EL EGR: PLANOS CON LAS INSTALACIONES PARA LA GESTIÓN DE RCD

In this chapter, we highlight challenges and opportunities for daily large-scale demand fore- casting in the retail industry. To this end, we present solution approaches for large-scale demand forecasting based on ML that are able to leverage large datasets. The considered datasets contain time series that are enriched with explanatory data. We also conduct a com- prehensive empirical evaluation that covers different aspects of the methods and focuses on specific characteristics of the use case.

The most important result is that ML methods are indeed a viable alternative to estab- lished approaches for large-scale retail forecasting. Our empirical evaluation offers evidence that a ML method is the best approach at every level of the organizational hierarchy as well as the article hierarchy. However, an analysis of different levels of data usage reveals that the performance of ML methods heavily depends on the data that is used during the training phase. Statistical time series models are the better option if not enough data is available. This is the case if the demand history covers less than 150 days. Moreover, if a ML model is fitted for each time series separately, it performs usually worse than the reference methods. If the models are fitted per time series, adding explanatory feature data can even lead to worse results as overfitting is more likely (e.g. curse of dimensionality). Hence, we conclude that a few hundred training samples are hardly enough to build a good prediction model based on ML. Those results partially support the observations of comparative studies that emphasize the competitiveness of time series models (e.g. Makridakis et al. (2018a,b)).

However, the flexibility of ML methods allows to pool data from multiple time series in order to increase the number of training samples substantially. Data pooling is reasonable in the considered application scenario because it can be expected that stores and articles share similar demand patterns but it is not certain that a method benefits from data pooling as each time series has also unique characteristics. ML methods perform at least as good as the reference methods if pooled training data without features is used. The inclusion of explanatory feature data provides an additional performance boost and makes ML methods superior. More data allows the models to separate noise from actual demand patterns which is in particular useful for special days where only a small number of observations is available per time series and it is difficult to define suitable adjustment rules. A rather short demand history of 150 days is sufficient to outperform the reference methods if data is pooled. While the performance of the reference methods stagnates rather quickly, the performance of ML methods continuously improves as more training data becomes available. Therefore, we come to the conclusion that only ML methods are able to gain more information from a growing amount of training data.

The fact that data pooling works very well is also favorable for the application of ML models because this means that only a smaller number of ML models needs to be optimized and maintained which makes the deployment of ML models in real-world applications much more feasible. In this context, our results also suggest that frequent re-fitting of the ML models that are trained with data across stores and articles might not be necessary as the

4.7. CONCLUSION 81

performance gains are limited, i.e., less than 3%. Hence, ML models need only to be trained every couple of months or even less frequent, which reduces the computational requirements dramatically.

In this context, we need to point out that leveraging the hierarchies of the application do- main is also a possibility to reduce computational costs. For certain combinations of top-level (e.g. store-category level) and bottom level (e.g. store-article level), it is reasonable to rely on top-down forecasts if other approaches are not feasible. In some cases, top-down predictions of ML models are even more accurate than direct predictions of the reference methods at the lower level. Forecasts at different levels of the hierarchy can also serve as a sanity check of the predictions at lower levels as uncertainty is pooled which makes aggregated predictions partially more reliable. This is especially useful as articles in certain categories of baked goods are substitutable and subject to high substitution rates. For ML methods, it can also be beneficial to augment the training set with samples from aggregated time series, which is another advantage for considering the hierarchies.

With respect to the evaluated methods, we notice that artificial neural networks (ANNs) outperform gradient boosted decision trees. However, the results are rather mixed with re- spect to the different types of ANNs, i.e., recurrent neural networks (e.g. LSTMs) and feed- forward neural networks (e.g. MLPs). The LSTMs perform best at the store-category level on the biggest dataset (dataset v1) while the results are comparatively poor on the store-article level using a much smaller dataset (dataset v2). The structure of a LSTM is rather complex, which makes overfitting more likely. Hence, LSTMs need either more training data to prevent this problem or the input sequences and features need to be designed more carefully.

