Given the following demand data, use exponential smoothing to forecast the demand in time period

$13$: $(16$ points$)$

Time $123456789101112$

Demand $1958211110994991171161039689103$

$($a$)$ Use the average over the $12$ time periods as the initial estimate $(!)$ and try different $\backslash$alpha values

$($from $0\mathrm{.}1$ to $0\mathrm{.}5$ with an increment of $0\mathrm{.}1).$ Which $\backslash$alpha value gives the best fit to the demand data?

$($b$)$ Use different values for the initial estimate $($e$.$g$.,$ average over the first $3,6,$ and $9$ time periods$)$

and reevaluate the $\backslash$alpha value that gives the best fit to the demand data. Is this $\backslash$alpha value influenced by

the initial estimate?