Abstract: There are many researches focusing on the optimum inventory solution in the supply chain that the order is committed at the lead time and the inventory shortage can be allowed. More specifically, the lead-time is assumed as a negative exponential distribution and the optimum inventory solution is conducted in one-manufacturer and one-customer supply chain. Or, the lead-time is assumed that it can be changed according to the time, and the inventory shortage can be allowed in one-manufacturer and one-customer supply chain. Or, only part of the inventory shortage can be backordered. In first case, the shortage needs so long time to be backordered in some times, and probably can't be backordered in the extreme condition.
In this paper, the lead-time is assumed that it can be controlled. Once the order from the customer is provided, the shortage can be immediately backordered fully at the end of the order time. With such simple assumptions, the research is easily extended into the two-echelon supply chain consisting of a single manufacturer and multiple customers. The optimization process can be described as follows. Firstly, the separate inventory models respectively relating to the manufacturer and the customers are formulated according to the manufacturing stock behavior and order stock behavior. Then the joint inventory model of the supply chain is provided by combining the separate models together. Finally, with the joint optimization method, the optimum parameters, i.e. the optimum reorder point, the optimum production time interval and the optimum order time are derived to guarantee the joint inventory model has the minimal inventory cost. Similarly, we also provide the optimum results for separate inventory models for comparison to show the advantage of the joint optimization method.
The simulation analysis shows that the result about the estimated lead-time is so different between the joint optimization method and the separate optimization method. The interval of the lead time is much narrower in the joint optimization method. With enlarging the value of the lead time, the reorder point increases consistently in linear relation. All the customers have the same following result, i.e. the reorder point is larger in the separate optimization method than in the joint optimization method. In addition, we also simulate the relation between the inventory cost and the reorder point for all the customers. All the customers have same quadratic curve shape and the optimum reorder point is existed and at the bottom of the quadratic curve. The optimum reorder point is unique and is existed. This result is evidently shown that the proposed joint inventory model and the related optimization method are correct. The research result of the paper is helpful for the researchers to study the inventory optimization problem in the more complicated supply chain and it can be useful for real-world supply chain operation and optimization management.

Key words: supply chain inventory, single-manufacturer multi-customers, Lead-time order, Shortage