A stochastic linear knapsack problem

Abstract

In this article we consider a stochastic version of the continuous linear knapsack problem, i.e., a model with a random linear constraint, and provide an efficient algorithm for solving it. An original problem Po is first transformed into a deterministic equivalent problem Po. Furthermore, by a change of variables, Po is transformed into P. Then, in order to solve P, a subproblem P(μ.) with positive parameter μ is introduced, and a close relation between P and P(μ) is clarified. Furthermore, an auxiliary problem PR(μ) of P(μ) with positive parameter R is introduced, and a relation between PR(μ) and P(μ) is also clarified. From these relations, a direct relation connecting PR(μ) with P is derived. An efficient algorithm based on this relation for solving P is proposed. It is shown that time complexity of the algorithm is O(n log n), where n is the number of items. Finally, some further research problems are discussed.