Mixed-Integer Linear Programming Approximations for the Stochastic Knapsack

We develop mathematical programming approximations to tackle the stochastic knapsack problem. In this problem, the decision maker considers items for which either weights or values, or both, are random. The aim is to select a subset of these items to be included into their knapsack. We study both static and dynamic variants of this problem: in the static setting, the decision about which items should be included in the knapsack is taken at the outset, before any random item value or weight is revealed; in the dynamic setting, items are received sequentially, and the decision about a particular item is made by taking into account previously observed values and weights. The knapsack has a given capacity, and if the total realised weight exceeds this capacity then a penalty cost is incurred for each unit of excess capacity utilised. The goal is to maximise the expected net profit. We tackle the case of normally distributed item weights and we show that our approach extends to the case in which item weights are correlated and follow a multivariate normal distribution. In addition, we show our approach represents an effective heuristic for the case in which item weights follow generic probability distributions. In an extensive computational study we demonstrate that our models are near-optimal and more scalable than other state-of-the-art approaches.