The bilevel knapsack problem with stochastic right-hand sides

We introduce the bilevel knapsack problem with stochastic right-hand sides, and provide necessary and sufficient conditions for the existence of an optimal solution. When the leader’s decisions can take only integer values, we present an equivalent two-stage stochastic programming reformulation with binary recourse. We develop a branch-and-cut algorithm for solving this reformulation, and a branch-and-backtrack algorithm for solving the scenario subproblems. Computational experiments indicate that our approach can solve large instances in a reasonable amount of time.