Abstract
Two stochastic knapsack problem (SKP) models are considered: the static broken knapsack problem (BKP) and the SKP with simple recourse and penalty cost problem. For both models, we assume: the knapsack has a constant capacity; there are n types of items and each type has an infinite supply; a type i item has a deterministic reward vi and a random weight with known distribution Fi. Both models have the same objective to maximize expected total return by finding the optimal combination of items, that is, quantities of items of each type to be put in knapsack. The difference between the two models is: if knapsack is broken when total weights of items put in knapsack exceed the knapsack's capacity, for the static BKP model, all existing rewards would be wiped out, while for the latter model, we could still keep the existing rewards in knapsack but have to pay a fixed penalty plus a variant cost proportional to the overcapacity amount. This paper also discusses the special case when knapsack has an exponentially distributed capacity.
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