Croston's Method Research Articles

Intermittent demand: Linking forecasting to inventory obsolescence

The standard method to forecast intermittent demand is that by Croston. This method is available in ERP-type solutions such as SAP and specialised forecasting software packages (e.g. Forecast Pro), and often applied in practice. It uses exponential smoothing to separately update the estimated demand size and demand interval whenever a positive demand occurs, and their ratio provides the forecast of demand per period. The Croston method has two important disadvantages. First and foremost, not updating after (many) periods with zero demand renders the method unsuitable for dealing with obsolescence issues. Second, the method is positively biased and this is true for all points in time (i.e. considering the forecasts made at an arbitrary time period) and issue points only (i.e. considering the forecasts following a positive demand occurrence only). The second issue has been addressed in the literature by the proposal of an estimator (Syntetos–Boylan Approximation, SBA) that is approximately unbiased. In this paper, we propose a new method that overcomes both these shortcomings while not adding complexity. Different from the Croston method, the new method is unbiased (for all points in time) and it updates the demand probability instead of the demand interval, doing so in every period. The comparative merits of the new estimator are assessed by means of an extensive simulation experiment. The results indicate its superior performance and enable insights to be gained into the linkage between demand forecasting and obsolescence.

Forecasting Non-normal Demand by Support Vector Machines with Ensemble Empirical Mode Decomposition

Non-Normal demand is the demand with infrequent demand occurrences or irregular demand sizes, which is very difficult to forecast. In this study, an ensemble empirical mode decomposition (EEMD) based support vector machines (SVMs) learning approach is proposed to forecast demand in these two cases. This approach is under a “decomposition-and-ensemble” principal to decompose the original non-normal demand series into several independent “smooth” and “continuous” subseries including a small number of intrinsic mode functions (IMFs) and a residue by EEMD technique. Then SVMs are used to model each of the subseries so as to achieve more accurate forecast respectively. Finally, the forecasts of all subseries are aggregated by a SVMs model to formulate an ensemble forecast for the non-normal demand series. Two data sets each has four artificial data series were used to test the effectiveness of the proposed approach. Empirical results demonstrate that the proposed ensemble learning approach outperforms the other forecasting methods such as SVMs , ARIMA and Croston method in terms of RMSE, MAPE, MdRAE and MASE.

Evaluation of forecasting error measurements and techniques for intermittent demand

This paper is a part and contribution to develop flexible and robust supply chain forecasting systems under changing market demands. It suggests new tools and models to evaluate forecasting error measurements. The paper especially studies slow moving or intermittent demand, when for items the forecasting time periods often have zero demand. For the difficult to forecast intermittent demand the Croston forecasting technique is mostly regarded as a better choice than single exponential smoothing. These two methods, Croston and single exponential smoothing, together with two modifications of the Croston method, are discussed and evaluated with real intermittent data. The apprehended performance of a forecasting technique is dependent of the chosen measurement of forecast errors. The main purpose is to examine and evaluate different forecasting error measurements. Traditional measurements of forecast errors are studied, mean absolute deviation ( MAD), mean square error ( MSE), etc. together with new suggested error and bias measurements: “periods in stock” ( PIS) and “number of shortages” ( NOS). PIS considers the time aspect of the forecast error, NOS considers the development of cumulated forecast error ( CFE). A complementary measure for descriptive statistics of time series is also suggested, mean average change. The conclusion, through principal components analysis ( PCA), is that a single measure of forecast errors does not present the total different dimensions of the errors; complementary error measures must be used.

