Authors: Li Li, Yanfei Kang, Fotios Petropoulos and Feng Li

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Summary:

Intermittent demand with several periods of zero demand is ubiquitous in practice. Over half of inventory consists of spare parts, in which demand is typically intermittent (Nikolopoulos Citation2021). Given the high purchase and shortage costs associated with intermittent demand applications, accurate forecasts could be coupled with improved inventory management in the field of manufacturing (Jiang, Huang, and Liu Citation2021), aerospace (Wang and Petropoulos Citation2016), retailing (Sillanpää and Liesiö Citation2018) and so on Balugani et al. (Citation2019) and Babai et al. (Citation2019).

What makes intermittent demand challenging to forecast is that there are two sources of uncertainty: the sporadic demand occurrence, and the demand arrival timing. Seminal work on intermittent demand forecasting by Croston (Citation1972) proposed to separately forecast the sizes of demand and the inter-demand intervals. Then some scholars followed this idea and put forward some developments. For example, Syntetos–Boylan Approximation (SBA) proposed by Syntetos and Boylan (Citation2005) delivered approximately unbiased estimates and constituted the benchmark in subsequently proposed methodologies for intermittent demand forecasting.

Syntetos, Boylan, and Croston (Citation2005) proposed a categorisation of demand patterns to facilitate the selection of Croston’s method (Citation1972) and SBA (Syntetos and Boylan Citation2005). A classification rule was expressed in terms of the average inter-demand interval and the squared coefficient of variation of demand sizes (Syntetos, Boylan, and Croston Citation2005). Kostenko and Hyndman (Citation2006) developed the SBC categorisation scheme (Syntetos, Boylan, and Croston Citation2005) and suggested a simple and more accurate rule, which has been widely used in the research of intermittent demand (Petropoulos and Kourentzes Citation2015; Spiliotis et al. Citation2021).

However, Croston’s method (Citation1972) and SBA update demand sizes and intervals, which leads to inapplicability in periods of zero demand when considering inventory obsolescence. To overcome this shortcoming, Teunter, Syntetos, and Zied Babai (Citation2011) proposed a new method called Teunter–Syntetos–Babai (TSB) to update the demand probability instead of the demand interval. TSB has been proved to have good empirical performance for the demands within linear and sudden obsolescence (Babai, Syntetos, and Teunter Citation2014).

The aforementioned forecasting methods for intermittent demand are all parametric methods, which estimate the parameters of a specific distribution. Instead, non-parametric intermittent demand methods directly estimate empirical distribution based on past data, with no need for any assumption of a standard probability distribution. The bootstrapping methods, and the overlapping and non-overlapping aggregation methods dominate the research field of non-parametric intermittent demand forecasting (Willemain, Smart, and Schwarz Citation2004; Hasni et al. “On the Performance of Adjusted Bootstrapping”  Citation2019; Hasni et al. “Spare Parts Demand Forecasting”  Citation2019; Boylan and Syntetos Citation2021; Boylan and Babai Citation2016).

In particular, temporal aggregation is a promising approach to intermittent demand forecasting, in which a lower-frequency time series can be aggregated to a higher-frequency time series. Latent characteristics of the demand, such as trend and seasonality, appear at higher levels of aggregation. Nikolopoulos et al. (Citation2011) first introduced temporal aggregation to intermittent demand forecasting and proposed the Aggregate-Disaggregate Intermittent Demand Approach (ADIDA). To tackle the challenge of determining the optimal aggregation level, Petropoulos and Kourentzes (Citation2015) considered combinations of forecasts from multiple temporal aggregation levels simultaneously. This approach is called the Intermittent Multiple Aggregation Prediction Algorithm (IMAPA). The overall results of their work suggested that combinations of forecasts from different frequencies led to improved forecasting performance.

Recently, some attention has been paid to applying machine learning approaches to improve forecasting accuracy for intermittent demand, such as neural networks (Lolli et al. Citation2017), support vector machines (Kaya and Turkyilmaz Citation2018; Jiang, Huang, and Liu Citation2021), and so on.

