DOCUMENT DE TRAVAIL Centre de recherche sur l aide à l évaluation et à la décision dans les organisations (CRAEDO)

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1 Publié par : Published by : Publicación de la : Édition électronique : Electronic publishing : Edición electrónica : Disponible sur Internet : Available on Internet Disponible por Internet : Faculté des sciences de l administration Université Laval Québec (Québec) Canada G1K 7P4 Tél. Ph. Tel. : (418) Fax : (418) Céline Frenette Vice-décanat à la recherche et au développement Faculté des sciences de l administration http :// rd@fsa.ulaval.ca DOCUMENT DE TRAVAIL A DISTANCE-BASED COLLECTIVE WEAK ORDERING Slim Ben Khelifa Jean-Marc Martel Centre de recherche sur l aide à l évaluation et à la décision dans les organisations (CRAEDO) Version originale : Original manuscript : Version original : Série électronique mise à jour : One-line publication updated : Seria electrónica, puesta al dia ISBN ISBN - ISBN

2 A Distance-Based Collective Weak Ordering Slim BEN KHELIFA and Jean-Marc MARTEL (1) Abstract A group decision-making approach can be seen as a two stage process. The first stage allows for multi-criteria evaluation of the alternatives and the second aims at deriving a collective weak ordering from the partial orderings supplied by the members after the first stage. The problem of combining the weak orderings to form a collective ranking is investigated. An axiomatic structure relating to the concept of distance between binary relations is developed. An algorithm for deriving the collective weak ordering is proposed. The underlying idea of this algorithm is to rank first the less dominated alternatives. Key words: Group Decision-Making, Multi-Criteria Decision Aid, Outranking Methods, Distance, Weak Ordering, Partial Ordering. (1) Faculté des Sciences de l Administration, Université Laval, Ste-Foy, Québec, PQ, G1K 7P4, CANADA

3 1. Introduction In most organizations, group problem solving and group decision problems are important aspects when dealing with strategic decisions involving environmental or siting issues. Decision-making in organizations requires the participation of managers who come from a wide range of departments and have different interests, backgrounds, experience, culture, etc. This diversity as well as the complexity of group decision-making, common to many group interactions, make reaching an agreement burdensome. Many studies (Coughlan and Armour, 1992) indicate that performance, in terms of satisfaction with the final recommendation, of the group is enhanced when a structured process is followed. A structured process is needed to promote critical thinking and efficient use of member s skills and knowledge. A group decision-making approach can be seen as a two stage process (figure 1). In the first stage, members are asked to rank alternatives (objects, actions, projects, candidates,...). These rankings are the results of the application of a multi-criteria decision aid (MCDA) method such as ELECTRE III (Roy, 1978), ELECTRE IV(Roy et Hugonnard, 1982) (when the coefficients of importance of the criteria are not taken into account) or PROMETHEE (Brans et al., 1984). The second stage consists of combining these rankings (partial weak orderings) into a collective or compromise ranking (a weak ordering or a partial weak ordering). Figure 1 A two stage process group decision-making Member 1... Member t... Member s First Stage Ranking Ranking of member 1... of member t... Ranking of member s Second stage Collective (compromise) ranking 1

