Write linear programming project – Operation Management
1. A linear programming problem cost minimization problem has objective function: Minimize X +Y. It has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100, X≥ 0, Y≥0.
1.1. Use QM for Windows to plot the feasible region. Paste image of Linear Programming Results window and Solution List window here.
1.2. Paste image of Graph window here.
Explain if each of the following statements below is true or false. Please make sure you provide a reason.
1.3. There are four corner points including (50, 0) and (0, 12.5).
1.4. The two corner points are (0, 0) and (50, 12.5).
1.5. The feasible region is triangular in shape, bounded by (50, 0), (33-1/3, 8-1/3), and (100, 0).
1.6. The graphical origin (0, 0) is in the feasible region.2. A manager must decide on the mix of products to produce for the coming week. Each unit of Product A requires three minutes per unit for molding, two minutes per unit for painting, and one minute for packing. Each unit of Product B requires two minutes per unit for molding, four minutes for painting, and three minutes per unit for packing. There will be 600 minutes available for molding, 600 minutes for painting, and 420 minutes for packing. Both products have contributions of $1.50 per unit.
2.1. Formulate the problem as a LP and write the linear programming model here. (Include definitions of decision variables, objective function and constraints.)
2.2. Solve the problem using QM for Windows. Paste image of Linear Programming Results window and Solution List window here.
2.3. What combination of A and B will maximize contribution, and what is the maximum possible contribution?
2.4. Write the problem in standard form. (Include definitions of decision variables, objective function and constraints.)
2.5. Write the initial simplex table and label it Simplex Table 2A.1.
Simplex Table
Basis CB
Zj
Cj – Zj
2.6. After creating Simplex Table 2A.1, find the entering variable using the Cj-Zj values and write it here.
2.7. What is the leaving variable? (Show MRR calculations.)
2.8. Write the elementary row operations for finding the new row __.
2.9. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
2.10. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
2.11. Write the new table and label it Simplex Table 2B.2.
Simplex Table
Basis CB
Zj
Cj – Zj
2.12. What is the entering variable?
2.13. What is the leaving variable? (Show MRR calculations.)
2.14. Write the elementary row operations for finding the new row __.
2.15. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
2.16. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
2.17. Write the new table and label it Simplex Table 2C.3.
Simplex Table
Basis CB
Zj
Cj – Zj
2.18. Is there an entering variable? If not, explain what this means.
3. A LP problem has three constraints: 2X + 10Y ≤ 100; 4X + 6Y ≤ 120; 6X + 3Y ≤ 90 and the non-negativity constraints. The objective is to Maximize X.
3.1. Write the problem in standard form. (Include definitions of decision variables, objective function and constraints.)
3.2. Solve the problem using QM for Windows. Paste image of Linear Programming Results window and Solution List window here.
3.3. Write the initial simplex table and label it Simplex Table 3A.1.
Simplex Table
Basis CB
Zj
Cj – Zj
3.4. After creating Simplex Table 3A.1, find the entering variable using the Cj-Zj values and type it here.
3.5. What is the leaving variable? (Show MRR calculations.)
3.6. Write the elementary row operations for finding the new row __.
3.7. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
3.8. If row __ changes, write the elementary row operations for finding the new row __. If row __ does not change, explain why it does not change.
3.9. Write the new table and label it Simplex Table 3B.2.
Simplex Table
Basis CB
Zj
Cj – Zj
3.10. Is there an entering variable for the next table? If so, what is the entering variable? If not, explain what this means.
3.11. What is the largest quantity of X that can be made without violating any of these constraints? Explain your answer.
4. Bryant’s Pizza, Inc. is a producer of frozen pizza products. The company makes a net income of $1.00 for each regular pizza and $1.50 for each deluxe pizza produced. The firm currently has 150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 pound of dough mix and 4 ounces (16 ounces= 1 pound) of topping mix. Each deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix. Based on the past demand, Bryant wants to make at least 50 regular pizzas and at least 25 deluxe pizzas. The problem is to determine the number of regular and deluxe pizzas the company should make to maximize net income.
4.1. Formulate this problem as an LP problem and write the linear programming model here. (Include definitions of decision variables, objective function and constraints.)
4.2. Solve using QM for Windows. Paste image of Linear Programming Results window and Solution List window here.
4.3. Explain your solution in words.
4.4. How much dough mix and topping mix are leftover?
5. A gold processor has two sources of gold ore, source A and source B. In order to keep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must not exceed $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. Ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton.
5.1. Formulate this problem as a LP and write the linear programming model here. (Include definitions of decision variables, objective function and constraints.)
5.2. Solve the problem in QM for Windows. Paste image of Linear Programming Results window and Solution List window here.
5.3. How many tons of ore from both sources must be processed each day to maximize the amount of gold extracted? Explain your answer.
5.4 What is the maximum amount of gold extracted? Explain your answer.
