We consider a two-stage serial inventory system whose cost structure exhibits economies of scale in both stages. In the system, stage 1 faces Poisson demand and replenishes its inventory from stage 2, and the latter stage in turn orders from an outside supplier with unlimited stock. Each shipment, either to stage 2 or to stage 1, incurs a fixed setup cost. We derive important properties for a given echelon-stock (r, Q) policy for an approximation of the problem where all states are continuous. Based on these properties, we design a simple heuristic algorithm that can be used to find a near-optimal (r, Q) policy for the original problem. Numerical examples are given to demonstrate the effectiveness of the algorithm.
We are concerned with the management of a two-echelon stochastic inventory system involving setup costs. Research on multi-echelon inventory systems started about four decades ago, when Clark and Scarf (1960) published their seminal paper that characterized the optimal control policy when only the leading stage dealing with a limitless supplier is charged with setup costs. When economies of scale appear in more than one stage, however, the characterization of optimal policies meets much tougher resistances. Subsequent research on multi-echelon stochastic inventory systems with economies of scale focused more on heuristic policies. For more information about these heuristics, the reader is referred to Axsater, 1993a and Axsater, 1993b, Chen and Zheng, 1994a and Chen and Zheng, 1994b, Federgruen and Zipkin (1984), Shang (2008), Chao and Zhou (2009), Olsson (2009), Hariga (2010), Shang and Zhou (2010), as well as the comprehensive survey papers by Chen (1999b) and Axsater (2003).
We study the following two-stage serial inventory system. Stage 1 directly faces Poisson demand and replenishes its inventory by ordering from stage 2, which in turn orders from an outside supplier with unlimited stock. Economies of scale are reflected in the ordering costs at both stages. That is, each shipment, either to stage 2 or to stage 1, incurs a fixed setup cost. We limit our search for the optimal policy to the (r, Q) type. Formally, stage 2 orders a fixed quantity Q2 from its immediate upper stream whenever its echelon inventory position (outstanding orders plus on-hand inventory at stage 2 and the entire inventory at or in-transit to all down-stream stages minus backorders at stage 1) drops to a re-order point r2. Stage 1 places an order with quantity Q1 whenever its inventory position (inventory in-transit plus on-hand inventory minus backorder) drops to r1. Policies of this type are widely used in practice because the more restricted order sizes accommodate easy packaging, transportation, and coordination.
We make the simplifying assumption that the leadtime from the outside supplier to stage 2 is zero. This is reasonable when stage 2 has a strong bargaining power over the outside supplier and the latter is physically close by. In deterministic inventory systems, this assumption enabled the development of 98%-effective policies (see, e.g., Roundy, 1985 and Roundy, 1986). In stochastic inventory systems, this assumption has been adopted by numerous researchers, including Naddor, 1956 and Naddor, 1963, Pressman (1977), Nahmias and Demmy (1981), Azoury and Brill (1986), and Hill (1999). Also under this assumption, Chen (1999a) established a lower bound for the long-run average cost of any feasible policy, and then provided a 94%-effective heuristic policy. The zero-supplier-leadtime assumption is known for two important consequences:
(1)
The optimal policy is nested, in the sense that stage 2 must send a shipment to stage 1 whenever it receives an order.
(2)
The optimal policy possesses the zero-inventory-ordering property; that is, stage 2 must have zero on-hand inventory when it orders.
We first demonstrate important properties of any given (r,Q) policy when applied to the continuous-state approximation of the original problem, and then achieve the optimization of policy parameters by reducing this approximated problem to two single-variable quasi-convex optimization problems. Next, based on the above properties, we design a simple heuristic algorithm that can be used to determine near-optimal policy parameters for the original problem. Our numerical experiments attest to the effectiveness of this algorithm. For most problems tested, the algorithm reaches 99%-effectiveness, and shows great improvements over Chen's method.
The rest of the paper is organized as follows. We present the two-stage model as well as some known results in Section 2. In Section 3, we present properties revolving around the echelon-stock (r,Q) policy in the continuous-state approximation of the original model. In Section 4, we provide a heuristic algorithm that can be used to compute near-optimal policy parameters for the original problem. We present some numerical examples in Section 5, and conclude the paper in Section 6. All mathematical proofs have been relegated to Appendix.
We focused on a two-stage serial inventory system where economies of scale apply to both stages. We established properties concerning both continuous-state approximation and the (r, Q) policy, and developed a heuristic algorithm for computing near-optimal control parameters for the system under management. Our numerical study verified the effectiveness of the algorithm. There is potential for our method to be generalized to cover inventory systems involving more stages and/or more intricate structures. Having much more complex relations among their various decision and performance variables, however, such systems pose other challenges.
