Lead time demand for simple exponential smoothing: an adjustment factor for the standard deviation
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Lead time demand for simple exponential smoothing: an adjustment factor for the standard deviation RD Snyder1, AB Koehler2* and JK Ord3 1
Monash University, Australia, 2Miami University, OH and 3Pennsylvania State University, PA, USA
A new simple formula is found to correct the underestimation of the standard deviation for total lead time demand when using simple exponential smoothing. The traditional formula for the standard deviation of lead time demand is to multiply the standard deviation for the one-period-ahead forecast error (estimated by using the residuals) by the square root of the number of periods in the lead time. It has been shown by others that the traditional formula signi®cantly underestimates variation in the lead time demand when the mean of the process is somewhat changing and simple exponential smoothing is appropriate. This new formula allows one to see readily the signi®cant size of the underestimation of the traditional formula and can easily be implemented in practice. The formula is derived by using a state-space model for simple exponential smoothing. Keywords: Lead time demand; exponential smoothing; prediction intervals; safety stock
Introduction The incorrect estimation of the variance for forecast error when simple exponential smoothing is used in computerized inventory control systems has been examined in many studies.1±3 Originally, Brown4 proposed estimating the standard deviation for the total lead time demand by multiplying the standard deviation, s, for the demand one-period-ahead by the square root of pthe number of periods, h, in the lead time (namely, s h). We call this estimate the `traditional formula.' This formula is appropriate if a stationary model is generating the time series, that is, if the smoothing constant for the simple exponential smoothing process is zero. However, the point of using exponential smoothing is to account for the changing mean or level of the time series (a nonstationary process). Hence, it is important to determine the effect of a nonzero smoothing constant on the variance of the total lead time demand. Previous studies1,2 have shown that this variance is signi®cantly underestimated. In this paper, we have developed a simple formula for computing the variance of total lead time demand that accounts for both the length of the lead time and the size of the smoothing constant in simple exponential smoothing. For the generating model, we use one of the two models studied by Muth.5 Both models assume constant variance for the error term. We choose the model with a single source of random error (SSOE) for several reasons. It is Correspondence: Dr AB Koehler, Department of Decision Sciences and Management Information Systems, Miami University, Oxford, OH 45056, USA. E-mail: [email protected]
directly equivalent to the ARIMA (0, 1, 1) model with no restriction on the correlations. Muth's other model has two (multiple) sources of
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