Q$4.$ An emergency room operates for $24$ hours and have the following number of tasks during the indicated time slots:

$1$

$2$

$3$

$4$

$5$

$6$

$8$:$00-12$:$0012$:$00-16$:$0016$:$00-20$:$0020$:$00-24$:$0024$:$00-4$:$004$:$00-8$:$00$

Number of tasks

$28$

$12$

$16$

$10$

$14$

$12$

There are three shifts starting at $8$:$00,16$:$00$ and $24$:$00.$ All nurses who start at these times can handle $2$ tasks per hour and work $8$ consecutive hours. However, some nurses ask to start at $12$:$00$ or $20$:$00$ and say that they can work more productively handling $3$ tasks per hour if they let them start at $12$:$00$ or $20$:$00$ and work $8$ consecutive hours. The emergency room pays $20005$ salary to all nurses except for those working $24$:$00-08$:$00$ shift, who earn $20\%$ more.

a$)$ Formulate an LP to minimize the total salaries paid. $(5$ pts$)$

b$)$ Suppose that each excess task costs $$50$ per time SPots. Revise the LP to minimize the weckly total cost of nurses. $(10$ pts$)$

c$)$ Suppose that it costs the company $$75$ for each task short per time slots. Revise the LP to minimize the weekly total cost of nurses. $(10$ pts$)$