On the bias of Croston’s forecasting method

Croston’s forecasting method (CR) has been shown to be appropriate in dealing with intermittent demand items. The method, however, suffers from a positive bias as discussed by Syntetos and Boylan [Syntetos, A.A., Boylan, J.E., 2005a. The accuracy of intermittent demand estimates. International Journal of Forecasting 21, 303–314] who proposed a modification (SB). Unfortunately, the modification ignores the damping effect on the bias of the probability that a demand occurs. This leads to overcompensation and a negative bias, which can in fact be larger than the positive bias of the original method. Syntetos [Syntetos, A.A., 2001. Forecasting for Intermittent Demand, Unpublished Ph.D thesis, Buckinghamshire Chilterns University College, Brunel University] proposed another modification (SY) that takes the damping effect into account, thereby reducing the bias. However, he eventually disregarded it from the empirical analysis, because of the analytical results that SY never dominates SB as well as CR when both bias and variance are considered. Levén and Segerstedt [Levén, E., Segerstedt, A., 2004. Inventory control with a modified Croston procedure and Erlang distribution. International Journal of Production Economics 90, 361–367] also proposed a modified Croston method (LS) and claimed it to be unbiased. We compare all four methods in a numerical study. Our results strengthen the finding from Boylan and Syntetos [Boylan, J.E., Syntetos A.A., 2007. The accuracy of a modified Croston procedure. International Journal of Production Economics 107, 511–517] that LS suffers from a much more severe bias that the other methods. They also confirm SB as the best method when the Mean Square Error is considered. However, SY has a much smaller average absolute bias of 1% compared to 5% for the SB method. From an inventory control point of view, this is an important advantage of the SY method, since biases distort calculations of the expected lead time demand as well as safety stock calculations. An additional advantage of the SY method is its robust performance over the range of parameter values that we considered. Based on these results, we suggest that the SY method should receive more consideration as an alternative to CR and SB.

Forecasting intermittent demand: a comparative study

Methods for forecasting intermittent demand are compared using a large data set from the UK Royal Air Force. Several important results are found. First, we show that the traditional per period forecast error measures are not appropriate for intermittent demand, even though they are consistently used in the literature. Second, by comparing the ability to approximate target service levels and stock holding implications, we show that Croston's method (and a variant) and Bootstrapping clearly outperform Moving Average and Single Exponential Smoothing. Third, we show that the performance of Croston and Bootstrapping can be significantly improved by taking into account that an order in a period is triggered by a demand in that period.

Adapting Wright's modification of Holt's method to forecasting intermittent demand

The challenge of accurately forecasting demand for spare parts is due to the intermittent nature of their demand. This study compares a recent modification of Croston's method and a method based on Holt's double exponential smoothing taking the firm's competitive priorities into consideration. Wright's modification of Holt's method is presented as a viable alternative when forecasting demand for spares especially when trend is present. Results indicate that firms focusing on minimizing inventory levels as a priority should consider forecasting using the Modified Croston's method. If their priority is high customer service, then the modified Holt's method is superior.

Lumpy demand forecasting using neural networks

The current study applies neural network (NN) modeling in forecasting lumpy demand. It is, to the best of our knowledge, the first such study. Our study compares the performance of NN forecasts to those using three traditional time-series methods (single exponential smoothing, Croston's method, and the Syntetos–Boylan approximation). We find NN models to generally perform better than the traditional methods, using three different performance measures. We also independently validate earlier findings that the Syntetos–Boylan approximation performs better than the Croston's and single exponential smoothing methods in lumpy demand forecasting.

A new approach of forecasting intermittent demand for spare parts inventories in the process industries

Accurate demand forecasting is of vital importance in inventory management of spare parts in process industries, while the intermittent nature makes demand forecasting for spare parts especially difficult. With the wide application of information technology in enterprise management, more information and data are now available to improve forecasting accuracy. In this paper, we develop a new approach for forecasting the intermittent demand of spare parts. The described approach provides a mechanism to integrate the demand autocorrelated process and the relationship between explanatory variables and the nonzero demand of spare parts during forecasting occurrences of nonzero demands over lead times. Two types of performance measures for assessing forecast methods are also described. Using data sets of 40 kinds of spare parts from a petrochemical enterprise in China, we show that our method produces more accurate forecasts of lead time demands than do exponential smoothing, Croston's method and Markov bootstrapping method.

The accuracy of a Modified Croston procedure

A modified Croston procedure to forecast intermittent demand is reviewed. The modified procedure, published in this journal in 2004, consists of a new method for estimating the mean demand and a smoothing method for estimating the variance of the forecasted demand rate. It is found that the smoothed variance estimate is based on an invalid measure of forecast accuracy. Moreover, it is shown that the method of forecasting mean demand produces biased forecasts. The accuracy of this method is assessed, using simulated data and the mean square error measure, and compared with Croston's method and single exponential smoothing. It is found that Croston's method is generally more accurate than its modification, particularly for strongly intermittent series.

