Optimal and Heuristic Echelon (r, nQ, T) Policies in Serial Inventory Systems with Fixed Costs

Abstract

This paper studies a periodic-review, serial inventory system in which echelon (r, nQ, T) policies are implemented. Under such a policy, each stage reviews its inventory in every T period and orders according to an echelon (r, nQ) policy. Two types of fixed costs are considered: one is associated with each order batch Q, and the other is incurred for each inventory review. The objective is to find the policy parameters such that the average total cost per period is minimized. This paper provides a method for obtaining heuristic and optimal policy parameters. The heuristic is based on minimizing lower and upper bounds on the total cost function. These total cost bounds, which are separable functions of the policy parameters, are obtained in two steps: First, we decompose the total cost into costs associated with each stage, which include a penalty cost for holding inadequate stock. Second, we construct lower and upper bounds for the penalty cost by regulating downstream policy parameters. To find the optimal solution, we further construct cost bounds for each echelon (a subsystem that includes a stage and all of its downstream stages) by regulating holding and backorder cost parameters. The echelon lower-bound cost functions, as well as the stage cost bounds, generate bounds for the optimal solution. In a numerical study, we find that the heuristic is near optimal when the ratio of the fixed cost to the holding cost at the most downstream stage is large. We also find that changing the optimal batch sizes may not affect the optimal reorder intervals or, equivalently, the delivery schedules under some conditions.