4 There are several methods that can be used for the compromise ranking problem. Borda (1781) proposed the method of weighted preferences. Each position p of the ranking is worth p points, with the winning candidate being the one with the lowest total score. Other distance-based models have been suggested by Armstrong et al. (1982), Blin (1976), Bogart (1975), Cook and Seiford (1978), Cook et al. (1986a-b), Kemeny and Snell (1962) and Roy and Slowinski (1993), among others. Apart from Bogart, Cook et al., and Roy and Slowinski, the investigation was limited to aggregate linear orderings into a linear compromise ranking (Blin; Cook and Seiford; among others), or quasi-linear orderings into a quasi-linear compromise ranking (Armstrong et al.; Kemeny and Snell; among others). Bogart generalized the model of Kemeny and Snell to accommodate partial orderings. His definition of partial orderings precludes the case of ties: x is either preferred or incomparable to y. He presented a set of natural axioms similar to those developed in Kemeny and Snell which any measure of distance should satisfy in order to be considered as a social choice function. The model presented in Cook et al.(1986a) allows a comparison of linear, weak and partial orderings. A partial ordering is represented by a pair of binary matrices; namely, an information matrix and a (transitive) preference matrix (Cook et al., 1986b). In both papers, a set of axioms characterizing the distance is presented and it is proven that a unique distance satisfying these axioms exists. However, the computational aspect for determining a compromise ranking has not been dealt with in either case. Roy and Slowinski (1993) state two sets of conditions (logical and significance conditions) for deriving a criterion of distance between a pair of partial weak orderings coming from different viewpoints. This criterion is taken into account in the selection of a variant of technical construction of a regional water supply system. There was no attempt to derive a compromise ranking. In this paper the theoretical concepts originated in Roy and Slowinski for deriving a measure of distance between binary relations linking alternatives in partial orderings has been adapted to form a new approach to the problem of determining a compromise weak ordering. In section 2 a set of conditions defining a distance is presented. An algorithm for deriving a 2

5 compromise ranking is presented in section 3 and an illustrative example is discussed in section The model Let X be a finite set of m alternatives (actions, projects, candidates,...) and (x, y) an ordered pair of alternatives of X 2 and let t the index identifying the s members of the group (t= 1, 2,..., s). Three binary relations shall be considered: preference, indifference, and incomparability. It is reasonable to consider that these relations verify the following set of conditions: Preference: x y y x ( x 1 y y x) where is an asymmetric relation; Indifference: x y where is a reflexive and symmetric relation; Incomparability: x? y where? is an irreflexive and symmetric relation. Definition 2.1. The relations {,,,?} constitute a partial preference structure in X if they verify the previously stated conditions, and if given two alternatives x and y of X, one and only one relation holds between x and y. This structure defines a partial ordering. As indicated earlier the individual rankings are partial orderings resulting from the application of an MCDA outranking method. It has to be noted that each ranking (partial ordering) is derived from the intersection of two weak orderings. In the first one alternatives are ranked in a decreasing order of their capability to outrank others. In the second ranking alternatives are ranked in an increasing order of their capability to be outranked by others. Definition 2.2 The intersection is carried out as follows: x y in the partial ordering if x is preferred to y in both weak orderings or if x is preferred to y in one of them and they are tied in the second; x y in the partial ordering if they are tied in either weak orders; x? y in the partial ordering if x is preferred to y in one weak ordering and the opposite is true in the other. 3

6 Let R and R be two partial orderings. There is a disagreement between R and R if and only if x and y of X are related differently in this pair of rankings. The intensity of the disagreement, denoted (.,.), depends on the type of binary relations linking the alternatives in the rankings. Since the partial preference structure defining partial orderings consists of four (4) binary relations, these measures can be represented in a table (table 2). There are sixteen (16) measures of disagreement of which four (4) are equal to zero, for there is no disagreement when the binary relations are identical. R' Table 2 Disagreement measure between a pair of partial orderings x R x y x y x? y x y y 0 (, ) (,?) (, ) x y (, ) 0 (,?) (, ) x? y (?, ) (?, ) 0 (?, ) x y (, ) (, ) (,?) 0 Roy and Slowinski (1993) present a set of conditions in order for a function g to be considered as a measure of distance between pairs of partial orderings. g is defined as follows: g(r, R )= d( x, y) where the value of d(x, y) depends of the nature of relations xy, relating x and y in R and R. For example, d(x, y)= (, ), if x y in R and x y in R. Following this direction a similar set of conditions is presented. It is then shown that (.,.) is a distance (not g as it is the case in Roy and Slowinski). The three following axioms are the metric properties. Axiom 1 (Nonnegativity) For every OU, {,,?, }, ( O, U) 0 iff ( O, U) > 0, for O U and ( O, U)= 0, for O U. 4

