Distance-based consensus methods: a goal programming approach

PERGAMON

Omega, Int. J. Mgmt. Sci. 27 (1999) 341±347

Distance-based consensus methods: a goal programming approach Jacinto GonzaÂlez-PachoÂn a, 1, Carlos Romero b, * a

Facultad de InformaÂtica, Universidad PoliteÂcnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain Escuela TeÂcnica Superior de Ingenieros de Montes, Universidad PoliteÂcnica de Madrid, 28040 Madrid, Spain

b

Received 1 May 1998; accepted 1 October 1998

Abstract Several authors have proposed a social choice function based upon distance-consensus between di€erent committee rankings. Under this framework, the total absolute disagreement between committees is minimised. The purpose of this paper is to formulate the underlying optimisation problem as a goal programming (GP) model. To do this, the following three GP formulations are proposed: (a) a linear weighting GP model, where consensus is established by the minimisation of the weighted aggregated disagreement, (b) a MINMAX GP model, where the consensus is de®ned as the minimisation of the maximum disagreement and (c) an extended GP model, which subsumes the two previous formulations as particular cases. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Group decisions; Goal programming; Multi-criteria; Social OR

1. Introduction In social choice theory, it is usual to conjecture the existence of a set of decision-makers or electoral committees (DM) that express their preferences with the help of an ordinal scale de®ned over the feasible set. In fact, despite the postulates of utility theory, the DM usually ®nds it easier to use an ordinal scale rather than an interval one. Coherently with these ideas, the search of a consensus through the de®nition of an ordinal ranking is a relevant problem in social choice theory. Some authors have addressed the problem of de®ning a distance function on the set of all rankings. Thus, from a minimum distance point of view, it is possible to determine the ordinal ranking closest to all decision-makers. For this purpose, two particular distance functions corresponding to metrics 1 and 2, re-

* Corresponding author. Fax: +34-91-543-9557; e-mail: [email protected]. 1 E-mail: jgpachon@®.upm.es

spectively, were used (see the book by Cook and Kress [1] for a general explication on the subject). Metric 1 was proposed by Cook and Seiford [2] to establish the distance (consensus) between two ordinal rankings. These authors demonstrated that this distance holds for a set of `reasonable' axioms, which may be considered an acceptable social choice function. From a computational point of view, they obtain the consensus-ordinal-ranking for metric 1 by solving a linear assignment problem. The link between the Borda±Kendall method and metric 2 was investigated by Cook and Seiford [3]. To solve the underlying optimisation problem in this case, they propose the minimum variance method. This method derives from Ref. [4] where the consensus-ordinal-ranking is obtained by solving a statistical estimation problem. Cook et al. [5] attempt to generalise the problem to a generic space de®ned for any metric p (1R p R 1). However, all the analytical e€orts once again focus on metrics 1 or 2. The reasons for the preponderance of

0305-0483/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 4 8 3 ( 9 8 ) 0 0 0 5 2 - 8

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these speci®c metrics in determining a consensus between ordinal rankings functions are as follows. 1. Consensus ranking obtained for both metrics have a clear statistical meaning. Thus, consensus derived from metric 1 can be statistically interpreted as the median ranking and as the mean ranking for metric 2. 2. From a computational point of view, speci®c algorithms are needed to obtain consensus rankings for both metrics. Thus, for metric 1 the determination of consensus ranking implies the resolution of a linear assignment problem. As for metric 2, the Borda±Kendall or the minimum variance methods, for instance, allow for the precise determination of the consensus ranking. This paper aims to support the view that p-metric spaces are a suitable framework to pose and to determine consensus-ordinal-rankings. The task will be undertaken with the use of goal programming (GP) to induce distance-based consensus methods. The idea of using GP to induce models has been recently suggested by Ignizio [6] and will be explored in this paper within the context of social choice problems. Although GP is very well known in the OR/MS literature, its possible application in the determination of consensus-ordinalrankings has been neglected with the exception of [2], where the use of minimum GP is suggested to address consensus problems although in a di€erent direction to the one followed in this paper. The paper is organised as follows. Section 2 presents the analytical framework connecting GP with the determination of consensus rankings. In Section 3 the di€erent GP solutions obtained are interpreted from a preferential point of view. Section 4 de®nes the properties holding for the di€erent solutions obtained. Section 5 illustrates, with the help of numerical examples, the ideas previously presented. The main conclusions derived from this research are presented in Section 6. In Appendix A, the properties stated in Section 4 are formally demonstrated.

