$\backslash$table$[[,$Past Sales,$],[$Month$,$Year $1,$Year $2,$Year $3,\backslash$table$[[$Current Sales$],[$This Year$]]],[$January$,352,317,338,364],[$February$,329,331,346,343],[$March$,365,344,383,391],[$April$,358,386,404,437],[$May$,412,423,431,458],[$June$,446,472,459,494],[$July$,420,415,433,468],[$August$,471,492,518,555],[$September$,355,340,309,387],[$October$,312,301,335,364],[$November$,567,629,594,662],[$December$,533,505,527,581]]($i$)$ Determine the seasonal factors for the $12$ months based on past sales.

$[3]$

$($ii$)$ After considering seasonal effects, apply the average method to forecast monthly sales of this year.

$[3]$

$($iii$)$ After considering seasonal effects, apply the moving$-$average method base on the most recent three months to forecast monthly sales of this year. $[3]$

$($iv$)$ After considering seasonal effects, apply the exponential smoothing method to forecast monthly sales this year. Use an initial estimate of $420$ and a smoothing constant of $0\mathrm{.}2.$

Now find optimal value of $$ for the model and compare the results.

$[6]$

$($v$)$ Compare the MAD, MSC and MAPE values obtained in parts $($ii$)-($iv$).$

$[1]$

$2$

$($vi$)$ Calculate the combined forecast for each month by averaging the forecasts for that month obtained in parts $($ii$)-($iv$).$ Then calculate MAD for these combined forecasts.

$[3]$

$($vii$)$ Based on these results, what is your recommendation for how to do the forecasts for next year?

$[2]$