7 Axiom 2 (Symmetry property) For every O U {,,?, }, ( O, U) = ( U, O). Axiom 3 (Triangle inequality) For every O, U, V {,,?, } ( O, V) + ( V, U) ( O, U). Let us now present the set of conditions that these measures of disagreement should satisfy. Condition 1 (C1) (,?) = (,?) and (, ) = (, ). This condition is natural since the preference relation opposite relations. and the inverted preference are Condition 2 (C2) (, ) + (, ) = (, ). This second condition indicates that the indifference relation lies between the preference and the inverted preference relations. Roy & Slowinski (1993) consider that (, ) + (, ) (, ) (, ) < (, ) (see figure 2). However, the strict inequality (, ) + is incompatible with the triangle inequality (axiom 3). Condition 3 (C3) (,?) (,?). This condition states that the disagreement between preference and incomparability and indifference and incomparability are not necessarily identical as is the case in Cook et al. (1986a) where it is presumed that (,?) and (,?) Condition 4 (C4) (,?) (, ). are equal. Let us go back to the first stage of the group decision process in order to explain why the disagreement between indifference and incomparability relations is greater than the disagreement between the indifference and the preference relations. As we mentioned earlier, a partial ordering result from the intersection of two weak orderings. Indifference in the partial ordering indicates that alternatives x and y are indifferent in both weak orderings. However, a preference in the partial ordering indicates that x is preferred to y in both weak orderings or preferred in one and indifferent in the other. Finally, incomparability is implied because x is preferred to y in one weak ordering and vice versa in the second. It can be noted then, that the disagreement between preference and indifference is less important than the disagreement between indifference and incomparability. In the former case there is at 5

8 least one and at most two different binary relations, tying x and y in the weak orderings; whereas, in the latter case there are exactly two different binary relations tying x and y in the weak orderings. As a result, we state that (,?) (, ). Roy and Slowinski (1993) consider only the limit case (,?) = (, ). Figure 2 Distances between binary relations? (,?) (?, ) (,?) (, ) (, ) (, ) Condition 5 (C5) (, ) = { (, ):, {,,?, }} Max O U O U. The preference and inverted preference relations are the most incompatible relations. Condition 6 (C6) ( O, U) > 0, for O U and ( O, U) = 0, for O = U where, {,,?, } O U. This condition states that the minimum distance is positive. We consider here that the minimum distance between a pair of distinct binary relations is one. It is worthwhile to note that from C3, C4, C5, and C6 it can be inferred that (, ) (,?) (?, ) (, ) 1. (2.1) Proposition If C1 to C6 is satisfied, then is a metric. (proof in appendix) 6

9 In order to assign values to these measures we assume that the difference between adjacent measures of (2.1) of disagreement are equal: (, ) (,?) = (,?) (?, ) = (?, ) (, ). From C1, C2, and C6, it can be inferred that (, ) =1 and (, ) and (?, ) : = 2. By solving the following linear system, we obtain the values of (,?) (,?) (?, ) = (?, ) 2 (,?) = (,?) (?, ) which is equivalent to (,?) 2 (?, ) = 2 (,?) (?, ) = 2 Hence, (,?) = 5/3 and (?, ) =4/3. Table 2-2 summarizes the values assigned to. Table 2-2 Values assigned to R x y x y x? y x y R' x y 0 1 4/3 1 x y 1 0 5/3 2 x? y 4/3 5/3 0 5/3 x y 1 2 5/ Algorithm for deriving a collective weak ordering At each iteration of the algorithm developed for deriving a compromise weak ordering an index of dominance is calculated for each alternative. At the first iteration, for instance, the index associated to x is φ ( x) 1 = φ ( x, y) (3.1) y x where φ s () t ( x, y) = (, R ( x, y)) t = 1 t and R () ( x, y) =? ( t ) if x y ( t ) if x y. ( t ) if x? y t if y x ( ) 7

