Forecasting of customer demands for production planning by local $k$-nearest neighbor models

Highlights

• Comprehensive empirical study utilizing a large dataset of industrial time series.

• Low forecast errors and low computation times compared to established benchmarks.

• High service levels and low safety stocks in an inventory simulation.

• Best results among all applied methods regarding short time series.

Abstract

Demand forecasting is of major importance for manufacturing companies since it provides a basis for production planning. However, demand forecasting can be a difficult task because customer demands often fluctuate due to several influences. Methods of nonlinear dynamics have shown promising results in numerous applications but they have mostly been neglected in the context of demand forecasting. This paper evaluates the forecasting performance of local $k$-nearest neighbor models, which base on the theory of dynamical systems, in a comprehensive empirical study utilizing a large dataset of industrial time series of the M3-Competition. After a broad literature review, the theoretical background is described. Subsequently, different parameter configurations and model selection strategies are compared. A locally constant mean and a locally constant median are compared to locally linear regression models with four different regularization methods and different parameter configurations. In this comparison, the locally constant mean and the locally linear ridge regression with high regularization parameters provide the best trade-offs between forecast accuracy and computation times. Finally, these models achieve a high performance regarding low forecast errors, short computation times as well as high service levels in an inventory simulation compared to established benchmark methods. In particular, they obtain the best results among all applied methods regarding short time series. Moreover, they achieve the lowest errors considering the original accuracy criterion of the M3-Competition. Hence, local $k$-nearest neighbor models can be regarded as a valid alternative for demand forecasting in an industrial context, accomplishing high forecast accuracy with short computation times.

Introduction

Accurate demand forecasts are essential for all companies in a supply chain since several decisions have to be made before the actual values of important variables are known. In addition to forecasts of resource and energy demands (Hahn et al., 2009, Weinert et al., 2011, Hong and Fan, 2016), electricity prices (Aggarwal et al., 2009, Weron, 2014), failure prognosis for maintenance planning (Peng et al., 2010, Sikorska et al., 2011, Fritzsche et al., 2014, Gao et al., 2015) or yield and quality predictions (Lieber et al., 2013, Colledani et al., 2014), forecasts of future customer demands are necessary. In particular, accurate demand forecasts are crucial in build to stock productions, for instance often occurring in the fast moving consumer goods industry. In this context, raw material orders as well as production programs base strongly on demand forecasts and customer demands are directly satisfied from finished goods inventories. Normally, forecasts are calculated on a weekly or monthly basis for a horizon of approximately one year (Fleischmann et al., 2015) based on time series of past customer orders (Tempelmeier, 2008). Due to various influencing factors, demand evolutions are often highly volatile. For example, the demand of a customer depends on the customer’s remaining product inventory, the number of available substitute products on the market, the success of marketing campaigns, the current economic, political, social or ecological situation as well as basic agreements (Wiendahl et al., 2007, Porter, 2008). Therefore, forecasting future customer demands can be a difficult task.

A typical demand planning process in a company consists of six steps: (i) data acquisition and preparation, (ii) statistical forecasting, (iii) judgmental forecasting, (iv) consensus forecasting, (v) release of the forecast as well as (vi) evaluation and monitoring (Kilger and Wagner, 2015, Petropoulos et al., 2016). In this context, the step of statistical forecasting is of particular importance since it provides forecasts by automatic algorithms that are used as a basis for the following steps. Therefore, the paper at hand focuses on statistical forecasting and neglects the other steps of demand planning. It has to be noted that this paper considers the forecasting of regular customer demand characterized by time series with integer values greater than zero. For applications regarding intermittent demand, which is characterized by a sporadic demand evolution including periods without a demand, such as often observed for spare parts, see (Syntetos and Boylan, 2001, Wallström and Segerstedt, 2010, Kourentzes, 2013, Kourentzes, 2014, Van Wingerden et al., 2014).

