A Variant of Knapsack Problem Efficient for Benders Decomposition

We introduce a variant of knapsack problem which deals with divisible items and combinatorial discounts. Here, a combinatorial discount means that if we buy some combination of items with certain amounts, we receive a discount. If some items are left over for one discount and another discount is available, both discounts will be applied. In this way, multiple discounts could be applied. We consider to apply Benders decomposition technique for this problem, and prove that we only need to consider a subset of the cone used in Benders decomposition. This implies that we can apply Benders decomposition efficiently to this problem. Results of preliminary numerical experiments are also shown.