Despite that intermittent demand forecasting has obtained some research achievements in recent decades (Nikolopoulos et al. Citation2011; Petropoulos and Kourentzes Citation2015; Kourentzes and Athanasopoulos Citation2021), there is still much scope for improvements (Nikolopoulos Citation2021). For example, limited attention has been given to combination schemes for intermittent demand forecasting. The literature indicates that forecast combination can improve forecast accuracy in modelling fast-moving time series (Bates and Granger Citation1969; De Menezes, Bunn, and Taylor Citation2000; Petropoulos et al. Citation2022; Li, Petropoulos, and Kang Citation2022). In this study, we aim to examine whether the forecast combination improves intermittent demand forecasts. The main contributions of our work are: (1) providing a discussion and comparison of forecast combination methods in the context of intermittent demand forecasting, (2) developing a feature-based combination framework for intermittent demand, which can determine optimal combination weights evaluated by the given error measure, and (3) improving the accuracy of both point and quantile forecasts to support real inventory decisions.

This paper focuses on forecast combinations for intermittent demand. We review a handful of forecasting methods, and investigate the performance of some existing forecast combination methods for intermittent demand. We introduce time series features and diversity to propose a generalised forecast combination framework, which can automatically determine the optimal combination weights. We conduct an empirical investigation based on real-life data to analyse the forecast accuracy and gain insights related to inventory decisions.

The results of point forecasts are measured by RMSSE, which focuses on the expectation. The proposed framework notably outperforms other combination methods and the best individual method, especially for the RAF dataset with highly intermittent series. Moreover, for M5 competition data, our methods achieve a competitive performance compared with the top three ranked methods in the M5 competition. In addition, the proposed framework can be regarded as a generalised pooling method customised for each time series by reducing the weights of some methods to minimal values. The empirical evaluation based on RAF and M5 datasets provides good evidence of the superiority and flexibility of the proposed framework. We acknowledge that our combination methods increase the computational time compared with individual methods. Decision makers should consider the trade-off between accuracy and computational cost in actual inventory management.

The proposed framework has also been applied to quantile forecast combinations, especially for high quantiles to estimate the right part of the demand distribution. We use SPL to measure the quantile forecasting performance and make it used in the optimisation objective. The examined results show that our methods can provide accurate forecasts of both central tendency and high quantiles, which directly connect with the inventory decision.

The good performance of our proposed framework can be attributed to: (i) defining an appropriate forecasting pool on the top of the framework, which consists of intermittent demand forecasting methods and traditional time series forecasting models, (ii) applying diversity or time series features to determine the optimal combination weights automatically, and (iii) applying to both point and quantile forecasts to support inventory decisions. The diversity and the features selected for intermittent demand are all effective inputs of the proposed framework. Extracting the diversity independent of historical data makes it more flexible for intermittent demand forecasting, especially when the training set is limited in positive demands. In addition, the features in FIDE are all easily understood. The two features focusing on the presence of recent demand are proved more critical for constructing the forecast combination model. These advantages of the proposed methods lead to broad application prospects in intermittent demand forecasting.

However, we recognise the lack of a comprehensive evaluation of inventory performance in the current study. Petropoulos, Wang, and Disney (Citation2019) combined financial, operational, and service metrics to form a holistic measure for inventory control objectives. Ducharme, Agard, and Trépanier (Citation2021) focused on stock-out events and proposed a novel metric called Next Time Under Safety Stock. The utility measures are essential to achieve a direct link between inventory holding costs and service levels in the production system. Such analysis needs to proceed based on restocking policies, which are not available for the RAF and M5 datasets without any background information of inventory. Future research should investigate the inventory performance of our proposed framework in the field of a specific inventory management problem. Another limitation of this paper is lacking an automatic procedure for choosing features for modelling FIDE. Several scholars have investigated selecting features automatically from a large number of features (Lubba et al. Citation2019; Theodorou et al. Citation2021). Although these approaches seem more general, they take over much computational time, and the selected features are often difficult to understand in the applications. Based on the results of our work, the nine features in FIDE are efficient and can be used as the benchmark pool of features for intermittent demand. In further research, we will study a standard procedure to select features automatically for the proposed framework, aiming to achieve both interpretability and computational efficiency.