; Macharis et al., 1998), giving an answer to question A1 (not to B1). Anyway, we think that a compensatory scheme is not suitable for group decision, because the natural heuristic based on majority rules combined with respect to significant minorities should be
17
preserved. Therefore, a compensatory scheme does not seem to be a good model of the SDMÕs preferences. In this sense, ELECTRE is the multicriteria decision methodology closest to this heuristic based on concordance and discordance principles, which it is often used when groups perform pairwise comparisons (Lootsma, 1997). However, the use of ELECTRE for building an aggregation model of the SDMÕs preferences (following Approach B) is not straightforward, because this model requires some numeric information which the SDM can not derive directly from member particular rankings. Besides, when the group members possess different values and informational systems, Roy (1996) recommends to modify the traditional ELECTRE methods incorporating these new characteristics.
3. Problem formulation Let A be the finite set of alternatives or potential actions considered by a group and evaluated on multiple criteria. The group is composed of a set M ¼ f1; 2; . . . ; N g of members, whose work is somehow controlled by a Supra Decision Maker. Let us suppose that a multicriteria aggregation method based on fuzzy preference relation is used for deriving the group members particular rankings. Let ri : AXA ! ½0; 1 be a valued binary relation which aggregates the preferences of the ith member on the multiple criteria describing the elements of A. Let Oi be a complete ranking of A derived using some procedure for exploiting ri . The problem is: from N pairs ðri ; Oi Þ, obtaining a ranking O which performs the best compromise according to the SDMÕs preferences. Remarks: • This problem arises when the facilitator (may be the SDM) uses fuzzy outranking approaches for obtaining individual rankings, as in ELECTRE III and PROMETHEE, but is not limited to these particular methods (Fodor and Roubens, 1994). • The model of the SDM plays a fundamental role, which has been neglected or simplified by
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previous approaches. Building this model is hard and different from a typical multicriteria decision model, because: (a) it should solve the probable conflict between Oi and ri , when the last one is not transitive (point B1 of previous section); (b) the imprecise and qualitative nature of the information provided by the individual rankings and fuzzy preference relations. • A good model of the SDMÕs way of thinking should be based on a widely accepted heuristic for democracy. In the following section, we present a method strongly inspired by ELECTRE III.
4. The new method ELECTRE-GD Our approach consists of the following steps: first, it is necessary to propose a way to solve the possible conflict between the information provided by individual rankings and fuzzy preference relations. Second, we will define a fuzzy outranking relation for the SDM, which should be exploited in some rational way in order to derive final group ranking. 4.1. Solving conflicts: the preference matrix As we have previously discussed, the basic idea is considering each member k as a criterion of a new multicriteria problem. Each pair of actions ða; a0 Þ 2 A A should be compared according to the point of view of criterion k. For this comparison, the SDM considers an information composed of two significant elements: the relative position of actions a and a0 in Ok and the values rk ða; a0 Þ and rk ða0 ; aÞ. Let uk : A ! N be a function defined as uk ðai Þ ¼ cardðBÞ þ 1; where B ¼ faj 2 A : aj is ranked worse than ai in Ok g. Note that if ai is the best in Ok , then uk ðai Þ ¼ cardðAÞ. If ai is the worst ranked, uk ðai Þ ¼ 1. Now, we need to introduce two thresholds which reflect some feelings of the SDM on the fuzzy preference relation rk . Suppose that k ð0 < k < 1Þ is such that if rk ða; a0 Þ P k the SDM agrees
with the assertion ‘‘in absence of other information this is a strong reason for thinking that a is at least as good as a0 since the point of view of member k’’. Let b another threshold parameter such that if rk ða; a0 Þ 6 k b, the SDM feels that ‘‘in absence of other information, action a does not seem at least as good as a0 from the point of view of member k’’. In the interval k b 6 rk 6 k the SDM is doubtful about the outranking aSa0 . Combining these intervals for rk ða; a0 Þ and rk ða0 ; aÞ we can distinguish nine different situations (zones): I II III IV V VI VII VIII IX
rk ða; a0 Þ P k; rk ða0 ; aÞ P k, rk ða; a0 Þ P k; k b < rk ða0 ; aÞ < k, rk ða; a0 Þ P k; rk ða0 ; aÞ 6 k b, rk ða0 ; aÞ P k; k b < rk ða; a0 Þ < k, k b < rk ða; a0 Þ < k; k b < rk ða0 ; aÞ < k, rk ða0 ; aÞ 6 k b; k b < rk ða; a0 Þ < k, rk ða0 ; aÞ P k; rk ða; a0 Þ 6 k b, rk ða; a0 Þ 6 k b; k b < rk ða0 ; aÞ < k, rk ða0 ; aÞ 6 k b; rk ða; a0 Þ 6 k b.
