The Exponential Smoothing Model Is Given By $$\hat { y } _ { t + 1 } = \alpha y _ { t } + ( 1 - \alpha ) \hat { y } _ { t }$$

The exponential smoothing model is given by $$\hat { y } _ { t + 1 } = \alpha y _ { t } + ( 1 - \alpha ) \hat { y } _ { t }$$ where $$\hat { y } _ { t + 1 } = \text { forecast of sales for period } t + 1$$ $$y _ { t } = \text { actual sales for period } t$$ $$\hat { y } _ { t } = \text { forecast of sales for period } t$$ $$\alpha = \text { smoothing constant, } 0 \leq \alpha \leq 1$$ This model is used to predict the future based on the past data values.
a. The observed values with the smoothing constant a = 0.45 are given in the below table. The third column of the table displays the forecast values obtained using the above model. The forecasted error $$y _ { t } - \hat { y _ { t } }$$ is calculated in the fourth column, and the square of the forecast error and the sum of squared forecast errors are given in fifth column. Construct this table in your spreadsheet model using the formula above. (Hint: The first forecast value is same as the observed value.)
Alpha = 0.45 $\begin{array}{ccccc}&&&&\text {Squared Forecast }\\\text { Day } & \text { Observed Value } & \text { Forecast } & \text { Forecast Error } & \text { Error } \\\hline 1 & 15 & 15.00 & 0.0000 & 0.0000 \\\hline 2 & 12 & 15.00 & -3.0000 & 9.0000 \\\hline 3 & 14 & 13.65 & 0.3500 & 0.1225 \\\hline 4 & 18 & 13.81 & 4.1925 & 17.5771 \\\hline 5 & 15 & 15.69 & -0.6941 & 0.4818 \\\hline 6 & 16 & 15.38 & 0.6182 & 0.3822 \\\hline 7 & 15.66 & 5.3400 & 28.5159 \\\hline 8 & 21 & 18.06 & -0.0630 & 0.0040 \\\hline 9 & 18 & 18.03 & -6.0346 & 36.4169 \\\hline 10 & 12 & 15.32 & 3.6809 & 13.5494 \\\hline 11 & 19 & 16.98 & 5.0245 & 25.2458 \\\hline 12 & 22 & 19.24 & -5.2365 & 27.4211 \\\hline 13 & 14 & 16.88 & 4.1199 & 16.9737 \\\hline 14 & 21 & 18.73 & 1.2660 & 1.6026 \\\hline 15 & 20 & 19.30 & -0.3037 & 0.0922 \\\hline& \text { Sum of the } \\&\text { squared forecast } \\177.39 & \text { error } \\\hline\end{array}$
b. The value of á is often chosen by minimizing the sum of squared forecast errors. Use Excel Solver to find the value of á that minimizes the sum of squared forecast errors.