Article

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

Université de Valenciennes; Université Paris-Nord

Journal of Heuristics (Impact Factor: 1.47). 08/1996; 2(2):147-167. DOI: 10.1007/BF00247210 - [Show abstract] [Hide abstract]

**ABSTRACT:**Knapsack Sharing Problem (KSP) is an NP-Hard combinatorial optimization problem, admitted in numerous real world applications. In the KSP, we have a knapsack of capacity $c$ and a set of $n$ objects, namely $N$, where each object $j$, $j = 1,\ldots, n$, is associated with a profit $p_j$ and a weight $w_j $. The set of objects $N$ is composed of m different classes of objects $J_i , $i = 1,\ldots,m$, and $N = \cup_{i=1}^m J_i$ . The aim is to determine a subset of objects to be included in the knapsack that realizes a max-min value over all classes. In this article, we solve the KSP using an approximate solution method based upon tabu search. First, we describe a simple local search in which a depth parameter and a tabu list are used. Next, we enhance the algorithm by introducing some intensifying and diversifying strategies. The two versions of the algorithm yield satisfactory results within reasonable computational time. Extensive computational testing on problem instances taken from the literature shows the effectiveness of the proposed approach.Computational Optimization and Applications 10/2002; · 1.28 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we propose an exact algorithm for the knapsack sharing problem. The proposed algorithm seems quite efficient in the sense that it solves quickly some large problem instances. The problem is decomposed into a series of single constraint knapsack problems; and by applying the dynamic programming and another strategy, we solve optimally the original problem. The performance of the exact algorithm is evaluated on a set of medium and large problem instances (a total of 240 problem instances). This algorithm is parallelizable and this is one of its important feature.Journal of Combinatorial Optimization 01/2002; 6(1):35-54. · 0.59 Impact Factor -
##### Article: Lagrangean heuristics combined with reoptimization for the 0–1 bidimensional knapsack problem

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**ABSTRACT:**First, this paper deals with lagrangean heuristics for the 0–1 bidimensional knapsack problem. A projected subgradient algorithm is performed for solving a lagrangean dual of the problem, to improve the convergence of the classical subgradient algorithm. Secondly, a local search is introduced to improve the lower bound on the value of the biknapsack produced by lagrangean heuristics. Thirdly, a variable fixing phase is embedded in the process. Finally, the sequence of 0–1 one-dimensional knapsack instances obtained from the algorithm are solved by using reoptimization techniques in order to reduce the total computational time effort. Computational results are presented.Discrete Applied Mathematics. 01/2006;

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