A Stochastic Knapsack Problem with Nonlinear Capacity Constraint

There exist various generalizations and stochastic variants of the NP-hard 0/1 knapsack problem [1,2]. The following model is considered here. A knapsack of capacity C is given, together with K classes of objects. The stochastic nature come into play since, in contrast to the classical knapsack, the objects belonging to each class become available randomly. The inter-arrival times are exponentially-distributed with means depending on the class and on the state of the knapsack. Each object has a sojourn time independent from the sojourn times of the other objects and described by a class-dependent distribution. The other difference with respect to the classical model consists is the following generalization. For k = 1;K, let nk be the number of objects of class k that are currently inside the knapsack; then, the portion of knapsack occupied by them is given by a nonlinear function bk(nk). When included in the knapsack, an object from class k generates revenue at a positive rate rk. The objects can be placed into the knapsack as long as the sum of their sizes does not exceed the capacity C. The problem consists in finding a policy that maximizes the average revenue, by accepting or rejecting the arriving objects in dependence of the current state of the knapsack. A-priori knowledge of structural properties of the (unknown) optimal policies is useful to find satisfactorily accurate suboptimal policies. The family of coordinate-convex policies is considered here. In this context, structural properties of the optimal policies are investigated. New insights into a criterion proposed in [3] to improve coordinate-convex policies are discussed and the greedy presented in [5] is further developed. Applications in Call Admission Control (CAC) for telecommunication networks are discussed. In this case, the objects are requests of connections coming from K different classes of users, each with an associated bandwidth requirement and a distribution of its duration