Recall the Tahoe Salt example earlier in the chapter with the historical sell-through demand from its retailers, shown in Table 7-1. The demand data are also shown in column B of Figure 7-7 (see associated spreadsheet Chapter 7-Tahoe-salt). Tahoe Salt is currently negotiating contracts with suppliers for the four quarters between the second quarter of Year 4 and the first quarter of Year 5.An important input into this negotiation is the forecast of demand that Tahoe Salt and its retailers are building collaboratively. They have assigned a team—consisting of two sales managers from the retailers and the vice president of operations for Tahoe Salt—to come up with this forecast. The forecasting team decides to apply each of the adaptive forecasting methods discussed in this chapter to the historical data. The goal is to select the most appropriate forecasting method and then use it to forecast demand for the next four quarters. The team decides to select the forecasting method based on the errors that result when each method is used on the 12 quarters of historical demand data.
Demand in this case clearly has both a trend and seasonality in the systematic component. Thus, the team initially expects Winter’s model to produce the best forecast.
1. Moving Average
The forecasting team initially decides to test a four-period moving average for the forecasting. All calculations are shown in Figure 7-7 (see worksheet Figure 7-7 in spreadsheet Chapter 7-Tahoe-salt) and are as discussed in the section on the moving-average method earlier in this chapter. The team uses Equation 7.9 to estimate level and Equation 7.10 to forecast demand.
As indicated by column K in Figure 7-7, the TS is well within the ± 6 range, which indicates that the forecast using the four-period moving average does not contain any significant bias. It does, however, have a fairly large MAD12 of 9,719, with a MAPE12 of 49 percent. From Figure 7-7, observe that
L12 = 24,500
Thus, using a four-period moving average, the forecast for Periods 13 through 16 (using Equation 7.10) is given by
F13 = F14 = F15 = F16 = L12 = 24,500
Given that MAD12 is 9,719, the estimate of standard deviation of forecast error, using a four-period moving average, is 1.25 X 9,719 = 12,149. In this case, the standard deviation of forecast error is fairly large relative to the size of the forecast.
2. Simple Exponential Smoothing
The forecasting team next uses a simple exponential smoothing approach, with a = 0.1, to forecast demand. This method is also tested on the 12 quarters of historical data. Using Equation 7.11, the team estimates the initial level for Period 0 to be the average demand for Periods 1 through 12 (see worksheet Figure 7-8). The initial level is the average of the demand entries in cells B3 to B14 in Figure 7-8 and results in
L0 = 22,083
The team then uses Equation 7.12 to forecast demand for the succeeding period. The estimate of level is updated each period using Equation 7.13. The results are shown in Figure 7-8.
As indicated by the TS, which ranges from -1.38 to 2.15, the forecast using simple exponential smoothing with a = 0.1 does not indicate any significant bias. However, it has a fairly large MAD12 of 10,208, with a MAPE12 of 59 percent. From Figure 7-8, observe that
L12 = 23,490
Thus, the forecast for the next four quarters (using Equation 7.12) is given by
F13 = F14 = F15 = F16 = L12 = 23,490
In this case, MAD12 is 10,208 and MAPE12 is 59 percent. Thus, the estimate of standard deviation of forecast error using simple exponential smoothing is 1.25 X 10,208 = 12,760. In this case, the standard deviation of forecast error is fairly large relative to the size of the forecast.
3. Trend-Corrected Exponential Smoothing (Holt’s Model)
The team next investigates the use of Holt’s model. In this case, the systematic component of demand is given by
Systematic component of demand = level + trend
The team applies the methodology discussed earlier. As a first step, it estimates the level at Period 0 and the initial trend. As described in Example 7-3, this estimate is obtained by running a linear regression between demand, Dt, and time, Period t. From the regression of the available data (see worksheet holts-regression), the team obtains the following:
L0 = 12,015 and T0 = 1,549
The team now applies Holt’s model with a = 0.1 and b = 0.2 to obtain the forecasts for each of the 12 quarters for which demand data are available (see worksheet Figure 7-9). They make the forecast using Equation 7.14, update the level using Equation 7.15, and update the trend using Equation 7.16. The results are shown in Figure 7-9.
As indicated by a TS that ranges from -2.15 to 2.00, trend-corrected exponential smoothing with a = 0.1 and b = 0.2 does not seem to significantly over- or underforecast. However, the forecast has a fairly large MAD12 of 8,836, with a MAPE12 of 52 percent. From Figure 7-9, observe that
L12 = 30,443 and T12 = 1,541
Thus, using Holt’s model (Equation 7.14), the forecast for the next four periods is given by the following:
In this case, MAD12 = 8,836. Thus, the estimate of standard deviation of forecast error using Holt’s model with a = 0.1 and b = 0.2 is 1.25 X 8,836 = 11,045. In this case, the standard deviation of forecast error relative to the size of the forecast is somewhat smaller than it was with the previous two methods. However, it is still fairly large.
4. Trend- and Seasonality-Corrected Exponential Smoothing (Winter’s Model)
The team next investigates the use of Winter’s model to make the forecast. As a first step, it estimates the level and trend for Period 0, and seasonal factors for Periods 1 through p = 4. To start, the demand is deseasonalized (see worksheet deseasonalized). Then, the team estimates initial level and trend by running a regression between deseasonalized demand and time (see worksheet winters-regression). This information is used to estimate the seasonal factors(see worksheet deseasonalized). For the demand data in Figure 7-2, as discussed in Example 7-4, the team obtains the following:
L0 = 18,439 T0 = 524 S1 = 0.47 S2 = 0.68 S3 = 1.17 S4 = 1.67
It then applies Winter’s model with a = 0.05, b = 0.1, g = 0.1 to obtain the forecasts. All calculations are shown in Figure 7-10 (see worksheet Figure 7-10). The team makes forecasts using Equation 7.17, updates the level using Equation 7.18, updates the trend using Equation 7.19, and updates seasonal factors using Equation 7.20.
In this case, the MAD of 1,469 and MAPE of 8 percent are significantly lower than those obtained with any of the other methods. From Figure 7-10, observe that
L12 = 24,791 T12 = 532 S13 = 0.47 S14 = 0.68 S15 = 1.17 S16 = 1.67
Using Winter’s model (Equation 7.17), the forecast for the next four periods is
In this case, MAD12 = 1,469. Thus, the estimate of standard deviation of forecast error using Winter’s model with a = 0.05, b = 0.1, and g = 0.1 is 1.25 X 1,469 = 1,836. In this case, the standard deviation of forecast error relative to the demand forecast is much smaller than with the other methods.
The team compiles the error estimates for the four forecasting methods as shown in Table 7-2. Based on the error information in Table 7-2, the forecasting team decides to use Winter’s model. It is not surprising that Winter’s model results in the most accurate forecast, because the demand data have both a growth trend as well as seasonality. Using Winter’s model, the team forecasts the following demand for the coming four quarters:
Second Quarter, Year 4: 11,902
Third Quarter, Year 4: 17,581
Fourth Quarter, Year 4: 30,873
First Quarter, Year 5: 44,955
The standard deviation of forecast error is 1,836.
Source: Chopra Sunil, Meindl Peter (2014), Supply Chain Management: Strategy, Planning, and Operation, Pearson; 6th edition.