Moreover, for both types of neural networks, it is beneficial to model the learning problem as a classification task instead of a regression problem. Transforming the forecasting problem to a classification task offers an additional level to address the uncertainty of a prediction. The prediction of a classification model, i.e., a probability distribution over the target classes, can be interpreted as a density forecast. For the typical encoding of a classification problem (i.e. 1-hot encoding), it is in particular recommendable to select the median of the distribution instead of the class having the highest probability, i.e., the mode of the distribution. However, an ordinal encoding of the target classes works slightly better and also allows to pick different quantiles of the predicted distribution.

The vast majority of the evaluation is concerned with single-step predictions because they are most relevant for the determination of ordering decisions. However, the experiments related to multi-step predictions indicate that the trained ML models are quite robust, i.e., the forecast errors are rather stable within a season (i.e. a week) and the ranking of the evaluated methods is constant over different horizons. Hence, it is reasonable to initially focus on single-step prediction before other forecasting steps are explored.

Part III

Decision Support

5

Daily Decision Support

The research presented in this chapter is based on joint work with Sebastian Müller, Moritz Fleischmann, and Heiner Stuckenschmidt. In particular, Section 5.3 is based on a paper titled

“A data-driven newsvendor: From data to decision”(Huber et al., 2019) and Section 5.4 is

based on a paper titled “Data-driven Inventory Management under Customer Substitution”. For both projects, I was particularly responsible for the aspects concerning forecasting and Machine Learning as well as the empirical evaluation.

A point forecast (e.g. expected value) does not reflect an optimal decision because fore- casts are stochastic, i.e., a point forecast is only a functional of a distribution, and costs for overages and underages are not always symmetric. Newsvendor models (Silver et al., 2017) are inventory models for determining the optimal quantities of perishable goods, taking into account demand uncertainty and costs. Hence, we study and propose data-driven solution approaches for newsvendor problems in this chapter.

5.1

Introduction

Demand uncertainty is a major challenge in supply chain management practice and research. An important remedy for demand risk is the deployment of safety stock. In order to set ap- propriate stock levels, many inventory models assume a specific demand distribution (Silver et al., 2017). Despite the theoretical insights generated, the distribution assumption is prob- lematic in real-world applications, as the actual demand distribution and its parameters are not known to the decision maker in reality and may even change over time (Scarf, 1958).

Recent advances in data storage and processing technologies have led companies to col- lect more and more data due to the promises of “big data”. The growing availability of large datasets may help overcome the issue of unknown demand distributions and improve the performance of inventory models in real-world situations. Data that are indicative of future demand provide an opportunity to make better-informed decisions. These data include exter- nal information that is available through the Internet and data from internal IT systems. While this potential is widely recognized (see e.g. Bertsimas and Kallus (2018)) and the adaption of data-driven decision making increases, many companies are still struggling to turn data into better decisions (Brynjolfsson and McElheran, 2016).

Extant literature is rather fragmented in that regard and proposes multiple alternative directions. We intend to contribute to a more holistic understanding of the potential of data-

driven inventory management. Hence, our main research question is: How to get from data to decision in newsvendor settings? To this end, we distinguish three levels on which data can be used to revise the traditional decision process (see Figure 5.1). We discuss how these levels are interrelated, and we quantify their respective impact in a real-life application.

Demand estimation Inventory optimization

data

inventory decision demand

distribution

(a) Separate estimation and optimization

Integrated estimation and optimization data

inventory decision

(b) Integrated estimation and optimization

Figure 5.1: The three levels of data-driven inventory management.