The information about preferences contained in rk should be aggregated with another one that follows from ranking Ok . It is not an easy task. Many techniques for rank ordering alternatives on the basis of fuzzy binary relations have been proposed (e.g. Fodor and Roubens, 1994). No one seems to be clearly better than others (cf. Bouyssou and Vincke, 1995). In our opinion, the main criticism concerns with the fact that the relative position of alternatives a and b in the ranking is not determined by rða; bÞ and rðb; aÞ (r denotes the fuzzy preference relation). Other alternatives strongly influence on their positions, and therefore the final ranking often does not respect the Decision MakerÕs preferences that should be derived from rða; bÞ and rðb; aÞ. This drawback can lead to violations of the crisp preference that could be obtained from r (rða; bÞ much greater than rðb; aÞ and however a is ranked worse than b). The SDM could consider five different situations concerning with uk : • uk ðaÞ uk ða0 Þ. Denotes the case in which action a is ranked as one of the best action, while a0 is one of the worst (this classification is defined by
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
•
• • •
the SDM which could take into account the opinion from DMk ). uk ðaÞ > uk ða0 Þ. Denotes when action a is ranked better than a0 , but the previous situation does not hold. uk ðaÞ ¼ uk ða0 Þ. Actions a and a0 are tied in the ranking, or their difference is negligible. uk ða0 Þ > uk ðaÞ. uk ða0 Þ uk ðaÞ.
It is not possible here to assess a quantitative meaning to the statements ‘‘one of the best ranked actions’’ and ‘‘one of the worst’’. It depends on the number of ranked actions and on the SDMÕs perception about the reliability of the method used for deriving a particular ranking. But roughly, we could use the top 15% and the bottom 15%, respectively, as reference levels by default. In practice, the interaction SDM-analyst should assess the correct levels. Combining the nine possible situations for rk and uk ðaÞ uk ða0 Þ, there does exist 45 combinations. They can be arranged in a matrix with nine rows and five columns. The SDM can express preferences about these events using preference binary relations P, Q, I, R. According to Roy (1996), they mean: P (strict preference). Corresponds to the existence of clear and positive reasons that justify significant preference in favor of one (identified) of the two actions. I (indifference). Corresponds to the existence of clear and positive reasons that justify equivalence between the two actions.
Q (weak preference). Corresponds to the existence of clear and positive reasons that invalidate strict preference in favor of one (identified) of the two actions, but that are insufficient to deduce either strict preference in favor of the other action or indifference between the two actions, not allowing either of the two preceding situation to be distinguished as appropriate. R (incomparability). Corresponds to an absence of clear and positive reasons that justify any of the three preceding relations. Most of these 45 judgements are trivial ones. We can propose, by default, the ‘‘preference matrix’’ in Table 1. Some positions in the ‘‘preference matrix’’ reflect a good coincidence between rk ða; a0 Þ and uk ðaÞ uk ða0 Þ (for instance see (3,1), (1,3)). Other positions like (3,5) and (7,1) reflect strong (although possible) discrepancy. In a decision aiding process, this matrix is a result of the interaction between an analyst and the SDM. If necessary, some doubtful elements may be adjusted (for instance, in positions (3,5) and (7,1) an incomparability statement should be considered). This adjustment does not depend on a particular group member, the preference matrix is an expression of the SDMÕs beliefs, his own feeling about the reliability of methods used for deriving the fuzzy preference relations and actorÕs ranking. Note that when the preference matrix has been assessed, given a pair ða; a0 Þ we have a correspondence between actors and elements of the preference matrix. That is, for a pair ða; a0 Þ and for the
