Order-Level Inventory Systems with Independent Stochastic Leadtimes - Zalkind, David (Management Science, 1978)
Inventory systems where leadtimes are stochastic and independent, so that orders are not restricted to arrive in the same sequence in which they were placed, can arise if the vendor with whom orders are placed is unreliable in processing them or if orders for the same item are placed with different vendors at different times. We present a procedure for finding the optimal order-level Z (stock on hand plus on order) for a pre-determined scheduling-period t (time interval between decisions about placing orders), that is, a tZ policy. Backordering of shortages is assumed. Most results are derived when both the demand distribution and the leadtime distribution are discrete and bounded. The main concept employed in the derivation is the use of the distribution of the number of outstanding orders to find the distribution of the amount of stock on order. It is then shown that for tZ policies, the solution can be interpreted as the weighted average of constant leadtime systems. Further results that may be helpful in constructing heuristic decision rules are given. The nonoptimality of the sZ policy (i.e., order to Z whenever inventory falls to s) for independent stochastic leadtime systems is also discussed.