Optimal control of serial inventory systems with fixed replenishment intervals.

We consider a single-item, periodic-review, serial inventory/production system, with linear inventory-holding and penalty costs. To facilitate shipment consolidation and capacity planning, we assume that the system has implemented fixed replenishment intervals; each stage is allowed to order only at given equidistant times. Further, for each stage except the most downstream one, the replenishment interval is assumed to be an integer multiple of the replenishment interval of the next downstream stage. This reflects the fact that the further upstream in a supply chain, the higher setup times and costs tend to be, and thus larger batches are desired. Our model with fixed replenishment intervals is a direct generalization of the serial model of Clark and Scarf (1960). For this generalized model, we prove the optimality of base-stock policies, we derive newsboy equations for the optimal base-stock levels, and we describe an efficient exact solution procedure for the case with mixed Erlang demands. Finally, we present extensions to assembly systems and to systems with a modified fill-rate constraint instead of backorder costs.

Subject classifications: inventory/production: multiechelon, stochastic demand, fixed replenishment intervals, optimal policies.

Area of review: Manufacturing, Service, and Supply Chain Operations.

1. Introduction

Two primary sources of costs in supply chains are capacity costs and material costs; hence, decisions affecting each of these should be made taking the other into account. Typically, capacity decisions are made for a longer term than material decisions; thus, capacity decisions are often made first. Material decisions follow, and themselves are often made sequentially, according to a hierarchical approach with two decision levels:

(i) A first level decides on such things as the form of batching and the batch sizes and replenishment intervals, where a multi-item, multiechelon view is taken. This enables these decisions to accommodate setups, capacity constraints, capacity flexibilities, capacity partitioning, and shipment consolidation. These decisions may be reviewed annually, for example;

(ii) A second level decides on reorder and base-stock levels, adapted on a daily or weekly basis (e.g., when procedures like exponential smoothing are used for demand forecasting). Here the batching rule is taken as given, and a single-item, multiechelon view can be incorporated.