Table 1 Preference matrix Zone I II III IV V VI VII VIII IX
uðaÞ uða0 Þ 0
aPa aPa0 aPa0 aPa0 aPa0 aPa0 aQa0 aPa0 aPa0
uðaÞ > uða0 Þ 0
aQa aPa0 aPa0 aIa0 aQa0 aPa0 aIa0 aIa0 aQa0
19
uðaÞ ¼ uða0 Þ 0
aIa aQa0 aQa0 a0 Qa aIa0 aQa0 a0 Qa a0 Qa aIa0
uða0 Þ > uðaÞ 0
a Qa aIa0 aIa0 a0 Pa a0 Qa aIa0 a0 Pa a0 Pa a0 Qa
uða0 Þ uðaÞ a0 Pa a0 Pa a0 Qa a0 Pa a0 Pa a0 Pa a0 Pa a0 Pa a0 Pa
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kth actor the analyst can find the corresponding element of the preference matrix, which will be used by the algorithm exposed in the next paragraph. 4.2. A fuzzy outranking relation for aggregating SDM’s preferences Once the preference matrix has been obtained, the definition of a valued binary relation following ELECTRE III is straightforward (e.g. Roy, 1990). 4.2.1. Preliminary definitions Definition 1. We say that action a outranks action a0 from the point of view of actor k (restricted outranking relation aSk a0 ) if and only if aPk a0 , aQk a0 , or aIk a, in agreement with the corresponding element of the preference matrix. Definition 2. An actor k is in concordance with the assertion aSG a0 , (SG means group outranking), if and only if aSk a0 . In the following CðaSG a0 Þ will denote the concordant coalition, the set of actors which are in concordance with aSG a0 . Definition 3. An actor k is in discordance with the assertion aSG a0 if and only if a0 Pk a. DðaSG a0 Þ will denote the discordant coalition, joining the actors which are in discordance with aSG a0 . Definition 4. An actor k belongs to veto coalition V ðaSG a0 Þ if and only if the two following conditions hold: (i) uk ða0 Þ uk ðaÞ, (ii) rk ða0 ; aÞ rk ða; a0 Þb ((ii) could be changed by a stronger condition like rk ða0 ; aÞ P k and rk ða; a0 Þ < k b). Definition 5. An actor k belongs to the incomparability coalition CðaRG a0 Þ if aRk a0 .
4.2.2. The concordance index For defining a comprehensive outranking relation SG , we take into account the fact that the role
of different actors (criteria) is not the same from the SDM point of view. As in ELECTRE methods, the importance of the jth criterion is taken into account by means of two effects: • its importance coefficient wj > 0, which is considered in the definition of a concordance degree; • its veto capacity. Following ELECTRE, in order to model the strength of the arguments in favor of the assertion aSG a0 , we define the concordance index as X wj ; j 2 CðaSG a0 Þ; ð1Þ Cða; a0 Þ ¼ 1=W where W ¼
X
wj ;
j 2 M 0;
M 0 ¼ fk 2 M : aSk a0 or a0 Sk ag: In group decision a statement of incomparability expresses the desire of no voting. Therefore, criteria belong CðaRG a0 Þ are excluded of M 0. 4.2.3. The power of veto coalition The power for determining a veto condition should be given by the number and importance of actors which belong to V ðaSG a0 Þ. In the following, we propose a model based on simple ideas taken from the natural heuristic of group pairwise comparisons. Let vj be the number of votes (for veto condition) which the SDM agrees assigning to jth member. Suppose that i denotes a middle, nonrelevant group member; then, without loss of generality, vi should be equal 1. Other values (2,3, or still more) are reserved to actors whose opinions are considered more relevant by the SDM. The number of votes for veto are close related to the number of votes for concordance wj , but they are not necessarily the same. By default, the analyst could assume identity. Let NV be the number of votes which the SDM considers necessary to become in a serious argument against aSG a0 . We define the discordance index as
