$-$Present the formula used and explain why you are using that formula

$-$Explain your constraints and other information in the parameter dialogue box

$-$Explain the results and answer the case questions

and answer each sub$-$letter

Automobile Alliance, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsized and luxury cars. One plant outside Detroit, Michigan, assembles two models from the family of midsized and luxury cars. The first model, the Family Thrillseeker, is a four$-$door sedan with vinyl seats, plastic interior, standard features, and excellent gas mileage. It is marketed as a smart buy for middle$-$class families with tight budgets, and each Family Thrillseeker sold generates a modest profit of $$3,600$ for the company. The second model, the Classy Cruiser, is a two$-$door luxury sedan with leather seats, wooden interior, custom features, and navigational capabilities. It is marketed as a privilege of affluence for upper$-$middle$-$class families, and each Classy Cruiser sold generates a healthy profit of $$5,400$ for the company.$$Rachel Rosencrantz, the manager of the assembly plant, is currently deciding the production schedule for the next month. Specifically, she must decide how many Family Thrillseekers and how many Classy Cruisers to assemble in the plant to maximize profit for the company. She knows that the plant possesses a capacity of $48,000$ labor$-$hours during the month. She also knows that it takes six labor$-$hours to assemble one Family Thrillseeker and $10\mathrm{.}5$ labor$-$hours to assemble one Classy Cruiser. Because the plant is simply an assembly plant, the parts required to assemble the two models are not produced at the plant. Instead, they are shipped from other plants around the Michigan area to the assembly plant. For example, tires, steering wheels, windows, seats, and doors all arrive from various supplier plants. For the next month, Rachel knows that she will only be able to obtain $20,000$ doors from the door supplier. A recent labor strike forced the shutdown of that particular supplier plant for several days, and that plant will not be able to meet its production schedule for the next month. Both the Family Thrillseeker and the Classy Cruiser use the same door parts, with four needed for the Family Thrillseeker and two for the Classy Cruiser. In addition, a recent company forecast of the monthly demands for different automobile models suggests that the demand for the Classy Cruiser is limited to $3,500$ cars. There is no limit on the demand for the Family Thrillseeker within the capacity limits of the assembly plant.

a$.$Formulate and solve a linear programming model to determine the number of Family Thrillseekers and the number of Classy Cruisers that should be assembled. Before she makes her final production decisions, Rachel plans to explore the following questions independently, except where otherwise indicated.

b$.$The marketing department knows that it can pursue a targeted $$500,000$ advertising campaign that will raise the demand for the Classy Cruiser next month by $20$ percent. Should the campaign be undertaken?

c$.$Rachel knows that she can increase next month$$s plant capacity by using overtime labor. She can increase the plant$$s labor$-$hour capacity by $25$ percent. With the new assembly plant capacity, how many Family Thrillseekers and how many Classy Cruisers should be assembled?

d$.$Rachel knows that overtime labor does not come without an extra cost. What is the maximum amount she should be willing to pay for all overtime labor beyond the cost of this labor at regular$-$time rates? Express your answer as a lump sum.$$

e$.$ Rachel explores the option of using both the targeted advertising campaign and the overtime labor$-$hours. The advertising campaign raises the demand for the Classy Cruiser by $20$ percent, and the overtime labor increases the plant$$s labor$-$hour capacity by $25$ percent. How many Family Thrillseekers and how many Classy Cruisers should be assembled using the advertising campaign and overtime labor$-$hours if the profit from each Classy Cruiser sold continues to be $50$ percent more than for each Family Thrillseeker sold?

f$.$ Knowing that the advertising campaign costs $$500,000$ and the maximum usage of overtime labor$-$hours costs $$1,600,000$ beyond regular time rates, is the solution found in part e a wise decision compared to the solution found in part a$?$

g$.$Automobile Alliance has determined that dealerships are actually heavily discounting the price of the Family Thrillseekers to move them off the lot. Because of a profit$-$ sharing agreement with its dealers, the company is not making a profit of $$3,600$ on the Family Thrillseeker but instead is making a profit of $$2,800.$ Determine the number of Family Thrillseekers and the number of Classy Cruisers that should be assembled given this new discounted profit.

h$.$The comp