Consider the following data for solving problems $1,2,3$& $5$:

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Month

Demand

Month

Demand

January

$89$

July

$223$

February

$57$

August

$286$

March

$144$

September

$212$

April

$221$

October

$275$

May

$177$

November

$188$

June

$280$

December

$312$

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$1.$ Using a four$-$month moving average, determine the one$-$step$-$ahead forecasts for July through December.

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$2.$ Using a four$-$month moving average, determine the two$-$step$-$ahead forecast for July through December. $($Hint: The two$-$step$-$ahead forecast for July is based on the observed demands in February through May.$)$

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$3.$ Compute the MAD for the forecasts obtained in problems $1$ and $2.$ Which method gave better results? Based on forecasting theory, which method should have given better results?

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$4.$ Determine the following:

a$.$ The value of $$consistent with $$in moving averages.

b$.$ The value of $$consistent with $.$

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$5.$ Use the arithmetic average of the first six months of data as a baseline to initialize the exponential smoothing.

a$.$ Determine the one$-$step$-$ahead exponential smoothing forecasts for August through December, assuming $=0\mathrm{.}20.$

b$.$ Compare the accuracy of the forecasts obtained in part $($a$)$ with the one$-$step$-$ahead six$-$month moving$-$average forecasts.

c$.$ Comment on the reasons for the result you obtained in part $($b$)$

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$6.$ Consider the following data:

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Year $1$

Demand

Year $2$

Demand

Year $3$

Demand

$1$

$12$

$1$

$16$

$1$

$14$

$2$

$25$

$2$

$32$

$2$

$45$

$3$

$76$

$3$

$71$

$3$

$84$

$4$

$52$

$4$

$62$

$4$

$47$

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a$.$ Using the data from years $2$ and $3,$ determine initial values of the intercept, slope, and seasonal factors for Winters$$s method.

b$.$ Assume that the observed dema

nd for the first quarter of year $4$ was $18.$ Using $,,$ and $,$ update the estimates of the series, the slope, and the seasonal factors.

c$.$ What are the forecasts made at the end of the first quarter of year $4$ for the remaining three quarters of year $4?$

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