Distributionally robust Weber problem with uncertain demand

Weber problem is an important model in facility location field, and it can be modeled as a stochastic problem when the future demand of customers is uncertain. By minimizing the maximal expectation of the objective on an ambiguity set, the distributionally robust optimization (DRO) can utilize the valuable information from historical data and thus it has become an attractive formulation for stochastic problems. In this paper, an extended moment-based DRO formulation and a polynomial-time algorithm are contributed to solving the Weber problem with uncertain demand. Specifically, by constructing a new ambiguity set, which is proved to contain the true distribution with high probability, an extended DRO formulation allowing positive semidefinite covariance matrix is first built for general stochastic problems. To obtain a more robust solution, the Weber problem is then reformulated into an equivalent stochastic variational inequality (SVI). Following the extended DRO formulation, a distributionally robust Weber problem (DRWP) is further developed with minimizing the expectation of a residual function of the SVI. The DRWP can be transformed to a semidefinite program (SDP) with an undesired exponential number of constraints. Through the adoption of a plane decomposition technique, a simple algorithm is proposed to solve the resulted SDP and obtain the optimal solution to DRWP in polynomial time. Some preliminary numerical results demonstrate the effectiveness of the proposed DRWP and algorithm.