An optimal policy for a two-echelon inventory/distribution system

This paper develops a simple optimal solution for a two-echelon serial inventory/distribution system, consisting of a warehouse and a retailer, under deterministic demand. In this system a single item provided by an outside supplier is stocked at a warehouse and then distributed to customers through a retailer. Two different solution procedures of a deterministic model for obtaining the minimum of the total cost of set ups, inventory holding, transportation are available in the literature. In both of them the batch (part of a lot) size from the warehouse to the retailer must be less than or equal to the minimum of the inventory capacity at the retailer and the capacity of the transport vehicle. In providing the optimal solution, the inability of the earlier procedure was demonstrated by the latter one. But the later policy seems to be a complicated one including many sub-cases. It appeared to be much more complex when the inequality case of the capacity constraint is true. In contrast, in this paper we develop a simple optimal solution policy of the problem using Karush–Kuhn–Tucker theorem by taking into account the transportation time, which is able to consider all cases much more easily. A comparative study of the solution procedures on results of 5 numerical examples is also carried out to justify our claim.