A production/remanufacturing inventory model with price and quality dependant return rate

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Abstract

Inventory management of produced, remanufactured/repaired and returned items has been receiving increasing attention in recent years. The available studies in the literature consider a production environment that consists of two shops. The first shop is for production and remanufacturing/repair, while the second shop is for collecting used (returned) items to be remanufactured in the first shop, where demand is satisfied from producing new and from remanufacturing/repairing returned items. Numerical and analytical results from these developed models suggested that a pure (bang–bang) policy of either no waste disposal (total remanufacturing) or no remanufacturing (pure production and total disposal) is the best strategy, while the mixed strategy (a mixture of production and remanufacturing) is the optimum case under certain limited assumptions. In practice, the quality of the returned items and the purchasing price that reflects this quality is what usually governs a collection (or return) policy of used items. Unlike those available models in the literature, this paper suggests that the flow of returned items is variable, and is controlled by two decision variables, which are the purchasing price for returned items corresponding to an acceptance quality level. Deterministic mathematical models are presented for multiple remanufacturing and production cycles.

Introduction

The flow of products in supply chains is from upstream to downstream, i.e., from the supplier’s supplier to the customer’s customer. Shorter product life cycles and changes in customers’ consumption behaviors resulted in faster product flows and subsequently faster generation of waste and depletion of natural resources (e.g., Beamon, 1999). This gave rise to the drive towards collecting and remanufacturing used/returned products to extend their useable lives and thus reduce waste and conserve natural resources. Furthermore, economical incentives enticed and later governmental legislations compelled companies to initiate product recovery (e.g., remanufacturing, repairing, recycling, etc.) programs. Like supply chain, reverse logistics manages the flow of products, however in the opposite direction for remanufacturing or other purposes (i.e., from downstream to upstream), and therefore managing inventory in reverse logistics has been stressed in several studies (e.g., Fleischmann et al., 1997, Minner, 2001).

Although reverse logistics is relatively a new term, initial attempts to address the inventory of remanufactured items or products dates back to the 1960s, with Schrady (1967) being the first to investigate a repair–inventory system. He developed an EOQ model for repairable items which assumes that the manufacturing and recovery (repair) rates are instantaneous with no disposal cost. Schrady (1967) assumed a single manufacturing batch and multiple repair batches. Nahmias and Rivera (1979) extended Schrady’s model to allow for a finite repair rate with the assumption of limited storage in the repair and production shops.

The production/remanufacturing inventory problem started taking a new direction in the 1990s. Richter, 1996a, Richter, 1996b investigated the EOQ model for stationary demand that is satisfied from producing items of a certain product using new materials and components, and from repairing used/returned items that are collected from the market at some rate. The production environment described in Richter, 1996a, Richter, 1996b consists of two shops; with the first shop is for production and recovery, while the second shop is for collecting used/returned items. Some of these collected used/returned items are disposed outside the second shop at a rate, which may display the ecological behavior of the producer. In a follow-up work, Richter (1997) extended the cost analysis of his earlier works (Richter, 1996a, Richter, 1996b) to show that a pure (bang–bang) policy of either no waste disposal (total repair) or no repair (total waste disposal) dominates a mixed strategy of waste disposal and repair. Richter and Dobos (1999) extended Richter’s earlier work by considering an integer nonlinear programming problem with similar findings as before. Dobos and Richter (2003) investigated a production/recycling system with constant demand that is satisfied by non-instantaneous production and recycling with a single repair and a single production batch per interval. In a follow-up paper, Dobos and Richter (2004) generalized their earlier model (Dobos & Richter, 2003) by assuming multiple repair and production batches in a time interval. However, Dobos and Richter (2004) implied that their model has limitations since pure strategies of either no waste disposal (total repair) or no repair (total waste disposal) are technologically infeasible, and suggested that a more general and meaningful model would be to consider the quality of returned items.

Recently, and along the same line of research, Dobos and Richter (2006) extended their previous work by considering the quality of returned items. They considered two strategies to manage the collection of used items: (1) repurchase all used items and reuse only a maximal proportion of them (strategy 1) or (2) buyback only a proportion of the used items and decide how much of them to reuse (strategy 2). In their models, Dobos and Richter (2006) assumed that the proportion of the demand returned to be reused is dependent on two inter-dependent decision variables, which are: (1) buyback proportion and (2) use proportion. The product of these variables (0  (buyback proportion) × (use proportion)  1) is the proportion of reusable items, which represents the return rate = (demand rate) × (buyback proportion) × (use proportion), and it was assumed by Dobos and Richter (2006) to be fixed at a value. That is, the buyback proportion and the use proportion vary, but their product is a fixed value. This assumption limits their model as it compared two strategies for a fixed rather than a variant return rate. In addition, and for the sake of argument, let us assume a case where no ecological constraints are considered and a decision on which strategy to adopt is solely based on economical feasibility. For such a case, if recycling (i.e., recovery) is expensive, then the strategy of pure production should be favored, which Dobos and Richter (2006) did not consider. In addition, in their model, Dobos and Richter (2006) assumed that a pure recycling/reuse strategy is more cost effective than a pure production strategy. In our opinion, this assumption limits the application of their model further, especially for the case when the cost of a pure recycling/reuse strategy is either equal to or more than a pure production strategy. This point will be discussed later in this paper. Furthermore, Dobos and Richter (2006) ignored the purchasing price of raw materials in the forward flow (production/remanufacture), and the purchasing price of collected used items in the backward flow (returns). Therefore, a major difference between the work of Dobos and Richter (2006) and the one presented herein is that this paper assumes the return rate of used items (a decision variable) is dependent on two decision variables, the purchasing price for returned items and its corresponding acceptance quality level.

