Optimal continuous order quantity (s,S) policies; the 45-degrees algorithm

The most recent optimization algorithm for (s,S) order policies with continuous demand was developed by Federgruen and Zipkin (1985). This was also the first efficient algorithm, which uses policy iteration instead of discretization. Zheng and Federgruen (1991) developed an even more efficient algorithm for computing discrete order quantity (s,S) inventory policies. Since the continuous case prohibits enumeration, this algorithm does not apply to continuous order quantity systems. In this paper an efficient algorithm for continuous order quantity (s,S) policies is developed. A marginal cost approach is used for determining the optimal s. Furthermore, we construct two aid functions (generated by the optimality conditions for s and S), and exploiting their special properties a simple and efficient algorithm is obtained. The algorithm converges monotonically, such that at every iteration a policy improvement is obtained. Since every iteration finds a local minimum of the expected average cost, the number of iterations is at most N, where N < infinity represents the number of local minimums. The algorithm also applies to discrete order quantity systems, in which case it basically reduces to the algorithm of Zheng and Federgruen (with the difference that in general our algorithm will take larger than unit steps, since we are not using enumeration).