2. Analytical framework Let us consider i = 1, 2, . . . , n decision-makers or electoral committees (DM) which have to vote on or to rank j = 1, 2, . . . , m alternatives. The following variables and parameters will be used throughout the paper: Rij=rank given to jth alternative by ith decisionmaker or electoral committee. Wi=number of members of the ith electoral committee, or weight (in¯uence) of ith decision-maker.

Rj=®nal rank of jth alternative. By convention, the maximum number m will be assigned to the `best' alternative and the minimum number 1 to the `worst' one. The social choice function based upon distance functions [1], optimises consensus by minimising total aggregated disagreement. This function can be formulated for a general metric p as follows: X 1=p n X m p p Uˆÿ W i jRij ÿ Rj j pe‰1, 1† …1† iˆ1 jˆ1

The minus signs in Eq. (1) indicate that the social choice function U is an increasing function, i.e. a utility function holding the postulate `more is better'. If we are looking for an ordinal ranking of alternatives, the following feasible set F can be de®ned: F ˆ fRj =1RRj Rm,

m X Rj ˆ …m ‡ 1†m=2g

…2†

jˆ1

The ordinal ranking of alternatives is obtained by maximising Eq. (1) over the feasible set in Eq. (2). This formulation allows ties in the optimal ordering. That is, the proposed optimisation problem can generate weak (ties) or strong (no ties) orderings. If no ties (strong ordering) are required, a subset of `ad hoc' constraints has to be added to the model (see example in Section 5). For p = 1, the social choice or utility function given by Eq. (1) turns into the following expression: U ˆ ÿ‰Max 8i ,8j …Wi jRij ÿ Rj jŠ

…3†

Maximising Eq. (1) or Eq. (3) over the feasible set it is not an easy computational task. For instance, Cook and Seiford [2] show how the maximisation of Eq. (1) can be reduced to a linear assignment problem for p = 1. However, it is straightforward to show how taking into account the following change of variables [7]: 1 nij ˆ ‰jRij ÿ Rj j ‡ …Rij ÿ Rj †Š 2

…4†

1 pij ˆ ‰jRij ÿ Rj j ÿ …Rij ÿ Rj †Š 2

…5†

the above optimisation problems can be reduced to GP formulations. Thus, if Eqs. (4) and (5) are considered, the maximisation of Eq. (1) subject to Eq. (2) is equivalent to the following Archimedean linear GP problem [8]: n X m X Min W pi …nij ‡ pij † p iˆ1 jˆ1

J. GonzaÂlez-PachoÂn, C. Romero / Omega, Int. J. Mgmt. Sci. 27 (1999) 341±347

subject to:

343

Subject to:

Rj ‡ nij ÿ pij ˆ Rij

Wi …nij ‡ pij †RD

8i , 8j

X2F

…6†

For the p = 1 model, Eq. (6) turns into a linear weighted GP formulation. It should be noted that given the mathematical structure of goals and constraints in Eq. (6), the solution derived from the `Simplex' application will always be integer. Hence, it is not necessary to introduce any condition to guarantee the integer character of Rj variables in the p = 1 case. It should also be noted that in the case of an even number of objects because of the integer character of the solutions of the model in Eq. (6), it is not possible to obtain a zero ranking (i.e. a ranking in which all the objects are tied). However, the GP formulation proposed in [2] imposes a set of constraints to avoid this kind of situation. It can also be straightforwardly shown that the maximisation of Eq. (3) subject to Eq. (2) is equivalent to the following MINMAX or Chebyshev GP problem [9]: Min D, subject to: Wi …nij ‡ pij †RD

