This paper provides a survey of the research in and an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. The main parts of the paper are a section on the review of the available solution methodology, both exact and heuristic, and a section on the annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research.

A survey and annotated bibliography of multiobjective combinatorial optimization

In this article, we propose a solution procedure for the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). A relaxed version of this problem in which the path does not have to be elementary has been the backbone of a number of solution procedures based on column generation for several important problems, such as vehicle routing and crew pairing. In many cases relaxing the restriction of an elementary path resulted in optimal solutions in a reasonable computation time. However, for a number of other problems, the elementary path restriction has too much impact on the solution to be relaxed or might even be necessary. We propose an exact solution procedure for the ESPPRC, which extends the classical label correcting algorithm originally developed for the relaxed (nonelementary) path version of this problem. We present computational experiments of this algorithm for our specific problem and embedded in a column generation scheme for the classical Vehicle Routing Problem with Time Windows. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 216–229 2004

An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems

This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint. The main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value, whose ratio is polynomially bounded, are known. In spite of this difficulty, one of the algorithms presented here is strongly polynomial. Applications to other problems are also discussed.

Approximation Schemes for the Restricted Shortest Path Problem

https://hal.archives-ouvertes.fr/hal-00158652/document

Min–max and min–max regret versions of combinatorial optimization problems: A survey

We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy i¾?the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing many important widely studied problems such as shortest paths, spanning tree, and many others), we can compute in polynomial time an i¾?-Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we show that the factor of 2 is tight for these problems, i.e., it is NP-hard to do better. We present further results for three or more objectives, as well as for the dual problem of computing a specified number kof solutions which provide a good approximation to the Pareto curve.

Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems

We study methods for approximating the set of Pareto optimal paths in multiple-objective, shortest-path problems. Known generalizations of standard shortest-path methods will compute this set, but can suffer from rapidly increasing computational and storage demands as problem size increases. In an effort to avoid such difficulties, we develop approximation methods that can estimate the Pareto optima to any required degree of accuracy. The approximation methods are "fully polynomial"; that is, they operate in time and space bounded by a polynomial in problem size and accuracy of approximation-the greater the accuracy, the more time required to reach a solution. We show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of NP-complete, single-objective problems.

Approximation of Pareto optima in multiple-objective, shortest-path problems

Min-max problems on matroids are NP-hard for a wide variety of matroids. However, greedy type algorithms have data independent worst case performance guarantees, andn-enumerative algorithms yielde-optimal solutions ifn is sufficiently close to the rank of the underlying matroid. Data dependent performance guarantees can be obtained for max-min problems over matroids.

Worst case analysis of greedy and related heuristics for some min-max combinatorial optimization problems

We show how certain bicriterion fractional programs can be reduced to a one parameter linear program and a series of one-dimensional maximizations. The resulting algorithm is easily implemented using the PARAROW option of MPSX, and has readily solved problems having up to 300 variables and 150 constraints. As an illustration, we present a simple finite procedure for maximizing the sum of two linear fractional functions subject to linear constraints.

Arthur Warburton

Papers

Approximation of Pareto optima in multiple-objective, shortest-path problems

Worst case analysis of greedy and related heuristics for some min-max combinatorial optimization problems

Parametric Solution of Bicriterion Linear Fractional Programs