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
0
dða; a Þ ¼
(P
V ðaSG a0 Þ
1
vj
. NV
P if V ðaSG a0 Þ vj 6 NV ; otherwise: ð2Þ
4.2.4. The role of incomparability coalition The credibility degree of the group outranking aSG a0 should decrease when the cardinal of CðaRG a0 Þ increase. When a pairwise comparison is performed by a group, a decision is considered valid only if an important part of the group votes effectively (50% is a usual threshold). In this sense, we propose to define the comparability index as: . P 8P P > w
w 0 0 > j j M CðaRG a Þ M 0 wj > < P P if M 0 wj P CðaRG a0 Þ wj ; rða; a0 Þ ¼ > 0 > > P P : if M 0 wj < CðaRG a0 Þ wj : ð3Þ Note that when the strength of the incomparability coalition is similar to the effective voting, r approaches zero. Its maximum value is 1, reached when CðaRG a0 Þ is empty. 4.2.5. ELECTRE-GD valued relation The precedent indexes allow us to define our fuzzy outranking relation for group decision. This is rG : A A ! ½0; 1; rG ða; a0 Þ ¼ Cða; a0 Þ ð1 dða; a0 ÞÞ rða; a0 Þ;
ð4Þ
where Cða; a0 Þ; dða; a0 Þ; rða; a0 Þ are given by (1)– (3). rG should be interpreted as a credibility value (for the SDM) of the assertion ‘‘a is at least as good as a0 ’’ for the group. When the veto and incomparability coalitions are empty, rG ða; a0 Þ ¼ Cða; a0 Þ. But this credibility value is reduced when some actors belong to above coalitions, and is reduced to zero when a veto condition holds and when the comparability index vanishes. The obvious influence of ELECTRE methodology on the present approach for group decisions permits us to call rG as ELECTRE-GD fuzzy binary relation.
21
5. An example and a comparison The next example is discussed in Brans et al. (1997) and Macharis et al. (1998) using PROMETHEE method for multicriteria group decision aid. We use the same problem for testing our proposal and comparing it with PROMETHEE. An additional electricity power plant has to be build somewhere in the European Union. The problem is to identify the best possible location. The countries members of the Union introduce six possible locations: A1: Italy A2: Belgium A3: Germany
A4: Sweden A5: Austria A6: France
A comprehensive description of each proposal has been made available by the proponents. Four decision makers have been appointed to take part in the selection process. Each of them represents a specific economic interest: DM1: DM2: DM3: DM4:
(EP) Energy production (ENV) Environment (FIN) Finance (TU) Trade Unions
For this application, the 11 following criteria are formulated: C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
MP POW CC MT VIL DAN SEC CO SOC TPT FIN
Manpower (number of engineers) Power in MW Construction cost Annual maintenance cost Villages to evaluate Danger for environment Security level CO emission Social impact Transport facilities to the plant Financial return
A consensus is reached on objective data for these criteria and the following evaluation table (Table 2) is obtained. In this case, all the decision makers agree on these objective data. This is not
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Table 2 Performances of the alternatives
Italy Belgium Germany Sweden Austria France
C1 MP
C2 POW
C3 CC
C4 MT
C5 VIL
C6 DAN
C7 SEC
C8 CO
C9 SOC
C10 TPT
C11 FIN
80 55 83 40 52 94
500 580 600 450 880 960
1000 250 450 1000 900 950
5.2 3 3.8 7.5 3 3.6
8 1 4 7 3 5
0.5 4 3.5 0 4.5 3.5
9 3 7 10 2 4
0 5 65 0 10 10
2 8 6 10 5 3
300 175 125 450 150 250
4200 900 850 900 750 2000
Table 3 Min/max options and weights
DM1 EP DM2 ENV DM3 FIN DM4 TU
C1 MP
C2 POW
C3 CC
C4 MT
C5 VIL
C6 DAN
C7 SEC
C8 CO
C9 SOC
C10 TPT
C11 FIN
MIN 1
MAX 2
