A garment factory produces shirts and pants for Kmart chain. The contract is such that

quality control is done before shipping and all products supplied to Kmart satisfying the

quality requirements would be accepted by the chain. The factory employs $20$ workers

in the cutting department, $50$ workers in the sewing department, and $14$ workers in the

packaging department. The garment factory works $8$ productive hours a day $($no idle

time during these $8$ hours$).$ There is a daily demand for at most $180$ shirts. The demand

for pants is unlimited. Each worker can participate only in one activity$-$ the activity to

which they are assigned. The table below gives the time requirements $($in minutes$)$ and

profit per unit for the two garments.

Amount $($minutes$)$ per operation

Cutting Sewing Packaging Profit per unit$($$$)$

Shirts $40402010$

Pants $20100208$

a$)$ Explain why a Linear Programming $($LP$)$ model would be suitable for this case study.

$[5$ marks$]$

b$)$ Formulate a LP model to help the factory management to determine the optimal daily

production schedule that maximises the profit while satisfying all constraints.

$[10$ marks$]$

c$)$ Use the graphical method to find the optimal solution. Show the feasible region and

the optimal solution on the graph. Annotate all lines on your graph. What is the optimal

daily profit for the factory?

$[10$ marks$]$

Note: you can use graphical solvers available online but make sure that your graph is

clear, all variables involved are clearly represented and annotated, and each line is clearly

marked and related to the corresponding equation.

d$)$ Find the range for the profit $($$$)$ per shirt $($if any$)$ that can be obtained without affecting

the optimal point of part $($c$).$

$[5$ marks$]$