Note: A Simple Heuristic for Serial Inventory Systems with Fixed Order Costs

Consider a continuous-review, serial inventory system with N stages. Material flows from stage N to stage N − 1, N − 1 to N − 2, etc., until stage 1, where stationary compound Poisson demand occurs. Let and denote the demand rate and the mean of demand size, and D t denote the demand over a time interval t. There is a constant lead time Lj for stage j . Define L̃i = ∑i j=1Lj . Inventory is controlled by echelon-stock R nQ policies. That is, for each stage j , as soon as the echelon inventory-order position (inventory on order+ inventory on hand+ inventory at or in transit to all downstream stages− backorders) is at or below the reorder point Rj , an order of the smallest integer multiple of base quantity Qj is placed to raise the inventory position above Rj . We assume that base quantities satisfy integer-ratio constraints, i.e., Qj = qjQj−1, qj ∈ +, j = 2 N , where + denotes the set of positive integers. There is a linear echelon holding cost with rate hj for stage j . Unsatisfied demand is fully backlogged and incurs a linear penalty cost with rate b. Define hj = ∑N i=j hi as the local holding cost rate for stage j . A fixed order cost kj occurs for each base quantity Qj ordered. The objective is to minimize the long-run average systemwide cost. When demand is deterministic, it is well known that there exists a simple heuristic that guarantees a 94% effective policy, i.e., the cost of this heuristic policy is at most about 6% more costly than any feasible policy. This heuristic includes two steps. The first step is to identify clusters by using cost ratios. (A cluster is a set that includes consecutive stages; see a formal definition below.) The second step solves an economic order quantity (EOQ) problem for each cluster. A heuristic policy is then obtained by converting these EOQ solutions into integer-ratio order quantities. One well-known integer-ratio solution is the so-called “powerof-two” policy (e.g., Maxwell and Muckstadt 1985, Roundy 1985, and Federgruen et al. 1992). We refer the reader to Zipkin (2000) for a detailed discussion of the deterministic model and the power-of-two policy. In the stochastic demand model, however, finding effective base order quantities is more involved. The difficulty comes from the fact that, unlike the deterministic model, the objective function (the average total cost) is not a sum of separable, convex functions of control variables. To resolve this, Chen and Zheng (1998) construct cost bounds to replace the average total cost in the objective function. These cost bounds are a sum of separable functions of base quantities. Thus, a standard clustering algorithm (e.g., Maxwell and Muckstadt 1985) can be applied to find the optimal solution for these revised problems. They propose heuristics by converting these optimal solutions to power-of-two base quantities. The objective of this note is to propose a new heuristic that employs the same steps as the heuristic for the deterministic model. As we shall see, this heuristic outperforms the existing ones in a numerical study.