An inventory model with dependent product demands and returns

doi:10.1016/S0925-5273(00)00080-3

International Journal of Production Economics

Volume 72, Issue 1, 30 June 2001, Pages 73-87

https://doi.org/10.1016/S0925-5273(00)00080-3 Get rights and content

Abstract

In this paper an inventory model for a single reusable product is investigated, in which the random returns depend explicitly on the demand stream. Further, the model distinguishes itself from most other research in this field by considering leadtimes and a finite planning horizon. We show that neglecting the dependency between demands and returns of products may lead to bad performance with respect to total average relevant costs. Additionally, our results enable us to determine the minimal recovery probability or the minimal length of the planning horizon for which reuse is profitable.

Introduction

A number of papers have studied inventory policies under consideration of random returns (see [1] for an overview), but most of these assume that the return process is independent of the demand process (see e.g. [2]). This type of modeling may be good for situations in which no information about a dependence structure is available or situations with many different sources for the returns. But there are also situations in which a model with a dependence structure between the demands and the returns seems to fit better. We have such situations in the case of rented or leased products, or if delivered items are returned to the original manufacturer only. We can also think about packaging and transportation materials.

In this paper we develop and investigate an inventory model for a single, reusable product in which the random returns depend explicitly on the demand stream. A similar situation is also discussed in [3], but without considering backorders. Additionally, they do not consider the leadtime for purchasing. Also Yuan and Cheung [4] consider a model with dependent returns, but without a purchasing leadtime. We relax this assumption and allow a positive purchasing leadtime.

Silver and Kelle [5] determine an optimal purchasing policy for reusable containers by transferring the stochastic model in a deterministic one. We use a Markov-chain approach in order to determine the optimal order-up-to policy with respect to total average relevant cost.

Although in many papers the long-term behavior of the inventory is investigated (see e.g. [6]) we consider a finite planning horizon. The reason for this is that nowadays, the life-cycle of products is getting shorter and shorter, because of fast changing trends and new developments. This can be observed, in particular, for electronic products. Therefore, inventory models for short-term control have to be investigated and preferably objective functions should be used where no steady-state assumptions are necessary.

In this paper we show that there is a great difference between a situation with independent demands and returns and a situation, where the returns depend on the demand. Therefore, neglecting the dependency of the returns on demands may lead to poor policy performance. We show how to compute the optimal order-up-to level for a situation with dependent returns. In such a situation the optimal inventory level is less than in the other case and the average relevant costs can be reduced by using the information about the dependence structure. The influence of the return probability and the length of the planning horizon on policy performance is investigated. Additionally, we compare a system with product returns with a system without product returns. Our results enable to answer questions regarding the profitability of product reuse in relation with return probability and time-in-market.

The paper is further organized as follows. In Section 2 we start with a detailed description of the model and its underlying assumptions. In Section 3 we develop an approximation to the total average costs, which we use as the objective function, followed by a numerical study (Section 4). Finally, we summarize our results and conclusions and give an outlook for further research.

Section snippets

Model definition and assumptions

We consider a single, reusable product. Because of fast changing trends in the market, this product is planned to be produced only for a limited planning horizon T which is the same as the time in market. Practical limitations are the reason for reviewing the inventory only periodically. We assume, that the length of one period is known and constant. Without loss of generality, we take the length of a period equal to one, and the periods are numbered by $t=1, 2,…,T$ .

We assume that the demands per

Determination of the objective function

In order to determine the average relevant costs in one period we first introduce the inventory position I_t and illustrate the differences between the dependent and the independent case. Then we are able to determine the optimization problem.

Numerical study

Before we discuss the influence of the recovery probability and the length of the planning horizon on the economic profitability of product reuse (Section 4.2), we first compare the case of dependent returns with the case of independent returns. In our numerical examples we limit ourselves on constant demand and return rates since the focus of our investigations are effects resulting from the dependency and not from dynamic demand and return patterns. The following base case scenario for the

Summary, conclusions and outlook

In this paper we have developed a periodic review inventory system with product returns that depend explicitly on the demand stream. The system, which includes leadtimes and a finite planning horizon, is controlled by a fill-up policy. We showed how both the optimal policy and the minimal average total relevant cost depend on the recovery probability. Also we indicated the influence of the planning horizon on the minimal total average costs. The influence of the planning horizon on the optimal

