Knapsack problem with probability constraints
- 383 Downloads
This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below.
KeywordsSemidefinite programming Knapsack problem Stochastic optimization Recourse
Unable to display preview. Download preview PDF.
- 1.Cohn, A.M., Barnhart, C.: The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. In: Proceedings from TRISTAN III. San Juan, Puerto Rico (1998)Google Scholar
- 2.D’Atri G., Di Rende A.: Probabilistic analysis of knapsack-type problems. Methods Oper. Res. 40, 279–282 (1980)Google Scholar
- 3.Dean, B.C., Goemans, M.X., Vondrak, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. In: FOCS ’04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 208–217. IEEE Computer Society, Washington, DC, USA (2004)Google Scholar
- 4.Fortet, R.: L’algèbre de Boole et ses applications en recherche opérationelle (1959)Google Scholar
- 5.Garey M.R., Johnson D.S.: Computer and Intractability. W. H. Freeman and Company, New York (1979)Google Scholar
- 8.Helmberg C., Rendl F.: Solving quadratic (0,1)- problems by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998)Google Scholar
- 10.Helmberg, C., Poljak, S., Rendl, F., Wolkowicz, H.: Combining semidefinite and polyhedral relaxations for integer programs. In IPCO, pp. 124–134 (1995)Google Scholar
- 17.Rendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Technical report, University of Klagenfurt, Universitaetsstrasse 65–67, Austria (2003)Google Scholar