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Inventory problems are studied for a long time and many authors have investigated models with shortage phenomenon. To reduce inventory holding cost or decaying items we can have a shortage period in witch a part of demand is backordered; Although we pay fines to pending costumers, as shortage period is extended a larger portion of them will refuse to wait until replenishment. So, calculating an optimal duration for on hand inventory and shortage period with respect to inventory costs has became an interesting research field. However, related studies are rarely appeared in supply chain researches, where an integrated inventory model for a supplier and a retailer is proposed; especially when a... 

The thesis has considered a two-echelon supply chain which consists of one central warehouse and multiple retailers. Assume all retailers and the warehouse follow a continuous review inventory model. The costumer’s demand comes to retailers in a Poisson process. The retailers replenish their stocks from the warehouse in an (S-1, S) inventory control policy and the warehouse replenishes its stocks in an (r, Q) inventory control policy from an external supplier with an infinite capacity. It may be more representative to model stockouts as lost sales when the retailers are in a competitive market or in monopoly markets all demand will be backorder. Therefore, if the market situation goes...