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Chapter 3

Linear Programming Simplex Method

Exercise 9 Solving Linear Programming Problems

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______1. An iterative process is a procedure, which is repeated over and over following a random pattern. ______ 2. An inequality relationship may be converted into an equality by the addition of a slack variable. ______ 3. An infeasible solution is characterized as one where at least one constraint is violated. ______ 4. The Cj column in a maximization simplex table indicates the profit per unit for only those variables in the product mix. ______ 5. The significance of the quantity column in the simplex table is that it always shows the right hand side values of the constraints. ______ 6. Artificial, slack, and surplus variables are added to inequality constraints in order to convert them to equalities. Artificial variables are also added to constraints which were equalities to begin with.

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______ 7. There are always more variables in a simplex problem than there are constraints. ______ 8. If an artificial variable appears in the product mix of a table, which is optimal, it is likely that somebody has defined two or more incompatible constraints. ______ 9. Shadow prices are identified with variables, which are included in the product mix. ______ 10. For any basic variable in the product mix of a simplex table, the value of Cj – Zj in the column headed by the basic variables will always be negative in a maximization problem and positive in a minimization problem.

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. (Answer is in red letter) ______1. An iterative process is a procedure, which is repeated over and over following a random pattern. True ______ 2. An inequality relationship may be converted into an equality by the addition of a slack variable. True ______ 3. An infeasible solution is characterized as one where at least one constraint is violated. True ______ 4. The Cj column in a maximization simplex table indicates the profit per unit for only those variables in the product mix. True ______ 5. The significance of the quantity column in the simplex table is that it always shows the right hand side values of the constraints. True ______ 6. Artificial, slack, and surplus variables are added to inequality constraints in order to convert them to equalities. Artificial variables are also added to constraints which were equalities to begin with. True

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______ 7. There are always more variables in a simplex problem than there are constraints. False ______ 8. If an artificial variable appears in the product mix of a table, which is optimal, it is likely that somebody has defined two or more incompatible constraints. False ______ 9. Shadow prices are identified with variables, which are included in the product mix. True ______ 10. For any basic variable in the product mix of a simplex table, the value of Cj – Zj in the column headed by the basic variables will always be negative in a maximization problem and positive in a minimization problem. True

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 1. Which of the following is not true for the simplex procedures? a. Each iteration results a new solution, which is as good as or better than the previous solution. b. The procedure indicates when the optimum solution has been reached. c. The procedure ensures that the optimum solution is reached in the minimum number of iteration. d. All of the above choices are correct. ______ 2. If for a given solution, a slack variable is equal to zero then, a. the solution is not feasible b. the solution is optimal c. the entire amount of resource has been used d. all of the above choices are correct

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 3. A variable, which is not included in the product mix column for a given solution is: a. always equal to zero b. never equal to zero c. called a basic variable d. none of the above choices are correct ______ 4. With simplex, the initial solution contains only: a. slack variable in the product mix b. artificial variables in the product mix c. slack and artificial variables in the product mix d. real variables in the product mix ______ 5. The procedure for solving a minimization problem with simplex is exactly the same as solving a maximization problem but with one exception, which is: a. the identification of the replaced row b. the identification of the optimum column c. the computation of the Cj – Zj values d. the computation of the replacing row

Chapter 3

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. _____ 6. In solving a problem where 100 different products (100 real variables) are produced using just two resources (2 slack variables), then the combination that will yield the optimal profit is: a. produce all 100 products if both resources are used to full capacity b. produce, at most 2 products c. produce only one product d. none of the above choices are correct ______ 7. In converting > constraint to equality one will a. add an artificial variable b. add a slack variable c. subtract a surplus variable and add an artificial variable d. subtract a surplus variable ______ 8. In solving maximization problem using simplex method, the optimum column is obtained by getting the column with a. the largest positive value in the Cj – Zj row b. the largest negative value in the Cj – Zj row c. zero value in the Cj – Zj row d. smallest positive value in the Cj – Zj rowariables

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 9. Which of the following in a simplex table indicates that an optimal solution for maximization problem has been formed? a. All the Cj – Zj values are negative or zero. b. All the Cj – Zj values are positive or zero. c. There are no more slack variables in the product mix. d. All the Cj – Zj values are zeros. ______ 10. Which of the following must equal to zero? a. Basic variables b. Solution mix variables c. Non-basic variables d. Objective function coefficients for artificial variables

