Production, Manufacturing and Logistics
The newsboy problem with resalable returns: A single period model and case study

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Abstract

We analyze a newsboy problem with resalable returns. A single order is placed before the selling season starts. Purchased products may be returned by the customer for a full refund within a certain time interval. Returned products are resalable, provided they arrive back before the end of the season and are undamaged. Products remaining at the end of the season are salvaged. All demands not met directly are lost. We derive a simple closed-form equation that determines the optimal order quantity given the demand distribution, the probability that a sold product is returned, and all relevant revenues and costs. We illustrate its use with real data from a large catalogue/internet mail order retailer.

Introduction

In many businesses, customers have the legal right to return a purchased product within a certain time frame. The money is then partially or fully reimbursed and the product can be resold if the quality is good enough and there still exists demand for it.

For several reasons, such returns are especially apparent in catalogue/internet mail order companies. First of all, customers buy via a catalogue or a portal such as the internet and thus do not get to see the physical product before making their purchase decision. Consequently, the product often turns out to be the wrong size or shape or the color differs slightly from that shown in the catalogue or on the internet. Second, the main attractiveness of this ‘distant shopping’, being the ease with which one can order products from home without going anywhere, simultaneously constitutes its main downside. It is easy to return products. After filling out a return form, the product is collected or can be returned via mail. The purchase price and often also the shipping costs are refunded. Moreover, the fact that one does not have to bring a product back personally makes the return process anonymous. For the catalogue/internet mail order retailer that motivated this study (see also the next section), return rates can be as high as 75%.

Since returned products can be resold (if they are undamaged and returned before the end of the selling season), returns should be taken into account when taking ordering decisions. In this paper, we will show how this can be done for the case that a single order is placed for each product. Of course, this order should arrive before the start of the season. We note that this single order case is not unrealistic. Large ordering lead times (e.g. due to production in south-east Asia) and short selling seasons (one summer or one winter) often force mail order retailers to order the entire collection long before the start of the season.

So, we analyze the problem of determining the optimal order quantity for a single order, single period problem. This problem is well known as the newsboy problem or the newsvendor problem, and has been studied extensively in the literature. See Khouja [3] and Silver et al. [7] for overviews. However, to the best of our knowledge, Vlachos and Dekker [8] are the only ones who include a return option for customers.

Before discussing the model and results of Vlachos and Dekker [8], it is important to stress that we focus on a return option for customers. A number of papers have appeared in the literature, e.g. Kodama [4] and Lee [5], that study the newsboy problem with a return option for retailers. In those papers, retailers can return (part of the) unsold items at the end of the selling season and receive a full or partial refund. By offering such a return option to retailers, a wholesaler/supplier can overcome the so-called double marginalization problem and increase supply chain profit. In this paper, we do not study a wholesaler–retailer model with a return option for the retailer. Indeed, in the case study on which our model is based (see the next section), the suppliers are manufacturers that do not accept returns.

The analysis of Vlachos and Dekker [8] is based on two very restrictive assumptions. The first is that a fixed percentage of sold products will be returned. As a result, part of the variability in the net demand is ignored. Since demand variability is a key factor in the analysis, this leads to sub-optimal ordering quantities. The second restrictive assumption is that products can be resold at most once. But if return rates are high, it is likely that products are resold more than once. Indeed, that often occurs at the mail order retailer that will be discussed in the next section.

In this paper we analyze the newsboy problem with resalable returns, but without these restrictive assumptions. Each sold product is returned with a certain probability. Products can be resold any number of times. We derive a simple formula that determines the optimal ordering quantity. Using real data of the mail order retailer, we illustrate the use of this formula for a large selection of products from a certain selling season. Furthermore, we compare the resulting order quantities to those that were proposed by Vlachos and Dekker [8] and to the orders the company would have placed using its ordering rule.

The remainder of this paper is organized as follows. In Section 2, we describe the case study that motivated this research. In Section 3, we present the mathematical model and discuss the assumptions. Section 4 reviews the approximate analysis of Vlachos and Dekker [8]. The exact analysis is presented in Section 5. Section 6 discusses the available data from the case study, which we use to illustrate the results. Our procedure for estimating the mean and variance of gross demand for every product is shown in Section 6.1. For our computations, we assume Normality of gross demand. In Section 6.2, we show that, given this assumption, the distribution of net demand is approximately Normal for all products in our data sets. Section 7 describes the computational experiments and in Section 7.1 we show and analyze the cumulative results over all products as well as the detailed results for a selected group of nine products. Finally, we summarize our findings and indicate directions for further research in Section 8.

Section snippets

Case study

This research is motivated by a case study at a large mail order retailer. This company sells a broad range of hardware and fashion products via a catalogue and to a lesser extent via the internet. By law, customers have the right to return products within 10 days after delivery. In practice, however, the company allows returns after this period and the bulk of returns arrives in the second and third week after delivery.

This research concentrates on the fashion products, since these all have a

Model and assumptions

We assume that there are no interdependencies between ordered items, which holds for the case study that was described in the previous section, and therefore analyze a single item model. There is a single replenishment opportunity at which Q products are ordered. Those products arrive before the start of the selling season. The total number of products demanded during the season, i.e. gross demand, is denoted by G. Its mean and standard deviation are denoted by μG and σG, respectively.

An approximation of the optimal order quantity

Before we determine the exact optimal order quantity in the next section, we first present and discuss an approximation that was proposed by Vlachos and Dekker [8]. They studied the same model as we do. However, their analysis is based on two additional assumptions that will be given below. So the resulting ordering quantity is only approximately optimal. In a later section, we will study its performance for a number of real life examples.

The two simplifying assumptions are as follows:

  • Products

The exact optimal ordering quantity

Recall that the approximately optimal ordering quantity, derived in the previous section, was based on two simplifying assumptions. Those assumptions were needed, because the focus was on gross demand rather than net demand. In this section, we will not use any simplifying assumptions, and use a ‘net demand approach’ to determine the exact optimal ordering quantity.

Recall that the expected net revenue pN includes the collection cost if a product is returned and the salvage revenue if a product

Data

For our computations, we use data provided by a large mail order company. See Section 2 for a short description of the company. Recall from that section that we focus (for each product) on the order that is placed after the previews of season demand and of the return probability become available, and consider this to be the only (newsboy) order that is placed. So, we will restrict the discussion to data that is relevant for placing this order.

There are two data sets. Set 1 consists of 4761

Computational experiments

Using the data in Set 2, we perform computational experiments to compare the exact optimal order quantity Q* to the approximate order quantity Q^ and to the order quantity Q (resulting from the order rule) currently used by the retailer. The current order quantity is equal to the expected net demand, i.e.,Q=μP(1-rk).Recall from Section 2 that Q is actually an order-up-to level instead of an order quantity, since the preview order on which we focus is the second order. However, as we argued

Conclusion

We derived a simple closed-form formula that determines the optimal order quantity Q* for a single period inventory (newsboy) problem with returns. Using real data from a large catalogue/internet mail order company, Q* was compared to an approximation Q^ proposed in a previous study and to the order quantity Q currently used by the company. It turned out that Q^ differs more than 10% from Q* in most cases. The associated profit reduction is generally smaller than 5%, but more than 10% in cases

Acknowledgements

The authors thank the two anonymous referees for their helpful comments. The research of Dr. Ruud H. Teunter has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

The research presented in this paper is part of the research on re-use in the context of the EU sponsored TMR project REVersed LOGistics (ERB 4061 PL 97-5650) in which the Erasmus University Rotterdam (NL), the Otto-von-Guericke Universität Magdeburg (D), the Eindhoven University of Technology

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