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2017•Pesquisa OperacionalOpen access

STOCHASTIC KNAPSACK PROBLEM: APPLICATION TO TRANSPORTATION PROBLEMS

Stefanie Kosuch, Marc Letournel, Abdel Lisser

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Abstract

In this paper, we study the stochastic knapsack problem with expectation constraint. We solve the relaxed version of this problem using a stochastic gradient algorithm in order to provide upper bounds for a branch-and-bound framework. Two approaches to estimate the needed gradients are studied, one based on Integration by Parts and one using Finite Differences. The Finite Differences method is a robust and simple approach with efficient results despite the fact that estimated gradients are biased, meanwhile Integration by Parts is based upon more theoretical analysis and permits to enlarge the field of applications. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.

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What this paper is about

In this paper, we study the stochastic knapsack problem with expectation constraint. We solve the relaxed version of this problem using a stochastic gradient algorithm in order to provide upper bounds for a branch-and-bound framework. Two approaches to estimate the needed gradients are studied, one based on Integration by Parts and one using Finite Differences. The Finite Differences method is a robust and simple approach with efficient results despite the fact that estimated gradients are biased, meanwhile Integration by Parts is based upon more theoretical analysis and permits to enlarge the field of applications. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.

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Available abstract

In this paper, we study the stochastic knapsack problem with expectation constraint. We solve the relaxed version of this problem using a stochastic gradient algorithm in order to provide upper bounds for a branch-and-bound framework. Two approaches to estimate the needed gradients are studied, one based on Integration by Parts and one using Finite Differences. The Finite Differences method is a robust and simple approach with efficient results despite the fact that estimated gradients are biased, meanwhile Integration by Parts is based upon more theoretical analysis and permits to enlarge the field of applications. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.

Key concepts: Knapsack problem, Mathematical optimization, Constraint (computer-aided design), Set (abstract data type), Mathematics, Upper and lower bounds, Computer science, Applied mathematics

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