Analysis of Serial Inventory Systems with Batch Ordering and Nested Replenishment Schedule

We consider a periodic-review, serial inventory system, in which each stage replenishes inventory according to an echelon-stock (r, nQ, T ) policy – a standard echelon-stock (r, nQ) policy executed every T periods. Demands are independent, identically distributed in different periods and occur at the most downstream stage. Two types of fixed costs are considered: setup cost and replenishment cost. The former is associated with each batch ordered; the latter is incurred in each order period. We study two models. The first (second), referred to as the T -model (Q-model), assumes that batch sizes (replenishment intervals) are fixed. The objective is to determine reorder points and replenishment intervals (batch sizes) such that the average total cost per period is minimized. For both models, we establish lower and upper bounds on the average inventory costs per period for each stage. These cost bounds, which lead to an optimization algorithm and close-to-optimal heuristics, are constructed based on a novel idea of developing bounds on the average penalty cost per period to each upstream stage for holding inadequate stock. We also discuss extensions with continuous-time models and alternative fixed cost assumptions. These analytical results are used to investigate system design issues. Among others, replenishment interval decisions should be made before batch size decisions. In addition, when setup costs are equal to replenishment costs, continuous-review (r,Q) policies result in lower average total cost than periodic-review (r, T ) policies. However, the cost benefit decreases when setup costs at downstream stages are large.