Elsevier

Computers & Operations Research

Volume 34, Issue 9, September 2007, Pages 2674-2694
Computers & Operations Research

Bound sets for biobjective combinatorial optimization problems

https://doi.org/10.1016/j.cor.2005.10.003Get rights and content

Abstract

In this paper we introduce the concept of bound sets for multiobjective discrete optimization. We prove general results on lower and upper bound sets for combinatorial optimization problems with multiple objectives. We present general algorithms for constructing lower and upper bound sets for biobjective problems and provide numerical results on five different problem types.

Introduction

Multiobjective combinatorial optimization problems can be formulated as follows:min{Cx:Axb,xZn}.

Here, C is a p×n objective function matrix, where ck denotes the kth row of C. A is a m×n matrix of constraint coefficients and bRm. Usually the entries of C,A, and b are integers. The feasible set X{Axb,xZn} may describe a combinatorial structure such as e.g. spanning trees of a graph, paths, matchings etc. We shall assume that X is a finite set. By Y=CX we denote the image of X under C in Rp.

A feasible solution x^X is called efficient (or Pareto optimal) if there does not exist xX such that ckxckx^ for all k=1,,p and ckx<ckx^ for some k, 1kp. In other words, no solution is at least as good as x^ for all objectives and strictly better for at least one.

Efficiency refers to solutions x in decision space. In terms of the criterion space, with objective vectors CxRp, we use the notion of nondominance: If x is efficient then Cx=(c1x,,cpx) is called nondominated (or also efficient). The set of efficient solutions is XEX, the set of nondominated objective vectors (or points) is YNY. Surveys of the state of the art in multiobjective combinatorial optimization are available in [1], [2].

For y1,y2Rp we shall write y1y2 if yk1yk2,k=1,,p, y1y2 if y1y2 but y1y2 and y1<y2 if yk1<yk2,k=1,,p. Furthermore, we shall use Rp{yRp:y0} and analogously Rp and R>p.

The sets XE and YN can be partitioned into two subsets, respectively. An efficient solution x^XE is called a supported efficient solution if there exists some λR>p such that λTCx^λTCx for all xX. The set of all supported efficient solutions is denoted XsE, the elements of XEXsEXnE are called nonsupported efficient solutions. Analogously, we obtain a partition of nondominated points YN=YsNYnN into supported and nonsupported nondominated points. Note that this definition implies that YnNint((convYN)+Rp) because there is no hyperplane with gradient in R>p supporting convYN at yYnN.

Finally, we need to introduce some general (convex analysis) notation for sets in Rp. Let SRp, then bdS,intS,clS denote the boundary, interior, and closure of S, respectively. Furthermore, convS denotes the convex hull of S, SN{sS:(s-Rp)S={s}} is called the nondominated frontier of S and ScRpS denotes the complement of S in Rp. S is called Rp-closed, if the set S+Rp{s+y:sS,yRp} is closed and Rp-bounded, if there exists s0Rp such that Ss0+Rp. Finally, S is called Rp-convex, if S+Rp is convex.

Lemma 1 will be instrumental to derive results on bound sets.

Lemma 1 Sawaragi [3]

Let SRp.

  • 1.

    If S is an Rp-closed and Rp-bounded set then SSN+Rp (S is called externally stable).

  • 2.

    SN=(S+Rp)N.

  • 3.

    SNbdS.

In single objective combinatorial optimization lower and upper bounds l,uR with lmin{cx:xX}u have been essential tools for the solution of NP-hard problems.

The theory of approximation is based on bounds. They allow the development of algorithms with performance ratios, guaranteeing heuristic solutions to be no more than a certain percentage from optimality. Theoretical investigations lead to results on hardness of approximation. We refer to [4] for an extensive list of results. Bounds are used in implicit enumeration techniques such as branch and bound or dynamic programming for solving combinatorial optimization problems. The performance of the algorithms depends on the quality of the bounds. From this experience, one must assume that bounds might be valuable tools for multiobjective combinatorial optimization, too. However, a different concept of “bounding” is needed to account for the different concept of optimality in multiobjective optimization.

Generally accepted and well known bounds in multiobjective optimization are the ideal point yI defined by ykIminxXckx;k=1,,pas lower bound and the nadir point yN defined by ykNmaxxXEckx;k=1,,pas upper bound on the value of any efficient point. It is obvious that ykIykykN for all yYN. The benefits of using ideal and nadir values as bounds are limited due to a number of reasons.

