Xavier Gandibleux

Approximative solution methods for multiobjective combinatorial optimization

In this paper we present a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO). First, we discuss questions related to approximation in this context, such as performance ratios, bounds, and quality measures. We give some examples of heuristics proposed for the solution of MOCO problems. The main part of the paper covers metaheuristics and more precisely non-evolutionary methods. The pioneering methods and their derivatives are described in a unified way. We provide an algorithmic presentation of each of the methods together with examples of applications, extensions, and a bibliographic note. Finally, we outline trends in this area.

Metaheuristics for multiobjective optimisation

I Methodology.- A Tutorial on Evolutionary Multiobjective Optimization.- 2 Bounded Pareto Archiving: Theory and Practice.- 3 Evaluation of Multiple Objective Metaheuristics.- 4 An Introduction to Multiobjective Metaheuristics for Scheduling and Timetabling.- II Problem-oriented Contributions.- 5 A Particular Multiobjective Vehicle Routing Problem Solved by Simulated Annealing.- 6 A Dynasearch Neighborhood for the Bicriteria Traveling Salesman Problem.- 7 Pareto Local Optimum Sets in the Biobjective Traveling Salesman Problem: An Experimental Study.- 8 A Genetic Algorithm for Tackling Multiobjective Job-shop Scheduling Problems.- 9 RPSGAe - Reduced Pareto Set Genetic Algorithm: Application to Polymer Extrusion.

Tabu Search Based Procedure for Solving the 0-1 MultiObjective Knapsack Problem: The Two Objectives Case

We consider in this paper the solving of 0-1 knapsack problems with multiple linear objectives. We present a tabu search approach to generate a good approximation of the efficient set. The heuristic scheme is included in a redu tion decision space framework. The case of two objectives is developed in this paper. TS principles viewed into the multiobjective context are discussed. According to a prospective way, several variations of the algorithm are investigate. Numerical experiments are reported and compared with available exact efficient solutions. Intuitive justifications for the observed empirical behavior of the procedure and open questions are discussed.

A Tabu Search Procedure to Solve MultiObjective Combinatorial Optimization Problems

Several studies have considered metaheuristics, especially simulated annealing, for solving combinatorial optimization problems involving several objectives. Yet, few works have been devoted to tabu search approaches. In this paper, we present a heuristic based upon tabu search principles to generate a good approximation of the set of the Pareto-optimal (efficient) solutions.

Two phase algorithms for the bi-objective assignment problem

In this paper, we present several algorithms for the bi-objective assignment problem. The algorithms are based on the two phase method, which is a general technique to solve multi-objective combinatorial optimisation (MOCO) problems. We give a description of the original two phase method for the bi-objective assignment problem, including an implementation of the variable fixing strategy of the original method. We propose several enhancements for the second phase, i.e., improved upper bounds and a combination of the two phase method with a population based heuristic using path relinking to improve computational performance. Finally, we describe a new technique for the second phase with a ranking approach, which outperforms all other tested algorithms. All of the algorithms have been tested on instances of varying size and range of objective function coefficients. We discuss the results obtained and explain our observations based on the distribution of objective function values.

Bound sets for biobjective combinatorial optimization problems

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.

Multiobjective Combinatorial Optimization — Theory, Methodology, and Applications

This chapter provides an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe their main characteristics, and review the main properties and theoretical results. One section is devoted to a brief description of the available solution methodology, both exact and heuristic. The main part of the chapter consists of an annotation of the existing literature in the field organized problem by problem. We conclude the chapter by stating open questions and areas of future research. The list of references comprises more than 400 entries.

GRASP for set packing problems

Abstract The principles of the Greedy Randomized Adaptative Search Procedure (GRASP) metaheuristic are instantiated for the set packing problem. We investigated several construction phases, and evaluated improvements based on advanced strategies. These improvements include a self-tuning procedure (using reactive GRASP), an intensification procedure (using path relinking) and a procedure involving the diversification of the selection (using a learning process). Two sets of various numerical instances were used to perform the computational experiments. The first set contains randomly generated instances, while the second includes instances relating to real problems in railway planning. No metaheuristic has previously been applied to this combinatorial problem. Consequently, we have discussed GRASP’s performances both in relation to lower/upper bounds and to the results obtained with Cplex when such results are available. Our analysis, based on the average performances observed, shows the impact of the suggested strategies, and indicates the configuration that produces the best results.

A two phase method for multi-objective integer programming and its application to the assignment problem with three objectives

In this paper, we present a generalization of the two phase method to solve multi-objective integer programmes with p>2 objectives. We apply the method to the assignment problem with three objectives. We have recently proposed an algorithm for the first phase, computing all supported efficient solutions. The second phase consists in the definition and the exploration of the search area inside of which nonsupported nondominated points may exist. This search area is not defined by trivial geometric constructions in the multi-objective case, and is therefore difficult to describe and to explore. The lower and upper bound sets introduced by Ehrgott and Gandibleux in 2001 are used as a basis for this description. Experimental results on the three-objective assignment problem where we use a ranking algorithm to explore the search area show the efficiency of the method.

Hybrid Metaheuristics for Multi-objective Combinatorial Optimization

Many real-world optimization problems can be modelled as combinatorial optimization problems. Often, these problems are characterized by their large size and the presence of multiple, conflicting objectives. Despite progress in solving multi-objective combinatorial optimization problems exactly, the large size often means that heuristics are required for their solution in acceptable time. Since the middle of the nineties the trend is towards heuristics that “pick and choose” elements from several of the established metaheuristic schemes. Such hybrid approximation techniques may even combine exact and heuristic approaches. In this chapter we give an overview over approximation methods in multi-objective combinatorial optimization. We briefly summarize “classical” metaheuristics and focus on recent approaches, where metaheuristics are hybridized and/or combined with exact methods.