This paper considers inventory models of the order-quantity/order-point type, or (Q,r) models. In general, the control parameters (Q and r) depend on both the demand process and the replenishment lead time. Although many studies have treated lead time as constant, focusing solely on demand, a (Q,r) model with stochastic lead time could be a building block in Supply Chain Management. Variability in lead time between successive stages is often what disturbs supply chain coordination. In a two-stage system with a constant demand rate, we will concentrate on lead time as a random variable, and develop two probabilistic models. In the first, lead time T is exogeneous. Lead time is made endogenous in the second stochastic model through an ‚expediting factor’, the constant of proportionality between random variables – the expedited lead time and ordinary lead time. For expedited orders, shorter-than-average lead time can be obtained at a cost. Similarly, longer mean lead times result in a rebate for the customer. The second model thus has three decision variables. For each model, we show that the expected cost per unit time is jointly convex in the decision variables and obtain the global minimizer. Numerical examples are given. Sensitivity analyses are conducted with respect to the cost parameters, and suggestions are made for future research.
