Case: Alpha Alliance, a large automobile manufacturing company, organises 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 in Victoria assembles two models  from the family of midsized and luxury cars. The first model, the Eagle, is a fourdoor 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 Eagle sold generates a modest profit of $3,500 for the company. The second model, the Silhouette, 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 Silhouette sold  generates a healthy profit of $5,500 for the company.

Mark Evans, the manager of the assembly plant, is currently deciding the  production schedule for the next month. Specifically, he must decide how many Eagles and how many Silhouettes to assemble in the plant to maximise profit for the company. He knows that the plant possesses a capacity of 50,000 labour-hours during the month. He also knows that it takes six labour-hours to assemble one
Eagle and eight and a half labour-hours to assemble one Silhouette. 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 in New South Wales to the assembly plant. For example, tires, steering wheels, windows, seats, and doors all arrive from various supplier plants. For the
next month, Mark knows that he will only be able to obtain 25,000 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 Eagle and the Silhouette 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 Silhouette is limited to 3,800 cars. There is no limit on the demand for the Eagle within the capacity limits of the assembly plant.

a. Formulate and solve a linear programming model to determine the number of Eagles and the number of Silhouettes that should be assembled. How much profit will this strategy generate? How many doors and labour-hours are required and unused? Before he makes his final production decisions, Mark plans to explore the following questions independently, except where otherwise indicated. Some of these problems do not require re-optimisation, and sensitivity analysis will be adequate. If a problem requires re-formulation, you must show and discuss the new model.

b. The marketing department knows that it can pursue a targeted $400,000 advertising campaign that will raise the demand for the Silhouette next  month by 25 percent. Should the campaign be undertaken? If it should, how much an additional profit will the campaign generate?

c. Mark knows that he can increase next month's plant capacity by using overtime labour. He can increase the plant's labour-hour capacity by 30 per cent. With the new assembly plant capacity, how many Eagles and how many Silhouettes should be assembled? How much profit will this strategy generate? How many doors and labour-hours are required and unused?

d. Based on part c, Mark knows that overtime labour does not come without an extra cost. What is the maximum amount he 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.

e. Mark explores the option of using both the targeted advertising campaign and the overtime labour-hours. The advertising campaign raises the demand for the Silhouette by 25 percent, and the overtime labour increases the plant's labour-hour capacity by 30 percent. How many Eagles and Silhouettes should be assembled using the advertising campaign and overtime labour-hours? How much profit will this strategy generate? How many doors and labour-hours are required and unused?

f. Knowing that the advertising campaign costs $400,000 and the maximum usage of overtime labour-hours costs $1,800,000 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 forecasted that the demand for Silhouette will decrease to 2,500 cars next month. To maintain the current market share, Mark has decided to offer the promotional price, which will reduce the marginal profit of Silhouette to $4,950. Mark fears that a new optimised solution will cause the flood of Eagle cars into the market due to the reduction of Silhouette's profit, he has decided that the number of Eagles assembled should not exceed 4,700 cars and that the production ratio of Eagle to Silhouette must not exceed 2:1. How many Eagles and Silhouettes should be assembled? How much profit will this strategy generate? How many doors and labour-hours are required and unused? What is the production ratio of Eagle to Silhouette based on the results?

h. The company has identified a new robotic machine that reduce the assembling time for each Eagle and Silhouette to four and six hours respectively. Mark has been asked to come up with a new production strategy based on the new assembling time. How many Eagles and Silhouettes should be assembled? How much profit will this strategy generate?

i. Based on part h, if the robotic machine costs $3,000,000, what should Mark recommend to the company? Why?

j. Mark now makes his final decision by combining all the new considerations described in parts f and i. Determine the numbers of Eagles and Silhouettes to be assembled, the new profit, and the amount of required and unused resources.

k. Based on part j, if an automotive-part supplier approaches Mark to sell their car doors which are compatible with Eagle and Silhouette with the price of $800 per door. Should Mark procure the doors from this supplier? If so, how many doors should be ordered? Why?