8i , 8j

Rj ‡ nij ÿ pij ˆ Rij X 2 F0

…7†

where D is the largest deviation or consensus measure of the DM's whose ranking presents a maximum disagreement with respect to the consensus achieved. In this case, because of the structure of the ®rst block of constraints, the solutions provided by the `Simplex' can be non-integer. Because of that, the feasible set F 0 of the model in Eq. (7) is F plus a condition to guarantee the integer character of Rj variables (see Table 2). The aforementioned analysis can be generalised with the help of the following utility function, which attempts to aggregate social choice functions, Eqs. (1) and (3), into the following general formulation: U ˆ ÿ ‰Max 8i ,8j …Wi jRij ÿ Rj j†Š 1=p X n X m ÿl W pi jRij ÿ Rj j p

8i , 8j

…8†

iˆ1 jˆ1

For l = 0, we have the utility function given by Eq. (3), which de®nes the consensus by minimising the disagreement. For l = 1, average aggregate consensus given by Eq. (1) is obtained. The GP formulation associated to Eq. (8) is the following extended GP model [10]: n X m X Min D ‡ l W pi …nij ‡ pij † p iˆ1 jˆ1

Rj ‡ nij ÿ pij ˆ Rij X 2 F0

8i , 8j

…9†

For l = 0, we have a MINMAX or Chebyshev GP model, whereas for l = 1, an Archimedean GP model is obtained. Finally, di€erent extended GP models are obtained for ®nite values of l.

3. Preferential interpretation of the consensus-ordinalrankings obtained The di€erent consensus-ordinal-rankings found in the preceding section can be interpreted from a preferential point of view as follows. First, when the consensus is de®ned by the social choice function, Eq. (1), or by its equivalent Archimedean GP formulation, Eq. (6), then the solution is obtained by minimising the weighted aggregated disagreement. Metric p acts as weight attached to the deviation vRijÿRjv. Thus, as p increases, more importance is given to the largest deviation (importance given to the minority group). In this way, if what is emphasised is the sum of individual disagreements or utilities, then metric 1 should be used [11]. In short, p = 1 represents a solution of maximum agreement but can by very biased with respect to the wishes of a particular DM. As p increases, the interest or point of view of the minority groups is more respected. In short, there is an additive structure of preferences in this kind of Archimedeam GP model [12, ch. 7]. Second, when consensus is de®ned by the social choice function, Eq. (3), or its equivalent MINMAX or Chebyshev GP formulation, Eq. (7), then solution is obtained by minimising the maximum disagreement. This kind of consensus gives maximum importance to the minority group, or in other words to the DM whose ranking is farthest away from the solution achieved. This solution can be called Rawls' consensus because of the perfect similarity between the underlying preference structure in Eq. (3) and the principles of justice introduced by Rawls [13, pp. 75±80]. Indeed, this author argues that the welfare of society depends upon the utility of the worst-o€ household. If we substitute society's welfare by DM's utility (agreement) and the worst-o€ household by DM's whose ranking presents a maximum disagreement with respect to the consensus achieved, then the commented analogy is concluded. In short, there is a Rawlsian or MAXMIN structure of preferences in these MINMAX or Chebyshev GP models [14]. Finally, the utility function in Eq. (8) or its equivalent extended GP formulation in Eq. (9) can be con-

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sidered the surrogate of a general social choice function. In fact as particular cases, Eq. (8) or Eq. (9) subsume the consensus given by the minimisation of the aggregated disagreement (l = 1) and the Rawls' consensus (l = 0).

4. Some properties on the proposed social choice process A social choice process frequently requires certain conditions. This section examines those conditions veri®ed by the solutions proposed in this paper. First of all, it should be noted that all the consensus-ordinal-rankings generated by the di€erent GP models formulated in Section 2 are compromise solutions, Therefore, as it is well illustrated in the literature, these solutions hold an array of important properties such as: (a) feasibility, (b) least group regret, (c) no dictatorship, (d) pareto optimality, (e) uniqueness, (f) asymmetry and (g) independence of irrelevant alternatives [15, pp. 71±74]. Besides the aforementioned properties, the following ones are also met by the consensus-ordinal-rankings obtained: De®nition 1 (neutrality). A social choice process is neutral if all alternatives are treated equally. More precisely, any permutation in the set of alternatives induces a corresponding permutation of the consensus ordinal ranking. Proposition 1. All consensus-ordinal-ranking based on distance model is neutral. De®nition 2 (anonymity). A social choice process is anonymous if all decision-markers or committees members are treated equally, i.e. their consensus-ordinalranking depends only on the numbers of decisionmakers having each ranking. Proposition 2. All consensus-ordinal-ranking based on distance model is anonymous. De®nition 3 (monotonicity). In a social choice process, when a DM moves an alternative in its ranking and

