This article examines a two commodity substitutable inventory system—two different brands of super computers under continuous review. The demand points for each commodity are assumed to form independent Poisson processes. The reordering policy is to place orders for both the commodities when the total net inventory level drops to any one of the prefixed levels with prescribed probability distribution. Lost sales are assumed during the stock out period. The lead time for a reorder is exponentially distributed with parameter(, depending on the size of the ordering quantity. The limiting probability distribution for the joint inventory levels is also evaluated. Various operational characteristics and total expected cost rate are derived. Numerical examples are provided to find optimal reorder quantity and band width .
TopIntroduction
In dealing with multi-commodity inventory system in a single location, the joint and coordinated reordering policy have been given more attention than that of individual reorder for each commodity separately. There are many advantages for joint replenishment, as they share setup costs, quantity discounts, utilize the same transport facilities, etc.
In the real life situation, the global sales agencies deal with rare electronic products like super computers with almost same configuration and identical functioning. This motivates the researcher to consider the substitutable commodities inventory control under joint reorder policy. In general these high tech machines are having substitutable nature because of its common number crunching behavior(General purpose computer). Keeping them in stock for sales purpose is high risk but yield high profit. Maintaining these “A” type items of ABC inventory classification, attracted many researchers in the past decade. We also assumed elongated production schedule and that the lead time distribution parameter depends on the number of items ordered.
In continuous review inventory systems, Ballintify (1964) and Silver (1974) have considered a can-order policy, which has a can order level and a reorder level. In this policy, an item must be ordered when its inventory level reaches the reorder level and when an item in the group is ordered, other items with inventory positions at or below their respective can-order levels are also ordered. Subsequently, many articles have appeared with models involving the above policy. Another article of interest is due to Federgruen et. al (1984), which deals with the general case of compound Poisson demands and non-zero lead time. A review of inventory models under joint replenishment is provided by Goyal and Satir (1989). Kalpakam and Arivarignan (1993) have introduced 
policy with a single reorder level 
defined in terms of the total number of items in the stock.
The work on methods to solve the joint replenishment problem throughout the years has been extensive. Readers are referred to the publications of Fung and Ma (2001), Goyal (1973, 1974, 1988), Goyal and Belton (1979), Kaspi and Rosenblatt (1991), Nilsson et al. (2007), Nilsson and Silver (2008), Olsen (2005), Silver (1976), Van Eijs (1993), Viswanathan (1996, 2002, 2007) and Wildeman et al. (1997) and references therein.
Xiao and Tiaojun et al. (2007) have developed a dynamic game model of a supply chain consisting of one manufacturer and one retailer to study the coordination mechanism and the effect of demand disruption on the coordination mechanism, where the market demand is sensitive to retail price and service. Shi and Kuiran and Xiao and Tiaojun (2008) analysed a supply chain consisting of a risk-neutral manufacturer selling a perishable product to a loss-averse retailer and presented the optimal ordering decision between the manufacturer and the retailer in a single period inventory with uncertain demand. Two types of contracts, buyback contract and markdown-price contract with the retailer’s loss aversion consideration are investigated, respectively.