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Article Excerpt
1. Introduction
Understanding how customers respond when they cannot find the product they are looking for is of paramount importance for both retailers and producers (Rajaram and Tang, 2001; Gruen et al., 2002; Colacchio et al., 2003). When a customer who demands a particular product cannot find it on the shelf, she may delay her purchase, decide not to buy the item, buy the item at another store, or substitute the item with another product (Campo et al., 2003). The substitution rate between two products is defined as the rate at which a customer arriving with the intention of purchasing a particular product purchases the other one when the product she intends to buy is not available. Recent studies have reported that on average 40% of consumers purchase a substitute product when they are faced with an out-of-stock situation (Gruen et al., 2002).
From the producers' perspective, product substitution may result in a partial loss due to a substitution to a lower-priced item (in the case of substitution to the same brand) or, more unfavorably, in a demand shift within the category to competitors' items (in the case of substitution to a different brand). Similarly, retailers can improve their profits by taking product substitution into account when making product range and inventory decisions about a category. More precisely, they can use substitution rates to determine the optimal mix of products, and the optimal inventory levels of products included in the range (see, e.g., van Ryzin and Mahajan (1999), Smith and Agrawal (2000), Netessine and Rudi (2003), Cachon et al. (2005) and Kok and Fisher (2007)).
If a product is not carried by the retailer, or, it is carried by the retailer but it is temporarily out of stock, customers will not be able to find that item in the store. Although both of these situations may lead to product substitution, the first case is due to a strategic product mix decision, whereas the latter is due to demand uncertainty and inventory decisions. In this study, we focus on stock-out-based substitutions. Understanding customer response in this setting is also instrumental in the management of product range. Furthermore, despite the adoption of various initiatives, empirical studies show that the measured stock-out rates in retailing are quite high. Studies reported by Andersen Consulting (1996) and Gruen et al. (2002) show that on average 8.3% of the stock keeping units that a typical retailer carries are not in stock at a particular moment in time.
Although inventory and product mix planning under product substitution problems are analyzed extensively in the literature, the number of studies that focus on the estimation of substitution rates is limited. Substitution rates are used as parameters in inventory and product range planning problems, and the objective of this study is to propose a practical method that can be used to estimate the substitution rates by using information that is readily available to retailers. Since inventory data is reported to be unavailable or highly inaccurate (DeHoratius and Raman, 2004), our proposed method does not rely on it and uses only point of sale (POS) data. Furthermore, no assumptions are made regarding the inventory control policy used by the retailer and the characteristics of the customer demand arrival process.
Our state-space-based method clusters POS intervals into states where each state corresponds to a specific substitution scenario that depends on the availability of the products under consideration. We then consolidate available POS data for each state, and estimate the substitution rates using the consolidated information.
The stock-out situations and resulting substitution actions affect the observed sales of a product in such a way that sales information no longer reflects the core demand of the product. Therefore, the estimation of demand from sales information can be considered to be a problem of the estimation of censored and truncated distributions from a right-censored sample (see, e.g., Greene (1997) and Allison (2002)). A number of studies examine estimators for such distributions (see, e.g., Mullahy (1986), Shaw (1988) and Grogger and Carson (1991)). However, in order for these estimation methods to perform accurately, the truncated or censored parts should be small. Furthermore, these methods do not give information about the substitution structure.
The closest study to the problem analyzed in this paper was presented by Anupindi et al. (1998). In that paper a maximum likelihood estimation method was presented to estimate the arrival rates and stock-out-based substitution rates in a setting with Poisson arrivals and the simultaneous replenishment of products. Furthermore, this method requires that inventory data be available all the time. A comparison of our method with the method proposed in Anupindi et al. (1998) shows that our method performs satisfactorily even when inventory-related information is not available.
The main contribution of this paper is the development of a method that can be used under very general conditions to estimate product substitution rates that are required by inventory and product range planning methods. Our method does not assume any arrival, substitution or inventory control structure, and it only requires POS data.
The outline of the paper is as follows. In Section 2, we describe our model and estimation method. In Section 3, we explain the implementation of our method for two different information availability situations. In Section 4, we provide a computational evaluation of our method's performance. In Section 5 we compare the performance of our method with that of the maximum likelihood estimation method presented in Anupindi et al. (1998). Finally, we present our concluding remarks in Section 6.
2. Model
2.1. Model description
We consider a retailer that stocks and sells N products in a category. Demand rates for these N products are [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.N], respectively. We assume that the retailer has access to the POS data for the period [0, T], i.e., information on the number of units sold of each product in each POS interval. Each POS interval is assumed to have a length of [tau] unit-time periods.
If customers cannot find their first-choice product on the shelf, the demand for that product will either spillover to another product according to a probabilistic substitution structure or will be lost. With probability 1 - [psi] customers do not substitute. Given that a customer decides to substitute, the probability of substituting product j for product i is [[beta].sub.ij]. Then, for an arriving customer, the probability of substituting product j for product i is [[alpha].sub.ij] = [psi][[beta].sub.ij]. If the second-choice product is not available either, then the demand is lost. In other words, we put one substitution attempt restriction, which is also imposed in Anupindi et al. (1998) and Smith and Agrawal (2000). When the service levels are reasonably high, the impact of the secondary level substitutions may be negligible; therefore, a model with the one substitution restriction can still generate reliable estimates of the primary substitution rates. Kok et al. (2006) state that it is also possible to approximate a multiple substitution attempt model with a single-attempt model by adjusting the parameters.
The most common customer choice model that operates under the one substitution attempt restriction is the market-share-based substitution model (see, Smith and Agrawal (2000), Netessine and Rudi (2003) and Kok and Fisher (2007)). In the market-share-based model, the substitute product is chosen according to the substitution probability matrix
[[alpha].sub.ij] =[psi][[lambda].sub.j]/[[SIGMA].sub.(l[member of]N\{i})[[lambda].sub.l]],I, j = 1,2, ..., N, and i[not equal to]j.
Smith and Agrawal (2000) define three additional choice models that operate under the single substitution attempt assumption:
Random:
[[alpha].sub.ij] = [psi]/N - 1, i, j = 1, 2, ..., N, and i[not equal to]j;
Substitution to adjacent product:
[[alpha].sub.1, 2] = [[alpha].sub.N, N - 1] = [psi]; [[alpha].sub.i, i - 1] = [[alpha].sub.i, i + 1] = [psi]/2, i = 2, ..., N - 1
and Substitution to a single item:
[[alpha].sub.ik] = [psi], k =...
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