5. Auto Assembly

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 10.5 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 part.

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 months plant capacity by using overtime

labor. She can increase the plants 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 plants 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 profitsharing 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 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 7.5 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 whether to undertake the

advertising campaign, whether to use overtime labor, the number of Family

Thrillseekers to assemble, and the number of Classy Cruisers to assemble?