Service network design with equilibrium-driven demands

We study a service network design problem in which the network operator wishes to determine facility locations and sizes in order to satisfy the demand of the customers while balancing the cost of the system with a measure of quality-of-service faced by the customers. We assume customers choose the facilities that meet demand, in order to minimize their total cost, including costs associated with traveling and waiting. When having demand served at a facility, customers face a service delay that depends on the total usage (congestion) of the facility. The total cost of meeting a customer’s demand at a facility includes a facility-specific unit travel cost and a function of the service delay. When customers all minimize their own costs, the resulting distribution of customer demand to facilities is modeled as an equilibrium. This problem is motivated by several applications, including supplier selection in supply chain planning, preventive healthcare services planning, and shelter location-allocation in disaster management. We model the problem as a mixed-integer bilevel program that can be reformulated as a nonconvex mixed-integer nonlinear program. The reformulated problem is difficult to solve by general-purpose solvers. Hence, we propose a Lagrangian relaxation approach that finds a candidate feasible solution along with a lower bound that can be used to validate the solution quality. The computational results indicate that the method can efficiently find feasible solutions, along with bounds on their optimality gap, even for large instances.