Single facility location on a network under mill and delivered pricing

Firms normally use either a mill price or a delivered price policy, depending on market conditions (type of good, transportation way, customers location, costs, etc). In this paper, the problem of selecting the best location for an entering firm in competition with some pre-existing firms, under each price policy, is studied on a network for the first time. With mill pricing, an equilibrium in price rarely exists and it is assumed that all competing firms set a common mill price for all customers. With delivered pricing, there exists a Nash equilibrium in price and it is assumed that the equilibrium price in each area is offered to the customers in that area. In both cases, we consider that customers buy from the cheapest facility and the same rules are used for tie breaking in the lowest cost. While the profit maximization problem for the entering firm always has optimal solutions under mill pricing, this problem might not have an optimal solution under delivered pricing. We show some discretization results and give procedures to find the full set of optimal, or ε-optimal, solutions to the problem under the two price policies. A comparison of results with the two price policies is given by using an illustrative example.