a$.$ Computation of the exponentially smoothed forecast:

Given data:

Week $1$: $35$ calls

Week $2$: $25$ calls

Week $3$: $40$ calls

Week $4$: $45$ calls

Week $5$: $35$ calls

Week $6$: $33$ calls

Alpha approximation: $0\mathrm{.}2$

Forecast for week $7$: $50$ calls

The formula for the exponentially smoothed forecast is:

$[$F$\_\{$t$+1\}=$ alpha$*$A$\_\{$t$\}+(1-$ alpha$)$F$\_\{$t$\}]$

where Ft$+1$ is the exponentially smoothed forecast for the next period, At is the actual demand for the current period, and Ft is the exponentially smoothed forecast for the current period.

Substituting the given values, we get:

$[$F$\_\{7\}=0\mathrm{.}233+(1-0\mathrm{.}2)50]$

$[$F$\_\{7\}=6\mathrm{.}6+0\mathrm{.}8]$

$[$F$\_\{7\}=7\mathrm{.}4]$

Therefore, the exponentially smoothed forecast for week $7$ is $7\mathrm{.}4$ calls.

b$.$ Reforecast each period using alpha $=0\mathrm{.}6$:

Given data:

Week $1$: $35$ calls

Week $2$: $25$ calls

Week $3$: $40$ calls

Week $4$: $45$ calls

Week $5$: $35$ calls

Week $6$: $33$ calls

Alpha approximation: $0\mathrm{.}6$

Forecast for week $7$: $50$ calls

The formula for the exponentially smoothed forecast with alpha $=0\mathrm{.}6$ is:

$[$F$\_\{$t$+1\}=0\mathrm{.}6$A$\_\{$t$\}+(1-0\mathrm{.}6)$F$\_\{$t$\}]$

Substituting the given values, we get:

$[$F$\_\{7\}=0\mathrm{.}633+(1-0\mathrm{.}6)50]$

$[$F$\_\{7\}=20\mathrm{.}8+0\mathrm{.}4]$

$[$F$\_\{7\}=21\mathrm{.}2]$

Therefore, the exponentially smoothed forecast for week $7$ using alpha $=0\mathrm{.}6$ is $21\mathrm{.}2$ calls.

c$.$ MAD $($Mean Absolute Deviation$)$ is a measure of the accuracy of the forecast. It calculates the average absolute deviation of the forecast from the actual demand. The lower the MAD, the more accurate the forecast.

To calculate the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}2,$ we need to first calculate the absolute deviation for each period:

Week $1$: Absolute deviation $=|35-50|=15$

Week $2$: Absolute deviation $=|25-50|=25$

Week $3$: Absolute deviation $=|40-50|=10$

Week $4$: Absolute deviation $=|45-50|=5$

Week $5$: Absolute deviation $=|35-50|=15$

Week $6$: Absolute deviation $=|33-50|=17$

Then, we calculate the MAD:

MAD $=(|35-50|+|25-50|+|40-50|+|45-50|+|35-50|+|33-50|)/6$

MAD $=(15+25+10+5+15+17+3)/6$

MAD $=118/6$

MAD $=19\mathrm{.}33$

Therefore, the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}2$ is $19\mathrm{.}33.$

To calculate the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}6,$ we need to first calculate the absolute deviation for each period:

Week $1$: Absolute deviation $=|35-50|=15$

Week $2$: Absolute deviation $=|25-50|=25$

Week $3$: Absolute deviation $=|40-50|=10$

Week $4$: Absolute deviation $=|45-50|=5$

Week $5$: Absolute deviation $=|35-50|=15$

Week $6$: Absolute deviation $=|33-50|=17$

Then, we calculate the MAD:

MAD $=(|35-50|+|25-50|+|40-50|+|45-50|+|35-50|+|33-50|)/6$

MAD $=(15+25+10+5+15+17+3)/6$

MAD $=118/6$

MAD $=19\mathrm{.}33$

Therefore, the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}6$ is $19\mathrm{.}33.$

d$.$ The MAD provides a measure of the accuracy of the forecast, with a lower MAD indicating a more accurate forecast. In this case, both the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}2$ and the MAD for the exponentially smoothed forecast with alpha $=0\mathrm{.}6$ are $19\mathrm{.}33.$ Therefore, the smoothing constant of $0\mathrm{.}2$ provides a more accurate forecast for week $7$ compared to the smoothing constant of $0\mathrm{.}6.$

However, the actual demand in week $7$ was $33$ calls, and the closest smoothing constant to the actual demand was $0\mathrm{.}2,$ providing a forecast of $33$ calls. Therefore, the smoothing constant of $0\mathrm{.}2$ provides the closest forecast to the actual demand in week $7.$

Final answer: a$.$ The exponentially smoothed forecast for each week is computed using the given values and the formula:

$[$F$\_\{$t$+1\}=$ alphaA$\_\{$t$\}+(1-$ alpha$)$F$\_\{$t$\}]$

where Ft$+1$ is the exponentially smoothed forecast for the next period, At is the actual demand for the current period, and Ft is the exponentially smoothed forecast for the current period.

Substituting the given values, we get:

$[$F$\_\{7\}=0\mathrm{.}233+(1-0\mathrm{.}2)*50]$

$[$F$\_\{7\}=6\mathrm{.}6+$