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Operations Research Letters 15 (1994) 85-93

Understanding the lead-time effects in stochastic inventory systems with discounted costs
Jing-Sheng Song
Graduate School ~?fManagement, University of California. lrvine, CA, USA

(Received 12 January 1993; revised 1 October 1993)

Abstract

This paper explores qualitative effects of lead-time uncertainty in a basiccontinuous-time single-item inventory model with Poisson demand and stochastic lead times. The objective is to minimize the infinite-horizon expected total discounted cost. Order costs are linear, so a base-stock policy is optimal. Traditionally and intuitively, people believe that a (stochastically) longer lead time results in a larger lead-time demand and therefore that the system should have a higher optimalbase-stock level. In fact, Song (1994) shows that this is true for models with the average cost criterion. Here, we show that this intuition is not always validated for models with the discounted cost criterion. We identify certain systems in which the lead times do have a monotonic impact on the optimal base-stock levels. Examples include systems with constant, geometric, gamma, or uniformlead-time distributions. In addition, we find that a shorter lead time does not necessarily result in a smaller optimal system cost.
Key words: Inventory; Discounted cost criterion; Stochastic lead time; Base-stock policy; Stochastic ordering; Laplace

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1. Introduction

We investigate qualitative lead-time effects on optimal inventory decisions and system performance in a basic dynamicinventory model with stochastic lead times and demands. Traditionally and intuitively, people believe that a longer lead time results in a larger lead-time demand and therefore that the system should have a higher optimal base-stock level. Our study reveals this intuition is not always true when the objective is to minimize the infinite-horizon total discounted cost. We identify certain systemsin which the lead times do have a monotonic impact on the optimal base-stock levels. In addition, we find that a shorter lead time does not necessarily result in a smaller optimal system cost. These results, we believe, will enhance our understanding of this basic inventory model, which in turn may help management gain insights into the design of inventory systems with stochastic lead times. Hereis the basic model: Demands form a Poisson process. Replenishment orders are placed with an outside supplier and are delivered after a stochastic lead time. We assume an exogenous, sequential supply process, as described in Zipkin [1] and Svoronos and Zipkin [2]. Here, "exogenous" means that the evolution of the supply system is independent of our demands and orders. This is an appropriate model0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0 1 6 7 - 6 3 7 7 ( 9 3 ) E 0 0 8 7 - 8

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J.-S. Song / Operations Research Letters 15 (1994) 85 93

when, for example, the supplier's operation is large enough relative to ours that our orders contribute little to the fluctuations in the supplier's total workload that determines the lead times of our orders."Sequential" means that we can think of the supplier's production and distribution system as a processing network which operates sequentially, so that orders never cross in time. Thus, the lead times in our model are dependent except in the constant-lead-time case. Also, the lead-time random variable represents the marginal lead-time distribution. All stockouts are backordered. We assume that the ordercosts are proportional to the quantity ordered and that there are linear holding and penalty costs for the inventory on hand and on backlog, respectively. We seek to minimize the infinite-horizon expected total discounted cost. It is well known that a stationary base-stock policy is optimal. (A base-stock policy is a policy described by a single critical number, called the base-stock level; an...