On One-For-One-Ordering Inventory Policies with State-Dependent Leadtimes

Abstract

This paper describes several one-for-one-ordering [continuous-review ( s , S ) policies, where s = S − 1] inventory models in which the time required for order replenishment, or leadtime, depends on the number of orders outstanding. Demand is assumed to be a Poisson random variable with a constant mean λ, and leadtime is assumed to be state dependent in either one of two ways: (1) the portion of the leadtime that corresponds to the actual filling of orders (i.e., the service time) is an exponentially-distributed random variable with distribution function B Tm ( t ) = 1 − e −μ( m ) t , where m is the number of outstanding orders just after the previous order has been filled (viz., an imbedded-Markov-chain approach); and (2) the instantaneous probability at an arbitrary point in time of an order being filled (i.e., a service completed) in an infinitesimal interval of time Δ t is μ( n )Δ t + 0(Δ t ), where n is the number of outstanding orders (viz., a birth-death approach). Several models are investigated for each type. The orders placed are assumed to go into a single-server queue, and queuing theory results are used to obtain the expected inventory costs as a function of S in order to obtain an optimal value of S . Some of the models are expanded to include costs dependent on a service-rate parameter and are then optimized with respect to both S and the service-rate parameter.