Approximability of the Two-Stage Knapsack problem with discretely distributed weights

The knapsack problem is a widely studied combinatorial optimization problem. It consists in choosing among a set of items a subset such that the total weight of the chosen items respects a given weight restriction (the capacity of the knapsack) while the total reward of the chosen items is maximized. The most common applications arise in fields where some capacity has to be respected (storage, transport, packing, network optimization...) or where the decision maker has to handle limited resources (recourse allocation, cutting stock problems...). However, knapsack problems also serve as subproblems in less obvious fields of application such as cryptography or finance. As in many applications the decision maker has to face uncertainty in the involved parameters, more and more studies are made on various settings of the Stochastic Knapsack problem, where some of the parameters are assumed to be random (i.e. not exactly known in the moment the (pre-)decision has to be made). In this paper we restrict our study to the case where the weights are assumed to be random. Moreover, we assume that the decision can be made in two stages: A pre-decision is made while the item weights are still unknown, i.e. the decision maker assigns some items to the knapsack without knowing their exact weights. In this first stage we obtain a certain reward for the added items. Then, once the weights of all items have come to be known, we can make a corrective decision (second stage): If additional items are added, the reward obtained for these items is smaller than it would have been in the first stage. And if items are removed, a penalty has to be paid that is naturally strictly greater than the received first-stage reward. The objective is to maximize the first-stage reward plus the expected second-stage gain,