To determine the exponential smoothing forecast for the next time period with a smoothing constant ([tex]\(\alpha\)[/tex]) of 0.40, follow these steps:
1. Identify the historical data and the smoothing constant:
- Profits: [tex]\( [16,012, 16,162, 14,955, 17,016, 18,663, 17,200, 19,142, 18,457, 20,242] \)[/tex]
- Smoothing constant: [tex]\(\alpha = 0.40\)[/tex]
2. Initialize the first forecast:
- The first forecast is set to the first profit value. Therefore, the first forecast is [tex]\(16,012\)[/tex].
3. Use the exponential smoothing formula to calculate the forecast for each subsequent period:
- Exponential Smoothing Formula: [tex]\( F_{t+1} = \alpha \cdot Y_t + (1 - \alpha) \cdot F_t \)[/tex]
where:
- [tex]\( F_{t+1} \)[/tex] is the forecast for the next period.
- [tex]\( Y_t \)[/tex] is the actual value at time [tex]\( t \)[/tex].
- [tex]\( F_t \)[/tex] is the forecast for the current period.
- [tex]\( \alpha \)[/tex] is the smoothing constant.
4. Calculate the forecast iteratively for each time period:
- For February 2014 (2nd month):
[tex]\[
F_2 = 0.40 \times 16,162 + (1 - 0.40) \times 16,012
\][/tex]
- For March 2014 (3rd month):
[tex]\[
F_3 = 0.40 \times 14,955 + (1 - 0.40) \times F_2
\][/tex]
- Continue this process until the last month’s data.
- The consecutive calculations would proceed step-by-step for each period, updating the forecast using the above formula.
5. Obtain the final forecast after iterating through all the months:
- After processing the data through all 9 months, the exponential smoothing forecast for the next time period (October 2014) would be determined.
Following these calculations, the resultant forecast value should be rounded to one decimal place.
6. Final Forecast:
The exponential smoothing forecast for the next time period is 18,994.8 (rounded to one decimal place).
Thus, the exponential smoothing forecast for the next time period (October 2014) is [tex]\(\$18,994.8\)[/tex].
