Decision Support
Cone contraction and reference point methods for multi-criteria mixed integer optimization

https://doi.org/10.1016/j.ejor.2013.03.006Get rights and content

Highlights

  • An interactive approach for a mixed integer multi-criteria optimization is introduced.

  • The DM makes pairwise comparisons of identified Pareto optimal points and gives reference points.

  • Assuming a quasi-concave value function pairwise comparisons are used to contract the cone of admissible objective vectors.

  • Numerical simulation tests indicate reasonably fast convergence.

  • Convergence is guaranteed for the pure integer case.

Abstract

Interactive approaches employing cone contraction for multi-criteria mixed integer optimization are introduced. In each iteration, the decision maker (DM) is asked to give a reference point (new aspiration levels). The subsequent Pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function. Thereby, the procedures solve a sequence of optimization problems with integer variables. In such a process, the DM provides additional preference information via pair-wise comparisons of Pareto optimal points identified. Using such preference information and assuming a quasiconcave and non-decreasing value function of the DM we restrict the set of admissible objective vectors by excluding subsets, which cannot improve over the solutions already found. The procedures terminate if all Pareto optimal solutions have been either generated or excluded. In this case, the best Pareto point found is an optimal solution. Such convergence is expected in the special case of pure integer optimization; indeed, numerical simulation tests with multi-criteria facility location models and knapsack problems indicate reasonably fast convergence, in particular, under a linear value function. We also propose a procedure to test whether or not a solution is a supported Pareto point (optimal under some linear value function).

Introduction

Multi-criteria optimization with integer variables has received relatively little attention in comparison with continuous problems during the past decades. Yet, studies on integer valued interactive approaches have advanced remarkably as indicated by reviews of Ehrgott and Gandibleux, 2000, Ehrgott, 2005, Köksalan, 2009, Alves and Clímaco, 2007 – moreover, the last reference reviews also the developments in multi-criteria mixed integer problems as well as provides references for earlier reviews. Recently much progress has been made also in the area of multi-objective metaheuristics and evolutionary algorithms with integer variables. For reviews see Ehrgott and Gandibleux, 2004, Gandibleux and Ehrgott, 2005.

Integer variables represent increased difficulty in the multi-criteria problems in the same way as in single objective problems. Thus numerous heuristic approaches have been put forward, often applicable only for a special type of problems, e.g. spanning tree, assignment or shortest path problems. Approaches have been suggested that work only in a subset of problems with integer variables – they deal with bi-criteria problems, with linear objective functions or with problems having a special structure. Some suggested approaches assume an implicit value function with some properties like quasiconcavity – as we do – and thus may guarantee that the best solution is found. The alternative is that the process is not aiming at converging to the best solution but, instead, represents a learning process, where the decision maker acquires insight into the set of non-dominated solutions.

Quite a significant number of papers present methods that aim at generating all Pareto optimal solutions first and choosing the most preferred solution thereafter. However, a large share of the Pareto optimal solutions may be uninteresting, while second best solutions may be dominated. Hence, it often does not pay off to generate the set of all Pareto optimal solutions. A favorable approach is to use an interactive procedure which is characterized by the alternating phases of the decision maker’s (DM) preference information elicitation and computation of candidate solutions. Thereby, only a small fraction of the Pareto optimal solutions needs usually be generated.

Korhonen et al. (1984) consider an interactive method for the multi-criteria problem of choosing from an explicit list of alternatives. Assuming an underlying quasiconcave and non-decreasing value function and using pair-wise comparison results they develop convex polyhedral cones whose points cannot be better than the vertex of the cone. In each iteration, such non-admissible cones are used to rule out alternatives in the list and a Pareto optimal choice is made from the remaining admissible set. Along the iterations, for any single alternative in the admissible set, Korhonen et al. (1984) solve a linear programming problem to test if the alternative can be ruled out. Iterations stop, if all alternatives are either identified or ruled out. At this point the best alternative identified is optimal.

Based on the result of Korhonen et al., 1984, Lokman et al., 2011 introduced an interactive method applicable for general multi-criteria integer programs. As both our methods and the approach of Lokman et al. (2011) rule out solutions employing convex cones, we discuss the advantages of our methods against their approach in the electronic appendix. Both Korhonen et al., 1984, Lokman et al., 2011 as well our approaches employ cone contraction to exclude solutions; for a review of cone contraction, see Kadzinski and Slowinski (2012).

The reference point method has been extensively used after its introduction by Wierzbicki (1980). In this article, we propose a reference point approach for a multi-criteria mixed integer or pure integer problem with an underlying quasiconcave value and non-decreasing function. In a special case, a linear value function is assumed.