Traditionally, the process of inventory optimization contains the two steps demand esti- mationand inventory optimization (see Figure 5.1a). The demand estimation problem and the inventory optimization problem are usually addressed separately in the literature. The objec- tive in the estimation problem is to minimize some estimation error (e.g. mean squared error). The problem of optimal inventory levels is usually not addressed by the forecasting and ML literature. Recently, the strict separation between estimation and optimization (and more gen- erally between Machine Learning (ML) and Operations Research (OR)) has been questioned by some authors (Prak et al., 2017; Bertsimas and Kallus, 2018; Huber et al., 2019). Instead of separately estimating the demand distribution and optimizing the inventory decision, the authors integrate both into a single optimization problem. Hence, we distinguish three levels of data-driven approaches in inventory management:

• The first level on which data can be exploited is demand estimation. The available data may contain information about future demand that can be extracted by suitable forecasting methods. These methods use historical demand data and other feature data (e.g. weekdays, prices, weather, and product ratings) to estimate future demand. The output of these models is a demand estimate together with historical forecast errors. If additional information can be extracted, the reduced demand risk results in more accurate decisions. There is a large set of established and refined forecasting meth- ods that use time series data in order to predict demand distributions. More recently, Machine Learning (ML) approaches have been successfully applied to numerous fore-

5.1. INTRODUCTION 87

casting problems (Thomassey and Fiordaliso, 2006; Carbonneau et al., 2008; Crone et al., 2011; Barrow and Kourentzes, 2018; Huber et al., 2019).

• On the second level, the inventory decision is optimized based on the demand forecast and the historical forecast errors. To this end, it is necessary to incorporate the re- maining uncertainty associated with the forecast. Traditionally, uncertainty is modeled through a demand distribution assumption (Silver et al., 2017). We call this approach

model-basedsince it explicitly models a demand distribution. However, this assump-

tion might be misspecified and lead to suboptimal inventory policies (Ban and Rudin, 2018). Instead of speculating about a parametric demand distribution, the assumption can be replaced by empirical data that are now available on large scale. This approach is called Sample Average Approximation (SAA) (Kleywegt et al., 2002; Shapiro, 2003) and we call it data-driven as it does not rely on a distribution assumption. In a multi- product case, this level also includes the consideration of substitution rates among the products.

• On the third level, demand estimation and optimization are integrated into a single model that directly predicts the optimal decision from historical demand data and fea- ture data, as depicted in Figure 5.1b (Beutel and Minner, 2012; Sachs and Minner, 2014; Bertsimas and Kallus, 2018; Ban and Rudin, 2018; Huber et al., 2019). This approach is also data-driven, as it does not require the assumption of a demand dis- tribution and works directly with data. We propose novel integrated estimation and optimization (IEO) approaches (see Figure 5.1b) that directly optimize the inventory decision based on data and do not rely on demand distribution assumptions.

From the existing literature, it is not yet clear whether and under which circumstances data-driven approaches are preferred to model-based approaches. Furthermore, the question of the conditions under which separate or integrated estimation and optimization is superior remains open.

To shed light on these questions, we focus on the newsvendor problem as the basic in- ventory problem with stochastic demand. We empirically analyze the effects of data-driven approaches on overall costs on the three levels. Moreover, we develop novel data-driven solution methods that combine modern ML approaches with optimization and empirically compare them to well-established methods.

In our approaches, we integrate Artificial Neural Networks (ANNs) and Decision Trees (DTs) into an optimization model. Most previous work on integrated estimation and opti- mization assumed the inventory decision to be linear in the explanatory features (Beutel and Minner, 2012; Sachs and Minner, 2014; Ban and Rudin, 2018). This assumption poses many restrictions on the underlying functional relationships. We extend this literature by integrating multiple alternative ML methods and optimization in order to avoid these strong assumptions and incorporate unknown seasonality, breaks, thresholds, and other non-linear relationships.

Recently, Oroojlooyjadid et al. (2016) and Zhang and Gao (2017) also used ANNs in this context.

We evaluate our solution approaches with real-world data from a large bakery chain in Germany. The company produces and sells a variety of baked goods. It operates a central production facility and over 150 retail stores. Every evening, each store must order products that are delivered the next morning. Reordering during the day is not possible. Most of the goods have a shelf life of only one day. Thus, leftover product at the end of the day is wasted, while stock-outs lead to lost sales and unsatisfied customers. Moreover, when customers cannot find their preferred product in stock, they might choose a similar product instead (Gruen et al., 2002; van Woensel et al., 2007).

