On a stochastic knapsack problem and generalizations

TLDR: A Monte Carlo approximation procedure is developed to solve SKPs with general distributions on the random returns and utilizes upper- and lower-bound estimators on the true optimal solution value in order to construct a confidence interval on the optimality gap of a candidate solution.

Abstract: We consider an integer stochastic knapsack problem (SKP) where the weight of each item is deterministic, but the vector of returns for the items is random with known distribution. The objective is to maximize the probability that a total return threshold is met or exceeded. We study several solution approaches. Exact procedures, based on dynamic programming (DP) and integer programming (IP), are developed for returns that are independent normal random variables with integral means and variances. Computation indicates that the DP is significantly faster than the most efficient algorithm to date. The IP is less efficient, but is applicable to more general stochastic IPs with independent normal returns. We also develop a Monte Carlo approximation procedure to solve SKPs with general distributions on the random returns. This method utilizes upper- and lower-bound estimators on the true optimal solution value in order to construct a confidence interval on the optimality gap of a candidate solution.