Summary Lecture Notes in Economics and Mathematical Systems

their practical application or results in inaccurate inventory control and consequently, loss of profit. Consequently, the following research questions will be addressed in this work: 1. How can external factors that influence demand be incorporated in inventory optimization to better align supply and demand? 2. Can forecasting and inventory optimization be integrated instead of performing two sequential tasks? 3. How can the time of the last sale be used to estimate unobservable lost sales and substitution behavior? 4. What are the optimal order quantities in a multi-product setting with an aggregated service level target and does the empirical decision maker behave profitmaximizing? 5. Are decision makers in real-world environments subject to the same behavioral biases as in laboratory experiments? 1.3 Outline This research aims to use the data available to retailers in order to better balance excess inventory and out-of-stock situations for perishable products. Using real data containing daily and hourly sales from 2008 on, the inventory replenishment decisions for fruit, vegetables, and bakery products of more than 60 stores were analyzed. Its contribution is to overcome limitations of existing research such as assumptions on the theoretical demand distribution not fitting the demand observations, to improve order decisions by taking additional information into account and to analyze the ordering behavior of a real decision maker from a behavioral perspective. We first review related work on unobservable lost sales estimation, assortment planning and behavioral operations management in Chap. 2. We then develop a data-driven model for single-period problems that integrates demand forecasting and inventory optimization in Chap. 3. The model is distribution-free and takes external variables such as price and weather into account that influence demand. A linear inventory function of the external variables is fitted to historical demand observations to determine its coefficients using Linear 4 1 Introduction Programming (LP). We compare the model to a time-series forecast and regression analysis in a numerical study and analyze the results based on real data. This chapter is based on Beutel and Minner (2012). In Chap. 4 we extend the model from Chap. 3 by assuming that the retail manager has no full demand information and only observes sales. If a stockout occurs, the lost sales have to be estimated. Otherwise, the retail manager would underestimate true demand in future periods and order too few units. We establish sales patterns from days with full availability based on Lau and Lau (1996) to estimate the demand and compare the data-driven model to other parametric and non-parametric approaches in a numerical study. This chapter is based on Sachs and Minner (2014). In Chap. 5 we consider a two-product model where customers either choose not to purchase a product if it is out-of-stock or purchase a substitute if their firstchoice is sold out. We determine the additional demand due to substitution from the sales patterns that we used in Chap. 4 for the unobservable lost sales estimates. We compare the performance of the model to a parametric approach based on Poissondistributed and real data. We analyze empirical decisions in Chap. 6. We first develop a normative model to determine optimal order quantities for an aggregated service level target over several products and identify elements of the optimal policy. We then compare the performance of the model to the decisions of a single manufacturer who supplies retail stores with bakery products. Finally, we investigate whether the decision maker is subject to behavioral biases that were found in laboratory experiments. This is joint work with Michael Becker-Peth (University of Cologne), Stefan Minner (Technische Universität München) and Ulrich W. Thonemann (University of Cologne). Chapter 7 concludes this thesis by summarizing the main findings. We herein state the limitations and areas for future research. Chapter 2 Literature Review In the following we will review literature on unobservable lost sales estimation, assortment planning and behavioral operations management. With our focus on retail, we exclude literature on assortment planning in production from our analysis. In contrast to retail where substitution is consumer-driven, demand substitution in production systems is usually controlled by the supplier side. If demand cannot be filled, downward substitution takes place and excess demand is satisfied from a superior product (see for example Hsu and Bassok 1999). In the assortment planning literature, three main types of substitution can be distinguished based on their causes. If substitution occurs as a response to a temporary out-ofstock situation, it is called stockout-based substitution. If customers substitute one product for another to realize savings from price differences, it is referred to as price-based substitution. Assortment-based substitution takes place when a product is not carried by the store at all and thus, a customer chooses another variant from the available assortment set instead of her favorite product. For a comprehensive review on the assortment planning problem and related aspects, the interested reader is referred to Kök et al. (2008) and Pentico (2008). 