Analysis of Two Commodity Markovian Inventory System with Lead Time free essay sample

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. 2001 Korean Society for Computational Applied Mathematics and Korean SIGCAM. 427 428 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 [1964] and Silver [1974] have considered a coordinated reo rdering policy which is represented by the triplet (S, c, s), where the three parameters Si , ci and si are speci? ed for each item i with si ? ci ? Si , under unit sized Poisson demand and constant lead time. We will write a custom essay sample on Analysis of Two Commodity Markovian Inventory System with Lead Time or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page 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 [1984], 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 [1989]. Kalpakam and Arivarignan [1993] have introduced (s, S) policy with a single reorder level s de? ned 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 [1994] have considered a two commodity continuous review inventory system without lead time. In their model, each demand is for one unit of ? rst commodity or one unit of second commodity or one unit of each of commodity 1 and 2, with pre? xed probabilities. Krishnamoorthy and Varghese [1994] have considered a two commodity inventory problem without lead time and with Markov shift in demand for the type of commodity namely †commodity-1†, †commodity-2† or †both commodities†. In this paper a two commodity inventory system with joint reorder level which triggers a reorder for both commodities with an exponentially distributed lead time is considered. The probability distribution of inventory level for both commodities, mean reorder rate and shortage rate in the steady state have been computed. The results are numerically illustrated. 2. Problem formulation Consider a two commodity inventory system with the maximum capacity Si units for i-th commodity (i = 1, 2). It is asssumed that demands for i-th Analysis of two commodity Markovian inventory system with lead time 429 Figure 1. Space of Inventory levels (0, S2 ) (S1, S2 ) (0, s) @ @ @ @ @ @ @ @ (0, 0) (s, 0) (S1 , 0) commodity are of unit size and having Poisson distribution with parameter ? i (i = 1, 2). The demand process of the two commodities are further assumed to be independent. The reordering policy is to place order for both the commodities when the total net available inventory is equal to s(? (Si ? s)/2) and the ordering quantity will be Qi(= Si ? s), i = 1, 2. The lead time is assumed to be distributed as negative exponential with parameter  µ(gt; 0). The demands that occur during stockout periods are lost. Let I(t) denote the net inventory level at time t. Then the process I = {(I1(t), I2(t)), t ? 0} has the state space E = {(i, j) | i = 0, 1, 2,  ·  ·  · , S1 and j = 0, 1, 2,  ·  ·  · , S2 }.