final customer, it forms a ‘multi-echelon’ inventory system. The echelon stock of a stock point equals all stock at this stock point, plus in-transit to or on-hand at any of its downstream stock points, minus the backorders at its downstream stock points.

The analysis of multi-echelon inventory systems that pervades the business world has a long history. Multi-echelon inventory systems are widely employed to distribute products to customers over extensive geographical areas. Given the importance of these systems, many researchers have studied their operating characteristics under a variety of conditions and assumptions. Since the development of the economic order quantity (EOQ) formula by Harris (1913), researchers and practitioners have been actively concerned with the analysis and modeling of inventory systems under different operating parameters and modeling assumptions .Research on multi-echelon inventory models has gained importance over the last decade mainly because integrated control of SCs consisting of several processing and distribution stages has become feasible through modern information technology. Clark and Scarf were the first to study the two-echelon inventory model. They proved the optimality of a base-stock policy for the pure-serial inventory system and developed an efficient decomposing method to compute the optimal base-stock ordering policy. Bessler and Veinott extended the Clark and Scarf model to include general arbores cent structures. The depot-warehouse problem described above was addressed by Eppen and Schrage who analyzed a model with a stockless central depot. They derived a closed-form expression for the order-up-to-level under the equal fractile allocation assumption. Several authors have also considered this problem in various forms. Owing to the complexity and intractability of the multi-echelon problem Hadley and Whitin recommend the adoption of single-location, single-echelon models for the inventory systems.

Sherbrooke considered an ordering policy of a two-echelon model for warehouse and retailer. It is assumed that stock outs at the retailers are completely backlogged. Also, Sherbrooke constructed the METRIC (multi-echelon technique for coverable item control) model, which identifies the stock levels that minimize the expected number of backorders at the lower-echelon subject to a bud get constraint. This model