Production, Manufacturing and Logistics
On optimal inventory control with independent stochastic item returns

https://doi.org/10.1016/S0377-2217(02)00592-1Get rights and content

Abstract

To a growing extent companies take recovery of used products into account in their material management. One aspect distinguishing inventory control in this context from traditional settings is an exogenous inbound material flow. We analyze the impact of this inbound flow on inventory control. To this end, we consider a single inventory point facing independent stochastic demand and item returns. This comes down to a variant of a traditional stochastic single-item inventory model where demand may be both positive or negative. Using general results on Markov decision processes we show average cost optimality of an (s,S)-order policy in this model. The key result concerns a transformation of the model into an equivalent traditional (s,S)-model without return flows, using a decomposition of the inventory position. Traditional optimization algorithms can then be applied to determine control parameter values. We illustrate the impact of the return flow on system costs in a numerical example.

Introduction

Recovery of used products and materials is receiving growing attention. Seeking alternatives to waste disposal and incineration, efforts are made to reintegrate used products into industrial production processes. Waste paper recycling and car parts remanufacturing have been around for a long while. Reusable packaging and electronic scrap recycling are more recent examples. Pressured by environmentally conscious customers and enhanced legislation, manufacturers are assuming responsibility for the entire life-cycle of their products. Take-back of products after use is a growing trend. In addition to enhanced environmental performance and a ‘green’ image, product recovery may also prove beneficial due to savings in material, manufacturing, and disposal costs. On the other hand, fairly high uncertainty with respect to timing, quantity, and quality aspects appears to be one of the major obstacles to efficient recovery of used products. From a logistics perspective the above reuse opportunities give rise to an additional goods flow from the user back to the sphere of producers. The management of this material flow opposite to the conventional supply chain flow is addressed in the rapidly expanding field of ‘reverse logistics’ (Fleischmann et al., 1997).

Inventory management is one of the areas concerned. The return flow of used products needs appropriate integration into material planning. The situation can be characterized as follows (see Fig. 1 adapted from Fleischmann et al., 1997).

A company serves demand for a certain product. Serviceable on-hand stock may be replenished by ordering new products from an external source, e.g., production or outside procurement. Additionally, used products are returned from the market. After being reconditioned to meet new product quality standards, e.g. by means of cleaning, repair, or remanufacturing, returned products are added to the serviceable inventory. Examples are manifold, including the management of reusable packaging (Kelle and Silver, 1989), leased copy machines (Muckstadt and Isaac, 1981), and car parts remanufacturing (Van der Laan and Salomon, 1997). The above framework differs from traditional inventory management situations in essentially two aspects. First, returns represent an exogenous inbound material flow causing a loss of monotonicity of stock levels. Second, two alternative supply sources are available for replenishing the serviceable inventory. While both aspects as such are not new to inventory theory it is their combination that leads to novel situations. Related models that are well established in inventory control literature include repair models on the one hand (Nahmias, 1981) and two-supplier-models on the other hand (Moinzadeh and Nahmias, 1988). However, neither of them captures the characteristics of the above setting appropriately. Focusing on item failure, repair models typically assume returns to be accompanied by simultaneous demand for replacement items. This strong correlation between demand and returns is not necessarily present in the product recovery environment sketched above. On the other hand, traditional two-supplier-models differ from the above scenario in that they typically address the tradeoff between a fast but expensive supplier and a cheaper but slower one. In product recovery however, the cheaper channel, i.e. recovery, often also is the faster one. Instead, it is its limited availability that implies the need for an additional source.

Several authors have proposed inventory models within the scope of the above setting (see, e.g., Silver et al., 1998, p. 503ff). A classification can be made into single versus two-level models, depending on an explicit distinction between used and serviceable items. Within the former class Cohen et al. (1980) assume that a fixed fraction of the products issued in a given period is returned after a fixed sojourn time in the market and may subsequently be reused. Optimality of a periodic review ‘order upto’ policy is shown when disregarding fixed costs and procurement leadtimes. Kelle and Silver (1989) extend this approach by allowing for fixed order costs and stochastic sojourn time in the market. They propose an approximation scheme transforming this model into a classical dynamic lotsizing problem. Buchanan and Abad (1998) assume that returns are a stochastic fraction of the number of items in the market for each period, which is equivalent with an exponentially distributed market sojourn time. The authors derive an optimal procurement policy depending on two state variables, namely the on-hand inventory and the number of items in the market.

In the two-level models control policies comprise decisions on new orders and repair activities. Simpson (1978) considers the tradeoff between material savings due to reuse versus additional inventory carrying costs and proves optimality of a three parameter critical number policy to control order, repair, and disposal for the discrete time case without fixed costs and leadtimes. Inderfurth (1997) shows this policy to be optimal also in the case of fixed and identical leadtimes for repair and procurement. In a single-level variant of this model, assuming any returned item to be repaired or disposed of immediately, a mixed ‘order upto––dispose down to’ policy is shown to be optimal provided the procurement leadtime equals the repair leadtime or exceeds it by at most one period.