leaves the relative standing of the others unchanged, if the alternative stands at least as well relative to each other candidate as before, then this process veri®es the monotone non-decreasing property. Proposition 3. All consensus-ordinal-ranking based on a distance model veri®es the monotone non-decreasing property. Formal proofs of these propositions can be found in Appendix A.

5. Numerical illustrations To illustrate the ideas presented in Sections 2±4, the subsequent example used by Guilbaud [16] and Hwang and Lin [17] to explain Condorcet functions will be used. Suppose that 60 voters vote among three political parties as follows: 23 votes: A P B P C 17 votes: B P C P A 2 votes: B P A P C 10 votes: C P A P B 8 votes: C P B P A where P represents a preference relationship. For p = 1 the linear weighted GP model shown in Table 1 is obtained. By solving this model the following solution is obtained: RA ˆ RB ˆ RC ˆ 2 This implies the following weak ordering: A I B I C, I being an indi€erence relationship. The measurement of the total disagreement or indicator of consensus for this solution is 120 units. That is, the minimum weighted average disagreement is equal to 120 units. Let us now consider a solution without ties (i.e. a strong ordering). In this case, the following set of con-

Table 1 Linear weighted goal programming formulation of a social choice problem Objective function MIN 23(n1+p1) + 17(n2+p2) + 2(n3+p3) + 10(n4+p4) + 8(n5+p5) + 23(n6+p6) + 17(n7+p7) + 2(n8+p8) + 10(n9+p9) + 8(n10+p10) + 23(n11+p11) + 17(n12+p12) + 2(n13+p13) + 10(n14+p14) + 8(n15+p15) Subject to: RA+n1ÿp1=3 RB+n6ÿp6=2 RC+n11ÿp11=1 RA+n2ÿp2=1 RB+n7ÿp7=3 RC+n12ÿp12=2 RB+n8ÿp8=3 RC+n13ÿp13=1 RA+n3ÿp3=2 RA+n4ÿp4=2 RB+n9ÿp9=1 RC+n14ÿp14=3 RA+n5ÿp5=1 RB+n10ÿp10=2 RC+n15ÿp15=3 1RRAR3 1RRBR3 1RRBR3 RA+RB+RC=6

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straints can be incorporated into the GP model: RA ÿ 3U1 r0,

RB ÿ 3U2 r0,

RC ÿ 3U3 r0

U1 ‡ U2 ‡ U3 ˆ 1: U1 , U2 , U3 INTEGER The above block of auxiliary constraints imposes that one of the political parties be placed in the ®rst place. By solving the corresponding augmented problem, the following solution is obtained: RA ˆ 3,

RB ˆ 2,

RC ˆ 1

This implies the following strong ordering: A P B P C. The new measurement of the total disagreement or indicator of consensus for this solution is 144 units. This solution coincides with the solution provided by the linear assignment problem proposed by Cook and Seiford [2]. For p = 1 the MINMAX or Chebyshev GP model shown in Table 2 is obtained. By solving this model, the following solution is obtained: RA ˆ 2,

RB ˆ 3,

RC ˆ 1

This implies the following strong ordering B P A P C. The indicator of consensus for this solution corresponds to a maximum disagreement of 23 units. It is interesting to note that for Rawls' consensus, where the largest deviation (disagreement of the DM farthest away from the consensus achieved) procures a minimum (23 units), there is a weighted average disagreement of 152 units. On the other hand, the solution of maximum average consensus (144 units) corresponds to a maximum disagreement of 34 units. This is the measurement of the clash between maximum average agreements (consideration of the majority) and minimum largest deviation (consideration of the minority). By applying the utility function in Eq. (8) or its surrogate and operational version in Eq. (9) to our example, only the weighted and the Chebyshev (Rawls) consensus are obtained. In other words, there are no intermediate solutions between the solution for which the aggregated and weighted interest of all DM is opti-

mised and the solution for which the interest of the minority group is given maximum importance.

