MANAGEMENt SCIENCE Yolo34. No.6. June 1988 Printed in u.s.A. A CONTINUOUS REVIEW MODEL FOR AN INVENTORY SYSTEM WITH TWO SUPPLY MODES* KAMRAN MOINZADEH ANDSTEVEN NAHMIAS Department of Management Science, University of Washington, Seattle, Washington 98195 Leavey School of Business, Santa Clara University, Santa Clara, California 95053 In this paper we develop an approximate model of an inventory control system in which there exist two options for resupply, with one having a shorter lead time. Because the optimal policy appears to be extremely complex, we consider a reasonable extension of the standard (Q, R) policy to allow for two different lot sizes QI and Q2, and two different reorder levels, R I and R2. Expressions for the expected on hand inventory and the expected backorders are developed and a procedure for determining the policy parameters is given. The model is validated by simula- tion, and calculations are included which compare the average annual cost with and without emergency ordering. (INVENTORY PRODUCTION; OPERATING CHARACTERISTICS; LEADTIME POL- ICY; ORDERING POLICY; STOCHASTIC MODELS) 1. Introduction I I I I I I I i I I I I I I In many large-scale inventory systems, options exist for expediting orders through the use of alternative modes of resupply. For example, orders that are normally shipped by sea or land, which have a relatively long lead time associated with resupply, could be shipped by air under emergency conditions at a higher cost but in a shorter lead time. Various versions of this problem have been studied. Barankin (1961) developed a model for a single planning period only and the two options available are a one-period and a zero-period lag only. Daniel (1962) treats an extension of this model to multiple planning periods and derives the form of an optimal policy assuming that the emer- gency order is bounded by a given constant. Neuts (1964) derives essentially the same results for a similar model. Fukuda (1964) extends Daniel's analysis to allow both expedited and regular orders to be placed simultaneously and includes a positive set-up cost for ordering. However, he still restricts attention to the case where the normal lead time is one period and the expedited lead time is zero periods. The most general analysis of this type is due to Whittmore and Saunders (1977) who construct a multi-period dynamic model and allow both the long and short lead times to be of arbitrary lengths. Unfortunately, the form of the optimal policy for their model is extremely complex, requiring the solution of a multi-state dynamic program. Furthermore, fixed costs of ordering are not present. They are able to obtain explicit results only for the case where the long and short lead times differ by one period only. Bulinskaya (1964), Veinott (1966), and Wright (1968) obtained similar results for this case. Related, but somewhat different models, have been considered by Allen and D'Esopo (1968), Gross and Soriano (1972), Hadley and Whitin (1962), and Rosenshine and Obee (1976). However, none of these studies, nor the ones above, consider the problem at the level of generality that we do. The goal of this study is to derive approximately optimal order policies that are relatively easy to compute and easy to implement for a general version of the problem. 761 I l I - . Accepted by Leroy B. Schwarz, former Departmental Editor, received February 1986. This paper has been with the authors 6 months for 2 revisions. 0025- I 909/88/3406/0761 $0 1.25 CopyrightQ 1988,The Instituteof ManagementSciences l