Question $3(30\%)$ A production firm is required by contract to provide $10,15,25,$ and

$20$ units of the same specification diesel engines at the end of each quarter of the year,

respectively. The factory's production capacity for each quarter and the cost of

producing each diesel engine are as shown in the table below. Additionally, if the

produced diesel engines are not delivered in the same quarter, a cost of $$0\mathrm{.}15$ dollar per

unit is incurred for storage and maintenance for each quarter of delay.

$($a$)$ Complete the below table. $4$ destination Supply $(7$ marks$)$

$($b$)$ Use northwest corner rules to find the initial solution.

$($c$)$ Prove that the initial solution from part $($a$)$ is not the optimal solution.

$($d$)$ Use the stepping stone method to find out the $1$st iteration.

$($e$)$ How to handle unbalanced transportation problems?

$(5$ marks$)$

$(6$ marks$)$

$(7$ marks$)$

$(5$ marks$)$

Question $4(30\%)$

Suppose you are the manager, you need to assign four jobs to four workers, the table

shows the cost of each worker to each job. Each worker can be assigned to one or two

jobs. We have recorded the cost of works in different jobs and you are expected to find

one perfect candidate for each job using the Hungarian Method.

$($a$)$ What type of problem is it$?$

$($b$)$ Convert the cost matrix to a proper form for use under the Hungarian Method. What

is the optimal solution?

$($c$)$ If Job $1$ cannot be assigned to Worker A$,$ and Worker D can only take Job $2$ or $4,$

what is the optimal solution?

$($d$)$ If the table shows the profit of each job, what is the optimal solution? $(10$ marks$)$

Question $3(30\%)$ A production firm is required by contract to provide $10,15,25,$ and

$20$ units of the same specification diesel engines at the end of each quarter of the year,

respectively. The factory's production capacity for each quarter and the cost of

producing each diesel engine are as shown in the table below. Additionally

Question $3(30\%)$ A production firm is required by contract to provide $10,15,25,$ and

$20$ units of the same specification diesel engines at the end of each quarter of the year,

respectively. The factory's production capacity for each quarter and the cost of

producing each diesel engine are as shown in the table below. Additionally, if the

Question $)$ A production firm is required by contract to provide $10,15,25,$ and $20$ units of the same specification diesel engines at the end of each quarter of the year, respectively. The factory's production capacity for each quarter and the cost of producing each diesel engine are as shown in the table below. Additionally, if the

produced diesel engines are not delivered in the same quarter, a cost of $$0\mathrm{.}15$ dollar per unit is incurred for storage and maintenance for each quarter of delay.

$\backslash$table$[[$Quarter$,$Production Capacity $($unit$),$Plant,$\backslash$table$[[$Unit$],[$cost $($$$)]]],[$I$,25,$A at Quarter I,$10\mathrm{.}8],[$II$,35,$B at Quarter II$,11\mathrm{.}1],[$III$,30,$C at Quarter III,$11],[$IV$,10,$D at Quarter IV$,11\mathrm{.}3]]$

$($a$)$ Complete the below table.

$(7$ marks$)$

$\backslash$table$[[$Plant$,$Production Cost $($$$)$ per unit,Production Capacity$],[$I$,$II$,$III,IV$],[$A$,10\mathrm{.}8,,,,],[$B$,$M$,,,,],[$C$,$M$,$M$,,,30],[$D$,$M$,$M$,$M$,,],[$Sales Quantity,,$15,,,]]$

$($b$)$ Use northwest corner rules to find the initial solution.

$(5$ marks$)$

$($c$)$ Prove that the initial solution from part $($a$)$ is not the optimal solution.

$(6$ marks$)$

$($d$)$ Use the stepping stone method to find out the $1$st iteration.

$(7$ marks$)$

$($e$)$ How to handle unbalanced transportation problems?

$(5$ marks$)$