2.1 Unobservable Lost Sales The existing literature distinguishes between parametric and non-parametric approaches. The former is based on theoretical demand distributions such as the normal in Nahmias (1994), who suggests to approximate mean and standard deviation from sales data given an order-up-to level inventory policy. The censored part of the right tail of the normal distribution is calculated by taking into account that the distribution function is symmetric about its mean. A major advantage of the normal distribution function is its wide applicability to large datasets due to the © Springer International Publishing Switzerland 2015 A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical Systems 680, DOI 10.1007/978-3-319-13305-8_2 5 6 2 Literature Review Law of Large Numbers. In contrast, a Poisson process is generally more suitable for discrete and small datasets as in Conrad (1976) or the compound Poisson in Springael and van Nieuwenhuyse (2005). But, as Agrawal and Smith (1996) emphasize, another important requirement is that the distribution chosen is also capable of capturing the effects of demand variation as present in retailing. They derive the parameters of the negative binomial distribution by matching the sample mean and the frequency of observing zero demand to the observed frequency of demand. Lau and Lau (1996) propose a nonparametric model that does not require any prior distributional assumptions based on the product limit method (Kaplan and Meier 1958) and daily sales patterns obtained from previous observations. Berk et al. (2007) use Bayesian updates for obtaining the parameter values of the negative binomial, Gamma, Poisson and normal distribution for the censored newsvendor problem. They rely on an approximation of the posterior distribution by matching the first two moments given that one parameter is known (e.g. mean or variance). Lu et al. (2006) consider Bayesian updates in the context of durable goods for a general distribution function and apply their findings to the normal distribution with known variance. Lu et al. (2008) analytically investigate the benefits from overstocking to learn about the true demand in a Bayesian setting. Tan and Karabati (2004) suggest an updating mechanism to achieve a desired service level by iteratively adjusting the inventory level. Jain et al. (2013) also use Bayesian updates, but additionally take the timing of sales transactions before a stockout occurs into account. 2.2 Assortment Planning Pentico (1974) is among the first to address the assortment planning problem with stochastic demand. A single-period newsvendor solution is obtained under the assumption that customer arrivals occur before any demand is filled. A “nocrossover” assumption prohibits stockout-based substitution. van Ryzin and Mahajan (1999) gain theoretical insights on the trade-off between inventory costs and product variety benefits. There is only assortment-based but no stockout-based substitution. Their analysis is restricted to cases where all variants offered have the same retail price-cost ratio. By adding variety to an assortment, total demand increases but comes at the expense of potential cannibalization effects if demand of the other items in the assortment decreases. Fast and costly replenishment strategies are desirable for fashion items for which demand is characterized by purchase behavior that depends on previous demand and therefore allows forecasting future sales once the season has started. In contrast, demand is assumed to be independent for casual items and scale economies should be leveraged by using large-scale store formats. 2.3 Assortment Planning with Stockout-Based Substitution 7 Another extension of van Ryzin and Mahajan (1999) is proposed by Maddah and Bish (2007) who additionally take pricing into account in a newsvendor setting with assortment-based substitution. Demand is represented by a multinomial logit (MNL) model with a mixed multiplicative and additive form. For a high customer arrival rate or large mean demand, the optimal prices in an assortment have equal profit margins. Topaloglu (2013) also builds on the model of van Ryzin and Mahajan (1999). He extends the model by varying the assortments offered over a selling period. Topaloglu (2013) sets up a nonlinear model to determine which products to offer in an assortment and for how long each product should be offered. He uses the MNL model in order to solve the nonlinear model. Miller et al. (2010) address the assortment planning problem for infrequently purchased goods. They assess the robustness of the optimal assortment for changing customer preferences. Therefore, they develop a MNL model with heterogeneous utilities as well as two other choice models. In addition to the retailer’s objective function with choice probabilities, they establish upper and lower bounds on the expected profit by assuming that the customer purchases the most respectively least profitable product. They apply adaptive