Stochastic Knapsack

Problem Definition This problem deals with packing a maximum reward set of items into a knapsack of given capacity, when the item-sizes are random. The input is a collection of n items, where each item i ∈ [n] := {1, · · · , n} has reward r i ≥ 0 and size S i ≥ 0, and a knapsack capacity B ≥ 0. In the stochastic knapsack problem, all rewards are deterministic but the sizes are random. The random variables S i s are independent with known, arbitrary distributions. The actual size of an item is known only when it is placed into the knapsack. The objective is to add items sequentially (one by one) into the knapsack so as to maximize the expected reward of the items that fit into the knapsack. As usual, a subset T of items is said to fit into the knapsack if the total size i∈T S i is at most the knapsack capacity B. A feasible solution (or policy) to the stochastic knapsack problem is represented by a decision tree. Nodes in this decision tree denote the current " state " of the solution (i.e. previously added items and the residual knapsack capacity) as well as the new item to place into the knapsack at this state. Branches in the decision tree denote the random size instantiations of items placed into the knapsack. Such solutions are called adaptive policies, to emphasize the fact that the items being placed may depend on previously observed outcomes. More formally, an adaptive policy is given by a mapping π : 2 [n] × [0, B] → [n], where π(T, C) denotes the next item to place into the knapsack when some subset T ⊆ [n] of items has already been added, and C = B − i∈T S i is the residual knapsack capacity. The policy ends when the knapsack overflows (i.e. the