10 The alternatives belonging to the first category X 1 (ranked first) are those for which φ 1 (.) are minimal and therefore less dominated. After eliminating from consideration the alternatives of X 1, the second iteration consists of computing the index φ 2 ( x ) and determining the alternatives belonging to the second category X 2. The algorithm progresses as follows (kth iteration): Step 1 For each alternative belonging to the set Y of non-ranked alternatives k Compute φ k ( x) = φ k ( x) ϑ( X " ) where ϑ( X ) = φ ( x, y) " ; x X k k iff φ k ( x) = Min { φ k ( z): z X / X " = 1 "} " = 1 ; y X " Step 2 Increment k (k=k+1); k Update the set of remaining alternatives Y, Y = X X If Y =, then stop else go to step 1. = 1 " " ; The weak ordering is obtained, at most, in m iterations. Besides, at most [s.(m)+m-2] additions are needed to compute the φ 1 (.) indices in the first iteration, and at most [m.(m- 1) / 2] subtractions are needed in the subsequent iterations to calculate φ 2 (.), φ 3 (. ), etc. 4. Numerical example This illustrative example is derived from a case study involving a committee composed of three members. The responsibility of the committee is to propose a ranking of dumping sites. Six sites are considered, X={x 1, x 2,..., x 6 }. Each member is asked to perform an outranking analysis and to suggest a ranking. These ranking are presented in figure 4. 8

11 Ranking of member 1 Figure 4 Members rankings x 2 x 5 x 4 x 3 x 6 x 1 Ranking of member 2 x 3 x 1 x 6 x 2 x 4 x 5 Ranking of member 3 x 1 x 6 x 3 x 2 x 5 x 4 At first glance it is clear that the members do not agree on how to rank the alternatives. However, x 1 appears to be a good compromise. Let us determine a weak ordering from these partial orderings. First iteration ( k=1, Y=X) Step 1 ( xvsx 1 2) ( xvsx) 1 3 () 1 ( 2) () 3 () 1 ( 2) () 3 φ 1 ( x 1 ) = (, ) + (, ) + (, ) + (, ) + (, ) + (, ) + ( xvsx 1 4) ( xvsx) 1 5 () 1 ( 2) () 3 () 1 ( 2) () 3 (,?) + (, ) + (, ) + (,?) + (,?) + (, ) + ( xvsx) 1 6 () 1 ( 2) () 3 (, ) + (, ) + (, ) =2+2+11/3+10/3+0=33/3. φ 1 ( x 2 ) =54/3; φ 1 ( x 3 ) =46/3; φ 1 ( x 4 ) =39/3; φ 1 ( x 5 ) =58/3; φ 1 ( x 6 ) =60/3; x1 X1 since φ1 ( x1) = Min { φ1 ( x), x Y}. 9

12 Step2 k=k+1; Y=X - X 1 ={x 2, x 3, x 4, x 5, x 6 }; Go to step 1. Second iteration X 2 ={x 4 }; Y={x 2, x 3, x 5, x 6 }; Third iteration X 3 ={x 3 }; Y={x 2, x 5, x 6 }; Fourth iteration X 4 ={x 2, x 5, x 6 }; Y =. The computational details of all the iterations are summarized in table 4. The resulting weak ordering is given in figure 4-2. Table 4 Computational details x 1 x 2 x 3 x 4 x 5 x 6 φ 1 (.) φ 2 (.) φ 3 (.) φ 4 (.) x /3 10/3 0 33/3 * x /3 42/3 30/3 18/3 * x / /3 34/3 18/3 * - x 4 11/3 2 10/ /3 28/3 * - - x 5 16/ /3 42/3 30/3 18/3 * x /3 42/3 30/3 18/3 * Figure 4-2 Collective weak ordering x 1 x 4 x 3 x 2,x 5,x 6 It is worthwhile to note that x 1 is ranked first as was argued above. On the other hand, one can wonder why the weak ordering is not based solely on φ 1 (.) since these indices can be utilized to rank the alternatives. The weak ordering is arrived at by ranking the alternatives 10