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A dynamic inventory system with recycling
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Modelling the influence of returns for an omni-channel retailer
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Citation Excerpt :
The finite horizon model coupled with a detailed terminal period cost structure, as compared to zero terminal cost assumption used in papers reviewed in Table 2, is another unique feature of this paper. Under these distinct and realistic elements of our model, we quantify the value of incorporating returns in inventory management as in Kelle and Silver (1989), Buchanan and Abad (1998), and Benedito and Corominas (2013), which is increasing in the return rate (Kiesmüller and Van der Laan, 2001; Zerhouni et al., 2013). First, return-smart retailers should reconsider their inventory replenishment decisions by incorporating return flows.
Retailers have to deal with increasing levels of product returns as the shares of e-commerce sales soars. With this increase, it is no longer feasible to dispatch returned products to outlets or landfills, hence retailers must re-evaluate them both to maximize profit and to minimize their environmental impact. Our objective is to study a retailer’s optimal inventory control policy under product returns to maximize expected profit which is the sales revenue minus the procurement, backorder, holding, and salvage costs incurred in a finite horizon. We model a period’s returns to be stochastically dependent on the previous period’s sales quantity. Using dynamic programming formulation, we solve for the optimal periodic review inventory policy and provide structural results on the optimal policy of the final period. Through numerical studies, we show that incorporating detailed sales-dependent returns could increase a retailer’s expected profit by 23%. Ignoring this dependency in determining the optimal inventory policy results with increased order frequency, higher levels of backorders and more leftovers which could eventually end up in a landfill, but above all could lead to a significant overestimation of the resulting profit.
Optimal pricing and inventory control strategy for a continuous-review system with product return
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We consider joint pricing and inventory control strategies for a stochastic inventory system with product return. Customer's demand and product return follow independent point processes with demand rate being price-dependent. Using a general verification theorem, we prove that an $(s^{⁎}, S^{⁎}, p^{⁎})$ policy is optimal, where the optimal price $p^{⁎} (x)$ is a quasi-concave function of inventory level with a nonpositive changeover point. Furthermore, we propose an algorithm to search for the optimal policy parameters and numerically study the effect of product return.
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An inventory control problem of a hybrid manufacturing/remanufacturing system under stochastic demand is studied. The paper proposes a successive substitution strategy between prime, budget and low budget segments of a market. Meaning a manufactured product may substitute for a remanufactured product at a discounted rate. The success of this proposal proportionally is dependent on the discount rate. Similarly, a once-remanufactured product may substitute for a twice-remanufactured product. In this proposal, a sales-dependent return is adopted to enhance returns prediction. This tactic utilizes a virtual stock of previous period sales instead of a used product stock. In such arrangements, only a fraction of customers accept keeping and returning their used product at the end of the product's lifetime. This is a push system because as soon as the company makes an order for remanufacturing, a core will be returned by a ready/potential customer and instantly will be pushed into the remanufacturing process. The system is formulated by a bi-objective Markov decision process considering profit and customer satisfaction criteria to provide optimal policy in the form of a lookup table. The heuristic policy is then characterized by utilizing five control parameters. This is a state-dependent threshold policy with a small deviation from the optimality. Then the effects of thresholds are investigated applying DOE and ANOVA. Finally, the performance of the successive substitution strategy using real data from the tire industry is examined. The results show that profitability, serviceability and core acquisition can be improved by using the proposed strategy.
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^☆: The research presented in this paper makes up part of the research on reuse in the context of the TMR project REVersed LOGistics financially supported by the European Union (ERB 4061 PL 97-650) in which apart from Eindhoven University of Technology take part the Erasmus University Rotterdam (NL), the Aristoteles University of Thessaloniki (GR), INSEAD (F), the Otto-von-Guericke-Universitaet Magdeburg (D) and the University of Piraeus (GR). The second author greatly acknowledges the financial support provided by the NWO.

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An inventory model with dependent product demands and returns☆

Abstract

Introduction

Section snippets

Model definition and assumptions

Determination of the objective function

Numerical study

Summary, conclusions and outlook

Quantitative models for reverse logistics: A review

European Journal of Operational Research

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

OR Spektrum

A dynamic inventory system with recycling

Naval Research Logistics Quarterly