Christophe Wilbaut

New convergent heuristics for 0-1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0-1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0-1 multidimensional Knapsack problem. Other encouraging results obtained for 0-1 MIP problems are also presented.

Iterative semi-continuous relaxation heuristics for the multiple-choice multidimensional knapsack problem

Recently several hybrid methods combining exact algorithms and heuristics have been proposed for solving hard combinatorial optimization problems. In this paper, we propose new iterative relaxation-based heuristics for the 0-1 Mixed Integer Programming problem (0-1 MIP), which generate a sequence of lower and upper bounds. The upper bounds are obtained from relaxations of the problem and refined iteratively by including pseudo-cuts in the problem. Lower bounds are obtained from the solving of restricted problems generated by exploiting information from relaxation and memory of the search process. We propose a new semi-continuous relaxation (SCR) that relaxes partially the integrality constraints to force the variables values close to 0 or 1. Several variants of the new iterative semi-continuous relaxation based heuristic can be designed by a given update procedure of multiplier of SCR. These heuristics are enhanced by using local search procedure to improve the feasible solution found and rounding procedure to restore infeasibility if possible. Finally we present computational results of the new methods to solve the multiple-choice multidimensional knapsack problem which is an NP-hard problem, even to find a feasible solution. The approach is evaluated on a set of problem instances from the literature, and compared to the results reached by both CPLEX solver and an efficient column generation-based algorithm. The results show that our algorithms converge rapidly to good lower bounds and visit new best-known solutions.

An iterative variable-based fixation heuristic for the 0-1 multidimensional knapsack problem

An iterative scheme which is based on a dynamic fixation of the variables is developed to solve the 0-1 multidimensional knapsack problem. Such a scheme has the advantage of generating memory information, which is used on the one hand to choose the variables to fix either permanently or temporarily and on the other hand to construct feasible solutions of the problem. Adaptations of this mechanism are proposed to explore different parts of the search space and to enhance the behaviour of the algorithm. Encouraging results are presented when tested on the correlated instances of the 0-1 multidimensional knapsack problem.

A hybrid quantum particle swarm optimization for the Multidimensional Knapsack Problem

In this paper we propose a new hybrid heuristic approach that combines the Quantum Particle Swarm Optimization technique with a local search method to solve the Multidimensional Knapsack Problem. The approach also incorporates a heuristic repair operator that uses problem-specific knowledge instead of the penalty function technique commonly used for constrained problems. Experimental results obtained on a wide set of benchmark problems clearly demonstrate the competitiveness of the proposed method compared to the state-of-the-art heuristic methods.

Three-stage hybrid-flowshop model for cross-docking

This paper deals with the optimization of a cross-docking system. It is modeled as a three-stage hybrid flowshop, in which shipments and orders are represented as batches. The first stage corresponds to the receiving docks, the second stage corresponds to the sorting stations, and the third stage corresponds to the shipping docks. The objective of the problem is to find a schedule that minimizes the completion time of the latest batch. In order to obtain good quality feasible solutions, we have developed several heuristic schemes depending on the main stage considered, and several rules to order the batches in this stage. Then, we propose a branch-and-bound algorithm that takes into account the decomposition of the problem into three stages. To evaluate the heuristics and to reduce the tree size during the branch-and-bound computation, we also propose lower bounds. Finally, the computational experiments are presented to demonstrate the efficiency of our heuristics. The results show that the exact approach can solve instances containing up to 9-10 batches in each stage (i.e., up to 100 jobs). In addition, our heuristics were evaluated over instances with up to 3000 jobs, and they can provide good quality feasible solutions in a few seconds (i.e., less than 2s per heuristic).

