Question $4.(20$ points$).$ Springfield Motors produces the Homer and, now has

introduced the Homer Mini. The Homer Mini is the miniature$-$sized equivalent of the

Homer, approximately the size of a Smart Fortwo. During their production, employees

will be utilizing the same workstations and labor hours, though the Homer Mini

incorporates $30\%$ new components compared to the Homer. For ease of

understanding, assume the manufacturing process for both vehicles consists of just

two tasks: general assembly $($with $180$ combined hours available per day for both

vehicles$)$ and engine fitting $($with $250$ combined hours available per day for both

vehicles$).$

Suppose that each Homer Mini brings in a profit of $$2,500,$ which necessitates $9$ hours

for general assembly, and another $9$ hours for engine fitting. Meanwhile, each Homer

contributes a profit of $$2,900,$ requires $15$ hours in general assembly, and $12$ hours for

engine fitting. Springfield Motors is keen on figuring out how many of each vehicle

model to produce to maximize its total profit.

$($a$)$ Develop a linear programming model for this production problem. Solve this

model using the graphical method. Report your optimal solution and objective

function value.

$($b$)$ Lisa, the production manager, wishes to produce more Homers under the

optimal solution. She wonders what the minimum profit must be $($per unit of the

Homer$)$ such that it is optimal to produce at least one more unit of the Homer

compared to the solution of Part $($a$).$

$($c$)$ Now, Lisa wishes to invest in new technology to produce only Homers while

achieving the same total profit as Part $($a$).$ Investing in this technology can

$($potentially$)$ reduce the general assembly time and$/$or the installation time

required for each unit of the Homer and$/$or the Homer Mini. However, the

available daily capacity and the profit per unit for both Homer and Homer Mini

will remain the same. The investment cost to reduce the general assembly time

$($and installation time$)$ for either the Homer or Homer Mini is $$1,000/$hour and

$$1,500/$hr$,$ respectively. What is the minimum investment required to achieve

Lisa$$s goal? $($In this part, we assume the production units can be non$-$integer.$)($given the information in the question also produce a polygonal graph for a$))$