# The construction of a new stadium for a football team in Italy

Question 1: 5 marks
Preliminary plans are under way for the construction of a new stadium for a football team in Italy. Officials have questioned the number and profitability of the luxury corporate boxes planned for the upper deck of the stadium.  Corporations and selected individuals may buy the boxes for € 100000 each. The fixed construction cost for the upper-deck area is estimated to be €1500000, with a variable cost of €500000 for each box constructed.
a.     Develop a model for the total cost. Let x represent the number of boxes. (1 mark)
b.     What is the breakeven point for the number of luxury boxed in the new stadium? (2 marks)
c.     Develop a model for the total profit if x participants enroll in the workshop. (1 mark)
d.     Preliminary drawings for the stadium show that space is available for the construction of up to 50 luxury boxes. Promoters indicate that buyers are available and that all 50 could be sold if constructed. What is your recommendation concerning the construction of luxury boxes? What profit is anticipated? (1 mark)
Question 2: 18 marks
GulfGolf makes two different types of golfing gloves: a regular model and a professional model. The firm has 900 hours of production time available in tis cutting and sewing department, 300 hours available in its finishing department and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table.
Assuming that the company is interested in maximizing the total profit contribution, answer the following: (Solve using 4 decimal places)
a.     What is the linear programming model for this problem? (4 marks)
b.     What is the standard form of the linear programming model? (1.25 marks)
c.     Solve graphically and find the extreme points. (6 marks)
d.     What is the optimal number of gloves that must be manufactured graphically (round off the optimal number to an integer). ( 3.25 marks)
e.     What is the total profit contribution the company can earn with the listed production quantities? (0.5 mark)
f.      How many hours of production time will be scheduled in each department? (2.25 marks)
g.     What is the slack time in each department? (0.75 mark)
Question 3: 13 marks
Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of United Oil and H shares of Huber Steel. The annual return for United Oil is €3 per share and the annual return for Huber Steel is €5 per share. United Oil sells for €25 per share and Huber Steel sells for €50 per share. The portfolio has €300,000 to be invested. The portfolio risk index (0.25 per share of United Oil and 0.05 per share of Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 5000 shares of United Oil. The linear programming formation that will maximize the total annual return of the portfolio is as follows:
Max     3U + 5H          Maximize total annual return
s.t
25U +    50H ≤ 300, 000      Funds available
0.25U + 0.05H ≤       700        Risk maximum
1U               ≤     5,000       United Oil maximum
U, H ≥ 0
a.     Solve this problem using the graphical solution procedure. (4.25 marks)
b.     Find the feasible solution region and optimal solution (round off the optimal solution to the nearest tens integer). (2.75 marks)
c.     Calculate the range of optimality for the objective function coefficient of U. (2 marks)
d.     Calculate the range of optimality for the objective function coefficient of H. (2 marks)
e.     Suppose the objective function coefficient of ‘U’ is decreased from 3 to 1.5 and ‘H’ is increased from 5 to 9. What is the new optimal solution? (2 marks)

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