Discrete Optimization
On weighting two criteria with a parameter in combinatorial optimization problems

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Abstract

Two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1. For special problem types, e.g., when one of the criteria is a bottleneck value, efficient algorithms are known that solve for a given value of the weighting parameter.

We transform the underlying enumeration method into a parametric algorithm solving simultaneously for all values of the weighting parameter. Efficient implementations are presented for combinatorial problems with criteria as balanced optimization, min–sum with min–max, and min–sum with balanced optimization, considering the spanning tree, the linear assignment and the single machine scheduling problem. Further the new algorithmic scheme can easily incorporate a trade-off of the criteria by means of penalty functions, again without consequences for the algorithm and its complexity order.

Highlights

► A parametric algorithm solves for all combinations of two criteria in a combinatorial problem. ► Efficient implementations treat criteria such as min–sum and min–max or balanced optimization. ► An algorithm of O(m log n) combines the length of a spanning tree and its longest edge. ► With penalty functions one can refine the trade-off between the criteria.

Introduction

Minoux (1989) presented a generic algorithm for a combinatorial problem where one minimizes the sum of a bottleneck objective and a sum objective. Earlier Halpern (1976) considered such objectives in certain location problems. Martello et al. (1984) considered the so-called balanced optimization criterion, where one minimizes the difference of the maximum and the minimum cost in a solution. Other combinatorial problems with two criteria are, e.g., constrained balanced optimization in Punnen and Nair (1999), machine scheduling in Liao and Huang, 1991, Huang and Yang, 2009 as well as spanning trees in Dell’Amico and Maffioli (2000).

When two criterion values z1(S) and z2(S) on a solution S are combined in a weighted sum, minimizing z(λ)  λz1(S) +  (1  λ)z2(S) for some λ   (0, 1), the optimal solution is called an ‘efficient’ solution (a further improvement in one of the criteria must lead to a worse value in the other criterion). Many algorithms resort to an enumeration: for candidate values of one criterion, one solves a constrained sub-problem with respect to the other criterion, which can be related to the ‘ε-constrained method’ in multi criteria optimization for the generation of ‘efficient’ solutions, see, e.g., Ehrgott (2005). Burkard et al. (1981) investigated the convex combination of the sum and the bottleneck objective in combinatorial optimization problems.

The subject of this paper is a ‘parametric algorithm’ that minimizes z(λ) for all λ  [0, 1] in a single enumeration with two phases. The first phase records for relevant efficient solutions the two criterion values; the second phase processes these values to determine optimal values for all λ  [0, 1] with a routine of Hershberger (1989). Remarkably, the time complexity can be the same as for one value of λ. The resulting optimal results for all λ  [0, 1], give a decision maker full information on the effect of changing the weighting parameter λ. This parametric algorithm requires two conditions, as formulated later. If both z1 and z2 are sum objectives, then the techniques of Eisner and Severance (1976) can be more appropriate.

We formulate specific implementations that reduce with respect to time complexity. First, the generic ‘parametric algorithm’ and its conditions are described in Section 2. It is then applied to a single machine scheduling problem in Section 3 and to two spanning tree problems in Section 4. In the first spanning tree a trade-off is made between the length of the tree and the range of these lengths; in the second tree we seek a trade-off between the length of the tree and the maximum edge cost, a problem considered in Punnen and Nair (1996) for a fixed value of λ. Finally, in Section 5 we discuss how a refined trade-off of the criteria is possible within our framework by using penalty functions. Section 6 summarizes the paper.

Section snippets

The parametric algorithm

Typically in combinatorial problems one considers elementary objects in a finite set E with which one may build ‘solutions’ satisfying certain properties. These solutions have different criterion values arising from given ‘cost’ functions on E. More precisely we have:

  • a finite ‘ground’ set E of elementary objects e  E;

  • one or more ‘cost’ functions, e.g., c: ER and w:ER;

  • a collection P(E) of ‘pre-solutions’ derived from E;

  • a set of feasible solutions F = {SS  P(E), S satisfies (problem-specific)

Parametric algorithms in pseudo-polynomial time

We apply the parametric algorithm to a single machine problem with two criteria; a problem solved by branch and bound for a single weighted sum in Gupta and Sen (1984), as well as Tegze and Vlach (1988) and by a pseudo-polynomial algorithm in Liao and Huang (1991).

There are n jobs j  J with due dates dj and non-preemptive processing times pj. Feasible solutions are permutation schedules S =  (j1, j2,  , jn). In a schedule it is not only undesirable that the completion time Cj(S) is later than dj,

Spanning tree problems

We consider two spanning tree problems in a graph (V, E) with ∣V = n and ∣E = m. In the first one there is a single cost function, the length ce on each edge e; the second one, studied in Section 4.2, involves two cost functions.

Transforming criteria

When confronted with multiple criteria, one may choose to use target values. We discuss the effect of two ways of doing so, using problems and algorithms of previous sections.

Summary

Some algorithms, as presented in literature for combinatorial problems with two criteria, solve for a single convex combination of the criteria by applying an enumeration. This enumeration can lead to an efficient algorithm if the problem satisfies two conditions. Under the same conditions such algorithms can be extended. Using a parameter λ for weighting the two criteria, one can solve as well for all values of λ  [0, 1], in the same complexity.

The parametric algorithm generates optimal values z

Acknowledgement

We are grateful for the constructive comments of the referees; they have led to an improved presentation.

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