A branch-and-Benders-cut algorithm for a bi-objective stochastic facility location problem

Sophie N Parragh,Fabien Tricoire,Walter J Gutjahr

doi:10.1007/s00291-020-00616-7

Abstract

In many real-world optimization problems, more than one objective plays a role and input parameters are subject to uncertainty. In this paper, motivated by applications in disaster relief and public facility location, we model and solve a bi-objective stochastic facility location problem. The considered objectives are cost and covered demand, where the demand at the different population centers is uncertain but its probability distribution is known. The latter information is used to produce a set of scenarios. In order to solve the underlying optimization problem, we apply a Benders’ type decomposition approach which is known as the L-shaped method for stochastic programming and we embed it into a recently developed branch-and-bound framework for bi-objective integer optimization. We analyze and compare different cut generation schemes and we show how they affect lower bound set computations, so as to identify the best performing approach. Finally, we compare the branch-and-Benders-cut approach to a straight-forward branch-and-bound implementation based on the deterministic equivalent formulation.

Highlights

Facility location problems play an important role in long-term public infrastructure planning
We describe how we set up the master linear program (LP) to use the L-shaped method for solving a weighted-sum problem inside of the lower bound computation scheme
We have defined a bi-objective facility location problem (BOSFLP), which considers both a deterministic objective and a stochastic one, where the value of the second objective is evaluated by sample average approximation

Summary

Introduction

Facility location problems play an important role in long-term public infrastructure planning. Prominent examples concern the location of fire departments, schools, post offices, or hospitals They are relevant in public (or former public) infrastructure planning decisions in “regular” planning situations: they are of concern in the context of emergency planning, e.g., relief goods distribution in the aftermath of a disaster or preparation for slow onset disasters such as droughts. Since facility location decisions are usually long-term investments, the uncertainty involved in the demand figures should already be taken into account at the planning stage. This implies that decision makers face a trade-off between clientoriented and cost-oriented goals Instead of combining these two usually conflicting measures into one objective function, it is advisable to elucidate their trade-off relationship. We incorporate stochastic information on possible realizations of the considered demand figures in the form of scenarios sampled from probability distributions

Objectives

Methods

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Conclusion