Mixed-Integer Linear Programming Approximations for the Stochastic Knapsack

We develop a novel mathematical programming approximation framework 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 seamlessly 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.

💡 Research Summary

The paper introduces a comprehensive mixed‑integer linear programming (MILP) approximation framework for both static and dynamic versions of the stochastic knapsack problem (SSKP and DSKP). Items have random weights (and possibly random revenues per unit weight), and the knapsack incurs a penalty for any capacity overrun while receiving a salvage value for unused capacity. The objective is to maximize expected net profit, which consists of expected revenue, minus expected overrun cost, plus expected salvage.

For the static problem, the authors rewrite the objective using the first‑order loss function L(y, ω)=E