Improved convergent heuristics for the 0-1 multidimensional knapsack problem

Abstract

At the end of the seventies, Soyster et al. (Eur. J. Oper. Res. 2:195-201, 1978) proposed a convergent algorithm that solves a series of small sub-problems generated by exploiting information obtained through a series of linear programming relaxations. This process is suitable for the 0-1 mixed integer programming problems when the number of constraints is relatively smaller when compared to the number of variables. In this paper, we first revisit this algorithm, once again presenting it and some of its properties, including new proofs of finite convergence. This algorithm can, in practice, be used as a heuristic if the number of iterations is limited. We propose some improvements in which dominance properties are emphasized in order to reduce the number of sub problems to be solved optimally. We also add constraints to these sub-problems to speed up the process and integrate adaptive memory. Our results show the efficiency of the proposed improvements for the 0-1 multidimensional knapsack problem.

A hybrid heuristic for the 0–1 Knapsack Sharing Problem

“…It is easy to see that if we solve exactly the reduced problem associated with a partial solution x then all the solutions dominated by x can be eliminated. The use of the pseudo-cut in the ILPH is based on this observation and the following proposition (a proof can be found in Hanafi and Wilbaut (2011)). …”

“…Due to a slow convergence in practice, the method was originally proposed for problems with few constraints. In Wilbaut and Hanafi (2009) and more recently in Hanafi and Wilbaut (2011), the authors present several enhanced and extended variants of the previous approach to solve 0-1 MIP. First, a MIP relaxation is introduced to refine the upper bounds and to diversify the search.…”

“…Section 3 briefly describes the general basic version of the ILPH as proposed in Hanafi and Wilbaut (2011) and Wilbaut and Hanafi (2009). We describe in Section 4 the new components to strengthen the ILPH: a local search to intensify around the solutions visited by the ILPH; the use of additional constraints to strengthen the mathematical formulation of the KSP and to reduce the gap between the upper and the lower bounds on the optimal value of the problem.…”

Hybrid metaheuristics in combinatorial optimization: A survey

“…They combine LP as well as MIP relaxations into a powerful set of heuristics. These are related to the linear programming based algorithm (LPA) for solving 0-1 integer programs by Soyster et al [129], that was further enhanced by Hanafi and Wilbaut [130]. The main principle of LPA is to solve, in a first step, the LP relaxation of the original problem.…”

An exact approach for the prioritization process of industrial influents in wastewater systems

“…Multidimensional knapsack problems can be solved by exact methods when they have small dimensions but encounter difficulties for medium and large instances (Hanafi and Wilbaut 2011). The difficulty to obtain good solutions increases with the number of constraints and with the correlation between the objective function and the constraint coefficients (Fréville and Hanafi 2005).…”