Convexity and Solutions of Stochastic Multidimensional Knapsack Problems with Probabilistic Constraints

In the multidimensional knapsack problem a set of items, each with a value and a multidimensional size, is given and we want to select a subset of them in such a way that the total value of the selected items is maximized while the total size satisfies some capacity constraint for each dimension. In this paper we assume that the sizes are independent random variables such that each size follows the same type of probability distribution, not necessarily with the same parameter. A joint probabilistic constraint is imposed on the capacity constraints and the objective function is the same as that of the underlying deterministic problem. We showed that the problem is convex, under some condition on the parameters, for special continuous and discrete distributions: gamma, normal, Poisson, and binomial, where the latter two discrete distribution functions are approximated by logconcave continuous distribution functions.