6. Conclusions and future research The formulation of the optimisation of a social choice function as GP models o€ers the following advantages: 1. From a computational point of view, it is easier to solve a linear weighted GP model than a linear assignment problem, such as the one suggested by Cook and Seiford for the p = 1 case. 2. A GP formulation allows for the generalisation of the social choice function structure. Thus, for each value of metric p, a consensus with a precise meaning is obtained (see Section 3). Possible consensus ranges from the optimisation of the sum of individual disagreements ( p = 1) to the optimisation of the maximum disagreement ( p = 1). 3. Consensus-ordinal-rankings generated by the proposed family of GP models have an important number of key properties. Those properties, amongst others, are those usually found in compromise solutions (see Section 4). Hence, it can be stated that considering all properties, the solutions provided by the GP models satisfy a sound axiomatic basis. Finally, the interest of giving a cardinal interpretation to the utility function or social choice functions introduced in the paper should also be remarked. This would make it possible not only to establish a strong or weak ordering of the feasible alternatives but also to measure the intensity with which one alternative is preferred to another.

Acknowledgements Thanks are given to the two reviewers for their comments and suggestions. Christine Mendez checked English language. A preliminary shorter version of this

Table 2 Chebysev goal programming formulation of a social choice problem Objective function MIN D Subject to: 23(n1+p1) ÿ DR0 17(n2+p2) ÿ DR0 2(n3+p3) ÿ DR0 10(n4+p4) ÿ DR0 8(n5+p5) ÿ DR0 Goals and constraints of Table 1

345

23(n6+p6) ÿ DR 0 17(n7+p7) ÿ DR 0 2(n8+p8) ÿ DR 0 10(n9+p9) ÿ DR 0 8(n10+p10) ÿ DR0 RA, RB, RC INTEGERS

23(n11+p11) ÿ DR 0 17(n12+p12) ÿ DR 0 2(n13+p13) ÿ DR 0 10(n14+p14) ÿ DR 0 8(n15+p15) ÿ DR 0

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paper was presented at the 14th International Conference on Multiple Criteria Decision Making held in Charlottesville, Virginia, USA, June 1998. Both authors are grateful for the ®nancial support of the Spanish `ComisioÂn Interministerial de Ciencia y TecnologõÂ a' under projects SEC97-1266 and AGF950014, respectively. C.R. also wants to acknowledge the ®nancial support of `ConsejerõÂ a de EducacioÂn y Ciencia' de la Comunidad de Madrid.

Let {(Ri1, Ri2, . . . , Rim)}i $ I and {(Ri1 0 , Ri2 0 , . . ., Rik 0 )gi2Is be the sets of rankings attributed by the decision makers of I and I s, respectively, to the set of alternatives in A. If (R1, R2, . . ., Rm) and (R1 0 , R2 0 , . . ., Rk 0 ) are the consensus (agreed) ranking for the groups of the decision makers I and I s, respectively, considering that 8k $ {1, 2, . . . , n}9t $ {1, 2, . . . , n} such that Rik j 0 ˆ Rs…it †j 8j $ {1, 2, . . ., m} and the form of the objective function, Eq. (1), we get Rj 0 = Rj8j $ {1, 2, . . ., m}, which proves the result.

Appendix A

A.3. Proof of Proposition 3

A.1. Proof of Proposition 1

Let (Ri1, Ri2, . . ., Rim) be the ranking attributed by the decision maker i $ I and (R1, R2, . . ., Rm) the consensus ranking for the group of the decision makers I.

