Simulation Of The Control Of Exponential Smoothing By Methods Used In Industrial Practice

Demands of customers for products and of the production for parts are being forecasted quite often in companies. The results are used extensively within the operational production planning and control by IT Systems like the SAP system. Hereby preferably methods based on exponential smoothing are being applied. Especially, in industrial practice it is expected that the pattern of the data change over time in such a way, that the parameters of the exponential smoothing have to be changed. In some companies thousands of items require forecasting, which is very laborious. With an adaptive method the parameters are modified in a controlled manner, as changes in the pattern occur. Two adaptive methods used in industrial practice are explained. Simulation experiments over a huge number of periods show when they should be used. 1. INITIAL SITUATION Forecasts of demands for quantities of products in subsequent periods are carried out several times in a planning process of a company. The fact that the module “Demand Planning” is the most frequently used module within Enterprise Resource Planning (ERP) or Production Planning and Control (PPS) systems in companies, emphasizes that. Other planning functions within ERP or PPS systems use forecasts as well – one example is the use of “alternative planning types”. In general forecasting methods are chosen for a certain forecasting model. The more accurately a forecasting model describes the actual demand series, the better the forecasting results are. Actual changes in industrial practice can be corrected by adapting the forecasting model and in consequence by changing the forecasting method. Due to the Wold decomposition (Wold 1938) a forecasting model consists – simply speaking – of a linear combination of a sequence of uncorrelated random numbers and a random addend. If there’s a level shift within the forecasting model where one (or more) nonzero coefficients (of the linear combination) are being changed to a different non-zero value, the known forecasting methods adapt in time. Their adaption speed is relatively high when using methods based on exponential smoothing. Such (forecasting) methods are preferably used in ERP or PPS systems and therefore generally in companies. Empirical studies (as Armstrong 2001) show that exponential smoothing outperforms on average complex or statistical sophisticated methods and only sometimes they are worse. If such a level shift is highly significant, there may be a strong manual adaption of the parameters required – depending on the settings of the exponential smoothing. Because of the enormous amount of forecasts in companies, the number of manual settings required might rise quite fast. Furthermore such a change is to be distinguished from a temporal change, especially from an outlier. That is the reason why so called adaptive methods have been developed where the parameters are being adapted automatically. For an overview see (Mertens and Rässler 2005). For the investigation the behaviour of the forecasting methods are simulated about a huge number of periods. Such simulations at the IPF show that the results occur by exponential smoothing of the first degree. Since the behaviour of this exponential smoothing is easier to understand as the ones for seasonal data or data with a trend, the paper focuses on this method. Proceedings 26th European Conference on Modelling and Simulation ©ECMS Klaus G. Troitzsch, Michael Möhring, Ulf Lotzmann (Editors) ISBN: 978-0-9564944-4-3 / ISBN: 978-0-9564944-5-0 (CD) 2. FORECASTING METHODS AND KEY FIGURES The simplest forecasting method based on exponential smoothing determines the forecast value in period t ( ) t p through: t t 1 t p p e − = + α ⋅ . α is the smoothing parameter and t t t e y p = − the forecasting error. Whereas t y represents the demand of period t; so: ( ) t t 1 t 1 p y 1 p − − = α ⋅ + − α ⋅ . This exponential smoothing of the first degree is suitable for demands which fluctuate around a mean value. This will be advanced in chapter “Simulations”. Before starting with this standard procedure the initial value of the forecast (for the first period) and the smoothing parameter have to be set. That is usually done by analysing a long series of previous demands. This demand series formulates along with a performance criterion an optimization problem. In literature many performance criteria are being proposed; refer to (De Gooijer and Hyndman 2006) for example. Usually an unbiased forecasting method is required. This means that the expected value of the forecasting errors is zero. The standard deviation of the forecasting errors allows a statement about the safety level with which forecasted demands will occur in the future. That is why it has been examined. In order to achieve steady key figures all forecasting methods are applied on very long demand series – at least over 2500 periods. 3. ADAPTIVE METHODS Although there is no consensus as to the most useful adaptive method, the most widely-used one was developed by Trigg and Leach (Trigg and Leach 1967). They apply exponential smoothing of the first degree to the forecasting error and to its absolute value. The forecasting error being ( ) t t 1 t 1 SE e 1 SE − − = φ⋅ + − φ ⋅ and the absolute value ( ) t t 1 t 1 SAE e 1 SAE − − = φ⋅ + − φ ⋅ with a common smoothing parameter ( ) φ . The tracking signal t t t SE TS SAE = is now used as smoothing parameter t α for calculating the forecasting value of time period t + 1; this method is called control (of the smoothing parameter). Trigg shows in (Trigg 1964) that the tracking signal recognizes a structural change within a demand series and for this he recommends a smoothing parameter of 0,1 φ = . The starting values of these two exponential smoothing methods should be low because of the expected mean forecasting errors of nearly zero. In detail 0 SE 0,05 = and 0 SAE 0,1 = have been chosen during the investigation for this article. Since this method delivers sometimes unstable forecasts, t α is restricted in various ways, s. (Whybark 1973) and (Dennis 1978). In another approach the smoothing parameter is adapted by the Kalman Filter, s. (Bunn 1981), (Enns et al. 1982), (Synder 1988), for weighted least squares s. (Young 1999, Young et al. 1999, Young 2003) and its using for exponential smoothing s. (Harvey 1990). For further approaches s. (Mentzer 1988), (Mentzer and Gomes 1994), (Pantazopoulos and Pappis 1996) and (Taylor 2004). These adaptive methods were investigated empirically, but no method is superior. Gudehus in (Gudehus and Kotzab 2009) presents an adaptive method which he implemented in a number of consulting projects but which is not analysed in research so far. Gudehus uses an adaptive calculated smoothing parameter ( ) t λ α which is calculated at the end of each period t in his exponential smoothing of the first degree for the dynamic mean value forecast in period t ( ) ( ) ( ) ( ) ( ) ( ) m m t t t 1 1 t t 1 λ λ λ = α ⋅λ − + − α ⋅λ − and the dynamic variance forecast in period t ( ) ( ) ( ) ( ) ( )2 2 m s t t t 1 t 1 λ λ = α ⋅ λ − − λ − ( ) ( ) ( ) 1 t s t 1 λ λ + − α ⋅ − With current variation ( ) ( ) ( ) m s t 1 t t 1 λ λ − υ = λ − and maximal acceptable variation max υ ( ) t λ α is calculated by