Heyman (1977) considers a continuous review model with independent stochastic demand and returns. Assuming instantaneous procurement and repair and disregarding fixed costs, he derives an optimal disposal level, which limit is the number of returns accepted. Muckstadt and Isaac (1981) extend this approach by considering fixed order costs and non-zero procurement and repair leadtimes. Assuming independent Poisson demand and return processes and lot-for-lot repair, an (s,Q) order policy is proposed. The values of the control parameters are determined via an approximation of the net inventory distribution. Van der Laan et al. (1996) propose an alternative approximation scheme for this model. Moreover, an additional disposal option is considered, for which several policies are compared numerically. Finally, Van der Laan et al. (1999) present a detailed analysis of different policies to control serviceable and recoverable stock in the above setting, taking into account non-zero leadtimes for both sources. In particular, a push- and a pull-driven recovery policy is considered. The authors indicate that defining an appropriate inventory position as a basis for replenishment decisions is non-trivial in case of a large difference between the leadtimes of the two sources.

We conclude from the above that inventory control in a product recovery context has received considerable attention among operations researchers. However, current results are fairly scattered. Moreover, optimality results typically involve rather strong assumptions such as absence of fixed costs and/or leadtimes. With the present paper we contribute to a more systematic analysis of this area. As a starting point, it appears to be useful to thoroughly analyze the impact of the material return flow. For this purpose, we consider a traditional single-item stochastic inventory model extended with a stochastic inbound flow. We derive a complete characterization of this model in terms of an optimal policy structure and an explicit expression of the cost function. Moreover, by means of a transformation, the machinery of classical inventory theory is made available for the case with return flows. Hopefully, these results may serve as a basis for analyzing more complex models including, e.g., two supply sources.

The remainder of this paper is organized as follows. Section 2 gives a formal definition of our model and introduces notation. Section 3 is central. We analyze the cost function for a given (s,S) order policy and show that the model can be transformed into an equivalent model without return flows. In Section 4 we prove optimality of an (s,S) policy for our model. Section 5 addresses computational issues, including calculation of optimal control parameters, and discusses some extensions. Section 6 summarizes our findings.

Section snippets

The model

Following the above motivation we consider a standard single-item stochastic inventory model extended with a stochastic inbound item flow. For the time being, assume returned items to be added to the serviceable stock immediately. This may be interpreted as a case of directly reusable items, such as e.g. reusable packaging or commercial returns due to excess ordering. Yet we note that this assumption is mainly for notational convenience. In analogy with standard inventory models a fixed

Analysis of the cost function

In this section we analyze the structure of the cost function in the return-flow model for a given (s,S) control policy. The findings are essential for the analysis in the remainder of the paper. The main result of this section is a transformation of the return-flow model into an equivalent standard inventory model without returns.

Assume that replenishment orders are controlled according to an (s,S) policy. In this case the stock level after ordering evolves as follows:xn+1=xn−dnifdn<xn−s,Selse.

Optimal policy structure

In this section we show that an (s,S) order policy is average cost optimal in the return-flow model. For traditional inventory models optimality of an (s,S) policy under the condition of backordering unsatisfied demand is well known. A remarkably direct proof has been given by Zheng (1991). The approach relies on considering a relaxed model including disposal, for which optimality of an (s,S) order policy follows from deriving a bounded solution of the optimality equation of the corresponding

Numerical example and extensions

Having shown optimality of an (s,S) policy structure, the question remains how to determine optimal values of the control parameters. However, in view of Proposition 1 this reduces to finding optimal parameter values in a standard (s,S) inventory model. Recall in this context that convexity of G(·) implies convexity of the transformed cost function H(·). Several optimization and approximation approaches for this problem have been proposed in inventory control literature, see e.g. Johansen (1997)

Conclusions

In this paper we have considered a stochastic inventory model encompassing random item returns. Returns are independent of past demand and may be directly reused. Alternatively, a fixed repair leadtime can be assumed. The model comes down to a variant of a standard stochastic inventory model where demand may be both positive or negative. We have shown that the model can be transformed into an equivalent model with non-negative demand only. Moreover, we have proven optimality of an (s,S) order

References (25)

  • D. Iglehart

    Optimality of (s,S) policies in the infinite-horizon dynamic inventory problem

    Management Science

    (1963)
  • K. Inderfurth

    Simple optimal replenishment and disposal policies for a product recovery system with leadtimes

    OR Spektrum

    (1997)
  • Cited by (123)

    • Modelling the influence of returns for an omni-channel retailer

      2023, European Journal of Operational Research
    • Remanufacturing of multi-component systems with product substitution

      2022, European Journal of Operational Research
    View all citing articles on Scopus
    View full text