Our approach is interactive incorporating the DM’s preference information in the search. Two kinds of preference information are employed: reference points (aspiration levels of objectives) and pair-wise comparisons which rank already identified Pareto points. Transitivity is assumed and employed to reduce the number of pair-wise comparisons. Preference information is utilized in two ways: First, it is used to rule out feasible solutions that cannot bring improvement over the solutions already found. Second, if a linear value function can be assumed, then preference information is also used to rule out solutions using the result of Kallio et al. (1980). We propose a simple and operational test employing binary variables to check if a candidate point is not ruled out. In each iteration, a new reference point is defined and an admissible Pareto optimal solution is found. Iterations continue until all solutions are ruled out or – as usual in the reference point approaches – until a satisficing solution is found. In the former case, an optimal solution is found for a multi-criteria problem with the underlying value function. Such convergence is achieved for pure integer problems with a finite Pareto set. For other problems, however, converge to an optimum may not be guaranteed and termination of iterations may occur when a satisficing Pareto point is found.

In case the underlying value function is linear, the optimal solution is a supported Pareto point; i.e. optimal under some linear value function. However, all Pareto points generated by our methods during the iterations need not be supported even if a linear value function is assumed. For this reason, we also propose a novel test to diagnose if a Pareto optimal solution is supported.

In computational simulation experiments we employ underlying true linear value functions which are used in the pair-wise comparisons and thereby in determining the ranking of solutions over the iterations. Tests with multi-criteria facility location models and knapsack problems show that the interactive iterations converge reasonably fast, in particular, when a linear value function is assumed.

The rest of the article is organized as follows. In Section 2 we define the multi-criteria mixed integer programming problem. Section 3 provides an outline of our reference point methods. Section 4 specifies the algorithmic steps in detail under a quasiconcave value function. In Section 5, we consider the case of a linear value function. A procedure is proposed in Appendix A as well to test if a Pareto point is supported. Section 6 presents numerical illustrations and Section 7 concludes.

Section snippets

The multi-criteria mixed integer problem

Consider a multi-criteria optimization problem with k objectives to be maximized. Let g  Rk denote the column vector of objectives. The underlying value function which the DM aims to maximize is u(g). At the outset the value function is not known, but we assume that u is quasiconcave and non-decreasing. As a special case we assume u = wg is a linear value function with a positive row vector w > 0 of weights.

Given a set F of feasible objective vectors g, the multi-criteria problem ismaxgFu(g).

We

Outline of our reference point methods

In each iteration τ, τ = 1, 2, 3, … , of the iterative procedure, a reference point g¯τ in the objective space is employed to compute a Pareto point gˆτ. The reference point g¯τ is provided by the DM and it reflects the DM’s aspiration level of objective values given the Pareto points gˆν, for ν < τ, found so far. The reference point g¯τ guides the computation of the subsequent Pareto point gˆτ. Furthermore, in the course of the interactive iterations we employ pair-wise comparisons of the

The case of a quasiconcave value function

In this section we specify the reference point method outlined in Section 3 in case a quasiconcave value function is assumed. We refer to this method by QC. In the procedure each iteration starts by the computation of a new Pareto point using the problem (5), (6), (7) in Section 4.3 below and the current set of admissible points defined by (3), (4) in Section 4.2. The new Pareto point is compared pair-wise with the previously generated points and a full ranking of them is created as discussed

The case of a linear value function

In the reference point methods LIN and its simplification SLIN the value function u(g) = w g is assumed linear with some weight vector w > 0. In this case, the optimal solution is a supported Pareto point. However, all Pareto points generated by LIN and SLIN need not be supported. Therefore, in case a test is desired, we provide a procedure in Appendix A to diagnose if a Pareto optimal solution is supported.

For LIN and SLIN, in each iteration τ pair-wise comparisons in Step 3 are carried out as in

Computational illustration

Our procedure is implemented using AMPL/MOSEK (Fourer et al., 2003). Numerical tests were carried out with pure integer problems. For numerical illustration, we use simulation tests on three versions of a facility location model (Alves and Clímaco, 1999) and on three versions of a multi-criteria knapsack problem (Lokman et al. (2011).

The facility location model concerns selection of the sites among n regions, for waste processing facilities. All the n regions are potential sites to locate a

Conclusions

In this article, three versions of a reference point method employing cone contraction were proposed for multi-criteria optimization with mixed integer variables. For the version QC, we assume that the underlying value function of the decision maker (DM) is quasiconcave and non-decreasing. As a special case, a linear value function is assumed for the versions LIN and its simplification SLIN.

In each interactive iteration, the DM provides a reference point (new aspiration levels), and a Pareto

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The authors are grateful to Murat Köksalan for a seminar talk which inspired us to initiate the work on our article.

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