This paper is a critical review of exponential smoothing since the original work by Brown and Holt in the 1950s. Exponential smoothing is based on a pragmatic approach to forecasting which is shared in this review. The aim is to develop state-of-the-art guidelines for application of the exponential smoothing methodology. The first part of the paper discusses the class of relatively simple models which rely on the Holt-Winters procedure for seasonal adjustment of the data. Next, we review general exponential smoothing (GES), which uses Fourier functions of time to model seasonality. The research is reviewed according to the following questions. What are the useful properties of these models? What parameters should be used? How should the models be initialized? After the review of model-building, we turn to problems in the maintenance of forecasting systems based on exponential smoothing. Topics in the maintenance area include the use of quality control models to detect bias in the forecast errors, adaptive parameters to improve the response to structural changes in the time series, and two-stage forecasting, whereby we use a model of the errors or some other model of the data to improve our initial forecasts. Some of the major conclusions: the parameter ranges and starting values typically used in practice are arbitrary and may detract from accuracy. The empirical evidence favours Holt's model for trends over that of Brown. A linear trend should be damped at long horizons. The empirical evidence favours the Holt-Winters approach to seasonal data over GES. It is difficult to justify GES in standard form-the equivalent ARIMA model is simpler and more efficient. The cumulative sum of the errors appears to be the most practical forecast monitoring device. There is no evidence that adaptive parameters improve forecast accuracy. In fact, the reverse may be true.KEY WORDS Bibliography--exponential smoothing Comparative methods-ARI MA, exponential smoothing Control c h a r t s 4 U S U M Evaluation-forecast monitoring systems, exponential smoothing, adaptive Exponential smoothing-adaptive, coefficient choice, higher-order, review, theory Seasonality-estimation, harmonics Tracking signal-methodology Use-inventory controlExponential smoothing methods are widely used in industry. Their popularity is due to several practical considerations in short-range forecasting. Model formulations are relatively simple.

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Most time series methods assume that any trend will continue unabated, regardless of the forecast lead time. But recent empirical findings suggest that forecast accuracy can be improved by either damping or ignoring altogether trends which have a low probability of persistence. This paper develops an exponential smoothing model designed to damp erratic trends. The model is tested using the sample of 1,001 time series first analyzed by Makridakis et al. Compared to smoothing models based on a linear trend, the model improves forecast accuracy, particularly at long leadtimes. The model also compares favorably to sophisticated time series models noted for good long-range performance, such as those of Lewandowski and Parzen.forecasting: time series

Citation for final published version:Syntetos, Argyrios, Zied Babai, M. and Gardner, Everette S. 2015. Forecasting intermittent inventory demands: simple parametric methods vs. ABSTRACTAlthough intermittent demand items dominate service and repair parts inventories in many industries, research in forecasting such items has been limited. A critical research question is whether one should make point forecasts of the mean and variance of intermittent demand with a simple parametric method such as simple exponential smoothing or else employ some form of bootstrapping to simulate an entire distribution of demand during lead time. The aim of this work is to answer that question by evaluating the effects of forecasting on stock control performance in more than 7,000 demand series. Tradeoffs between inventory investment and customer service show that simple parametric methods perform well, and it is questionable whether bootstrapping is worth the added complexity.

This paper evaluates a variety of automatic monitoring schemes to detect biased forecast errors. Backward cumulative sum (cusum) tracking signals have bccn recommended in previous rcsearch to monitor exponential smoothing models. This research shows that identical performance can be had with much simpler tracking signals. The smoothed-error signal is recommended for CI = 0.1, although its performance deteriorates badly as r is increased. For higher c( values, the simple cusum signal is recommended. A tracking signal based on the autocorrelation in errors is recommended for forecasting models other than exponential smoothing, with one exccption. If thc time series has a constant variance, the backward cusum should give better results. KEY WORDS Tracking signals Monitoring forecasts Quality control Cusum Exponential smoothing Simulation AutocorrelationIn most forecasting systems, it is highly desirable to automatically monitor forecast errors to ensure that the system remains in control. For example, if a non-seasonal forecasting model is applied to a time series with unsuspected seasonality, biased errors will occur. When a trend develops in a time series being forecasted by simple exponential smoothing, the forecasts will lag. If the trend remains constant, the simple exponential smoothing model will lag the time series to infinity. Most forecasting models with fixed parameters will lag stepchanges in the mean, trend, or seasonality components of a time series. These problems need to be detected as quickly as possible to enable the forecasting model to be refitted to the data or changed to a more appropriate model. There are at least three warning signs when a forecasting system goes out of control. The first indicator is thc cumulative sum (cusum) of the forecast errors, which can be computed and tested in several different ways. The cusum should fluctuate around zero when the system is in control. If biased errors occur, the cusum will depart from zero. The second indicator is an estimate of the mean forecast error, which will also depart from zero when biased errors occur. The third indicator is the first-order autocorrelation in forecast errors. Since biased errors tend to have the same sign, the existence of any significant positive autocorrelation indicates lack of control.