MIN 1
MIN 1
0
0
0
0 MIN 1
0 MIN 1
0
0 MIN 1
MAX 1 MAX 1
0
0
0 MIN 1
0
0 MIN 1 MAX 1
MIN 1 MIN 1 0
0
0
0
0
0
0
0 MAX 1
0 MIN 1
0 MAX 3
compulsory, each decision maker may have his own evaluations. The decision makers assign weights of relative importance to each criterion. Each decision maker is not necessarily interested in the 11 criteria. A weight equal to zero will be assigned to the criteria considered as non relevant. In addition, each decision maker has the freedom to maximize or to minimize each criterion. These individual choices are presented in Table 3. It is obvious that each decision maker is not interested in the complete sets of criteria. We notice that the selected criteria are completely in agreement with the concerns of the decision makers. We also notice that DM1 (EP) wants to minimize manpower (MP) in order to reduce the costs while DM4 (TU) wants to maximize it for creating as much job as possible. This is of course an extreme situation but it illustrates how opposite point of view can generate strong conflicts between the decision makers. According to the ELECTRE III methodology (e.g. Ostanello, 1984; Roy, 1990) the following choices of indifference and strict preference
0 MAX 1
0 0
0
thresholds associated to each criterion were made (Table 4). According to the additional information pointed out before, for each decision maker we applied ELECTRE III to construct a fuzzy outranking relation. After, we used the genetic algorithm presented in Leyva L opez and Fernandez Gonzalez (1999) for exploiting the outranking relation and deriving a final ranking of the alternatives in Table 4 Indifference and strict preference thresholds to each criterion Criterion
Name
Thresholds p,q
C1
MP
C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
POW CC MT VIL DAN SEC CO SOC TPT FIN
p ¼ 10 for DM4 q ¼ 10 for DM1 and DM3 p ¼ 300 q ¼ 50; p ¼ 500 q ¼ 1; p ¼ 6 – p¼5 – p¼5 q¼2 p ¼ 100 –
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27 Table 5 Credibility matrix and final ranking associated to DM1: energy production A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.86 0.94 0.91 0.86 0.71
0.45 1 0.67 0.47 0.69 0.57
0.6 0.87 1 0.47 0.73 0.49
0.69 0.71 0.77 1 0.71 0.71
0.37 0.71 0.45 0.45 1 0.71
0.55 0.69 0.71 0.47 0.87 1
Credibility matrix: ðA2 A5Þ > A3 > A6 > A4 > A1; final ranking, k0 ¼ 0:689. Table 6 Credibility matrix and final ranking associated to DM2: environment
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.32 0.50 1 0.3 0.35
0.75 1 0.50 0.75 0.43 0.5
0.75 0.77 1 0.75 0.7 0.6
0.68 0.3 0.43 1 0.28 0.32
0.75 1 0.50 0.75 1 0.75
0.75 0.93 0.75 0.75 0.85 1
Credibility matrix: ranking, k0 ¼ 0:699.
A4 > A1 > A2 > A5 > A3 > A6;
final
Table 7 Credibility matrix and final ranking associated to DM3: Finance
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.5 0.5 0.45 0.5 0.33
0.62 1 0.28 0.69 0.33 0.67
0.82 1 1 0.72 0.35 0.67
0.83 0.83 0.33 1 0.33 0.83
0.76 1 0.83 0.84 1 0.83
0.98 0.5 0.5 0.38 0.5 1
Credibility matrix: ranking, k0 ¼ 0:619.
A1 > A6 > A2 > A4 > A3 > A5;
final
decreasing order of preferences. Tables 5–8 show the obtained results. As in Macharis et al. (1998), the best choice is not the same for each decision maker. DM1 DM2 DM3 DM4
(EP): Belgium (ENV): Sweden (FIN): Italy (TU): Germany
The classification of the alternatives for each decision maker can be seen better in Table 9. The next step is to solve a new multicriteria decision problem, with the same set of actions, but now
23
Table 8 Credibility matrix and final ranking associated to DM4: trade unions
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0.5 0.9 0.5 0.5 0.75
0.5 1 1 0.5 0.62 0.56
0.43 0.43 1 0.5 0.44 0.35
0.7 0.75 0.6 1 0.5 0.5
0.5 0.94 1 0.5 1 0.75
0.62 0.7 0.75 0.5 0.65 1
Credibility matrix: ranking, k0 ¼ 0:699.
