Social Optimal Location of Facilities with Fixed Servers,Stochastic Demand,and Congestion

We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk‐in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed‐integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.