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __b____ 1. Which of the following is not true for the simplex procedures? a. Each iteration results a new solution, which is as good as or better than the previous solution. b. The procedure indicates when the optimum solution has been reached. c. The procedure ensures that the optimum solution is reached in the minimum number of iteration. d. All of the above choices are correct. __c____ 2. If for a given solution, a slack variable is equal to zero then, a. the solution is not feasible b. the solution is optimal c. the entire amount of resource has been used d. all of the above choices are correct

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __d____ 3. A variable, which is not included in the product mix column for a given solution is: a. always equal to zero b. never equal to zero c. called a basic variable d. none of the above choices are correct __c____ 4. With simplex, the initial solution contains only: a. slack variable in the product mix b. artificial variables in the product mix c. slack and artificial variables in the product mix d. real variables in the product mix __d____ 5. The procedure for solving a minimization problem with simplex is exactly the same as solving a maximization problem but with one exception, which is: a. the identification of the replaced row b. the identification of the optimum column c. the computation of the Cj – Zj values d. the computation of the replacing row

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __d___ 6. In solving a problem where 100 different products (100 real variables) are produced using just two resources (2 slack variables), then the combination that will yield the optimal profit is: a. produce all 100 products if both resources are used to full capacity b. produce, at most 2 products c. produce only one product d. none of the above choices are correct ___C___ 7. In converting > constraint to equality one will a. add an artificial variable b. add a slack variable c. subtract a surplus variable and add an artificial variable d. subtract a surplus variable __b____ 8. In solving maximization problem using simplex method, the optimum column is obtained by getting the column with a. the largest positive value in the Cj – Zj row b. the largest negative value in the Cj – Zj row c. zero value in the Cj – Zj row d. smallest positive value in the Cj – Zj rowariables

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ___a___ 9. Which of the following in a simplex table indicates that an optimal solution for maximization problem has been formed? a. All the Cj – Zj values are negative or zero. b. All the Cj – Zj values are positive or zero. c. There are no more slack variables in the product mix. d. All the Cj – Zj values are zeros. ___d___ 10. Which of the following must equal to zero? a. Basic variables b. Solution mix variables c. Non-basic variables d. Objective function coefficients for artificial variables

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 1.

Maximize: Z = 4x1 + 5x2 subject to: 3x1+ 6x2 ≤ 36 4x1 + 2x2 ≤ 36 x1 + x2 ≤ 8 x1,x2 ≥ 0

2.

Maximize: Z = 6x + 3y subject to: 2x+ 4y ≤ 30 4x + 2y ≤ 30 2x + 3y ≥ 18 x1,x2 ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C1 C. Adding the appropriate slack, surplus, and artificial variables. 1.

Maximize: Z = 4x1 + 5x2 subject to: 3x1+ 6x2 ≤ 36 4x1 + 2x2 ≤ 36 x1 + x2 ≤ 8 x1,x2 ≥ 0

Z = 4x1 + 5x2 + 0S1 + 0S2 + 0S3 3x1+ 6x2 + 1S1 + 0S2 + 0S3= 36 4x1 + 2x2 + 0S1 + 1S2 + 0S3 = 36 x1 + x2 + 0S1 + 0S2 + 1S3 = 8

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C2 C. Adding the appropriate slack, surplus, and artificial variables.

2. Maximize: Z = 6x + 3y ubject to: 2x+ 4y ≤ 30 4x + 2y ≤ 30 2x + 3y ≥ 18 x1,x2 ≥ 0

Z = 6x + 3y + 1S1 + 0S2 + 0S3 + 10A 2x+ 4y + 1S1 + 0S2 + 0S3 + 0 A3 =30 4x + 2y + 0S1 + 1S2 + 0S3 + OA3 = 30 2x + 3y + 0S1 + 0S2 - 1S3 +1A3 = 18

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C3 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 3.

Minimize: C = 8x + 4y subject to: 10x+ 25y ≥ 100 y≤4 x,y ≥ 0

4.

Minimize: C = 4x + 5y subject to: x+ 6y = 12 x≤4 y≤ 8 y≥5 x,Y ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C3 C. Adding the appropriate slack, surplus, and artificial variables. 3. Minimize: C = 8x + 4y subject to: 10x+ 25y ≥ 100 y≤4 x,y ≥ 0

C = 8x + 4y + 0S1 + 0S2 + 10 A1 10x+ 25y + 1S1 + 0S2 + 1A1 = 100 0x + y - 0S1 + 1S2 + 0A1 = 4

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C4 C. Adding the appropriate slack, surplus, and artificial variables.