final
A3 > A2 > A6 > A1 > A4 > A5;
Table 9 Group performances. Classification of sites according to each new criterion Italy Belgium Germany Sweden Austria France
DM1:EP
DM2:ENV DM3:FIN
DM4:TU
6 1 3 5 2 4
2 3 5 1 4 6
4 2 1 5 6 3
1 3 5 4 6 2
working with the preference matrix of Section 4.1. Usually, the values of parameters k and b needed for defining the nine zones of rk should be determined by the interaction analyst – SDM – DMk ; because in this test problem we cannot access to this information, we consider for b the value 0.15 and for k the credibility threshold k0 obtained when the genetic algorithm of Leyva and Fernandez (see Appendix A) derives the ranking of DMk . On this basis, knowing the ranking and the fuzzy outranking relation previously obtained for each Decision Maker, assuming by default the preference matrix of Table 1, and using Definition 1 of point 4.2, we can derive the restricted outranking relations Sk , which are shown in Tables 10–13. Table 10 Restricted outranking relation associated to DM1
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 1 1 1 1 1
0 1 0 0 0 0
0 1 1 0 1 0
0 1 1 1 1 1
0 1 0 0 1 0
0 1 1 0 1 1
Note: ðAi; AjÞ ¼ 1 means ‘‘Ai outranks Aj’’ ðAi; AjÞ ¼ 0 means ‘‘Ai no outranks Aj’’.
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Table 11 Restricted outranking relation associated to DM2
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0 0 1 0 0
1 1 0 1 0 0
1 1 1 1 1 0
0 0 0 1 0 0
1 1 0 1 1 0
1 1 1 1 1 1
Table 12 Restricted outranking relation associated to DM3 A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 0 0 0 0 0
1 1 0 0 0 1
1 1 1 1 0 1
1 1 0 1 0 1
1 1 1 1 1 1
1 0 0 0 0 1
Table 13 Restricted outranking relation associated to DM4 A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
1 1 1 0 0 1
0 1 1 0 0 0
0 0 1 0 0 0
1 1 1 1 0 1
1 1 1 1 1 1
0 1 1 0 0 1
For example, the position (2,3) of Table 10 associated to DM1 is calculated of the following way: From Table 5 we have r1 ðA2; A3Þ ¼ 0:87 and r1 ðA3; A2Þ ¼ 0:67 with k ¼ 0:689. Then the comparison between A2 and A3 lies in the zone II of Section 4.1. Additionally, we can see from Table 9, first column, that u1 ðA2Þ > u1 ðA3Þ. On these basis and accepting the preference matrix of Table 1 we concluded that
With this information, the global analysis of ELECTRE-GD can be performed. Tables 14 and 15 contain the information about concordance and discordance applying expressions (1) and (2) of Section 4.2. Following (Macharis et al., 1998), criteria (actors) have the same weight (0.25 each one). In Table 16, the ELECTRE-GD valued relation given by expression (4) is pointed out. In this example, the comparability index is 1, and therefore it does not affect rG . For example, the position (2,3) in the concordance matrix is Cð2; 3Þ ¼ 0:75 which follows from expression (1) because DM1, DM2 and DM3 are in favor of the assertion A2SG A3.
Table 14 Concordance matrix A1 A2 A3 A4 A5 A6
A2S1A3: Hence the cell (2,3) in Table 10 is cellð2; 3Þ ¼ 1:
A2
A3
A4
A5
A6
0.50 1 0.25 0.25 0.0 0.25
0.50 0.75 1 0.50 0.50 0.25
0.50 0.75 0.50 1 0.25 0.75
0.75 1 0.50 0.75 1 0.50
0.50 0.75 0.75 0.25 0.50 1
Table 15 Discordance matrix
A1 A2 A3 A4 A5 A6
A1
A2
A3
A4
A5
A6
0 0.0 0.50 0.0 0.50 0.50
0.50 0.0 0.0 0.50 0.0 0.0
0.0 0.0 0.0 0.0 0.50 0.0
0.0 0.0 0.50 0.0 0.0 0.50
0.50 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.50 0.0
A1
A2
A3
A4
A5
A6
1 0.50 0.25 0.50 0.12 0.25
0.25 1 0.25 0.12 0.0 0.25
0.50 0.75 1 0.50 0.25 0.25
0.50 0.75 0.25 1 0.25 0.37
0.37 1 0.50 0.75 1 0.50
0.50 0.75 0.75 0.25 0.25 1
Note: NV ¼ 2.
