Question $1$: $(6$ marks$)$

A district manager of an important chain that sells electronic products is currently analysing why sales figures for its local outlets within the district are different among them $($some outlets are performing better than others in terms of annual sales figures$).$ She selects $10$ random outlets located in different localities within the district and considers the specification: for each outlet in the sample: yi measures total annual sales $($thousand dollars$),x1i$ is the number of competitor outlets in the locality where the outlet is located, $x2$i measures the local population $($millions$)$ and $x3$i indicates annual marketing expenditures in each sample outlet $($thousand dollars$).$

$\backslash$table$[[\backslash$table$[[$Total annual sales$],[($Thousand dollars$)]],\backslash$table$[[$Number of competitor$],[$outlets$]],\backslash$table$[[$Local population$],[($Millions$)]],\backslash$table$[[$Annual marketing$],[$expenditures$],[($Thousand dollars$).]]],[290,61,x2,x3$

Question $2$: $(6$ marks$)$

An office has five workers A$,$ B$,$ C$,$ D and E$,$ and five tasks have to be performed J$1,$ J$2,$ J$3,$ J$4$ and J$5.$ Workers differ in efficiency and tasks differ in their intrinsic difficulty. Each job will be assigned to one worker. Time each worker would take to complete each task is given in the effectiveness matrix below. How should the tasks be allocated to each worker to minimize the total overall cost?

$\backslash$table$[[,$J$,$J$,$J $3,$J$,$J $5],[$A$,18,36,9,33,45],[$B$,12,6,21,3,30],[$C$,24,33,30,21,33],[$D$,36,57,36,69,63],[$E$,27,15,21,18,30]]$

a$.$ Construct the reduced matrix and then the modified matrix to reach the optimal matrix showing the steps of the Hungarian algorithm. $(4$ marks$)$

b$.$ What is the optimal assignment $($one job per worker$)$ which will minimize total processing time? $(1$ mark$)$

c$.$ Obtain the optimal $($minimum$)$ total cost. $(1$ mark$)$