We consider a production-inventory system that consists of n stages. Each stage has a finite production capacity modelled by an exponential server. The downstream stage faces a Poisson demand. Each stage receives returns of products according to independent Poisson processes that can be used to serve demand. The problem is to control production to minimize discounted (or average) holding and backordering costs. For the single-stage problem (n=1), we fully characterize the optimal policy. We show that the optimal policy is base-stock and we derive an explicit formula for the optimal base-stock level. For the general n-stage problem, we show that the optimal policy is characterized by state-dependent base-stock levels. In a numerical study, we investigate three heuristic policies: the base-stock policy, the Kanban policy and the fixed buffer policy. The fixed-buffer policy obtains poor results while the relative performances of base-stock and Kanban policies depend on bottlenecks. We also show that returns have a non-monotonic effect on average costs and strongly affect the performances of heuristics. Finally, we observe that having returns at the upstream stage is preferable in some situations.
The importance of product returns is growing in supply chains. Customers often can return products a short time after purchase, due to take-back commitments of the supplier. For instance, the proportion of returns is particularly important in electronic business where customers cannot touch a product before purchasing it. Customers might also return used products a long time after purchase. This type of return has increased in recent years due to new regulations on waste reduction, especially in Europe. Some industries also encourage returns for economical and marketing reasons. Though different in nature, these two types of returns are similar from an inventory control point of view since they constitute a reverse flow which complicates decision making.
The inventory control literature on product returns is quite abundant (see e.g. Fleischmann et al., 1997, Ilgin and Gupta, 2010 and Zhou and Yu, 2011). However, most of the literature focusses on single-echelon systems with infinite production capacity. In this paper, we fill this gap by considering a n-stage production/inventory system with finite production capacity and product returns at each stage (see Fig. 1). The flow of returns at the finished good (FG) inventory may result from remanufacturing, recycling, repairing or simply returning new products. The flows of returns at the work-in-process (WIP) inventories can also result from disassembly operations. For instance, the Kodak company reuses only some parts of cameras like circuit board, plastic body and lens aperture ( Toktay et al., 2000).
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Fig. 1.
A two-stage production/inventory system with returns.
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More precisely, we adopt a queueing framework to model production capacity. Items are produced by servers one by one and each unit requires a random lead-time to be produced. We assume that each stage consists of a single exponential server and an output inventory. The downstream stage faces a Poisson demand. Each stage receives returns of products, according to independent Poisson processes, that can be used to serve demand. The problem is then to control production at each stage, in order to minimize discounted/average holding and backordering costs. We also study the single-stage problem which has not been studied in the literature. In what follows, we review the literature on single-echelon and multi-echelon systems with returns, before presenting in detail our contributions.
The literature on single-echelon systems is quite mature. Heyman (1977) considers an inventory system with independent Poisson demand and Poisson returns. Unsatisfied demands are backordered. Heyman assumes zero lead-times and linear costs for both manufacturing and remanufacturing. These strong assumptions imply that the optimal production policy is a make-to-order policy and that the optimal disposal policy is a simple threshold policy: when the inventory level exceeds a certain disposal threshold R, every returned item is disposed upon arrival. An explicit expression for the optimal disposal threshold is also derived. For a lost sale problem with exponential service times, Poisson demand and returns, Zerhouni et al. (in press) investigate the impact of ignoring dependency between demands and returns.
Fleischmann et al. (2002) consider a similar setting with deterministic manufacturing lead-time and fixed order cost. Again, remanufacturing lead-time and remanufacturing costs are neglected. They extend results standing for a system without returns by showing that the optimal policy is (s,Q) for the average-cost problem. For the periodic review version with a stochastic demand either positive or negative in each period, Fleischmann and Kuik (2003) show the average-cost optimality of an (s,S) policy. Simpson (1978) and Inderfurth (1997) consider a periodic-review problem where returns are held in a separate buffer until they are remanufactured or disposed of. When the remanufacturing lead-time is equal to the production lead-time and the costs are linear, they show that a three-parameter policy is optimal.
