A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards

Abstract

In this paper, we extend the knapsack problem to include more realistic situations by treating the rewards (or values) associated with each item included in the solution as random variables with distributions that are known (or may be estimated) rather than known integers, as in the usual formulation. We propose a dynamic programming solution methodology where the usual real-valued return function is replaced by a preference ordering on the distributions of returns from the items selected. In addition to extending previous solutions to the knapsack problem, we demonstrate the selection of a preference ordering criterion and illustrate the conditions required of the ordering to guarantee optimality of the procedure. A sample problem is shown to demonstrate the algorithm, and results of computational experience with 459 problems of varying sizes and parameters are presented.