ANALYSIS OF TWO COMMODITY MARKOVIAN INVENTORY SYSTEM WITH LEAD TIME N. ANBAZHAGAN AND G. ARIVARIGNAN
Abstract. A two commodity continuous review inventory system with independent Poisson processes for the demands is considered in this paper. The maximum inventory level for the i-th commodity is ﬁxed as Si (i = 1, 2). The net inventory level at time t for the i-th commodity is denoted by Ii (t), i = 1, 2. If the total net inventory level I(t) = I1 (t) + I2 (t) −2) −2) drops to a preﬁxed level s [≤ (S12 or (S22 ], an order will be placed for (Si − s) units of i-th commodity(i=1,2). The probability distribution for inventory level and mean reorders and shortage rates in the steady state are computed. Numerical illustrations of the results are also provided. AMS Mathematics Subject Classiﬁcation : 60J27, 90B05 Keywords and phrases : Inventory models, Joint replenishment, Stochastic lead times, Markov process.
1. Introduction With the fast expansion of activities in Business and Industrial sectors, many inventory systems are increasingly found to operate with more than single commodity. These systems unlike those dealing with single commodity, involve more complexities in the reordering procedures. In the modelling of such systems, initially models were proposed with independently established reorder points. But in situations where several products compete for common storage space or share the same transport facility or are procured from the same source, the above method overlooks potential savings associated with joint ordering and hence may not be optimal. Received November 29, 1999. Revised August 7, 2000. The work was carried out under a Major Research Project funded by University Grants Commission, India. c 2001 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 427
Anbazhagan and Arivarignan
The modelling of multi-item inventory system under joint replenishment has been receiving considerable attention for the past three decades. In continuous review inventory systems, Ballintfy  and Silver  have considered a coordinated reordering policy which is represented by the triplet (S, c, s), where the three parameters Si , ci and si are speciﬁed for each item i with si ≤ ci ≤ Si , under unit sized Poisson demand and constant lead time. In this policy, if the level of i-th commodity at any time is below si , an order is placed for Si − si items and at the same time, any other item j(= i) with available inventory at or below its can-order level cj , an order is placed so as to bring its level back to its maximum capacity Sj . Subsequently many articles have appeared with models involving the above policy and a more recent article of interest is due to Federgruen, Groenvelt and Tijms , which deals with the general case of compound Poisson demands and nonzero lead times. A review of inventory models under joint replenishment is provided by Goyal and Statir . Kalpakam and Arivarignan  have introduced (s, S) policy with a single reorder level s deﬁned interms of the total number of items in the stock. The supply is assumed to be instantaneous. This policy avoids separate ordering for each commodity. Since a single processing of orders for both commodities has some advantages in situation wherein procurement is made from the same supplies, items are produced on the same machine, or items have to be supplied by the same transport facility. Krishnamoorthy, Iqbal Basha and Lakshmy  have considered a two commodity continuous review inventory system without lead time. In their model, each demand is for one unit of ﬁrst commodity or one unit of second commodity or one unit of each of commodity 1 and 2, with preﬁxed probabilities. Krishnamoorthy and Varghese  have considered a two commodity inventory problem without lead time and with Markov shift in demand for the type of commodity namely...