4. Minimize: C = 4x + 5y subject to: x+ 6y = 12 x≤4 y≤ 8 y≥5 x,Y ≥ 0

C = 4x + 5y + 0S2 + 0S3 + 0S4 +10A1 +10A4 x+ 6y + 0S2 + 0S3 + 0S4 +1A1 + 0A4 = 12 x + 0y + 1S2 + 0S3 + 0S4 +0A1 + 0A4 = 4 0x + y + 0S2 + 1S3 + 0S4 +0A1 + 0A4 = 8 0x + y + 0S2 + 0S3 - 1S4 +0A1 + 1A4 = 5

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C5 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 5.

Minimize: C = 0.05x + 0.06y subject to: 5x+ 25y ≥ 50 25x + 10y ≥ 100 10x + 10y ≥ 60 36x + 20y ≥ 180 x,y ≥ 0

6.

Maximize: Z = 4A + 5B subject to: 3A+ 4B ≤ 150 A+2B ≤ 60 A ≥ 25 A,B ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C6 C. Adding the appropriate slack, surplus, and artificial variables. 5. Minimize: C = 0.05x + 0.06y subject to: 5x+ 25y ≥ 50 25x + 10y ≥ 100 10x + 10y ≥ 60 36x + 20y ≥ 180 x,y ≥ 0 C = 0.05x + 0.06y + 0S1 +0S2 + 0S3 + 0S4 +10A1 + 10A2 + 10A3 +10A4 5x+ 25y - 1S1 +0S2 + 0S3 + 0S4 +1A1 + 0A2 + 0A3 +0A4 = 50 25x + 10y + 0S1 -1S2 + 0S3 + 0S4 +0A1 + 1A2 + 0A3 +0A4 = 100 10x + 10y + 0S1 +0S2 - 1S3 + 0S4 +0A1 + 0A2 + 1A3 +0A4 = 60 36x + 20y + 0S1 +0S2 + 0S3 - 1S4 +0A1 + 0A2 + 0A3 +1A4 = 180

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C6 C. Adding the appropriate slack, surplus, and artificial variables. 6.

Maximize: Z = 4A + 5B subject to: 3A+ 4B ≤ 150 A+2B ≤ 60 A ≥ 25 A,B ≥ 0

Z = 4A + 5B + 0S1 +0S2 + 0S3 + 10A 3A+ 4B + 1S1 +0S2 + 0S3 + 0A = 150 A+ 2B + 0S1 + 1S2 + 0S3 + 0A = 60 A + 0B + 0S1 +0S2 - 1S3 + 1A = 25

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C7 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 7.

Minimize: C = 2x1 + 5x2 subject to: 3x1+ 2x2 = 30 x1 ≤ 5 x2 ≥ 10 x1,x2 ≥ 0

8.

Minimize: C = 2x1 + 5x2 + 9x3 subject to: 2x1+ 2x2 + x3 = 20 2x1+ 4x2 + 5x3 ≤ 20 x1 + 2x2 ≥ 6 x2 ≥ 10 x1,x2, x3 ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C7 C. Adding the appropriate slack, surplus, and artificial variables. 7.

Minimize: C = 2x1 + 5x2 subject to: 3x1+ 2x2 = 30 x1 ≤ 5 x2 ≥ 10 x1,x2 ≥ 0 C = 2x1 + 5x2 3x1+ 2x2 -1S1 +0S2 + 0S3 + 1A1 + 0A3 = 30 x1 + 0x2 +0S1 +1S2 + 0S3 + 0A1 + 0A3 = 5 0x1 + x2 +0S1 +0S2 - 1S3 + 0A1 + 1A3 = 10

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C8 C. Adding the appropriate slack, surplus, and artificial variables. 8. Minimize: C = 2x1 + 5x2 + 9x3 subject to: 2x1+ 2x2 + x3 = 20 2x1+ 4x2 + 5x3 ≤ 20 x1 + 2x2 ≥ 6 x2 ≥ 10 x1,x2, x3 ≥ 0 C = 2x1 + 5x2 + 9x3 + 0S2 + 0S3 + 0S4 + 10A1 + 10A3 + 10A4 2x1+ 2x2 + x3 + 0S2 + 0S3 + 0S4 + 1A1 + 0A3 + 0A4 = 20 2x1+ 4x2 + 5x3 + 1S2 + 0S3 + 0S4 + 0A1 + 0A3 + 0A4 = 20 x1 + 2x2 +0x3 + 0S2 - 1S3 + 0S4 + 0A1 + 1A3 + 0A4 = 6 0x1 + x2 +0x3 + 0S2 + 0S3 - 1S4 + 0A1 + 0A3 + 1A4 = 10

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C9 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 9.

