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 Adventure, 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 Adventure sold generates a modest profit of [i.] for the company. The second model, the Leisure 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 Leisure Cruiser sold generates a healthy profit of [ii.] 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 Adventures and how many Leisure Cruisers to assemble in the plant to maximize profit for the company. She knows that the plant possesses a capacity of [iii.] labour-hours during the month. She also knows that it takes [iv.] labour-hours to assemble one Family Adventure and [v.] labour-hours to assemble one Leisure 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 [vi.] doors from the door supplier. A recent labour 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 Adventure and the Leisure Cruiser use the same door parts, with [vii.] needed for the Family Adventure and [viii.] for the Leisure Cruiser.
In addition, a recent company forecast of the monthly demands for different automobile models suggests that the demand for the Leisure Cruiser is limited to [ix.] cars. [x.]
Tasks (total 90 marks)
a. Formulate and solve a linear programming model to determine the number of Family Adventures and the number of Leisure Cruisers that should be assembled, using both graphical method and Solver.
Before she makes her final production decisions, Rachel plans to explore the following questions independently, except where otherwise indicated.
The marketing department knows that it can pursue a targeted $500,000 advertising campaign that will raise the demand for the Leisure Cruiser next month by 20 percent. Should the campaign be undertaken?
Rachel knows that she can increase next month’s plant capacity by using overtime labour. She can increase the plant’s labour-hour capacity by 25 percent. With the new assembly plant capacity, how many Family Adventures and how many Leisure
Cruisers should be assembled?
Rachel knows that overtime labour does not come without an extra cost. What is the maximum amount she should be willing to pay for all overtime labour beyond the cost of this labour at regular-time rates? Express your answer as a lump sum.
Rachel explores the option of using both the targeted advertising campaign and the overtime labour-hours. The advertising campaign raises the demand for the Leisure Cruiser by 20 percent, and the overtime labour increases the plant’s labour-hour
capacity by 25 percent. How many Family Adventures and how many Leisure Cruisers should be assembled using the advertising campaign and overtime labour- hours if the profit from each Leisure Cruiser and each Family Adventure remain the same?
Knowing that the advertising campaign costs $500,000 and the maximum usage of overtime labour-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?
Automobile Alliance has determined that dealerships are actually heavily discounting the price of the Family Adventures to move them off the lot. Because of a profit-sharing agreement with its dealers, the company only makes 70% of the estimated profit on the Family Adventures. Determine the number of Family Adventures and the number of Leisure Cruisers that should be assembled given this new discounted profit.
The company has discovered quality problems with the Family Adventures by randomly testing Adventures at the end of the assembly line. Inspectors have discovered that in over 60 percent of the cases, two of the four doors on an Adventure do not seal properly. Because the percentage of defective Adventures determined by the random testing is so high, the floor foreman has decided to perform quality control tests on every Adventure at the end of the line. Because of the added tests, the time it takes to assemble one Family Adventure has increased by 1 hour. Determine the number of units of each model that should be assembled given the new assembly time for the Family Adventure.
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 Leisure 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 Leisure Cruisers if the decrease in profit is not more than $2,000,000.
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 Adventures to assemble, and the number of Leisure Cruisers to assemble?
Identify the slacks and binding constraints for the solution in j. Provide a sensitivity report for the solution in j. Explain the meaning of each item in the report to the board of directors.