Stochastic binary problems with simple penalties for capacity constraints violations

This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly NP-hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and arbitrary random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. ? This research is supported by an Actions de Recherche Concertées (ARC) project of the Communautée française de Belgique. Michael Poss is a research fellow of the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA). Martine Labbé also acknowledges support from “Ministerio de Ciencia e Innovacion” of Spain through the research project MTM2009-14039-C06. B. Fortz Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles, Brussels, Belgium E-mail: [email protected] M. Labbé Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles, Brussels, Belgium. E-mail: [email protected] F. Louveaux Department of Business Administration, University of Namur (FUNDP), Namur, Belgium. E-mail: [email protected] M. Poss Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles, Brussels, Belgium. E-mail: [email protected]