Stochastic Quadratic Knapsack with Recourse

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. Four different model formulations are proposed, based on the introduction of probability constraints. The first one is a two-stage quadratic knapsack with recourse (2QKR), and serves as the base on which we develop the three other models. The second one is a 2QKR with a probability constraint on the capacity of the knapsack on the first stage. The third one is a 2QKR where we introduce a probability constraint on the capacity of the knapsack in the second stage. Finally, the last model is also a 2QKR which uses probability constraints on the capacity constraints of both stages. 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.