$1.($i$)$ One person using a hammer will take $5$ days to nail new shingles on the roof of a $300$ m$2$ house.

$($ii$)$ Two persons using a hammer but working together $($one holds a shingle, the other swings the hammer$)$ will take $2$ days to do the same job.

$($iii$)$ One person using a nail gun will take $1$ day to do the same job.

a$.$ Calculate the labour productivity $($in m$2$ per worker per day$)$ for each of the above cases.

b$.$ Calculate the $\%$ change in productivity due to method change, i$.$e$.,$ from $($i$)$ to $($ii$).$

c$.$ Calculate the $\%$ change in productivity due to using technology, i$.$e$.,$ from $($i$)$ to $($iii$).$

$(11$ marks: $1,1,1,1,1,3,3)$

$2.$ The usage of item #$14-46-506$: $4$ ft$.$ Supersaver fluorescent lamps in the maintenance storeroom of ERCO Worldwide in Saskatoon in the first ten months of a given year are displayed below.

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 $10$

Feb $10$

Mar $66$

Apr $32$

May $34$

Jun $18$

Jul $24$

Aug $9$

Sep $14$

Oct $48$

Nov

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 $32$

May $34$

Jun $18$

Jul $24$

Aug $9$

Sep $14$

Oct $48$

MAD

$(7$ marks: $3,4)$

$4.$ For planning production in the medium term, auto producers forecast the total demand for new cars sold in Canada. The following data are number of new cars sold in Canada $($in thousands$)$ during each quarter of $2014$ to $2017($Source: Statistics Canada$$s CANSIM series V$42169911).$ Use Excel and the Annual Averages method to do the following and include a screenshot of your Excel results below:

Quarter $2014201520162017$

$1366378411429$

$2561576596628$

$3528539528563$

$4436447449450$

a$.$ Compute the seasonal relative for each quarter.

b$.$ Forecast the sales of new cars in each quarter of $2018.$