Solutions to the Assignment: Beginning With Linear Programming 1. 2. 3. 4. 5. 6. 7. 8. Corner points at (0, 6), (0, 0), (5, 0), (3, 4), & (2, 5). The region is bounded. 9. Corner points at (0, 16), (4, 8), (10, 2), & (14, 0). The region is unbounded. 10. Corner points at (0, 0), (4, 4), & (8, 12). The region is unbounded. 11. Corner points at (2, 1), (3, 6), (5, 4), & (5, 2). The region is bounded. 12. No feasible region. 13. 14. 15. 16. 17. 18. Maximize @ (7,9) = 84 z = 3x + 7y = 3(7) + 7(9) = 21 + 63 = 84 19. Minimize @ (0,8) = 32 z = 7x + 4y = 7(0) + 4(8) = 0 + 32 = 32 20. Maximum value is P = 30 @ x = 4 and y = 2. 21. Minimum value is z = 14 @ x = 4 and y = 2. No maximum value since the feasible region is unbounded. 22. Maximum value is P = 260 @ x = 2 and y = 5. 23. Minimum value is z = 140 @ x = 14 and y = 0. No maximum value since the feasible region is unbounded. 24. Minimum value is P = 20 @ x = 0 and y = 2. Maximum value is P = 150 @ x = 5 and y = 0. 25. No Feasible Region; therefore, no Minimum or Maximum. 26. Let A = the number of days to operate plant A Let B = the number of days to operate plant B Minimize Cost, C = 1000A + 900B Subject to 20A + 25B >= 200, 60A + 50B >= 500, A >= 0, B >= 0 Minimum Cost, C = $8333 @ (8.333,0) 27. Let b = the number of buses Let v = the number of vans Minimize Cost, C = 1200b + 100v Subject to 40b + 8v >= 400, 3b + v <= 36, b >= 0, v >= 0 Minimum Cost, C = $3600 @ (0,36) 28. Let C = the amount invested in the CD's Let M = the amount invested in the mutual fund Maximize Return, R = 0.05C + 0.09M Subject to C + M <= 60000, C <= 10000, C - 2M >= 0, C >= 0, M >= 0 Maximum Revenue, R = $5400 @ (0,60000) 29. Let x = the number of gallons produced by the old process Let y = the number of gallons produced by the new process Maximize Profit, P = 60x + 20y Subject to 20x + 5y <= 16000, 40x + 20y <= 30000, x >= 0, y >= 0 Maximum Profit, P = $64000 @ (0,3200) 30. Let A = the number of bags of brand A Let B = the number of bags of brand B Maximize Nitrogen, N = 8A + 3B Subject to 4A + 4B >= 1000, 2A + B <= 400, A >= 0, B >= 0 Maximum Nitrogen, N = $2000 @ (250,0) Assignment: Beginning With Linear Programming Complete each problem on graph paper. For submission, scan and attach a PDF file in the dropbox. You may use Excel if you already know how to graph inequalities in Excel. Show all work to receive credit. Graph each linear inequality. 1. y <= x - 1 2. 3x- 2y > 6 3. x >= -4 4. 6x + 4y >=24 5. 5x <= -2y 6. x >= 0 7. y < 0 Graph each system of linear inequalities, find the feasible region, and state whether the feasible region is bounded or unbounded. 8. 2x + y <= 10, x + 2y <= 12, x + y <= 7, x >= 0, y >= 0 9. 2x + y >= 16, x + 2y >= 14, x + y >= 12, x >= 0, y >= 0 10. 2x - y <= 4, x - y <= 0, x <= 8, x >= 0 11. 5x - y >= 9, x + y <= 9, -x + 3y >= 1, x <= 5 12. 2x + y >= 24, 4x + 3y <= 48, x <= 9 Find the corner points of each feasible region, if any exist. 13. 2x - 3y <= 6, -x + 4y <= 4, x >= 0, y >= 0 14. x + y <= 640, 1.5x + 2.75y <= 1080, x >= 0, y >= 0, and comment on your solution to this one. 15. 9x - 3y <= 6, 4x + 2y >= 8, y<= 6 16. x + y >= 2, 6x + 4y <= 36, 4x + 2y <= 20, x >= 0, y >= 0 17. 4x + 3y <= 240, x + 2y <= 140, x >= 0, y >= 0 For each set of corner points, maximize or minimize the objective function. 18. Maximize z = 3x + 7y, if the corner points are (0, 0), (0, 12), (7, 9), and (10, 0). 19. Minimize z = 7x + 4y, if the corner points are (0, 8), (0, 12), (12, 0), and (4, 3). Given the objective function and constraints, graph the feasible region, then maximize and/or minimize the objective function. 