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Consider a single-item, periodic review, infinite-horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. Orders can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting). Every T periods, one observes the current stock level and orders deliveries for the next T periods, thus incurring a fixed setup cost. The goal is to find a review period T and an ordering policy that minimize the long run expected average cost per period. Flynn and Garstka (Flynn, J., S. Garstka. 1990. A dynamic inventory model with periodic auditing. Opns. Res. 38 1089–1103.) characterize an optimal ordering policy when T is fixed and study a myopic policy whose cost is often close to the optimal cost. This paper covers the problem of selecting T. We prove an optimal review period T* exists, characterize its properties, and present methods for its computation. We also study an approximation to T* based on the myopic policy of our earlier paper and a crude but simple approximation expressing T* in terms of the two-thirds power of the model parameters. Analytic results (where the coefficient of variation of demand is small) and computational experiments suggest both approximations perform well when demands are normal.

Consider a single-item, periodic review, stationary inventory model with stochastic demands, proportional ordering costs, and convex holding and shortage costs, where shortages are backordered and Veinott's well known terminal condition holds. Orders can be scheduled for any period, but the actual inventory level is determined every T periods through an audit. This leads to a dynamic programming model where stage n contains periods (n − 1)T + 1 through nT. For both discounted and averaging criteria, a simple rule optimally describes the orders for the T periods of a stage as a function of the state (beginning inventory level) and the cumulative T-period order. The latter is optimally determined by a base stock policy with two base stock levels: one for the final stage, another for the rest. (The horizon may be finite or infinite.) Methods are presented for computing optimal policies, together with bounds on the costs of (suboptimal) myopic policies. Models with proportional costs and continuous demands are studied in detail. Computational experiments indicate that myopic policies perform quite well for such models. The selection of a best review period T is covered briefly. Applications of our model include just in time settings where audit decisions play a negligible role.

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