Exponential smoothing: The state of the art—Part II

In Gardner [Gardner, E. S., Jr. (1985). Exponential smoothing: The state of the art. Journal of Forecasting 4, 1–28], I reviewed the research in exponential smoothing since the original work by Brown and Holt. This paper brings the state of the art up to date. The most important theoretical advance is the invention of a complete statistical rationale for exponential smoothing based on a new class of state-space models with a single source of error. The most important practical advance is the development of a robust method for smoothing damped multiplicative trends. We also have a new adaptive method for simple smoothing, the first such method to demonstrate credible improved forecast accuracy over fixed-parameter smoothing. Longstanding confusion in the literature about whether and how to renormalize seasonal indices in the Holt–Winters methods has finally been resolved. There has been significant work in forecasting for inventory control, including the development of new predictive distributions for total lead-time demand and several improved versions of Croston's method for forecasting intermittent time series. Regrettably, there has been little progress in the identification and selection of exponential smoothing methods. The research in this area is best described as inconclusive, and it is still difficult to beat the application of a damped trend to every time series.

Managing lumpy demand for aircraft spare parts

This paper deals with effective forecasting methods for typically lumpy demand for aircraft spare parts, and analyzes the behavior of forecasting techniques when dealing with lumpy demand. Twenty forecasting techniques are considered and tested and historical data from Alitalia are used to analyze and compare their performance. The results demonstrate that item lumpiness is the dominant parameter and show that demand forecasting for lumpy items is a complex problem; results from previous studies are not very accurate. The best approaches are found to be weighted moving averages, the Croston method, and exponentially weighted moving average models.

On the stock control performance of intermittent demand estimators

The purpose of this paper is to assess the empirical stock control performance of intermittent demand estimation procedures. The forecasting methods considered are the simple moving average, single exponential smoothing, Croston's method and a new method recently developed by the authors of this paper. We first discuss the nature of the empirical demand data set (3000 stock keeping units) and we specify the stock control model to be used for experimentation purposes. Performance measures are then selected to report customer service level and stock volume differences. The out-of-sample empirical comparison results demonstrate the superior stock control performance of the new intermittent demand forecasting method and enable insights to be gained into the empirical utility of the other estimators.

On the categorization of demand patterns

The categorization of alternative demand patterns facilitates the selection of a forecasting method and it is an essential element of many inventory control software packages. The common practice in the inventory control software industry is to arbitrarily categorize those demand patterns and then proceed to select an estimation procedure and optimize the forecast parameters. Alternatively, forecasting methods can be directly compared, based on some theoretically quantified error measure, for the purpose of establishing regions of superior performance and then define the demand patterns based on the results. It is this approach that is discussed in this paper and its application is demonstrated by considering EWMA, Croston's method and an alternative to Croston's estimator developed by the first two authors of this paper. Comparison results are based on a theoretical analysis of the mean square error due to its mathematically tractable nature. The categorization rules proposed are expressed in terms of the average inter-demand interval and the squared coefficient of variation of demand sizes. The validity of the results is tested on 3000 real-intermittent demand data series coming from the automotive industry.

Stochastic models underlying Croston's method for intermittent demand forecasting

Croston's method is widely used to predict inventory demand when it is intermittent. However, it is an ad hoc method with no properly formulated underlying stochastic model. In this paper, we explore possible models underlying Croston's method and three related methods, and we show that any underlying model will be inconsistent with the properties of intermittent demand data. However, we find that the point forecasts and prediction intervals based on such underlying models may still be useful. Copyright © 2005 John Wiley & Sons, Ltd.

The accuracy of intermittent demand estimates

Intermittent demand appears sporadically, with some time periods showing no demand at all. In this paper, four forecasting methods, Simple Moving Average (SMA, 13 periods), Single Exponential Smoothing (SES), Croston's method, and a new method (based on Croston's approach) recently developed by the authors, are compared on 3000 real intermittent demand data series from the automotive industry. The mean signed and relative geometric root-mean-square errors are shown to meet the theoretical and practical requirements of intermittent demand, as do the Percentage Better and Percentage Best summary statistics based on these measures. These measures are subsequently applied in a simulation experiment. The out-of-sample comparison results indicate superior performance of the new method. In addition, the results show that the mean signed error is not strongly scale dependent and the relative geometric root-mean-square error is a well-behaved accuracy measure for intermittent demand.

