Optimal State-Dependent Pricing Policies for a Class of Stochastic Multi-Unit Rental Systems

This paper models a k-unit rental system with Poisson arrivals, exponential service times and no queue. Denoting the number of units that are out on rent as the state of the system, the state-dependent pricing model formalizes the intuitive notion that when most units are idle the rental rate should be low and when most units are out on rent the rental rate should be higher than the average. A computationally efficient algorithm based on a nonlinear programming formulation of the problem is provided for determining the optimal state-dependent rental rates (prices). The procedure ultimately reduces to the search on a single variable in an interval to determine the unique intersection point of a concave increasing function and a linear decreasing function. The algorithm takes, on the average, only about 1/2 second per problem on the IBM 360/65 (FORTRAN G Compiler). A discrete optimal control approach to the problem is shown to result in essentially the same procedure as the nonlinear programming formulation. Several properties of the optimal state-dependent prices are given. Comparisons of the optimal values of the objective function for the state-dependent and state-independent pricing policies show that the former is, on the average, only about 0.7% better than the latter which may explainpartly the reason why state-dependent pricing is not prevalent in many rental systems. Potential generalizations of the model are discussed.