Fit a model to the data using each of the following techniques and forecast the November usage in each case. $($Show your work in the tables below$)$

a$.$ Three$-$month moving average.

b$.$ Five$-$month moving average.

c$.$ Exponential smoothing with smoothing constant $=0\mathrm{.}1.($Start with $10$ units as forecast for Feb.$)$

d$.$ Exponential smoothing with smoothing constant $=0\mathrm{.}3.($Start with $10$ units as forecast for Feb.$)$

e$.$ Linear trend $($regression$).$

f$.$ plot the two moving average forecasts and the actual $($in one graph$)($in Excel$),$ the two exponential smoothing forecasts and the actual $($in one graph$),$ and the linear trend and the actual $($in one graph$)($three graphs altogether. Plot the graphs in Excel and then copy and paste them into this Word document.$)$

Just by observing the plots, which of the above techniques would you use to forecast the usage of fluorescent lamps and why? $($Hint: The plot overall closest to actual demand will be most accurate.$)$

g$.$ Alternatively, compute the MAD for each forecasting technique and determine the most accurate technique.

Table $1$: For parts a to e

Month

Usage

MA$3$

Forecast

MA$5$

Forecast

ES$0\mathrm{.}1$

Forecast

ES$0\mathrm{.}3$

Forecast

Trend Forecast

Jan

$10$

$10+0\mathrm{.}1(10-10)=10\mathrm{.}0$

$10+0\mathrm{.}3(10-10)$

$=10\mathrm{.}0$

Feb

$10$

$10+0\mathrm{.}1(10-10)=10\mathrm{.}0$

$10+0\mathrm{.}3(10-10)$

$=10\mathrm{.}0$

Mar

$66$

$10+0\mathrm{.}1(10-10)=10\mathrm{.}0$

$10+0\mathrm{.}3(10-10)$

$=10\mathrm{.}0$

Apr

$32$

$28\mathrm{.}7$

$10+0\mathrm{.}1(66-10)=15\mathrm{.}6$

$10\mathrm{.}0+0\mathrm{.}3(66-10)$

$=26\mathrm{.}8$

May

$34$

$36\mathrm{.}0$

$15\mathrm{.}6+0\mathrm{.}1(32-15\mathrm{.}6)=17\mathrm{.}2$

$26\mathrm{.}8+0\mathrm{.}3(32-10)$

$=33\mathrm{.}4$

Jun

$18$

$44\mathrm{.}0$

$30\mathrm{.}4$

$17\mathrm{.}2+0\mathrm{.}1(34-17\mathrm{.}2=18\mathrm{.}9$

$33\mathrm{.}4+0\mathrm{.}3(34-10)$

$=40\mathrm{.}6$

Jul

$24$

$28\mathrm{.}0$

$32\mathrm{.}0$

$18\mathrm{.}9+0\mathrm{.}1(18-18\mathrm{.}9)=18\mathrm{.}8$

$40\mathrm{.}6+0\mathrm{.}3(18-10)$

$=43$

Aug

$9$

$25\mathrm{.}3$

$34\mathrm{.}8$

$18\mathrm{.}8+0\mathrm{.}1(24-18\mathrm{.}8)=19\mathrm{.}3$

$43+0\mathrm{.}3(24-10)$

$=47\mathrm{.}2$

Sep

$14$

$17\mathrm{.}0$

$23\mathrm{.}4$

$19\mathrm{.}3+0\mathrm{.}1(9-19\mathrm{.}3)=18\mathrm{.}3$

$47\mathrm{.}2+0\mathrm{.}3(9-10)$

$=46\mathrm{.}9$

Oct

$48$

$15\mathrm{.}7$

$19\mathrm{.}8$

$18\mathrm{.}3+0\mathrm{.}1(14-18\mathrm{.}3)=17\mathrm{.}9$

$46\mathrm{.}9+0\mathrm{.}3(14-10)$

$=48\mathrm{.}1$

Nov

$23\mathrm{.}7$

$22\mathrm{.}6$

$17\mathrm{.}9+0\mathrm{.}1(48-17\mathrm{.}9)=20\mathrm{.}9$

$48\mathrm{.}1+0\mathrm{.}3(48-10)$

$=59\mathrm{.}5$

Table $2$: for part g

Month

Usage

$|$MA$3$ error$|$

$|$MA$5$ error$|$

$|$ES$0\mathrm{.}1$ error$|$

$|$ES$\mathrm{.}3$ error$|$

$|$Trend error$|$

Jan

$10$

Feb

$10$

Mar

$66$

Apr

$32$

$3\mathrm{.}3$

May

$34$

$-2$

Jun

$18$

$-26$

$-12\mathrm{.}4$

Jul

$24$

$-4$

$-8$

Aug

$9$

$-16\mathrm{.}3$

$-25\mathrm{.}8$

Sep

$14$

$-3$

$-9\mathrm{.}4$

Oct

$48$

$32\mathrm{.}3$

$28\mathrm{.}2$

MAD