The market area problem with two plants

This paper considers a two-plant production and distribution problem where the production cost of each plant depend on the amount produced. Demand is price inelastic and uniformly distributed on the plane. Transport costs are directly proportional to the straight-line travel distance from each plant. The constant of proportionality may differ between plants. Two cases are considered: (i) the two plants are owned by a single supplier, and (ii) the two plants are operated by competing firms. The profit-maximizing solution, when convexity assumptions are imposed on the production cost functions, determines the amount produced in each plant and defines the marketing region for each plant. It is shown that the solution for the single supplier case is identical to the solution in the competitor case and that the solution for each case is unique.