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 w j and a profit p j . 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.
Similar content being viewed by others
References
E. Balas and E. Zemel, “An algorithm for large zero-one knapsack problems,” Operations Research, vol. 28, pp. 1130–1154, 1980.
P. Chu and J.E. Beasley, “A genetic algorithm for the multidimensional knapsack problem,” Journal of Heuristics, vol. 4, pp. 63–86, 1998.
G.B. Dantzig, “Discrete variable extremum problems,” Operations Research, vol. 5, pp. 266–277, 1957.
D. Fayard and G. Plateau, “An algorithm for the solution of the 0–1 knapsack problem,” Computing, vol. 28, pp. 269–287, 1982.
A. Freville and G. Plateau, “The 0–1 bidimensional knapsack problem: Toward an efficient high-level primitive tool,” Journal of Heuristics, vol. 2, pp. 147–167, 1997.
P.C. Gilmore and R.E. Gomory, “The theory and computation of knapsack functions,” Operations Research, vol. 13, pp. 879–919, 1966.
M. Hifi, “Exact algorithms for large-scale unconstrained two and three staged cutting problems,” Computational Optimization and Applications, vol. 18, pp. 63–88, 2001.
E. Horowitz and S. Sahni, “Computing partitions with applications to the knapsack problem,” Journal of ACM, vol. 21, pp. 277–292, 1974.
S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, Chichester: England, 1990.
S. Martello and P. Toth, “A new algorithm for the 0–1 knapsack problem,” Management Science, vol. 34, pp. 633–644, 1988.
S. Martello and P. Toth, “An upper bound for the zero-one knapsack problem and a branch and bound algorithm,” European Journal of Operational Research, vol. 1, pp. 169–175, 1977.
S. Martello, D. Pisinger, and P. Toth, “New trends in exact algorithms for the 0–1 knapsack problem,” European Journal of Operational Research, vol. 123, pp. 325–332, 2000.
S. Martello, D. Pisinger, and P. Toth, “Dynamic Programming and strong bounds for the 0–1 Knapsack Problem,” Management Science, vol. 45, pp. 414–424, 1999.
R. Morabito and M. Arenales, “Performance of two heuristics for solving large scale two-dimensional guillotine cutting problems,” INFOR, vol. 33, pp. 145–155, 1995.
D. Pisinger, “An exact algorithm for large multiple knapsack problems,” European Journal of Operational Research, vol. 114, pp. 528–541, 1999.
D. Pisinger and P. Toth, “Knapsack Problems,” in D.-Z. Du and P. Pardalos (eds.), Handbook of Combinatorial Optimization, Kluwer Academic Publishers, vol. 1, 1998, pp. 299–428.
D. Pisinger, “Core problems in knapsack algorithms,” Operations Research, vol. 47, pp. 570–575, 1999.
S. Sadfi, Méthodes adaptatives et méthodes exactes pour des problèmes de knapsack linéaires et non linéaires, Thesis, LRI, Université d'Orsay, 1999.
J.M. Valéro de Carvalho and A.J. Ridrigues, “An LP-based approach to a two-stage cutting stock problem,” European Journal of Operational Research, vol. 84, pp. 580–589, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hifi, M., Mhalla, H. & Sadfi, S. Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem. J Comb Optim 10, 239–260 (2005). https://doi.org/10.1007/s10878-005-4105-5
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10878-005-4105-5