Notes:

$1.$ Problems $2-5$ require the use of a spreadsheet application like Excel. The relevant Excel Refresher will be held either in class on June $27,2019$ or at any office hour on June $26-28.$

$2.$ Display your final answers to two decimal places, but carry the full value in the calculations, i$.$e$.,$ do not round the numbers while performing calculations.

$3.$ Data for this problem can be downloaded from Canvas.

Problem $1$

Tarheels R$$ Us sells UNC merchandise in Chapel Hill. Having UNC gear in stock to sell to visitors is cruhcial, so the shop keeps inventory on hand and replenishes weekly. One such item are t$-$shirts. In order to have the appropriate inventory on hand, it is important to forecast weekly sales and determine the inventory levels accordingly at the beginning of each week. Your help is needed to evaluate alternative forecasting methods. The following table contains the sales for t$-$shirts for the previous $17$ weeks.

Week Sales Week Sales

$131010580$

$233011650$

$339012640$

$440513690$

$549514700$

$651015740$

$752516765$

$852017805$

$954318-$

a$.$ Use a $3-$week simple moving average to forecast sales for weeks $4$ to $18.$ Compute the Mean Absolute Deviation $($MAD$)$ and the Mean Square Error $($MSE$)$ of the forecasts for weeks $4$ to $17.$

b$.$ You suspect that recent sales data contains more useful information than older data. Accordingly, use a $3-$week weighted moving average method with weights of $0\mathrm{.}7,0\mathrm{.}2$ and $0\mathrm{.}1($most recent to least recent$)$ for weeks $4$ to $18.$ Again, compute the MAD and MSE of the forecasts for weeks $4$ to $17$ with the weighted moving average method.

c$.($i$)$ Using MAD as the measure, compare the performance of the two forecasting methods $($moving average, weighted moving average$).$ Which forecasting method is better? $($ii$)$ If you use MSE as the measure $($instead of MAD$)$ to compare the performance of the two forecasting methods $($moving average, weighted moving average$),$ which forecasting method is better?

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Problem $3$

Tarheels R$$ Us would also like to explore a more sophisticated forecasting method. They want you to consider exponential smoothing.

a$.$ Using simple exponential smoothing, compute three sets of forecasts for the sales of Tarheels R$$ Us t$-$shirts for weeks $2$ to $18,$ corresponding to $\backslash$alpha values of $0\mathrm{.}3,0\mathrm{.}6$ and $0\mathrm{.}8.$ Assume the forecast for week $1$ is equal to the actual sales in week $1.$

b$.$ To visually see the difference between forecasts and actual demand, plot the actual demand as well as the three simple exponential smoothing forecasts computed in $($a$)$ for weeks $1$ to $17$ on the same graph. Identify which value of $\backslash$alpha produces the most responsive forecast? What is the implication of this finding?

c$.$ Calculate the MAD and Mean Absolute Percentage Error $($MAPE$)$ using the forecast errors from weeks $2-17$ for each of these three sets of forecasts. Which $\backslash$alpha value seems to be most appropriate and explain why.

Problem $4$

The manager of Tarheels R$$ Us believes that their business is growing over time. Note that simple exponential smoothing may fail to capture this increasing trend. You have learned that the double exponential smoothing method accounts for such trends in the data.

a$.$ To verify the manager$$s belief, use the double exponential smoothing method to forecast sales for weeks $2$ to $18,$ assuming S$1=250$ and T$1=15$ and set $\backslash$alpha $=0\mathrm{.}8$ and $\backslash$beta $=0\mathrm{.}2.$

b$.$ Create a plot showing the actual sales, the double exponential smoothing forecast, and the forecast obtained using simple exponential smoothing in Problem $2$ with $\backslash$alpha $=0\mathrm{.}8$ for weeks $1$ to $17$ on the same graph.

c$.$ Compute MAD and MAPE for weeks $2$ through $17$ for the double exponential smoothing forecast and compare them with simple exponential smoothing with $\backslash$alpha $=0\mathrm{.}8$ from problem $2.$ Based on your calculations, is single or double exponential smoothing a better approach in this specific case? Why?

d$.$ Determine the least squares regression line relating demand to time period. Use week number as the independent variable and the sales as the dependent variable. Forecast the sales for week $18$ by using the regression line. Compare this forecast to the forecast obtained in part $($a$)$ for sales in week $18.$ Based on this, explain the difference between the two methods $($regression$,$ double exponential smoothing$).$

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Problem $5$

The Jersey Store is a store that exclusively sells sports jerseys. These tend to have significant seasonality depending on the sport. For instance, the sales for basketball jerseys peak in spring months, whereas more football jerseys are usually sold in the fall. It makes it difficult to forecast quarterly demand for such products having seasonality. The sales manager of The Jersey Store needs your help in this regard.

The following table shows the sales $($in thousands$)$ of their basketball jerseys from $2012$ to $2015$ by quarter.

Quarter $2012201320142015$

Winter $738840864774$

Spring $1050118210981020$

Summer $9013296108$

Fall $252384336234$

Total $2130253823942136$

a$.$ Determine the slope