Simultaneous price, location, and capacity decisions on a line of time-sensitive customers

Consider a monopolist who sells a single product to time-sensitive customers located on a line segment. Customers send their orders to the nearest distribution facility, where the firm processes (customizes) these orders on a first-come, first-served basis before delivering them. We examine how the monopolist would locate its facilities, set their capacities, and price the product offered to maximize profits. We explicitly model customers' waiting costs due to both shipping lead times and queueing congestion delays and allow each customer to self-select whether she orders or not, based on her reservation price. We first analyze the single-facility problem and derive a number of interesting insights regarding the optimal solution. We show, for instance, that the optimal capacity relates to the square root of the customer volume and that the optimal price relates additively to the capacity and transportation delay costs. We also compare our solutions to a similar problem without congestion effects. We then utilize our single-facility results to treat the multi-facility problem. We characterize the optimal policy for serving a fixed interval of customers from multiple facilities when customers are uniformly distributed on a line. We also show how as the length of the customer interval increases, the optimal policy relates to the single-facility problem of maximizing expected profit per unit distance.