Continuous-Review (s, S) Inventory Models with State-Dependent Leadtimes

Abstract

This paper describes the development of a continuous-review (s, S) inventory model with complete backordering and state-dependent, stochastic leadtimes. The demand on the inventory system is assumed to be a Poisson process, and the order-filling portion of the replenishment leadtime is allowed to depend on the number of outstanding orders (i.e., to be state dependent). The inventory state probabilities which are required in order to optimize the cost function with respect to s and S can be obtained via a queuing analysis on the order-filling process. Orders are assumed to arrive at a single-channel service facility, where the service may be state dependent in either one of two ways: (1) the instantaneous probabilities at an arbitrary point in time of an order being filled (i.e., a service completed) in an infinitesimal interval of length Δt is μ(m)Δt + o(Δt), where m is the number of outstanding orders (instantaneous state dependence); and (2) the order-filling time is an exponentially distributed random variable with distribution function Bn(t) = 1 − e−μ(n)t, where n is the number of outstanding orders just after the previous order has been filled (departure-point state dependence). The μ(·) used to illustrate the methodology is two state; that is, μ(·) = μ1 if only one order is outstanding and μ if more than one is outstanding. The appropriate queuing model describing the ordering process turns out to be a Poisson input, batch-service model with batch size Q = S − s. The instantaneous state-dependent model is analyzed using a Chapman-Kolmogorov approach, while the departure-point state-dependent model utilizes an imbedded-Markov-chain approach. The computational aspects of the models are illustrated for a specific set of values of the appropriate parameters.