20. Maximize P = 5x + 5y, subject to 2x + y <= 10, x + 2y <= 8, x >= 0, y >= 0. 21. Minimize and maximize z = 2x + 3y, subject to 2x + y >= 10, x + 2y >= 8, x >= 0, y >= 0. 22. Maximize z = 30x + 40y, subject to 2x + y <= 10, x + y <= 7, x + 2y <= 12, x >= 0, y >= 0. 23. Minimize and maximize z = 10x + 30y, subject to 2x + y >= 16, x + y >= 12, x + 2y >= 14, x >= 0, y >= 0. 24. Minimize and maximize P = 30x + 10y, subject to 2x + 2y >= 4, 6x + 4y <= 36, 2x + y <= 10, x >= 0, y >= 0. 25. Minimize and maximize P = 3x + 5y, subject to x + 2y <= 6, x + y <= 4, 2x + 3y >= 12, x >= 0, y >= 0. For each application, write the objective function and constraints. 26. Manufacturing: Production Scheduling A furniture company has two plants that produce the lumber used in manufacturing tables and chairs. In one day of operation, plant A can produce the lumber required to manufacture 20 tables and 60 chairs, and plant B can produce the lumber required to manufacture 25 tables and 50 chairs. The furniture company needs enough lumber to manufacture at least 200 tables and 500 chairs. It costs $1,000 to operate plant A for one day and $900 to operate plant B for one day. How many days should each plant be operated to produce a sufficient amount of lumber at a minimum cost? What is the minimum cost? 27. Transportation The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 40 students, requires 3 chaperones, and costs $1,200 to rent for the trip. Each van can transport 8 students, requires 1 chaperone, and costs $100 to rent for the trip. Since there are 400 students on the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 400 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs? 28. Investment An investor has a total of $60,000 to invest in a mutual fund and in a CD. The CD yields 5% per year and the mutual fund yields an average of 9% per year. The mutual fund requires a minimum investment of $10,000 and the investor requires that twice as much should be invested in CD's as in the mutual fund. How much should be invested in CD's and how much should be invested in the mutual fund in order to maximize the return? What is the maximum return? 29. Pollution Control Because of new federal regulations on pollution, a chemical plant introduced a new and more expensive process to supplement or replace an older process used in the production of a particular chemical. The older process emitted 20 grams of sulfur dioxide and 40 grams of particulate matter into the atmosphere for each gallon of chemical produced. The new process emits 5 grams of sulfur dioxide and 20 grams of particulate matter into the atmosphere for each gallon of chemical produced. The company makes a profit of 60 cents per gallon on the old process and 20 cents per gallon on the new process. If the government allows the plant to emit no more than 16,000 grams of sulfur dioxide and 30,000 grams of particulate matter into the atmosphere daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the daily profit? 30. Plant Nutrition A fruit grower can use two types of fertilizer in his orange grove, brand A and brand B. The amounts (in pounds) of nitrogen, phosphoric acid, and chlorine in a bag are given in the table. Tests indicate that the grove needs at least 1,000 pounds of phosphoric acid and at most 400 pounds of chlorine. Nitrogen Phosphoric Acid Chlorine Brand A (in pounds per bag) 8 4 2 Brand B (in pounds per bag) 3 4 1 If the grower wishes to maximize the amount of nitrogen added to the grove, how many bags of each mix should be used? How much nitrogen will be used