A hybrid heuristic for the 0-1 Knapsack Sharing Problem

A new hybrid method based on ILPH and QPSO is proposed and validated on the KSP.The proposed approach can be easily adapted to other variants of knapsack problems.New valid constraints are used to speed up the reduced problems solved inside ILPH.A local search is incorporated in ILPH as an intensification process.QPSO starts with the best solutions provided by ILPH where infeasibility is allowed. The Knapsack Sharing Problem (KSP) is a variant of the well-known NP-hard knapsack problem that has received a lot of attention from the researches as it appears into several real-world problems such as allocating resources, reliability engineering, cloud computing, etc. In this paper, we propose a hybrid approach that combines an Iterative Linear Programming-based Heuristic (ILPH) and an improved Quantum Particle Swarm Optimization (QPSO) to solve the KSP. The ILPH is an algorithm conceived to solve 0-1 mixed integer programming. It solves a series of reduced problems generated by exploiting information obtained through a series of linear programming relaxations and tries to improve lower and upper bounds on the optimal value. We proposed several enhancements to strengthen the performance of the ILPH: (i) New valid constraints are introduced to speed up the resolution of reduced problems; (ii) A local search is incorporated as an intensification process to reduce the gap between the upper and the lower bounds. Finally, QPSO is launched by using the k best solutions encountered in the ILPH process as an initial population. The proposed QPSO explores feasible and infeasible solutions. Experimental results obtained on a set of problem instances of the literature and other new harder ones clearly demonstrate the good performance of the proposed hybrid approach in solving the KSP.

Iterative Relaxation-Based Heuristics for the Multiple-choice Multidimensional Knapsack Problem

The development of efficient hybrid methods for solving hard optimization problems is not new in the operational research community. Some of these methods are based on the complete exploration of small neighbourhoods. In this paper, we apply iterative relaxation-based heuristics that solves a series of small sub-problems generated by exploiting information obtained from a series of relaxations to the multiple---choice multidimensional knapsack problem. We also apply local search methods to improve the solutions generated by these algorithms. The method is evaluated on a set of problem instances from the literature, and compared to the results reached by both Cplex solver and an efficient column generation---based algorithm. The results of the method are encouraging with 9 new best lower bounds among 33 problem instances.

New hybrid matheuristics for solving the multidimensional knapsack problem

In this paper we propose new hybrid methods for solving the multidimensional knapsack problem. They can be viewed as matheuristics that combine mathematical programming with the variable neighbourhood decomposition search heuristic. In each iteration a relaxation of the problem is solved to guide the generation of the neighbourhoods. Then the problem is enriched with a pseudo-cut to produce a sequence of not only lower, but also upper bounds of the problem, so that integrality gap is reduced. The results obtained on two sets of the large scale multidimensional knapsack problem instances are comparable with the current state-of-the-art heuristics. Moreover, a few best known results are reported for some large, long-studied instances.

Scatter Search for the 0–1 Multidimensional Knapsack Problem

The evolutionary metaheuristic called scatter search has been applied successfully to optimization problems for several years. In this paper, we apply the scatter search technique to the well-known 0–1 multidimensional knapsack problem. We propose a new relaxation-based diversification generator, which produces an initial population with elite solutions. The computational results obtained for a set of classic and correlated instances clearly show that (1) this generator can also be used as a heuristic for solving the multidimensional knapsack problem; (2) using the population produced by our generator as a starting point for the scatter search algorithm leads to better performance. We also enhance the scatter search algorithm by integrating memory and by using adapted intensification phases. Overall, the results are interesting and competitive compared to other population-based algorithms, such as genetic algorithms.

Hybrid Variable Neighbourhood Decomposition Search for 0-1 Mixed Integer Programming Problem

Abstract In this paper we propose new hybrid heuristics for the 0-1 mixed integer programming problem, based on the variable neighbourhood decomposition search principle and on exploiting information obtained from a series of relaxations. In the case of a maximization problem, we add iteratively pseudo-cuts to the problem in order to produce a sequence of lower and upper bounds of the problem, so that integrality gap is reduced. We validate our approaches on the well-known 0-1 multidimensional knapsack problem, in which the general-purpose CPLEX MIP solver is used as a black box for solving subproblems generated during the search process. The results obtained with these methods are comparable with the current state-of-the-art heuristics on a set of large scale instances.