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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.

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$?$

h$.$The company has discovered quality problems with the Family Thrillseeker by randomly testing Thrillseekers at the end of the assembly line. Inspectors have discovered that in over $60$ percent of the cases, two of the four doors on a Thrillseeker do not seal properly. Because the percentage of defective Thrillseekers determined by the random testing is so high, the floor foreman has decided to perform quality control tests on every Thrillseeker at the end of the line. Because of the added tests, the time it takes to assemble one Family Thrillseeker has increased from $6$ hours to $7\mathrm{.}5$ hours. Determine the number of units of each model that should be assembled given the new assembly time for the Family Thrillseeker.

i$.$ The board of directors of Automobile Alliance wishes to capture a larger share of the luxury sedan market and therefore would like to meet the full demand for Classy Cruisers. They ask Rachel to determine by how much the profit of her assembly plant would decrease as compared to the profit found in part a$.$ They then ask her to meet the full demand for Classy Cruisers if the decrease in profit is not more than $$2,000,000.$

j$.$ Rachel now makes her final decision by combining all the new considerations described in parts f$,$ g$,$ and h$.$ What are her final decisions on whet