Maximize: Z = 8x1 + 6x2 + 2x3 subject to: 3x1+ 2x2 + 6 ≤ 24 4x1+ 3x2 + 6x3 = 48 x1,x2, x3 ≥ 0

10.

Minimize: C = 4x + 5y + 3z subject to: 4x+ 2y + 3z = 100 x ≥ 20 y≤6 z ≥8 x,y, z ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C9 C. Adding the appropriate slack, surplus, and artificial variables. 9. Maximize: Z = 8x1 + 6x2 + 2x3 subject to: 3x1+ 2x2 + 6 ≤ 24 4x1+ 3x2 + 6x3 = 48 x1,x2, x3 ≥ 0 Z = 8x1 + 6x2 + 2x3 + 0S1 + 10A2 3x1+ 2x2 + 6 + 1S1 + 0A2 = 24 4x1+ 3x2 + 6x3 + 0S1 + 1A2 = 48

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C10 C. Adding the appropriate slack, surplus, and artificial variables. 10. Minimize: C = 4x + 5y + 3z subject to: 4x+ 2y + 3z = 100 x ≥ 20 y≤6 z ≥8 x,y, z ≥ 0

C = 4x + 5y + 3z + 0S2 + 0S3 + 0S4 + 10A1 +10A2 +10A4 4x + 2y + 3z + 0S2 + 0S3 + 0S4 + 1A1 + 0A2 + 0A4 = 100 1x + 0y + 0z - 1S2 + 0S3 + 0S4 + 0A1 +1A2 +0A4 = 20 0x + 1y + 0z + 0S2 + 1S3 + 0S4 + 0A1 +0A2 +0A4 = 6 0x + 0y + 1z + 0S2 + 0S3 - 1S4 + 0A1 +0A2 +1A4 = 8

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C11-C12 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 11.

Minimize: C = 6x1 + 16x2 subject to: x1+ x2 = 400 x1= 150 x2 ≥ 200 x1,x2, ≥ 0

12. Minimize: C = 1000x + 1500y subject to: 20x+ 20y ≥ 160 30x+ 60y ≥ 300 x≥ 2 x,y, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C11 C. Adding the appropriate slack, surplus, and artificial variables. 11. Minimize: C = 6x1 + 16x2 subject to: x1+ x2 = 400 x1= 150 x2 ≥ 200 x1,x2, ≥ 0 C = 6x1 + 16x2 + 0S3 + 10A1 + 10A2 + 10A3 1x1+ 1x2 + 0S3 + 1A1 + 0A2 + 0A3 = 400 1x1+ 0x2 + 0S3 + 0A1 + 1A2 + 0A3 = 150 0x1+ 1x2 - 1S3 + 0A1 + 0A2 + 1A3 = 200

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C12 C. Adding the appropriate slack, surplus, and artificial variables. 12. Minimize: C = 1000x + 1500y subject to: 20x+ 20y ≥ 160 30x+ 60y ≥ 300 x≥ 2 x,y, ≥ 0 C = 1000x + 1500y + 0S1 + 0S2 + 0S3 + 10A1 +10A2 +10A3 20x + 20y - 1S1 + 0S2 + 0S3 + 1A1 +0A2 +0A3 = 160 30x + 60y + 0S1 - 1S2 + 0S3 + 0A1 +1A2 +0A3 = 300 1x + 0y + 0S1 + 0S2 - 1S3 + 0A1 +0A2 +1A3 = 2

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C12-C14 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 13. Minimize: C = 10x1 + 20x2 subject to: x1+ x2 ≥ 10 3x1 + x2 ≤ 12 x1,x2, ≥ 0

14. Minimize: C = 120x1 + 100x2 subject to: 2x1+ x2 ≤ 18 5x1+ 4x2 ≥ 60 x1,x2, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C13 C. Adding the appropriate slack, surplus, and artificial variables. 13. Minimize: C = 10x1 + 20x2 subject to: x1+ x2 ≥ 10 3x1 + x2 ≤ 12 x1,x2, ≥ 0

C = 10x1 + 20x2 + 0S1 + 0S2 + 10A1 1x1+ 1x2 - 1S1 + 0S2 + 1A1 = 10 3x1+ 1x2 + 0S1 + 1S2 + 0A1 = 12

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C14 C. Adding the appropriate slack, surplus, and artificial variables. 14. Minimize: C = 120x1 + 100x2 subject to: 2x1+ x2 ≤ 18 5x1+ 4x2 ≥ 60 x1,x2, ≥ 0

