Determining Market Areas Captured by Competitive Facilities: A Continuous Equilibrium Modeling Approach

Many existing models concerning locations and market areas of competitive facilities assume that customers patronize a facility based on distance to that facility, or perhaps on a function of distances between the customer and the different facilities available. Customers are generally assumed to be located at certain discrete demand points in a two‐dimensional space, or continuously distributed over a one‐dimensional line segment. In this paper these assumptions are relaxed by employment of a continuum optimization model to characterize the equilibrium choice behavior of customers for a given set of competitive facilities over a heterogeneous two‐dimensional space. Customers are assumed to be scattered continuously over the space and each customer is assumed to choose a facility based on both congested travel time to the facility and on the attributes of the facility. The model is formulated as a calculus of variations problem and its optimality conditions are shown to be equivalent to the spatial customer‐choice equilibrium conditions. An efficient numerical method using finite element technique is proposed and illustrated with a numerical example.