Social optimal location of facilities with fixed servers, stochastic demand and congestion

We present two capacity choice scenarios for the socially optimal location of facilities with fixed servers, stochastic demand and congestion. Walk-in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems are motivating examples of such facilities. The choice of locations for such facilities influences not only distances for users traveling to the facilities but also user waiting times at the facilities. In contrast to most previous research, we explicitly embed both customer travel and delay costs in the objective function and solve the location-allocation problem as well as choose service capacities for each open facility simultaneously. The choice of capacity for a facility that is viewed as a queueing system could mean choosing a service rate for the ‘servers ’ (scenario 1) or choosing the number of servers (scenario 2). We are able to 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 widely available optimization software