The service system design problem aims to select the location and capacity of service facilities, and the assignment of customers to them in order to minimize the setup, access, and waiting time costs. We address the case when there is uncertainty about the demand for service, considering a service system that can be modeled as an independent network of M/M/1 queues. Robust optimization, with both budget and ball uncertainty sets, is used when the demand rate is unknown, but the arrival pattern can still be reasonably approximated as a Poisson process. Furthermore, we use distributionally robust optimization with a Wasserstein ambiguity set to address the case when the demand distribution is estimated from a limited sample. We are able to reformulate both the robust and distributionally robust problems as mixed-integer second-order cone programs that can be solved directly on commercial solvers. Extensive numerical experiments on benchmark test instances are conducted to compare the proposed approaches and to investigate the effect of problem size and parameters.
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