مسئله توزیع آزاد پسر روزنامه فروش با بازده فروش مجدد

We study the case of a catalogue/internet mail order retailer selling style goods and receiving large numbers of commercial returns. Returned products arriving before the end of the selling season can be resold if there is sufficient demand. A single order is placed before the season starts. Excess inventory at the end of the season is salvaged and all demands not met directly are lost. Since little historical information is available, it is impossible to determine the shape of the distribution of demand. Therefore, we analyze the distribution-free newsboy problem with returns, in which only the mean and variance of demand are assumed to be known. We derive a simple closed-form expression for the distribution-free order quantity, which we compare to the optimal order quantities when gross demand is assumed to be normal, lognormal or uniform. We find that the distribution-free order rule performs well when the coefficient of variation (CV) is at most 0.5, but is far from optimal when the CV is large.

In many countries, customers have the legal right to return unused purchases within a specified number of days after purchase, especially in the case of distant selling. The original purchase cost is then partially or fully reimbursed. As a result of this right, retailers selling products via mail order or over the internet generally have to deal with large volumes of returns. Because the sales process is remote, customers cannot see, feel and try the actual product, which often leads to a wrong decision. We will concentrate on the example of a mail order/internet retailer selling fashion products, since these have particularly high return rates. However, the model that we propose and our findings are generally applicable to other types of style goods as well (e.g., toys, personal computers or consumer electronics). Common reasons for returning fashion products are a wrong size, a change of mind (remorse) or the fact that the actual color differs slightly from the displayed one. The presence of return flows changes inventory control significantly (Fleischmann et al., 1997). First, the retailer has little control over the return flow in terms of quantity, quality and timing. Second, ordering decisions and processing of returned products have to be coordinated, since returned products can be resold in most cases. The higher the return rates, the more important these factors become. In the case of a mail order/internet retailer that we consider, return rates are usually larger than 18% and can be as high as 74% for specific fashion products (Mostard and Teunter, 2005). The management of return flows has received growing attention in the past decade. The whole of logistic activities to collect, disassemble and recover (parts of) used products or materials for the purpose of recapturing value or proper disposal is known as reverse logistics (Revlog Website, 1999 and Rogers and Tibben-Lembke, 1999). Much research in this field has been dedicated to the implications of return flows for the areas of distribution planning, inventory control and production planning. Fleischmann et al. (1997) have performed a review of the mathematical models that have been proposed in this context. However, the vast majority of the proposed inventory control models for reverse logistics concentrate either on returns that need extensive recovery (e.g., repair or remanufacturing) or on end-of-life products destined for recycling. In our case, the returned products are generally in an as-good-as-new condition and can be resold directly after testing and possibly repackaging provided there is enough demand and they are returned before the end of the selling season (in the case of seasonal products). We consider the inventory control problem for the case of a mail order/internet retailer selling fashion products. Besides the high return rates, seasonality, large supply lead times and lack of data are three important factors in this case that complicate inventory control. These are all related to the type of product, fashion clothing. We will next discuss each factor separately. Fashion products are seasonal: Fisher and Raman (1996) note that most fashion apparel companies introduce a completely new product line every season which must be designed and produced in time to be sold during a concentrated retail selling season. They also point out that the costs of excess inventory that must be sold below purchase cost at the end of the season and of lost sales due to stockouts are high in the apparel industry because of unpredictable demand and a complex supply chain. The supply lead times of fashion products are usually long: Therefore, retailers have to settle their entire season's order quantities well before they have an opportunity to observe actual sales performances ( Mantrala and Raman, 1999). Because of this single-order opportunity, it is natural to use a newsboy-type model for analyzing our problem. However, the standard newsboy model does not allow for returns. Only recently, Vlachos and Dekker (2003) first studied ordering policies for single-period products with returns. They extend the newsboy problem while making two simplifying assumptions. First, they assume that products can be resold only once. Second, they assume that a fixed percentage of sold products is returned and resalable. Considering several different return options, based on different handling of the returned products, they derive optimal order quantities for the various models resulting from these options. By numerical experiments, they show that the optimal classical newsboy quantity is far from optimal when return rates are high. In a following study, Mostard and Teunter (2005) argue that the two assumptions underlying the model of Vlachos and Dekker lead to a suboptimal order quantity. In practice, products can be returned and resold several times during a season, which contradicts the first assumption. Moreover, due to the second assumption, part of the variability in the number of (resalable) returns, given gross demand, is ignored. Mostard and Teunter drop these assumptions. Taking a net demand approach, they derive a simple closed-form equation that determines the optimal order quantity given the gross demand distribution, the probability that a sold product is returned, and all relevant revenues and costs. Using real data from a large mail order company, they compare this optimal order quantity to both the order quantity proposed by Vlachos and Dekker and to the company's order quantity. The former generally differs more than 10% from the optimal order quantity, while the associated reduction in profit is generally small but can be large in specific cases. The latter turns out to be far from optimal. There is a lack of historical data of fashion products: In order to determine the order quantity of a certain product, retailers need reasonable estimates of the return rate and the distribution of demand. But for obtaining reliable estimates, one needs historical data. Due to the fact that the collection of fashion products that retailers carry is renewed every season, there are usually no historical data on these products. Therefore, the limited historical data on similar products belonging to the same product group have to be used. However, since consumer preferences are highly unpredictable ( Jain and Paul, 2001), even comparisons of similar products in successive seasons are unlikely to lead to accurate forecasts. Therefore, firms often gather additional data for the products of the current season by asking a number of experts for their estimate and by distributing a preview catalogue among a selected group of customers prior to the season and offering a discount for pre-season orders. These additional data can be combined with the historical data on similar products in various ways to obtain forecasts for the mean and standard deviation of demand (see, e.g., Fisher and Raman, 1996; Mostard and Teunter, 2005), but are usually insufficient to estimate the shape of the demand distribution. Although there are studies in inventory control dealing with higher moments (e.g., Lau and Zaki, 1982), to the best of our knowledge, there is no literature on estimating more than the first two moments of the exact distribution function of total season demand in a similar context as discussed here, where only limited empirical data are available. Several authors, e.g. Fisher and Raman (1996) and Kurawarwala and Matsuo (1996), acknowledge the lack of demand data on fashion products and other style goods for forecasting demand. We therefore explore the distribution-free newsboy model in this paper. Distribution-free means that only the first two moments of demand are assumed to be known. Gallego and Moon (1993) first studied the distribution-free newsboy problem. They prove the optimality of Scarf's ordering rule for this problem. This rule finds the order quantity that maximizes expected profit against the worst possible demand distribution with a certain mean and variance (Scarf, 1958). The maximum amount that can be gained by knowing the complete demand distribution is shown to be negligible for most practical problems. This is shown for a variety of cases: the recourse case (in which there is one more chance to order after the initial order is placed), the fixed ordering cost case, the random yield case and the multi-product case. Since the paper of Gallego and Moon, the distribution-free approach has been adopted in a number of other studies (e.g., Alfares and Elmorra, 2005, Gallego et al., 2001, Moon and Choi, 1995, Moon and Choi, 1997, Moon and Choi, 1998, Moon and Gallego, 1994, Moon and Silver, 2000, Ouyang and Chang, 2002, Silver and Moon, 2001 and Wu and Ouyang, 2001). Numerical examples in these papers all show that Scarf's ordering rule is near optimal in a distribution-free setting and that it is robust. Furthermore, it is computationally simple and easy to understand, which makes it valuable in practice. A number of papers (e.g., Fisher et al., 2001, Hausman and Sides, 1973 and Kurawarwala and Matsuo, 1996; Murray and Silver, 1966) have addressed the forecasting and inventory management problem of style goods, which is apparent in many industries, such as the catalogue seller business. Moreover, the single-period problem (also referred to as the newsboy or newsvendor problem), in which there is a single order opportunity, has been studied widely in the literature (e.g., Abdel-Malek et al., 2004, Khouja, 1999, Lau, 1980, Petruzzi and Dada, 1999, Schweitzer and Cachon, 2000 and Silver et al., 1998). However, none of these papers addresses the problem covered here, where the focus is on determining the optimal order quantity for style goods when only the first two moments of demand are known and a substantial percentage of products delivered to customers is returned and available for resale. In this article, we apply the distribution-free approach to the single-period problem with returns. We compare the resulting order quantity and corresponding expected profit to the optimal order quantity and expected profit (in the case that the complete demand distribution is known). In this way, we determine the value of additional demand information, i.e. the extra profit that can be gained by knowing the complete distribution of demand instead of its first two moments. We will do this for a wide range of parameter values and under the assumption that gross demand is either normal, lognormal or uniform. The remainder of this paper is organized as follows. In Section 2, we introduce the notation and the assumptions underlying our distribution-free newsboy model. The distribution-free order quantity is derived in Section 2.1. In Section 4, we compare the distribution-free order quantity and expected profit to the optimal order quantity and expected profit for a wide range of parameters. But in order to compute the optimal order quantity and associated expected profit we need to know the distribution of net demand. Therefore, in Section 3, we will first examine the shape of the net demand distribution under the assumption that gross demand is normal, lognormal or uniform, to see whether net demand follows the same type of distribution. Finally, we present our main findings and conclusions in Section 5.

