Article
The 01 bidimensional knapsack problem: Toward an efficient highlevel primitive tool
Université ParisNord
Journal of Heuristics (Impact Factor: 1.14). 08/1996; 2(2):147167. DOI: 10.1007/BF00247210 ABSTRACT
Efficient codes exist for exactly solving the 01 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.
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.

 "Variable fixing is all the more efficient as the quality of the bounds is better [15] [30] [18]. To get good bounds, many authors have studied dual solvings of (P ) using surrogate duality [17] [14], composite duality [25] [21] [22], lagrangean duality [14] [35], LP relaxation [2] [8] and other heuristics [9] [19]. "
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 onedimensional 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. 
 "This problem, called the multidimensional (or multi constraint) knapsack problem is reviewed in [2] [18]. The bidimensional knapsack problem, a special case where the number of constraints is limited to two knapsack constraints, is investigated in Freville and Plateau [5]. Most problems assume that the weight and profit parameters are deterministic constants. "
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ABSTRACT: In the binary single constraint Knapsack Problem, denoted KP, we are given a knapsack of fixed capacity c and a set of n items. Each item j, j = 1,...,n, has an associated size or weight wj and a profit pj. The goal is to determine whether or not item j, j = 1,...,n, should be included in the knapsack. The objective is to maximize the total profit without exceeding the capacity c of the knapsack. In this paper, we study the sensitivity of the optimum of the KP to perturbations of either the profit or the weight of an item. We give approximate and exact interval limits for both cases (profit and weight) and propose several polynomial time algorithms able to reach these interval limits. The performance of the proposed algorithms are evaluated on a large number of problem instances. 
 "Les références [5] [10] ne présentent cependant que des résultats sur des instances de petites tailles. Enfin, Fréville et Plateau ont proposé une heuristique, qui combine des calculs de coefficients duaux avec des fixations de variables, uné elimination de contraintes et une procédure d'´ enumération, pour résoudre efficacement des instancesà 2 contraintes [13] "
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ABSTRACT: We present, in this article, a hybrid approach for solving the 0{1 multidimensional knapsack problem (MKP). This approach combines linear programming and Tabu search. The resulting algo rithm improves on the best result on many wellknown hard bench marks.
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