conjoint analysis to online purchase data in order to obtain individual product utilities and form consideration sets for each customer. By comparing results on customer preferences to a retailer’s market share, they show that their estimates are reasonable. They find that increased heterogeneity leads to higher profit uncertainty but at the same time allows the shift of customers to more profitable products, thus resulting in higher expected profits. Sauré and Zeevi (2013) study a dynamic assortment planning problem where the retailer learns about customer preferences by varying the set of products offered. There is only limited shelf space and the retailer must select which products to offer. They address the trade-off between learning out about customer preferences versus offering the best set of products (when no learning takes place anymore). They suggest policies to quickly find the best set of products and how to identify products as suboptimal that should not be carried. 2.3 Assortment Planning with Stockout-Based Substitution Smith and Agrawal (2000) develop a base-stock inventory model with stockoutbased substitution that determines the optimal assortment to be carried as well as inventory levels subject to a service level constraint. They show how further constraints such as shelf space can be incorporated into their approach. A logit choice model is used to determine substitution probabilities. Demand is also dependent on the inventory policy. In a comparison of single and multiple substitution attempts, they find that the more items are stocked the smaller is the effect of allowing for more substitution attempts since the probability of finding a suitable item approaches one. 8 2 Literature Review Mahajan and van Ryzin (2001b) extend the model of Smith and Agrawal (2000) by introducing dynamic consumer substitution where the number of substitution attempts is not restricted and substitution rates depend on the availability of substitutes in a given assortment. They model demand and substitution as a general choice process. The profit-maximization problem is solved with a stochastic sample path gradient algorithm which is compared to heuristic policies. The setting with dynamic substitution is compared to static substitution where demand is independent of the current on-hand inventory levels. An important finding of their analysis is that the profit function is not quasi-concave in inventory levels. Furthermore, larger amounts of popular items and fewer amounts of unpopular items should be stocked in an inventory system with substitution compared to a traditional newsvendor. Kök and Fisher (2007) develop a practice-motivated approach to determine the optimal assortment from sales data. Given their focus on products with long shelf life and high service level, the demand function is obtained from loglinear regression, ignoring unobservable lost sales. Parameters for assortment-based substitution are estimated from stores with varying assortments calibrated on fullassortment stores. The approach is then extended to possible out-of-stock situations. Stockout-based substitution rates are derived from individual store sales data using the expectation-maximization (EM) algorithm. Input data required include time of purchase, customer arrivals at different levels and number of product units sold. Other factors influencing purchase behavior such as price, weather and promotional activities are also incorporated. Finally, an iterative heuristic combined with a local search algorithm is applied to solve the assortment optimization problem. They find that stores should aim at higher inventory levels of goods with high demand variance thus hedging against potential lost sales. The amount of inventory to be carried of products with large case sizes depends on the available shelf space. Hopp and Xu (2008) formulate an attraction model with a factor for each product that depends on quality and price. Multiple substitution attempts are modeled by a static approximation as a simplification of the dynamic substitution approach of Mahajan and van Ryzin (2001b). Different settings of price, service and assortment competition are studied. In a duopoly with price, service and assortment competition, product variety diminishes compared to a monopoly in order to avoid price competition whereas the total number of products and thus inventory level increases. Honhon et al. (2010) consider the assortment planning problem with stockoutbased substitution. Demand is classified into different customer types whereas each type has a certain ranking of purchase preferences. Prices remain fixed in this model. The optimal assortment is determined for a fixed proportion of each customer type and a heuristic is provided for the more general case with random proportions. They find that the optimal set of assortment possesses a certain structure in terms of newsvendor fractiles and underage cost. Yücel et al. (2009) combine assortment planning with the supplier selection problem in the presence of quality issues and dynamic substitution behavior. For each of these aspects, a cost function is included in the overall objective functio