Linear Programming Project Using QM Software

Management Science

Linear Programming Project

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Student Number:

Student Name:

Dr. Alok Baveja

Management Science (52:620:321)

Spring 2015

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. (15 Points Total)

1.1. Use QM for Windows to plot the feasible region. Paste image of Linear Programming

Results window and Solution List window here. (5 points)

1.2. Paste image of Graph window here. (2 points)

Explain if each of the following statements below is true or false. Please make sure you provide a

reason. (2 points each)

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. (25 Points Total)

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 points)

Tip: It may help to put the problem data in the form of a table before formulating the

problem.

2.2. Solve the problem using QM for Windows. Paste image of Linear Programming Results

window and Solution List window here. (3 points)

2.3. What combination of A and B will maximize contribution, and what is the maximum

possible contribution? (1 point)

Management Science

Linear Programming Project

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2.4. Write the problem in standard form. (Include definitions of decision variables, objective

function and constraints.) (2 points)

2.5. Write the initial simplex table and label it Simplex Table 2A.1. (2 points)

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. (1 point)

2.7. What is the leaving variable? (Show MRR calculations.) (1 point)

2.8. Write the elementary row operations for finding the new row __. (1 point)

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. (1 point)

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. (1 point)

2.11. Write the new table and label it Simplex Table 2B.2. (2 points)

Simplex Table

Basis CB

Zj

Cj - Zj

Management Science

Linear Programming Project

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2.12. What is the entering variable? (1 point)

2.13. What is the leaving variable? (Show MRR calculations.) (1 point)

2.14. Write the elementary row operations for finding the new row __. (1 point)

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. (1 point)

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. (1 point)

2.17. Write the new table and label it Simplex Table 2C.3. (2 points)

Simplex Table

Basis CB

Zj

Cj - Zj

2.18. Is there an entering variable? If not, explain what this means. (1 point)

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. (25 Points Total)

3.1. Write the problem in standard form. (Include definitions of decision variables, objective

function and constraints.) (3 points)

3.2. Solve the problem using QM for Windows. Paste image of Linear Programming Results

window and Solution List window here. (3 points)

Management Science

Linear Programming Project

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3.3. Write the initial simplex table and label it Simplex Table 3A.1. (2 points)

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. (2 points)

3.5. What is the leaving variable? (Show MRR calculations.) (2 points)

3.6. Write the elementary row operations for finding the new row __. (2 points)

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. (2 points)

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. (2 points)

3.9. Write the new table and label it Simplex Table 3B.2. (3 points)

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. (2 points)

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Linear Programming Project

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3.11. What is the largest quantity of X that can be made without violating any of these

constraints? Explain your answer. (2 points)

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. (15 Points Total)

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 points)

4.2. Solve using QM for Windows. Paste image of Linear Programming Results window and

Solution List window here. (4 points)

4.3. Explain your solution in words. (4 points)

4.4. How much dough mix and topping mix are leftover? (3 points)

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. (20 Points Total)

5.1. Formulate this problem as a LP and write the linear programming model here. (Include

definitions of decision variables, objective function and constraints.) (6 points)

5.2. Solve the problem in QM for Windows. Paste image of Linear Programming Results

window and Solution List window here. (6 points)

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. (4 points)

5.4 What is the maximum amount of gold extracted? Explain your answer. (4 points)