Given the following demand data, use exponential smoothing to fore$-$ cast the demand in time period $13$:

Time $123456789101112$ Demand $1058211110994991171161039689103$

$($a$)$ Use the average over the $12$ time periods as the initial estimate $($S$0)$ and try different $\backslash$alpha values $($from $0\mathrm{.}1$ to $0\mathrm{.}9$ 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?