Coordination and control of multi-echelon inventory systems

Abstract

Efficient management of multi-echelon systems requires coordination between replenishment decisions, transportation planning and inventory control. In this context, we have studied three inventory models with stockpoints having different order/transportation moments. The first model we have studied is a single-item two-echelon serial inventory system with stochastic demand. The system is centrally controlled and ordering decisions are based on echelon order-up-to policies. The review period of the upper echelon is an integer multiple of the review period of the lower echelon. We assume that linear holding and backorder costs are charged as well as fixed ordering costs. We derive an exact analytical expression for the objective function to be used to determine optimal policy parameters and review periods. In a numerical study we illustrate that there may be several combinations of optimal review periods and that under high fixed ordering costs both stockpoints have the same order frequency. Additionally, we identify parameter settings under which the materials that arrive to the system are immediately pushed to downstream stages. In literature, many authors do not consider upstream stock unavailability when determining fixed ordering costs. We test the impact of this simplifying assumption on fixed ordering costs and illustrate when it is justified. Secondly, we consider an assemble-to-order model with a single item assembled from two components. One of the components has a long lead time, high holding cost and short review period as compared to the other one. We assume that net stocks are reviewed periodically, customer demand is stochastic and unsatisfied demand is backordered. Such a system cannot be solved to optimality by existing methods in literature since the review periods are not nested. Instead, we analyze the system under two different policies and show how to determine the policy parameters minimizing average holding and backorder costs. First, we consider a pure base stock policy, where orders for each component are placed such that the inventory position is raised up to a given base stock level. In contrast to this, only the orders for one component follow this logic while the other component’s orders are synchronized in case of a balanced base stock policy. Through mathematical analysis, we come up with the exact long-run average cost function and we show the optimality conditions for both policies. In a numerical study, the policies are compared and the results suggest that the balanced base stock policy works better than the pure base stock policy under low service levels and when there is a big difference in the holding costs of the components. The balanced base-stock policy is applicable only when sum of the review period and the lead time of the expensive component is larger than the sum for the other component. To find an efficient control policy for the reverse case, we characterize the optimal policy structure of the system by using dynamic programming under certain parameter assumptions. Then, we suggest an efficient heuristic ordering policy based on the optimal ordering behaviour. In the third model, we consider a one-warehouse multi-retailer inventory system with a time-based shipment consolidation policy at the warehouse. This means that retailer orders are consolidated and shipped to groups of retailers periodically. Customer demand is compound Poisson distributed and unsatisfied demand at each stock point is backordered and allocated on a first-come-first-served basis. The system is centralized and inventory levels are reviewed continuously. This means that the warehouse has access to point of sale data and inventory information at the retailers. All retailers apply order-up-to replenishment policies, whereas the warehouse, which replenishes from an outside supplier/manufacturer, uses a batch ordering policy. We derive exact probability distributions of the inventory levels at the retailers and use these to obtain exact expressions of the long-run average holding and backorder costs, average inventory levels and service levels. Based on the analytical properties of the objective function, we construct an exact optimization procedure for determining the optimal reorder levels and shipment intervals both for single item and multi-item cases.