Forecasting for the ordering and stock-holding of spare parts

A modern military organization like the UK's Royal Air Force is dependent on readily available spare parts for in-service aircraft in order to maximize operational capability. A large proportion of spare parts are known to have an intermittent or slow-moving demand pattern, presenting particular problems as far as forecasting and inventory control are concerned. In this paper, we use extensive demand and replenishment lead-time data to assess the practical value of forecasting models put forward in the literature for addressing these problems. We use an analytical method for classifying the consumable inventory into smooth, irregular, slow-moving and intermittent demand patterns. Recent forecasting developments are compared against more commonly used methods across the identified demand patterns. One recently developed method, a modification to Croston's method referred to as the approximation method, is observed to provide significant reductions in the value of the stock-holdings required to attain a specified service level for all demand patterns.

Evaluation of forecasting methods for intermittent parts demand in the field of aviation: a predictive model

Owing to the sporadic nature of demand for aircraft maintenance repair parts, airline operators perceive difficulties in forecasting and are still looking for superior forecasting methods. This paper deals with techniques applicable to predicting spare parts demand for airline fleets. The experimental results of 13 forecasting methods, including those used by aviation companies, are examined and clarified through statistical analysis. The general linear model approach is used to explain the variation attributable to different experimental factors and their interactions. Actual historical data for hard-time and condition-monitoring components from an airlines operator are used, in order to compare different forecasting methods when facing intermittent demand. The results confirm the continued superiority of the weighted moving average, Holt and Croston method for intermittent demand, whereas most commonly used methods by airlines are found to be questionable, consistently producing poor forecasting performance. We have, however, devised a new approach to forecasting evaluation, a predictive error-forecasting model which compares and evaluates forecasting methods based on their factor levels when faced with intermittent demand. A simple example is presented to illustrate the performance of the mathematical model. It is suggested that these findings may be applicable to other industrial sectors, which have similar demand patterns to those of airlines. Scope and purpose Demand forecasting is one of the most crucial issues of inventory management. Forecasts, which form the basis for the planning of inventory levels, are probably the biggest challenge in the repair and overhaul industry, as the one common problem facing airlines throughout the world is the need to know the short-term part demand forecast with the highest possible degree of accuracy. The high cost of modern aircraft and the expense of such repairable spares as aircraft engines and avionics constitute a large part of the total investment of many airline operators. These parts, though low in demand, are critical to operations and their unavailability can lead to excessive down time costs. Most airline materials managers deal with intermittent demand, which tends to be random and has a large proportion of zero values. In an effort to achieve this, the study has presented a model that could be of great benefit to airline operators and other maintenance service organisations. It will enable them to select in advance the appropriate forecasting method that better meets their cyclical demand for parts. This approach is consistent with the purpose of this study, which aims to compare different forecasting methods when faced with intermittent demand.

Forecasting sales of slow and fast moving inventories

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the sporadic nature of demand and (2) allows the average variability to change over time. The Croston approach is critically appraised in this paper. Corrections are made to underlying theory and modifications are proposed to overcome certain implementation difficulties. A parametric bootstrap approach is outlined that integrates demand forecasting with inventory control. The approach is illustrated on real demand data for car parts.

On the bias of intermittent demand estimates

Forecasting and inventory control for intermittent demand items has been a major problem in the manufacturing and supply environment. Croston (Operational Research Quarterly 23 (1972) 289), proposed a method according to which intermittent demand estimates can be built from constituent events. Croston's method has been reported to be a robust method but has shown more modest benefits in forecasting accuracy than expected. In this research, one of the causes of this unexpected performance has been identified, as a first step towards improving Croston's method. Certain limitations are identified in Croston's approach and a correction in his derivation of the expected estimate of demand per time period is presented. In addition, a modification to his method that gives approximately unbiased demand per period estimates is introduced. All the conclusions are confirmed by means of an extended simulation experiment where Croston's and Revised Croston's methods are compared. The forecasting accuracy comparison corresponds to a situation of an inventory control system employing a re-order interval or product group review.

Forecasting intermittent demand in manufacturing: a comparative evaluation of Croston's method

Intermittent demand appears at random, with many time periods having no demand. Manufacturers perceive the forecasting of intermittent data to be an important problem. In practice, the standard method of forecasting intermittent demand is single exponential smoothing, although some production management texts suggest the lesser-known alternative of Croston's method [Croston J.D., 1972, Forecasting and stock control for intermittent demands, Operational Research Quarterly, 23(3), 289-303]. We compared the two methods, using artificial data created to violate Croston's assumptions and real-world data from industrial sources. We conclude that Croston's method is robustly superior to exponential smoothing and could provide tangible benefits to manufacturers forecasting intermittent demand.