Frequently, manufacturing companies have to forecast customer demands for thousands of products simultaneously. Hence, automatic forecasting methods have to be applied to calculate big amounts of forecasts. While simple forecasting methods, such as moving averages, simple exponential smoothing or linear regression methods are still often used in industrial practice (Küsters et al., 2006), more sophisticated methods already proved applicability in many practical applications as well as in forecasting competitions (Hyndman, 2020). Autoregressive integrated moving average (ARIMA) models (Hyndman and Khandakar, 2008, Babai et al., 2013, Box et al., 2015, Svetunkov and Boylan, 2019) and complex variants of exponential smoothing (Gardner, 2006, Hyndman et al., 2008, Ferbar Tratar et al., 2016, Kück et al., 2016b, Sbrana and Silvestrini, 2014, Sbrana and Silvestrini, 2019) are established methods that have shown potential in several studies, such as the M3-Competition (Makridakis and Hibon, 2000). In addition, machine learning methods, such as artificial neural networks (Crone et al., 2008, Crone and Kourentzes, 2010, Adeodato et al., 2011, Montavon et al., 2012, Kourentzes, 2013, Kourentzes et al., 2014) or support-vector machines (Cortes and Vapnik, 1995, Crone et al., 2008, Lu, 2014) have shown promising results in current studies, such as the NN3-Competition (Crone et al., 2011). However, these methods generally need long computation times and they carry the risk of overfitting when they are applied fully automatically. Moreover, some of these methods need long time series of past customer demands in order to fit their parameters in a suitable way. While demand forecasting can commonly be considered as a big data problem of calculating many forecasts for several products at regular intervals, it is nevertheless often the case that the individual demand histories only comprise a few years. Hence, computationally efficient methods that achieve a stable forecasting performance and do not need long demand histories to fit their parameters suitably would be beneficial.

Despite the variety of methods that have been applied to forecast industrial time series, local $k$-nearest neighbor models have been mostly neglected so far. Based on a modeling by nonlinear dynamical systems, these methods are able to reconstruct the dynamical properties of a demand evolution as well as influencing factors out of a scalar time series of past customer demands. These so-called methods of nonlinear dynamics have been successfully applied to conduct different tasks in production and logistics systems, such as modeling and control of production systems (Scholz-Reiter et al., 2002, Papakostas et al., 2009) or forecasting sporadic spare parts demand (Nikolopoulos et al., 2016). Furthermore, small-scale studies covering one-step ahead forecasts with small sets of exemplary customer demand time series have already shown promising results of local $k$-nearest neighbor forecasting models of nonlinear dynamics to forecast regular monthly customer demand (Mulhern and Caprara, 1994, Kück and Scholz-Reiter, 2013, Kück et al., 2014). However, the exemplary results of these studies can only be seen as an indication that these methods could perform well to forecast industrial time series and this assumption has to be proven in a comprehensive study.

The contributions of this paper are the following:

• Proposition of local $k$-nearest neighbor models for regular monthly demand forecasting: The paper proposes local $k$-nearest neighbor models as a novel class of methods to forecast regular monthly customer demands of a manufacturing company.

• Parameter comparison and model selection strategies: Two different versions of $k$-nearest neighbor models are compared, namely locally constant models and locally linear regression models. While standard parameters are used for delay coordinate embedding, different regularization methods with respective regularization parameters are compared regarding their suitability to forecast industrial time series. Moreover, different model selection strategies are proposed.

• Evaluation regarding forecast accuracy: The forecasting performance of two pre-parameterized versions of the local $k$-nearest neighbor models as well as different individual model selection strategies are evaluated in a comprehensive empirical study utilizing 808 monthly industrial time series of the M3-Competition. As a first criterion of forecasting performance, the methods are evaluated in rolling-origin experiments with forecasts from multiple origins across short, medium and long horizons in comparison to several established benchmark methods. In addition to the evaluations in terms of mean forecast errors, Friedman and Nemenyi tests are conducted to assess significant differences between the performances of the local $k$-nearest neighbor models and the benchmark methods.

• Evaluation regarding computation times: In contrast to the usual practice in other studies, this paper also evaluates the computation times of the forecasting methods. In this context, mean times per parameter optimization and per forecast are compared.

• Evaluation regarding inventory performance: Another criterion that is often neglected in other studies is the inventory performance achieved based on calculated forecasts. This paper evaluates the local $k$-nearest neighbor models and the benchmark methods considering achieved service levels and needed safety stocks in an inventory simulation.

The remainder of this paper is structured as follows: Section 2 provides an overview of the existing literature regarding applications of local $k$-nearest neighbor forecasting models. Section 3 explains the theoretical background. After a parameter comparison and a comparison of different model selection strategies in Section 4, Section 5 comprises a comprehensive empirical study to assess the forecasting performance of the local $k$-nearest neighbor models in terms of forecast errors and computation times. Section 6 applies the forecasting methods in an inventory simulation and evaluates their performance regarding inventory metrics. Section 7 concludes the paper.