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We study a joint facility location and budget planning problem in a competitive market under random utility maximization (RUM) models. The objective is to locate new facilities and make decisions on the budgets (or costs) to spend on the new facilities, aiming to maximize an expected captured customer demand, assuming that customers choose a facility among all available facilities according to a RUM model. We examine two RUM frameworks in the discrete choice literature, namely, the additive and multiplicative RUM. While the former has been widely used in facility location problems, we are the first to explore the latter in the context. We show that, under the additive RUM framework, the resulting cost optimization problem becomes highly non-convex and may have several local optimum solutions. In contrast, the use of the multiplicative RUM brings several advantages to the competitive facility location problem. More precisely, we show that the cost optimization problem under the multiplicative RUM can be solved efficiently by a general convex optimization solver, or can be reformulated as a conic quadratic program and handled by a conic solver available in some optimization tools such as CPLEX or GUROBI. Furthermore, we consider a joint location and cost optimization problem and propose three approaches to solve the problem, namely, an equivalent conic reformulation, a multi-cut outer-approximation algorithm, and a local search heuristic. We provide numerical experiments based on synthetic instances of various sizes to evaluate the performances of the proposed algorithms in solving the cost optimization and joint location and cost optimization problems.

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