From an optimization perspective, the problem can be represented by a newsvendor model, and the available point-of-sales data can be used to calculate forecasts. We consider two types of newsvendor problems: First, we study a single-product newsvendor problem where the order quantity of each product is optimized independently. Second, we study a multi-product newsvendor problem with substitution where the order decisions of substi- tutable products are interdependent. We apply our data-driven methods to the problems and compare their performance to the performance of well-established approaches. To summa- rize, our key contributions include the following:

• We investigate the process from data to decision in a single-product and a multi-product newsvendor setting.

• We identify and conceptualize three levels of data-driven approaches in inventory man- agement.

• We propose novel data-driven integrated estimation and optimization (IEO) approaches for both newsvendor problems.

• We investigate the impact of the three levels on the overall performance in newsvendor problems.

• We compare our methods to well-established approaches on the three levels and show that data-driven methods outperform their model-based counterparts on our real-world dataset in most cases.

• We quantify the value of ML methods that exploit additional feature data. • We quantify the value of considering substitution and uncertainty.

The remainder of this chapter is organized as follows. In the next section, we provide an overview of related literature (see Section 5.2). We propose and evaluate solution approaches for the single-product newsvendor in Section 5.3 and for the multi-product newsvendor with substitution in Section 5.4. We summarize our findings in Section 5.5.

5.2. RELATED WORK 89

5.2

Related Work

In this section, we review the OR literature related to demand uncertainty and demand sub- stitution in inventory models. We refer to Section 4.2 for a review of literature concerning demand estimation using ML.

5.2.1 Demand Uncertainty

Most inventory management textbooks assume that the relevant demand distribution and its parameters are exogenously given and known (Silver et al., 2017). For a review of newsvendor-type problems, see Qin et al. (2011). In this section, we review the literature on inventory problems in which the demand distribution is unknown. More specifically, we focus on Robust Optimization, Sample Average Approximation (SAA), and Quantile Regres- sion (QR).

One approach that needs only partial information on demand distributions is robust opti- mization (Ben-Tal et al., 2009). Scarf (1958) studies a single period problem in which only the mean and the standard deviation of the demand distribution are known. He then opti- mized for the maximum minimum (max-min) profit for all distributions with this property. Gallego and Moon (1993) further analyzed and extended it to a setting where reordering is possible. Bertsimas and Thiele (2006) and Perakis and Roels (2008) provide more insights into the structure of robust inventory problems. The main drawback of robust optimization is its limitation to settings with very risk-averse decision makers. For most real-world applica- tions, robust optimization is overly conservative. For our analysis, we focus on methods that minimize expected costs instead of the max-min objective.

A data-driven method with a wider range of applications is Sample Average Approx- imation (SAA) (Kleywegt et al., 2002; Shapiro, 2003). Here, the demand distribution as- sumptions are replaced by empirical data. Levi et al. (2007) analyze the SAA solution of a newsvendor model and its multi-period extensions. The authors calculate bounds on the number of observations that are needed to achieve similar results compared to the case with full knowledge of the true demand distribution. These bounds are independent of the actual demand distribution. More recently, Levi et al. (2015) showed that the established bound is overly conservative and does not match the accuracy of SAA obtained in simulation studies. Therefore, they develop a tighter bound that is distribution specific. In this work, we pro- vide empirical support for the good performance of SAA and compare the results of diverse methods.

Instead of using sequential estimation and optimization, integrating both steps into a sin- gle optimization model has been suggested (Bertsimas and Kallus, 2018). Beutel and Minner (2012) incorporate a linear regression function for demand into their newsvendor model. The authors test their approach on simulated data and actual retail data. The model was later ex- tended to situations with censored demand observations (Sachs and Minner, 2014). Ban and Rudin (2018) propose an algorithm that is equivalent to the one in Beutel and Minner (2012),

in addition to a kernel optimization method. Furthermore, the authors show several properties of the algorithm and test it with empirical data in a newsvendor-type nurse staffing problem. Oroojlooyjadid et al. (2016) and Zhang and Gao (2017) integrate a neural network into a newsvendor model and compare it to several other approaches from the literature. However, they do not distinguish the effects of estimation, optimization, and integrated estimation and