Stochastic Knapsack Problem: Application to Transportation Problems

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.

A Hybridized Multi-Objective Memetic Algorithm for the Multi-Objective Stochastic Quadratic Knapsack Problem

“…These bounds are used in the branch-and-bound framework. Kosuch et al (2017) studied the stochastic knapsack problem with expectation constraints and proposed to solve the problem with the relaxed version of this problem using a stochastic gradient algorithm to provide upper bounds for a branch-and-bound framework. Blado & Toriello (2021) considered a two-stage stochastic multiple knapsack problem together with a set of possible disturbances with known probability of occurrence and proposed two branch-and-price approaches to solve it.…”