Let A = {a1, a2, . . . , am} be a set of alternatives put to the vote among a group of n decision makers or committees, I = {i1, i2, . . . , im}. Let As={a1 0 , a2 0 , . . . , ak 0 } the set output by applying a change of order or permutation, s, to the elements of A. That is, 8ak 0 there is an index j 2 f1, 2, . . . , mg such that ak 0 ˆ as…

j†

Let {(Ri1, Ri2, . . ., Rim)}i $ I and {(Ri1 0 , Ri2 0 , . . . , Rik 0 )}i $ I be the sets of rankings attributed by the members of I to the alternatives in A and As, respectively. If (R1, R2, . . . , Rm) and (R1 0 , R2 0 , . . . , Rk 0 ) are the consensus (agreed) ranking for the groups of the decision makers A and A s, respectively, considering that 8k $ {1, 2, . . ., m}9j $ {1, 2, . . . , m} such that Rik 0 = Ris( j)8i $ {1, 2, . . . , n} and the form of the objective function, Eq. (1), we get Rk 0 = Rs( j), which proves the result. A.2. Proof of Proposition 2 Let A = {a1, a2, . . . , am} be a set of alternatives put to the vote among a group of n decision makers or committees, I = {i1, i2, . . . , im}. Let I s={i1 0 , i2 0 , . . . , ik 0 } be the set output by applying a reordering or permutation, s, to the elements of I. That is, 8ik 0 there is an index t 2 f1, 2, . . . , ng such that ik 0 ˆ is…t†

Suppose that decision maker i moves an alternative aj in its ranking and leaves the relative standing of the others unchanged obtaining the new ranking (Ri1 0 , Ri2 0 , . . . , Rik 0 ). The relationship between the new ranking and the old one is the following: let Rij 0 be the new ranking for the alternative aj, 8k $ {1, 2, . . . , m}. If Rik > Rij 0 If Rik RRij 0

then Rik 0 ˆ Rik , then Rik 0 ˆ Rik ÿ 1

With this new ranking, a new consensus, (R1 0 , R2 0 , . . ., Rk 0 ), is obtained. In the social choice function we have the same terms, except for Rik when RikRRij 0 , where vRikÿRkv = vRik 0 + 1 ÿ Rkv, so Rk 0 = Rkÿ1. Using the last equation, we conclude that the alternatives stands at least as well relative to each other candidate as before.

References [1] Cook WD, Kress M. Ordinal information and preference structures: decision models and applications. Englewood Cli€s, NJ: Prentice-Hall, 1992. [2] Cook WD, Seiford LM. Priority ranking and consensus formation. Manage Sci 1978;24:1721±32. [3] Cook WD, Seiford LM. The Borda±Kendall consensus method for priority ranking problems. Manage Sci 1982;28:621±37. [4] Kendall M. Rank correlation methods. New York, NY: Hafner, 1962. [5] Cook WD, Kress M, Seiford LM. A general framework for distance-based consensus in ordinal ranking model. Eur J Oper Res 1996;96:392±7.

J. GonzaÂlez-PachoÂn, C. Romero / Omega, Int. J. Mgmt. Sci. 27 (1999) 341±347 [6] Ignizio JP. Goal programming its premise, past present and future. ESIGMA Meeting, Barcelona, Spain, 1997. [7] Charnes A, Cooper WW. Goal programming and multiple objective optimization: Part 1. Eur J Oper Res 1977;1:39±54. [8] Ignizio JP. Goal programming and extensions. Lexington, MA: D.C. Heath, 1976. [9] Ignizio JP, Cavalier TM. Linear programming. Englewood Cli€s, NJ: Prentice-Hall, 1994. [10] Romero C. Extended lexicographic goal programming: a unifying approach. Third International Conference on Multi-Objective Programming and Goal Programming (MOPGP98), Quebec, Canada, 1998. [11] Yu PL. A class of solution for group decision problems. Manage Sci 1973;19:936±46.

347

[12] Romero C. Handbook of critical issues in goal programming. Oxford: Pergamon 1981. [13] Rawls J. A theory of justice. Oxford: Oxford University Press, 1972. [14] Tamiz M, Jones D, Romero C. Goal programming for decision making: an overview of the current state-of-theart. Eur J Oper Res 1998, in press. [15] Yu PL. Multiple-criteria decision making. New York: Plenum Press, 1985. [16] Guilbaud GT. Theories of the general interest, and the logical problem of aggregation. In: Lazarsfeld PF, Henry NW, editors. Reading in mathematical social sciences. Chicago: Science Research Associated, 1966:262±307. [17] Hwang C-L, Lin M-J. Group decision making under multiple-criteria. In: Lecture notes in economics and mathematical systems, vol. 281. Berlin: Springer, 1987.