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Fig 4, while those of Groups 2 and 3 are similar. It can be seen from Fig. 4 that the ACFs of all lags are within the upper and lower limits, meaning that the errors are random. However, the ACF of lag 1 in Fig. 5 exceeds the upper limit. This indicates that autocorrelations do exist in the errors and that the errors are not random. From the ACFs, we can conclude that the decomposition model is adequate for forecasting the demands of Groups 1, 4, 5, 6, 7, and 8, but inadequate for forecasting those of Groups 2 and 3. Therefore, the ARIMA model is applied to Groups 2 and 3.
From the original time series of the demand of Group 3 (in Fig. 3), and the ACFs of its original series (in Fig. 6), it can be interpreted that the original series has a trend, and a high value of ACF of lag 12 indicates the existence of seasonality.^{2 }Hence, a nonseasonal firstdifference to remove the trend and a seasonal firstdifference to remove the strong seasonal spikes in the ACFs are tested. Fig. 7 shows the ACFs of the ARIMA (p,1,q)(P,1,Q)^{12 }model after applying the first difference. The non seasonal plot indicates that there is an exponential decay and one significant ACF of lag 2. Thus, the AR(1) and MA(1) process denoted by ARIMA (1,1,1)(0,1,0)^{12 }is identified. The ACFs of the
Estimated Autocorrelations for Original Series
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Fig 6. ACFs of the actual demand for Group 3.
Estimated Autocorrelations for 1 Nonseasonal Differences 1 Seasonal Differences
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Fig 7. ACFs after first differencing for Group 3.
ScienceAsia 27 (2001)
residuals after applying this ARIMA model shown in Fig. 8 reveals that there is a high value of ACF of lag 12. Therefore, the AR(1) and MA(1) process for the seasonal part or ARIMA (1,1,1)(1,1,1)^{12 }can be identified. The ACFs of the residuals generated from this model are shown in Fig. 9. Since all ACFs are within the two significant limits, the ARIMA (1,1,1)(1,1,1)^{12 }model is adequate.
Using the Statgraphic program, the model coefficients can be determined. The demand forecast for Group 3 is presented in Eq. 1.
F X X X X X X X X e e e t t t t t t t t t t 1 . 1 9 7 0 . 1 9 7 0 . 6 5 1 2 6 + 0 . 1 0 7 1 8 + 0 . 0 8 9 8 2 0 . 7 6 3 3 2 1 2 1 2 1 3 1 4 2 4 2 5 2 6 1 1 2 1 3 0 5 4 4 0 8 0 4 5 5 9 2 0 5 4 5 7 4 1 0 6 6 9 9 0 7 1 5 4 2 9 3 4 7 8 1 . . . . . .
(1)
where
F t i s t h e d e m a n d f o r e c a s t f o r p e r i o d t X t i s t h e a c t u a l d e m a n d f o r p e r i o e_{t }is the forecast error for period t d t
Similarly, the forecasting model for Group 2 is ARIMA (3,0,0)(3,0,0).^{12 }The demand forecast of Group 2 is presented in Eq. 2.
Estimated Residual ACF
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Fig 8. ACFs of the residuals of ARIMA (1,1,1)(0,1,0)^{12 }for Group 3.
model
Estimated Residual ACF
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Fig 9. ACFs of the residuals of ARIMA (1,1,1)(1,1,1)^{12 }for Group 3.
model