Table 16 Credibility matrix
A2PA3 and by Definition 1 it follows that
A1 1 0.50 0.50 0.50 0.25 0.50
A1 A2 A3 A4 A5 A6
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
Note that DM4 does not belong to the concordant coalition CðA2SG A3Þ but also he does not belong to veto coalition V ðA2SG A3Þ because u4 ðA3Þ is not much greater than u4 ðA2Þ (see column 4 in Table 9). Hence dðA2; A3Þ ¼ 0:0 (position (2,3) in Table 15). With this information and applying expression (4) we fill the cell (2,3) of Table 16 with: rG ðA2; A3Þ ¼ 0:75:
Thus, we have three actors with important objections when the ranking provided by PROMETHEE is analyzed. However, in the ranking derived by ELECTRE-GD, we only observe the last discrepancy. This is an obvious improvement, because the SDM can hardly support an ordering in which 75% of total number of members have strong disagreement.
6. Conclusions
Finally we obtained the group ranking using the genetic algorithm proposed by the authors in Leyva L opez and Fern andez Gonz alez (1999), exploiting the fuzzy relation presented in Table 16. A2 > A1 > A4 > A3 > A6 > A5;
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k0 ¼ 0:49:
ð5Þ Our result partially agrees with PROMETHEE, which gives the ranking: A2 > A3 > A1 > A4 > A6 > A5: ð6Þ We can observe a significant difference in the relative positions of A3 and (A1, A4). It is produced by the veto effect included in our proposal which is not considered by PROMETHEE. Note that: (a) CðA3; A1Þ ¼ CðA1; A3Þ ¼ 0:5, (b) CðA3; A4Þ ¼ CðA4; A3Þ ¼ 0:5, (c) dðA3; A1Þ ¼ 0:5 because DM3 belongs to V ðA3SG A1Þ, (d) dðA3; A4Þ ¼ 0:5 because DM2 belongs to V ðA3SG A4Þ, however, (c1) dðA1; A3Þ ¼ 0, (d1) dðA4; A3Þ ¼ 0 because any actor does not belong to the respective veto coalition. From points (c) and (d) follow that the values of rG ðA3; A1Þ and rG ðA3; A4Þ are affected, and therefore rG ðA1; A3Þ > rG ðA3; A1Þ þ b and rG ðA4; A3Þ > rG ðA3; A4Þ þ b conditions which explain why, in the ordering derived from rG using the genetic algorithm of Leyva and Fernandez, A1 and A4 are ranked better than A3. Other arguments in favor of our proposal could be the following: Consider the number of actors that have significant disagreement with some positions of a particular ranking. We can see that DM3 strongly disagrees with A3 > A1; DM2 strongly disagrees with A3 > A4; DM1 strongly disagrees with A1 > A5.
ELECTRE-GD is a natural extension of the ELECTRE III approach to collaborative group decision, using a genetic algorithm with very good properties for exploiting the fuzzy outranking relation, which is derived from ELECTRE ideas of concordance, discordance, veto and incomparability. A crucial point of our proposal is the existence of a Supra Decision Maker, a special actor (may be a single person, a small group of stakeholders) with authority for establishing consensus rules and priority information on the set of group members. ELECTRE-GD is in fact a way of modeling the SDMÕs preferences. The proposed method does not have structural properties which limit its application. The analyst can use ELECTRE III, PROMETHEE, or other method based on build up a fuzzy preference relation, in order to obtain the group members particular rankings. After, these pieces of information are used in the process of modeling the SDMÕs preferences. In this sense, ELECTRE-GD can be seen as a complement of these methods for group decisions. ELECTRE-GD performed very well in some examples in the sense of quality of solution as well in the sense of the effort needed for modeling. Once the individual rankings have been obtained this effort is probably greater than one from other simpler methods, but not more than one from a typical application of ELECTRE III to a multicriteria decision problem. Our proposal performs better than PROMETHEE in some test problems, probably because a compensatory scheme is not always well suited for group decisions, where veto effects are often very important. ELECTRE-GD works with the natural heuristic used by collaborative groups for making reasonable or consensus agreements, based on universally accepted majority rules combined
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J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
with the necessary observance of significant minorities.
