Probabilistic Set Covering Location Problem in Congested Networks

This paper focuses on designing a facility network, taking into account that the system may be congested. The objective is to minimize the overall fixed and service capacity costs, subject to the constraints that for any demand the disutility from travel and waiting times (measured as the weighted sum of the travel time from a demand to the facility serving that demand and the average waiting time at the facility) cannot exceed a predefined maximum allowed level (measured in units of time). We develop an analytical framework for the problem that determines the optimal set of facilities and assigns each facility a service rate (service capacity). In our setting, the consumers would like to maximize their utility (minimize their disutility) when choosing which facility to patronize. Therefore, the eventual choice of facilities is a user-equilibrium problem, where at equilibrium, consumers do not have any incentive to change their choices. The problem is formulated as a nonlinear mixed-integer program. We show how to linearize the nonlinear constraints and solve instead a mixed-integer linear problem, which can be solved efficiently.