In this Chapter, the author considers a continuous review inventory system (both perishable and non-perishable) with Markovian demand. The operating policies are (s, S) and (0,M) policy, that is the maximum inventory level at lower echelon is Sand whenever the inventory drops to s, an order for Q(= S - s) units is placed at the same time in the higher echelon, the maximum inventory level is fixed as M(= nQ: n = 1, 2, ….) and has an instantaneous replenishment facility from an abundant supply source. The demands that occur directly to the distribution centre are called direct demands. The arrival process for the direct demand follows Poisson process. The demand process to the retailer node is independent to the direct demand process and follows Poisson process. The demands that occur during stock out period are enter into the orbit of finite size. The joint probability distribution of the inventory level at lower echelon, higher echelon and the number of customer in the orbit is obtained in the steady state case.
Top1. Introduction
The management of multi-echelon inventory systems is a crucial part of supply chain operations. Therefore, it is not surprising that there already exists a large body of literature in this field. A lot of results are available for serial systems, assembly systems and divergent systems under stochastic demands. A lot of attention has been dedicated to policy evaluation as well as optimization of policy parameters under centralized and de-centralized decision making. The supply chain is traditionally characterized by a forward flow of materials and products and backward flow of information. Over the last two decades, researchers and practitioners have primarily investigated the various process of supply chain individually.
Hadley and Whitin (1963) provided an excellent account of inventory control theory and applications. A Complete review was provided by Benita M. Beamon (1998). Recently however, there has been increasing attention placed on performance, design and analysis of the supply chain as a whole. HP's (Hewlett Packard) Strategic Planning and Modeling (SPaM) group initiated this kind of research in 1977. From practical stand point the supply chain concept arose from a number of changes in the manufacturing environment, including the rising costs of manufacturing, the shrinking resources of manufacturing bases, shortened product life cycles, the leveling of planning field within manufacturing, inventory driven costs (IDC) involved in distribution and the globalization of market economics. Within manufacturing research, the supply chain concept grow largely out of two-stage multi-echelon inventory models, and it is important to note that considerable research in this area is based on the classic work of Clark and Scarf (1960).
Kalpakam and Arivarignan (1988) introduced multiple reorder level policy with lost sales in inventory control system. A complete review on this development was recorded by Federgruen (1993). Recent developments in two-echelon models may be found in Q. M. He, and E. M. Jewkes (2000) and Axsater. S (1993). A continuous review (s, S) policy with positive lead times in two echelon Supply Chain was considered by Krishnan. K. (2007). Rameshpandy. M. et. al (2014) and Satheeshkumar. R. et. al. (2014) considered the supply chain Model with retrial demands.
This Chapter deals with a simple supply chain that is modeled as a single warehouse and multiple retailer system handling a single product. In order to avoid the complexity, at the same time without loss of generality, we assumed identical demand pattern at each node, we consider a two level supply chain inventory system. It consists of one warehousing facility and one retailer. We assumed that the demands to the Distribution Centre follow Poisson process with parameter λD>0. The direct demand gets Q units at a time. The demands initiated at retailer node follow Poisson process with parameter λ(> 0). The demand to the retailer node requires single item at a time. The lead times are exponentially distributed with parameter µ(> 0). The retailer follows (s, S) policy to maintain inventory and the distributor follow (0, nQ) policy for maintaining inventory. The items are perishable& Non- perishable in nature, and it if it is perishable, it is assumed that the items are perishes only at the retailer node. The life time of an item is exponentially distributed with parameter γ (> 0) .The unsatisfied customers are treated as retrial customers and they are waiting in the orbit with finite capacity N. The repeated customers from the orbit (with capacity i) are entered into the system with rate iλ(> 0). The arriving demands finds the empty stock and the orbit is full are considered to be lost.