Appendix A In this appendix, we explain some elements of the genetic algorithm which allows us to exploit a known fuzzy outranking relation with the purpose of constructing a prescription for the multicriteria ranking problem. A potential solution of a ranking problem is represented as an ordinal representation. In general, a potential solution is a ranking of the set of actions by decreasing order of preference. These actions (known as genes in GAs) are joined together forming a string of values (known as chromosome). Any symbol in this string is referred to as an allele (Goldberg, 1989; Michalewicz, 1996). The chromosome is represented as the string of n-ary alphabet where n is the number of actions into the decision problem. In such a representation, each action is coded into n-ary form. Actions are then linked together to produce one long n-ary string or chromosome. An action coded with aki value in the ith entry of the string means that the action coded with aki value is ranking in the ith place of the ordering and aki is preferred to akj if i < j, where aki 2 A ¼ fa1 ; a2 ; . . . ; an g, i ¼ 1; 2; . . . ; n, and ½k1 ; k2 ; . . . ; kn is a permutation of ½1; 2; . . . ; n. Each individual is associated with a number k ð0 6 k 6 1Þ which will be connected with the credibility level of a crisp outranking defined on the set of genes. The fitness of an individual with credibility level k is calculated according to a given fitness function. The approach for defining individualÕs fitness involves separating the single fitness measure into two, one is called fitness and the other is called unfitness. We chose the fitness function f of an individual p with credibility level k as follows: Let p ¼ ak1 ak2 ; . . . ; akn be the schematic representation of an individualÕs chromosome and suppose that given aki and akj , two actions such that rðaki ; akj Þ P k and rðakj ; aki Þ 6 k b ðb > 0, representing a threshold level), we accept that ‘‘aki outranks akj ’’ ðaki S k akj Þ and ‘‘akj does not outrank aki ’’ ðakj nS k aki Þ. In this case into the
crisp outranking relation generated by k; SAk , a presumed preference favoring aki holds. Then: f ðpÞ ¼ jððaki ; akj Þ : aki nS akj and akj nS aki i ¼ 1; 2; . . . ; n 1; j ¼ 2; 3; . . . ; n; i < jÞj where ½k1 ; k2 ; . . . ; kn is a permutation of ½1; 2; . . . ; n. f(p) is the number of incomparabilities between pairs of actions ðaki ; akj Þ into the individual p ¼ ak1 ak2 ; . . . ; akn in the sense of the crisp relation SAk . Note that the quality of solution increases with decreasing fitness score. The unfitness u of an individual p measures the amount of unfeasibility (in relative terms) and we chose to define it as: uðpÞ ¼ jððaki ; akj Þ : aki S akj and akj nS aki i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n; i > jÞj. u(p) is the number of preferences between actions into the individual p which are not ‘‘wellordered’’ in the sense of SAk . An individual P is feasible if uðpÞ ¼ 0 and infeasible if uðpÞ > 0. Defining the unfitness taking the zero minimum value if and only if the solution is feasible seems a natural approach. Each individual P can then be represented by a triad of values f ; u and k. We are interested in: (i) Individuals whose unfitness function value is equal to zero. This assures us that the ordering represented by the individual is transitive; this is one of two characteristics that should be exhibited by all prescription (solution) of ranking problems (Vanderpooten, 1990). (ii) Individuals whose fitness function value is equal (or near) to zero. This objective improves the comparability of S on A. (iii) Individuals whose credibility level k is near to 1. This indicates us that the ordering represented by the individual with credibility level k is more trusty whenever the fitness and unfitness function values are zero or near to zero. In practice, the requirement connected to fitness function does not permit that k values approach to 1 because in this case, we could have many incomparable genes. Then, we use a genetic search for solving the multiobjective problem
J.C. Leyva-Lopez, E. Fernandez-Gonzalez / European Journal of Operational Research 148 (2003) 14–27
Min u; f Max k; Rs ; k 2 ½0; 1 k P k0 ; where Rs is a strict total order of A.
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