Dynamic pricing with duration consideration in rental business

We consider the basic problem of a service firm that offers a product for rental to its customers for a fixed rental duration. Many firms prefer a predetermined rental duration to simplify the rental/return process and offer a fixed duration, such as two days, to appeal to most of its potential clients. As customer demand is normally stochastic and dependent on the rental rate, the firm has to determine not only its initial stock level but also the rental rate for its product to maximize its total profit. We present a two-stage discrete-time model to examine the rental problem. In stage one, the total time horizon is divided into N days and returned units are added to the stock at the beginning of each day. In stage two, the first day is divided into K time periods, each period is a small time interval Δt. At each time period, we dynamically adjust the rental rate according to the stock level inventory on hand and expected rental demand. We obtain the optimal pricing policy for this problem and show the existence and uniqueness of the optimal price policy. We demonstrate the expected optimal revenue is concave in the inventory level through numerical analysis. Meanwhile, we analyze the effects of the potential market size and price sensitive index by numerical studies. We find the optimal solution of the deterministic demand and show the optimal revenue is concave in the inventory on hand. Finally we show the deterministic case is the upper bound of stochastic problem and the periodical fixed price policy is asymptotical optimal as the inventory and the customer arrival go infinity.