Heuristic and exact algorithms for the max–min optimization of the multi-scenario knapsack problem

We are concerned with a variation of the standard 0–1 knapsack problem, where the values of items differ under possible S scenarios. By applying the ‘pegging test’ the ordinary knapsack problem can be reduced, often significantly, in size; but this is not directly applicable to our problem. We introduce a kind of surrogate relaxation to derive upper and lower bounds quickly, and show that, with this preprocessing, the similar pegging test can be applied to our problem. The reduced problem can be solved to optimality by the branch-and-bound algorithm. Here, we make use of the surrogate variables to evaluate the upper bound at each branch-and-bound node very quickly by solving a continuous knapsack problem. Through numerical experiments we show that the developed method finds upper and lower bounds of very high accuracy in a few seconds, and solves larger instances to optimality faster than the previously published algorithms.