Arnaud Fréville

The multidimensional 0–1 knapsack problem: An overview

Abstract The multidimensional 0–1 knapsack problem is one of the most well-known integer programming problems and has received wide attention from the operational research community during the last four decades. Although recent advances have made possible the solution of medium size instances, solving this NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. This paper surveys the main results published in the literature. The focus is on the theoretical properties as well as approximate or exact solutions of this special 0–1 program.

An efficient tabu search approach for the 0-1 multidimensional knapsack problem

Abstract In this paper, we describe a new approach to tabu search (TS) based on strategic oscillation and surrogate constraint information that provides a balance between intensification and diversification strategies. New rules needed to control the oscillation process are given for the 0–1 multidimensional knapsack (0–1 MKP). Based on a portfolio of test problems from the literature, our method obtains solutions whose quality is at least as good as the best solutions obtained by previous methods, especially with large scale instances. These encouraging results confirm the efficiency of the tunneling concept coupled with surrogate information when resource constraints are present.

An efficient preprocessing procedure for the multidimensional 0–1 knapsack problem

Abstract The multidimensional 0–1 knapsack problem, defined as a knapsack with multiple resource constraints, is well known to be much more difficult than the single constraint version. This paper deals with the design of an efficient preprocessing procedure for large-scale instances. The algorithm provides sharp lower and upper bounds on the optimal value, and also a tighter equivalent representation by reducing the continuous feasible set and by eliminating constraints and variables. This scheme is shown to be very effective through a lot of computational experiments with test problems of the literature and large-scale randomly generated instances.

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.

The Multidimensional 0-1 Knapsack Problem—Bounds and Computational Aspects

The multidimensional 0-1 knapsack problem (MKP) is a resource allocation model that is one of the most well-known integer programming problems. During the last few decades, an impressive amount of research on the 0-1 knapsack problem has been published in the literature, and efficient special-purpose methods have become available for solving very large-scale instances. However, the multidimensional case has received less attention from the operational research community. Although recent advances have made solving medium size instances possible, solving the NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. This paper surveys the principal results published in the literature concerning both the problems theoretical properties and its approximate or exact solutions. The paper focuses on the more recent results—for example, those relevant to surrogate and composite duality, new preprocessing approaches creating enhanced versions of leading commercial software, and efficient metaheuristic-based methods.

A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem

Abstract This paper presents a preprocessing procedure for the 0–1 multidimensional knapsack problem. First, a non-increasing sequence of upper bounds is generated by solving LP-relaxations. Then, a non-decreasing sequence of lower bounds is built using dynamic programming. The comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal, or to fix a subset of variables to their optimal values. In addition, a heuristic solution is obtained. Computational experiments with a set of large-scale instances show the efficiency of our reduction scheme. Particularly, it is shown that our approach allows to reduce the CPU time of a leading commercial software.

An exact search for the solution of the surrogate dual of the 0-1 bidimensional knapsack problem

Abstract The surrogate dual of the 0–1 bidimensional knapsack problem is exactly solved by an algorithm with a modified dichotomic search. The primal (or dual) optimality is proved with a finite number of iterations. A lot of numerical experiments show the efficiency of our method: its reduced number of iterations is revealed to be independent of the size of the instances.

The 0-1 bidimensional knapsack problem: Toward an efficient high-level primitive tool

Efficient codes exist for exactly solving the 0-1 knapsack problem, which is a common primitive structure in relaxation and decomposition techniques for the solution of general models. We suggest moving to a higher primitive level by using the bidimensional knapsack, which can be used to enhance linear programming or Lagrangean type classical relaxations.With the ultimate aim of providing an exact and efficient solution to the bidimensional knapsack problem, we describe here a heuristic approach based on surrogate duality. In particular, we consider the usefulness of a specific preprocessing phase before a possible enumerative phase.Extensive numerical experiments, based on test problems from the literature as well as randomly generated instances, show that our code compares favorably with the GP procedure developed by Gavish and Pirkul for the multidimensional case.

Municipal Solid Waste Collection: An Effective Data Structure For Solving The Sectorization Problem With Local Search Methods

This paper deals with the crucial sectorization problem regarding household waste collection. Our purpose is to construct a fixed number of sectors which should be balanced with respect to daily to...