Mathematics - Algebra - Inequalities - Linear Programming

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bound or unbounded.
22. x + y > -2
3x - y < 6

26. 4x - 3y < 12
5x + 2y < 10
x > 0, y > 0

30. 3x + 4y > 12
2x - y > -2
0 < y < 3
x > 0

Formulate each of the following exercises as a linear programming problem and then solve.
2. NB manufactures two models of fax machines: A and B. Each model A costs $100 to make, and each model B costs $150. The profits are $30 for each model A and $40 for each model B. If the total number of fax machines demanded for per month does not exceed 2500 and the company has earmarked no more than $600,000/month for manufacturing costs, how many units of each model should NB make each month in order to maximize its monthly profit? What is the optimal profit?

8. A financier plans to invest up to $500,000 in two projects. Project A yields a return of 10 % on the investment whereas project B yields a return of 15% on the investment. Because the investment in project B is riskier than the investment in project A, the financier has decided that the investment in project B should not exceed 40% of the investment. How much should she invest in each project in order to maximize the return on her investment? What is the maximum return?

10. A farmer plans to plant two crops, A and B. The cost of cultivating crop A is $40/acre whereas that of crop B is $60/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If he expects to make a profit of $150/acre on crop A and $200/acre on crop B, how many acres of each crop should he plant in order to maximize his profit? What is the optimal profit?

14. A.N. manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires 1.5 square yds of plush, 30 cubic feet of stuffing, and 5 pieces of trim. Each Saint Bernard requires 2 square yards of plush, 35 cubic feet of stuffing, and 8 pieces of trim. The profit for each Panda is $10 and the profit for each Saint Bernard is $15. If 3600 square yards of plush, 66,000 cubic feet of stuffing, and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize profit? What is the maximum profit?

Solve each linear programming problem by the method of corners.

12. Maximize P = 3x -4y
subject to x + 3y < 15
4x + y < 16
x > 0, y > 0

20. Maximize C = 2x + 5y
subject to 4x + y > 40
2x + y > 30
x + 3y > 30
x > 0, y > 0

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method.

2. x y u v P | Constant
1 1 1 0 0 | 6
1 0 -1 1 0 | 2
3 0 5 0 1 30

8. x y z s t u v P | Constant
1 0 0 2/5 0 -6/5 -8/5 0 | 4
0 0 0 -2/5 1 6/5 8/5 0 | 5
0 1 0 0 0 1 0 0 | 12
0 0 1 0 0 0 1 0 | 6
0 0 0 72 0 -16 12 1 | 4920

Solve each linear programming problem by the simplex method.

12. Maximize P = 5x + 3y
subject to x + y < 80
3x < 90
x > 0, y > 0

14. Maximize P = 5x + 4y
subject to 3x + 5y < 78
4x + y < 36
x > 0, y > 0
18. Maximize P = 3x + 3y + 4z
subject to x + y + 3z < 15
4x + 4y + 3z < 65
x > 0, y > 0, z > 0

20. Maximize P = x + 2y - z
subject to 2x + y + z < 14
4x + 2y + 3z < 28
2x + 5y + 5z < 30
x > 0, y > 0, z > 0