To calculate the simple exponential smoothing forecast for the 4th week using an alpha (α) of 0.4, we will follow these steps:
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Determine the initial forecast (F1) for Week 1. It's common to use the value of Week 1 as the initial forecast: \( F_1 = 6.00 \)
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Calculate the forecast for subsequent weeks using the formula: \[ F_t = \alpha \cdot Y_{t-1} + (1 - \alpha) \cdot F_{t-1} \] where \( F_t \) is the forecast for time \( t \), \( Y_{t-1} \) is the actual value from the previous week (week \( t-1 \)), and \( F_{t-1} \) is the forecast for the previous week.
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Calculate the forecasts for Week 2 and Week 3:
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For Week 2: \[ F_2 = \alpha \cdot Y_1 + (1 - \alpha) \cdot F_1 \] \[ F_2 = 0.4 \cdot 6.00 + 0.6 \cdot 6.00 = 2.4 + 3.6 = 6.00 \]
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For Week 3: \[ F_3 = \alpha \cdot Y_2 + (1 - \alpha) \cdot F_2 \] \[ F_3 = 0.4 \cdot 4.00 + 0.6 \cdot 6.00 = 1.6 + 3.6 = 5.20 \]
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Calculate the forecast for Week 4: \[ F_4 = \alpha \cdot Y_3 + (1 - \alpha) \cdot F_3 \] \[ F_4 = 0.4 \cdot 3.00 + 0.6 \cdot 5.20 = 1.2 + 3.12 = 4.32 \]
Thus, the simple exponential smoothing forecast for the 4th week is 4.32.