How to Improve the Realism of Forecast Error (1995)

Range forecasts are often based on the assumption that forecast errors follow the normal, bell curve pattern of dispersion. If the assumption is true, then actual results will not fall outside the forecasted range with any significant frequency. If it is not, then seemingly very unlikely results will occur. In this paper, we offer a method for making range forecasts which are broader than what are dictated by the bell curve assumption. BELL CURVE PATTERN VS. THE EXTENDED TAIL CURVE PATTERN When forecast errors follow the bell curve pattern of dispersion, the pattern is as shown in the middle column of Table 1. (table 1 omitted) The values of z measure the number of standard deviations from the mean. For example, the table says that 4.560% of the forecast errors will be greater than two standard deviations from the mean in absolute value--2.280% of which are two or more standard deviations below the mean and another 2.280% which are two or more standard deviations above the mean. If there is a bell curve pattern and the equation gives a forecast of 1,000 units with a standard error of the regression of 40 units, then we can expect that the actual results will be less than 920 or greater than 1,080 in only 4.560% of all cases. As shown in Table 1, with this bell curve pattern, errors more than three standard deviations in absolute value are quite rare--the frequency is 0.280% for three or more, 0.008% for four or more, and 0.001% for five or more. When forecast errors follow the extended tail curve pattern of dispersion, then the pattern will be as shown in the right most column of Table 1. For example, the table says that 5.229% (2.614% + 2.614%) of the forecast errors will be greater than two standard deviations from the mean in absolute value. With this extended tail curve pattern, errors more than three standard deviations in absolute value are still unusual but not exceedingly rare--the frequency is 1.362% (0.681% + 0.681%) for three or more. Also, 0.401% of the forecast errors are more than four standard deviations in absolute value and 0.125% are five or more standard deviations in absolute value. A feature of the extended tail curve is its flexibility. It has a parameter which we call b. This parameter allows for different extended tail patterns of forecast error. Table 2 shows some typical values. (table 2 omitted) For example, if the value of b is 1.0, then 1.403% of the forecast errors will be more than three standard deviations in absolute magnitude. (We have developed an automatic LOTUS spreadsheet template that estimates the value of the b parameter for any set of raw data. A step-by-step procedure to prepare forecast by using the template appears later in the paper.) THE EXTENDED TAIL CURVE PATTERN IN FORECASTING APPLICATIONS We illustrate the use of the extended tail curve pattern in two forecasting applications: (1) Inventory management, and (2) anticipating service times. Inventory Management An important decision in inventory management is the number of units that should be on hand when an order is placed. If the number of units demanded is less than or equal to the number of units on hand, no shortage occurs. On the other hand, if the demand exceeds the number of units on hand, there is a shortage. Suppose that the demand has a mean of 100 units and a standard deviation of 10 units. If the number of units on hand is set at 130 units, it means that inventory is maintained to three standard deviations above the mean demand, i.e. equal to a z value of 3.0. We are concerned with two quantities. The first is the percentage of cases in which there is a shortage, i.e. when the demand is above 130. Another way of saying this is that the percentage of cases in which z will be above 3.0. If demands follow the bell curve pattern this percentage will be only 0.140% (0.280%/2). On the other hand, if demands follow the extended tail curve with a b parameter value of 0. …