The stochastic knapsack problem

The problem of packing a knapsack of integer volume F with objects from K different classes to maximize profit is studied. Optimization is carried out over the class of coordinate convex policies. For the case of K=2, it is shown for a wide range of parameters that the optimal control is of the threshold type. In the case of Poisson arrivals and of knapsack and object volumes being integer multiples of each other, it is shown that the optimal policy is always of the double-threshold type. An O(F) algorithm to determine the revenue of threshold policies is also given. For the general case of K classes, the problem of the optimal static control where for each class a portion of the knapsack is dedicated is considered. An efficient finite-stage dynamic programming algorithm for locating the optimal static control is presented. Furthermore, variants of the optimal static control which allow some sharing among classes are also discussed