We derived a simple closed-form formula that determines the order quantity for the distribution-free single-period (newsboy) inventory problem with returns in which only the mean and variance of gross demand are known. In order to account for the returns, the distribution-free order quantity was derived using a net demand approach. We compared the distribution-free order quantity to the optimal order quantity under the assumption that the gross demand distribution is either normal, lognormal or uniform. In order to be able to determine the optimal order quantity, we assumed that the net demand follows the same type of distribution as gross demand. This assumption was validated in Section 3 by comparing the simulated cdf of net demand to the normal, lognormal and uniform cdf with the same mean and standard deviation for a large number of examples. Using wide ranges of the relevant parameters, we compared the distribution-free and optimal order quantities and their respective expected profits. It turned out that for a small coefficient of variation (CV=0.1), the distribution-free order quantity differs around 1% (positive or negative) from the optimal order quantity in most cases, while the associated differences in expected profits are negligible for all three distributions. When the coefficient of variation is 0.5, the distribution-free order quantity is often far from optimal (up to +20% from the optimal order quantity), especially for the lognormal and uniform distribution and when the relative profit margin is small. However, the loss in expected profit is still small, around 1% on average for the lognormal and uniform distributions and even less for the normal distribution. When the coefficient of variation is greater than or equal to 1, the distribution-free order quantity is far from optimal and often also results in a considerable loss in expected profit (of up to 74%). Based on these results, we recommend the following to firms that face returns and have to determine single-period order quantities based on limited available data. For products with a coefficient of variation of gross demand (CV) of at most 0.5, just estimate the mean and standard deviation of gross demand and apply the distribution-free order rule in (5). For products with a CV of more than 0.5, try to estimate the entire distribution function and determine the optimal order quantity using (2). However, as pointed out in the Introduction, it is usually impossible to obtain a reliable estimate of the entire distribution function. In this case, there is no other option than to use the distribution-free ordering rule and to incur the large loss in expected profit.