To determine the forecast for period number 8 using exponential smoothing with a smoothing constant (alpha) of 0.7, we need to follow specific steps, taking into account the provided data and the given forecast.
### Step-by-Step Solution:
1. Identify the known values:
- Customer loads for the past 7 periods (Fridays) are:
[tex]\[ 49, 55, 57, 59, 56, 61, 62 \][/tex]
- Forecast for period 7 (F7) is 59 customers.
- Smoothing constant ([tex]\(\alpha\)[/tex]) is 0.7.
2. Identify the formula for exponential smoothing:
The exponential smoothing forecast for the next period (F8) is:
[tex]\[
F_{t+1} = \alpha \cdot A_t + (1 - \alpha) \cdot F_t
\][/tex]
Where:
- [tex]\(F_{t+1}\)[/tex] is the forecast for the next period.
- [tex]\(\alpha\)[/tex] is the smoothing constant.
- [tex]\(A_t\)[/tex] is the actual value in the current period.
- [tex]\(F_t\)[/tex] is the forecast for the current period.
3. Insert the known values into the formula:
- [tex]\(A_7 = 62\)[/tex] (actual customers in period 7)
- [tex]\(F_7 = 59\)[/tex]
- [tex]\(\alpha = 0.7\)[/tex]
[tex]\[
F_8 = 0.7 \cdot 62 + 0.3 \cdot 59
\][/tex]
4. Calculate the forecast for period 8:
First, calculate the products:
[tex]\[
0.7 \cdot 62 = 43.4
\][/tex]
[tex]\[
0.3 \cdot 59 = 17.7
\][/tex]
Then, add these two products together to get the forecast for period 8:
[tex]\[
F_8 = 43.4 + 17.7 = 61.1
\][/tex]
### Final Result:
The forecast for period number 8 is 61.1 customers.
