Competitive Facility Location with Selfish Users and Queues

AbstractIn an optimization process, firms or institutions frequently overlook the impact of their decisions on a population of users, thus leading to a poor performance of the system. This flaw can be corrected by explicitly embedding into the mathematical framework the rational reaction of the users. However, the resulting complexity yields formulations that cannot be solved by commercial software, hence the need for mathematical tools adapted to this situation. In our work, a step in that direction was achieved for a facility location problem where users individually maximize their utility, taking into account travel time, queueing delays, and the probability of accessing the service provided at one of the facilities. The techniques that we developed are not specific to this model (i.e., they can be adapted to other applications sharing similar properties).In a competitive environment, we consider the problem faced by a service firm that makes decisions with respect to both the location and service levels of its facilities, taking into account that users patronize the facility that maximizes their individual utility, expressed as the sum of travel time, queueing delay, and a random term. This situation can be modelled as a bilevel program that involves discrete and continuous variables as well as linear and nonlinear (convex and nonconvex) functions. We design for its solution an algorithm based on piecewise linear approximation as well as a matheuristic that exploits the very structure of the problem.The online appendix is available at https://doi.org/10.1287/opre.2018.1781.