13 according to an increasing order of the φ 1 (.) is x 1 x 4 x 3 x 2 x x. This 5 6 ranking implies some discrimination between the members. In fact, the anonymity property is not satisfied when ranking x 2, x 5, and x 6. As can be seen from figure 4 member 1 ranks x 2 before x 5, and x 5 before x 6 ; member 2 ranks x 5 before x 6, and x 6 before x 2 ; member 3 ranks x 6 before x 2, and x 2 before x 5. Such an outcome is referred to as a cycle or an effet Condorcet. In order to comply to the anonymity property, the three alternatives have to be indifferent. This is the case when applying the algorithm. In the ranking corresponding to φ 1 (.) the preference structure of member 1 concerning x 2, x 5, and x 6 prevails. 5. Conclusion We have described a structured approach to be used in group decision-making settings in order to develop a collective weak ordering. This approach is a two stage process. The first stage allows for a multi-criteria evaluation of each alternative by members. The outcomes of this stage are the partial (weak or complete) orderings of the group members. The second stage aims at deriving a collective weak (complete) ordering. The approach developed is based on a measure of distance. This approach is different from the previous approaches dealing with compromise ranking in which the underlying idea is to find a collective ranking as close as possible, in terms of agreement, to the members rankings. Instead, we were concerned with finding a ranking which disagrees less with the binary relations relating pairs of alternatives. From this perspective it appears that the previous approaches are global approaches because the focus is on the ranking as a whole. However, our approach works in a divide and conquer fashion and therefore can be considered as a local approach where the focus is on alternatives. While previous approaches did not address the elaboration of an algorithm to derive a compromise ranking from partial orderings, we have proposed such an algorithm which seems to comply to the anonymity property. 11

14 Appendix Proof of proposition of section 3 is a metric iff axioms 1-3 are satisfied. By C6, axiom 1 is satisfied since the minimum distance is 1. On the other hand, for every binary relation U {,,?, }, ( U, U ) and y are ranked in the same way in R and R. = 0 since there is no disagreement when x Axiom 2 is also fulfilled: For every O U {,,?, }, ( O, U) = ( U, O) since (i) the disagreement is identical whether O is relating x and y in R and U in R or vice versa and (ii) there is no discrimination between members. Axiom 3 is satisfied in any case. Case 1 (Indifference) I.1. (, ) + (, ) (, ) I.2. (, ) + (, ) (, ) I.3. (, ) + (,?) (,?) I.4. (, ) + (, ) (, ) I.5. (, ) + (, ) (, ) I.6. (, ) + (, ) (, ) I.7. (, ) + (,?) (,?) from C3 & C6. I.8. (, ) + (, ) (, ) from C1 & C6. I.9. (,?) + (?, ) (, ) I.10. (,?) + (?, ) (, ) from C4 & C6. 12

15 I.11. (,?) + (?,?) (,?) I.12. (,?) + (?, ) (, ) I.13. (, ) + ( 1, ) (, ) I.14. (, ) + ( 1, ) (, ) I.15. (, ) + (,?) (,?) I.16. (, ) + (, ) (, ) same as I.10. same as I.8. from C1, C3 & C6. Case 2 (Preference) P.1. (, ) + (, ) (, ) P.2. (, ) + (, ) (, ) P.3. (, ) + (,?) (,?) needs to be proven. P.4. (, ) + (, ) (, ) from C2. P.5. (, ) + (, ) (, ) P.6. (, ) + (, ) (, ) P.7. (, ) + (,?) (,?) P.8. (, ) + (, ) (, ) P.9. (,?) + (?, ) (, ) from C3, C4 & C6. P.10. (,?) + (?, ) (, ) P.11. (,?) + (?,?) (,?) P.12. (,?) + (?, ) (, ) needs to be proven. P.13. (, ) + (, ) (, ) P.14. (, ) + (, ) (, ) P.15. (, ) + (,?) (,?) P.16. (, ) + (, ) (, ) from C5 & C6. from C5 & C6. The same line of reasoning can be applied to verify the case of inverted preference ( ). 13

