Sanguineti, Marcello and Cello, Marco and Gnecco, Giorgio and Marchese, Mario
Optimality Conditions For A Nonlinear Stochastic Knapsack Problem.
In: AIRO 2012, September 4th7th, 2012, Vietri sul Mare, Italy
p. 16.
(2012)
Full text not available from this repository.
Abstract
We investigate a nonlinear stochastic knapsack problem with application in Call Admission Control (CAC) with two classes of users, preliminary studied in [1,2]. Among possible stochastic nonlinear generalizations [3.4] of the NPhard 0/1 knapsack problem, we consider the following model. One has a knapsack of capacity C and K classes of objects. The objects belonging to each class become available randomly. The interarrival times are
exponentiallydistributed with means depending on the class and on the state of the knapsack. The sojourn time of each object is independent of the others and described by a classdependent distribution. When included in the knapsack,
an object from class k generates revenue at a positive rate rk. The occupied portion of knapsack is given by a nonlinear function bk(nk), where, for k=1, . . . K, nk is the number of objects of class k currently inside. The objects can be inserted as long as the sum of their sizes does not exceed the capacity C.
The stochastic nonlinear 0,1programming problem consists in deciding either acceptance or rejection of the arriving objects in dependence of the current state
of the knapsack, in such a way to maximize the average revenue. The functions used to generate such decisions are called \policies". We focus on coordinateconvex policies. We provide an algorithm which generates all coordinateconvex policies satisfying three different necessary conditions for optimality. Then we derive exact expressions of the cardinalities of such three sets of policies. Finally, we give conditions under which these cardinalities
are significantly smaller than the cardinality of the set of all coordinateconvex policies.
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