An auto part manufacturer sells six di$3$erent parts to automobile companies: spot lights $($A$),$ break

lights $($B$),$ headlights $($C$),$ thermostat $($D$),$ fan ditch $($E$),$ and battery $($F$).$ Demand is expected to be within

$2,000,1,500,1,800,1,200,1,000$ and $1,000$ units of each of the auto parts, respectively. The table that

follows summarizes the revenues and material costs for each product:

Auto Parts Produced

A B C D E F

Price per Unit $200120180200430260$

Material Cost per Unit $3525404517060$

Each of these auto parts needs to be tested before being sold. Specifically, there are three di$3$erent

tests the protocol requires each part passes, before completing the quality check. Appropriate testing

equipment is required for each test: T$1,$ T$2$ and T$3.$ Equipment T$1$ and T$2$ are available $100$ hours per

week each, while T$3$ is available only $80$ hours per week. The time $($in minutes$)$ required per device for

each of the three tests is reported in the following table:

Electronic Devices Produced

A B C D E F

Test on T$1731261817$

Test on T$225321517$

Test on T$3513292$

Specialized personnel is required to operate each of the test equipment. The variable labour costs are

$16$ per hour for equipment T$1,12$ per hour for T$2,$ and $20$ per hour for T$3.$

A$)$ Considering the manufacturer intends to maximize profit, write a linear program for this problem

and find a solution with Excel Solver. Reflect on whether to impose a constraint on the nature of

the decision variables.

$[10$ marks$]$

B$)$ Suppose that the manufacturer is considering increasing only the available time for testing

equipment T$2.$ What would the increase in the profit be if the available time went up to $110,120,$

$130?$ Can you generalise your answer for any increase $100+$t where t is between $1$ and $10?$ Should

the car part manufacturer consider increasing the available time for testing equipment T$3,$

instead?

$[5$ marks$]$

C$)$ How would your solution change if the manufacturer only produced spot lights $($A$),$ break lights

$($B$),$ headlights $($C$)$ and thermostat $($D$)?$ Exaplain your answer.

$[5$ marks$]$

D$)$ Explain why imposing the constraint that the decision variables must be integer is di$3$erent

$($conceptually and mathematically$)$ from rounding up a solution found without integer constraints.

$[5$ marks$]$