An adaptive stochastic knapsack problem

We consider a stochastic knapsack problem in which the event of overflow results in the problem ending with zero return. We assume that there are n types of items available where each type has infinite supply. An item has an exponentially distributed random weight with a known mean depending on its type and the item’s value is proportional to its weight with a given factor depending on the item’s type. We have to make a decision on each stage whether to stop, or continue to put an item of a selected type in the knapsack. An item’s weight is learned when placed to the knapsack. The objective of this problem is to find a policy that maximizes the expected total values. Using the framework of dynamic programming, the optimal policy is found when n=2 and a heuristic policy is suggested for n>2.