  • 1.

    yI is difficult to obtain, if min{ckx:xX} is NP-hard, as it involves the solution of p NP-hard problems.

  • 2.

    yN is difficult to obtain if p>2 even if the single objective problems min{ckx:xX} are polynomially solvable. Furthermore, heuristics to compute an estimate y˜N may underestimate yN, i.e. yield y˜NyN, rendering the heuristic nadir point useless as an upper bound for YN. See [5], [6] for discussions on these problems.

  • 3.

    Last but not least, ideal and nadir values are “far away” from most nondominated points. This is due to the usually large number of nondominated points in YN and the conflicting nature of the objectives, whereas yI and yN show the combined best and worst values for all objectives over the range of YN.

Not much else has been done. Villarreal and Karwan [7] propose the use of sets of lower and upper bound vectors to limit the search space in dynamic programming. Local ideal and nadir points are used for bounding the nondominated sets of subproblems in nodes of the branch and bound tree, see e.g. [8], [9], [10]. Finally, Kaliszewski [11] proposes parametric bounds in interactive problems.

In this paper, following our preliminary investigation in [12], [13], we propose a new approach to bounding strategies for (MOCO) problems. Our earlier paper has also been the foundation for further investigation in [14].

We generalize the notion of a bound value to that of a bound set (in parallel to the fundamental transition from unique optimal value to nondominated set), thus incorporating the nature of multiple objectives into the definition of bound sets. Furthermore, we demonstrate that bound sets are not only easier to compute than ideal and nadir points, but also yield better results, i.e. are closer to nondominated points.

The rest of the paper is organized as follows. In Section 2 we give the definitions of lower and upper bound sets and prove some general results. In Section 3 we present techniques for computing lower and upper bound sets for polynomially solvable and NP-hard combinatorial problems. First, general techniques are given, these are followed by specifications for the assignment problem, the knapsack problem, the set covering problem, the set packing problem and the travelling salesman problem. For these five problems we provide numerical results in Section 4. We use several measures to determine the quality of the bound sets and show typical instances graphically.

Section snippets

Definitions and general results

We begin with the definition of lower and upper bound sets. Consider a multiobjective combinatorial optimization problem.

Definition 1

Let Y¯YN.

  • 1.

    A lower bound set L for Y¯ is an Rp-closed and Rp-bounded set LRp such that Y¯L+Rp and L(L+Rp)N.

  • 2.

    An upper bound set U for Y¯ is an Rp-closed and Rp-bounded set URp such that Y¯cl[(U+Rp)c] and U(U+Rp)N.

Note that although the first condition of the definition of a lower bound set seems to be the intuitive formalization of the idea “L is below Y¯” this

Constructing bound sets for MOCO

In this section we present techniques for constructing lower and upper bound sets for problems with two objectives. The general idea to construct bound sets is by using the weighted sum scalarized problemminxXλc1x+(1-λ)c2x.

We distinguish the cases where the single objective problems min{ckx:xX} are polynomially solvable, respectively, NP-hard. Note that the distinction applies to the single objective case. The biobjective case is usually NP-hard even for such problems as assignment, shortest

Numerical results

We have performed numerical tests on all problem types for which instances were available in the MOCO numerical instances library [19] at the time of writing. In this section we present these results, mainly to illustrate that the use of bound sets can be beneficial in multiobjective combinatorial optimization. To measure the quality of lower and upper bound sets, we used a number of measures.

Let L and U be an upper and lower bound set for an instance of a biobjective combinatorial optimization

Conclusions

We have introduced the concept of bound sets for biobjective combinatorial optimization problems. Motivated by the fact that (MOCO) problems are very difficult in general, yet appear often in practical applications, we duplicated the transition from one optimal solution in single objective optimization to a set of efficient solutions in the multicriteria case. We gave a general scheme for computing a lower bound set and proved some general results. With numerical tests we illustrated that good

Acknowledgements

We would like to thank Anthony Przybylski, PhD student at the University of Nantes, and Cameron Mellor, BE student at the University of Auckland, for their assistance in computing the quality measures and the bound sets for the TSP, respectively.

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