Deriving the cost function for a class of three-echelon inventory system with N-retailers and one-for-one ordering policy

A three-echelon inventory system with two warehouses and N-retailers is considered in this paper. Transportation times are constant and retailers face independent Poisson demand. Replenishments are one-for-one. The lead time of a retailer is determined not only by the constant transportation time but also by the random delay incurred due to the availability of stock at the warehouses. Axsäter [2] evaluated the value of the expected two-echelon inventory system cost with the same assumptions, but in this paper, we add a warehouse as third echelon and therefore this may lead to having one more delay in shipment which was incurred in the new warehouse. We obtained the cost function for the inventory system and tested it by several examples.