16 Case 3 (Incomparability) N.1. (?, ) + (, ) (?, ) N.2. (?, ) + (, ) (?, ) same as P.12. N.3. (?, ) + (,?) (?,?) N.4. (?, ) + (, ) (?, ) same as N.2. N.5. (?, ) + (, ) (?, ) from C3 & C6. N.6. (?, ) + (, ) (?, ) N.7. (?, ) + (,?) (?,?) N.8. (?, ) + (, ) (?, ) from C1, C5 & C6. N.9. (?,?) + (?, ) (?, ) N.10 (?,?) + (?, ) (?, ) N.11. (?,?) + (?,?) (?,?) N.12. (?,?) + (?, ) (?, ) N.13. (?, ) + (, ) (?, ) N.14. (?, ) + (, ) (?, ) N.15. (?, ) + (,?) (?,?) N.16. (?, ) + (, ) (?, ) from C1, C3 & C6. same as N.8. Let us prove that P.3. and P.12. hold. P.3. (, ) + (,?) (,?): From C5, we have that (, ) (,?) which implies by C2 that (, ) = (, ) + (, ) (,?). C1 and C4 imply that (, ) + (,?) (, ) + (, ) (,?); therefore, (, ) + (,?) (,?). P.12. (,?) + (?, ) (, ) From C1 & C4, it can be implied that 14

17 (, ) = (, ) + (, ) (,?) + (,?); C3 implies that (,?) + (,?) (,?) + (?, ) ; with C1, (, ) (,?) + (?, ). QED References Armstrong, R. D., Cook, W. D. & Seiford, L. M. (1982): Priority Ranking and Consensus Formation, Management Science, Vol. 28, Arrow, K. J. (1951) (2 nd edition 1963): Social Choice and Individual Values, Wiley, New York. Blin, J. M. (1976): A Linear Assignment Formulation for the Multiattribute Decision Problem, RAIRO, Vol. 10, Bogart, K. P. (1975): Preference Structures II: Distance Between Transitive Preference Relations, Journal of Applied Mathematics, Vol. 29, Borda, J.-Ch. (1781): Mémoire sur les élections au scrutin, Mémoires de l Académie des Sciences, Paris. Brans, J. P., Mareshal, B. & Vincke, Ph. (1984) : PROMETHEE: A New Family of Outranking Methods in Multicriteria Analysis, in J. P. Brans (ed.), Operational Research 84, Elsevier Science Publishers North-Holland, Condorcet, M. J. A. M. Caritat, Marquis De (1785): Essai sur l application de l analyse à la probabilité des décisions rendues à la pluralité des voies, Imprimerie Royale, Paris. Cook, W. D., Kress, M. & Seiford, L. M. (1986a): An Axiomatic Approach to Distance on Partial Ordering, RAIRO, Vol. 10, Cook, W. D., Kress, M. & Seiford, L. M. (1986b): Information and Preference in Partial Orders: A Bimatrix Representation, Psychometrika, Vol. 51, Cook, W. D. & Seiford, L. M. (1978): Priority Ranking and Consensus Formation, Management Science, Vol. 24, Coughlan, B. A. K. & Armour, C. L. (1992): Group Decision-Making for Natural Resources Management Applications, Resource Publication 185, U.S. Department of the Interior, Fish & Wildlife Service. Kemeny, J. G. & Snell, L. J. (1962): Preference Ranking: An Axiomatic Approach, in Mathematical Models in the Social Sciences, Ginn, New York, Roy, B. (1978): ELECTRE III: algorithme de classement basé sur une représentation floue des préférences en présence de critères multiples, Cahiers du CERO 20, Roy, B. & Hugonnard, J.C. (1982): Classement des prolongements de métro en banlieue parisienne (présentation d une méthode orginale), Cahiers du CERO, 23, Roy, B. & Slowinski, R. (1993): Criterion of Distance Between Technical Programming and Socio-Economic Priority, RAIRO, Vol. 27,