The inventory management research in the reverse logistics context is not limited to the studies surveyed above. Other researchers have developed models along the same lines as Schrady, 1967, Richter, 1996a, but with different assumptions. Examples of recent works, including, but not limited to, are those of Teunter, 2001, Teunter, 2004, Inderfurth et al., 2005, Konstantaras and Papachristos (2006), El Saadany and Jaber, 2008, Jaber and Rosen, 2008. These works and those surveyed in earlier paragraphs all assumed a constant return rate and ignored the factors that govern this rate. In practice, the purchasing price of a collected used (returned) item with a certain quality governs the usefulness of the remanufacture/repair process. For example, if the returned items are expensive or they have a poor quality, the whole remanufacture/repair process might not be economically feasible and therefore pure production with no returns might be the optimum solution. Although several researchers called for the need to differentiate the returned units according to their quality (Behret and Korugan, 2009, Blackburn et al., 2004, Bloemhof-Ruwaard et al., 1995, Grubbström and Tang, 2006, Reimer et al., 2000, Smith et al., 1996), there has been no work that modeled the collection rate of used items as price and quality dependent.

This paper extends the models developed in Dobos and Richter, 2003, Dobos and Richter, 2004 by assuming that the collection rate of used/returned items is dependent on the purchasing price (decision variable 1) and the acceptance quality level (decision variable 2) of these returns. This is done by incorporating a price–quality demand function, adopted from Vörös (2002), to model the collection rate of returned items. Vörös (2002) presented demand as a decreasing and increasing exponential functions of price and quality. Vörös (2002) integrated these functions into one function that describes the forward flow of a product, i.e., from the inventory system to the market, where demand increases as selling price (quality) decreases (increases). Vörös’s demand function describes a general and known behavior that is well documented in the literature (e.g., Kalish, 1983, Teng and Thompson, 1996). Since this paper considers the price and quality in the reverse flow, therefore, the logic of the model presented in Vörös (2002) is switched. That is, in the reverse flow, the flow of used/returned items increases as the purchasing price increases, and decreases as the corresponding acceptance quality level increases.

In recent years, some researchers provided clearer definitions to the terms repair, reconditioning, remanufacturing, and recycling. For example, De Brito and Dekker (2004) differentiated between the terms repair and remanufacturing by industry. They suggested that if only a part of the product deteriorates, then recovery options like repair or part replacement or retrieval are considered. King, Burgess, Ijomah, and McMahon (2006) defined the term repair as the correction of specified faults in a product, where the quality of repaired products is inferior to those of remanufactured and reconditioned. This paper adopts the term “remanufacturing”, which refers to repairing, reconditioning, refurbishing or remanufacturing.

In this paper, production, remanufacture, and waste disposal EPQ (economic production quantity) type models are developed and analyzed, where a manufacturer serves a stationary demand by producing new items of a product as well as by remanufacturing collected used/returned items. In these developed models, the return rate of used items is modeled as a demand-like function of purchasing price and acceptance quality level of returns. The model developed herein is a decision tool that helps managers in determining the optimum acceptable acquisition quality level and its corresponding price for used items that are collected for recovery purposes and that minimizes the total system cost.

The remainder of this paper is organized as follows. The next section, Section 2, is for assumptions, notations and description of the production/remanufacturing inventory system that will be investigated in this paper. Section 3 is for mathematical modeling. Section 4 is for numerical examples and discussion of results. This paper summarizes and concludes in Section 5.

Section snippets

Assumptions

This paper assumes: (1) finite production and remanufacturing rates, (2) remanufactured items are as good as new, (3) demand is known, constant and independent, (4) lead time is zero, (5) a single product case, (6) no shortages are allowed, (7) unlimited storage capacity is available and (8) infinite planning horizon

Decision variables

    P

    purchasing price for a single returned item as a percentage of the cost of raw materials required to produce a new item of the product (0 < P < 1)

    q

    acceptance quality level of returned

Mathematical modelling

The models developed in this section extend the models of Dobos and Richter, 2003, Dobos and Richter, 2004 by assuming the return rate of used items follows a demand-like function dependent on two decision variables which are the purchasing price, P, and the acceptance quality level, q, for these returned items. In addition, this paper accounts for the cost of raw materials required to produce a single new unit of the product, Cn, where the monetary value of the purchasing price for a returned

Numerical examples

This section provides four numerical examples to illustrate the behaviors of Models I and II and to draw some conclusions.

Richter, 1997, Teunter, 2001, and Dobos and Richter, 2003, Dobos and Richter, 2004 concluded that the optimal inventory holding strategy in the production–recycling model they discussed is a pure strategy (bang–bang strategy). That is, it is either buyback all used/returned items for remanufacture/recycle with no production option, or produce new items with no buyback or

Summary and conclusions

This paper extended upon the production, remanufacturing/repair and waste disposal model of Dobos and Richter, 2003, Dobos and Richter, 2004 by assuming a variable return rate of used items that follows a demand-like function of purchasing price and acceptance quality level of returns. Two mathematical models were developed. The first assumes a single remanufacturing cycle and a single production cycle, with the second being a generalized version of the first assuming multiple remanufacturing

Acknowledgements

The authors thank the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Deanship of the Faculty of Engineering, Architecture and Science at Ryerson University for supporting this research. The authors thank the anonymous reviewers for their positive comments and suggestions that improved the presentation of the paper.

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