C = 120x1 + 100x2 + 0S1 + 0S2 + 10A2 1x1+ 1x2 + 1S1 + 0S2 + 0A2 = 18 3x1+ 1x2 + 0S1 - 1S2 + 1A2 = 60

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-C15 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 15. Minimize: Z = 2x1 + 1x2 subject to: 2x1+ 3x2 ≥ 12 3x1+ 4x2 ≤ 21 6x1+ 5x2 ≤ 30 x1,x2, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C15 C. Adding the appropriate slack, surplus, and artificial variables. 15. Minimize: Z = 2x1 + 1x2 subject to: 2x1+ 3x2 ≥ 12 3x1+ 4x2 ≤ 21 6x1+ 5x2 ≤ 30 x1,x2, ≥ 0 Z = 2x1 + 1x2 + 0S1 + 0S2 + 0S3 + 10A1 1x1+ 1x2 - 1S1 + 0S2 + 0S3 + 1A1 = 12 1x1+ 0x2 + 0S1 + 1S2 + 0S3 + 0A1 = 21 0x1+ 1x2 + 0S1 + 0S2 + 1S3 + 0A1 = 30

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D D. Solve the following problems using simplex method: 1. Ignacio Furniture Co. makes two types of playhouses: a standard model and a deluxe model. The playhouse are sold to the independent dealers at a profit of P200/standard and P300/ deluxe. A standard requires 30 man-hours for assembly, and 20 man-hours for painting and finishing and 10 man-hours for inspecting. A deluxe requires 75 man-hours for assembly, 25 manhours for painting and finishing, and five (5) man-hours for inspecting. A production run generally has 15000 man-hours available for assembly, 6500 man-hours available for painting and finishing and 2500 man-hours for inspecting. Determine the maximum profit and optimal values of the decisions and slack variables.

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Solution to Exercise 9-D1 D. Solve the following problems using simplex method: 1 Maximize: Z = 200x1 + 300x2 subject to: Assembly 30x1+ 75x2 ≤ 15,000 Painting/finishing 20x1+ 25x2 ≤ 6,500 inspecting 10x1+ 5x2 ≤ 2,500 x1,x2, ≥ 0

Let x1 = standard model x2 = deluxe model

Z = 200x1 + 300x2 + 0S1 + 0S2 + 0S3 30x1+ 75x2 + 1S1 + 0S2 + 0S3 = 15,000 20x1+ 25x2 + 0S1 + 1S2 + 0S3 = 6,500 10x1+ 5x2 + 0S1 + 0S2 + 1S3 = 2,500

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Initial Tableau Cont.

Soln.

Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

0

S1

15,000

30

75

1

0

0

0

S2

6,500

20

25

0

1

0

0

S3

2,500

10

5

0

0

0

Zj

0

0

0

0

0

0

200

300

0

0

0

Cj - Zj

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Initial Tableau Cont.

Soln.

Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

ETR

0

S1

15,000

30

75

1

0

0

200

0

S2

6,500

20

25

0

1

0

260

0

S3

2,500

10

5

0

0

0

500

Zj

0

0

0

0

0

0

200

300

0

0

0

Cj - Zj

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Second Tableau 300

0

0

0

X1

X2

S1

S2

S3

Cont.

Soln.

300

X1

200

2/5

1

1/75

0

0

0

S2

1,500

10

0

-1/3

1

0

0

S3

1,500

8

0

-1/15

0

0

Zj

60,000

120

300

4

0

0

80

o

-4

0

0

Cj - Zj

Qty.

200

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Second Tableau 300

0

0

0

X1

X2

S1

S2

S3

ETR

Cont.

Soln.

300

X1

200

2/5

1

1/75

0

0

500

0

S2

1,500

10

0

-1/3

1

0

150

0

S3

1,500

8

0

-1/15

0

0

187.5

Zj

60,000

120

300

4

0

0

80

o

-4

0

0

Cj - Zj

Qty.

200

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Third (Final) Tableau Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

Cont.

Soln.

300

X1

140

0

1

11/75

-1/25

0

200

X1

150

1

0

-1/30

1/10

0

0

S3

300

0

0

13/5

-1/80

0

Zj

72,000

200

300

112/3

8

0

Cj - Zj

-72,000

0

0

-112/3

-8

0

The solution is optimal: X2 = 140 X1= 150

Max Z = 200x1 + 300x2 = 72,000

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D2 D. Solve the following problems using simplex method: 2. To make one unit of product x requires three (3) minutes in Dept. I and one (1) minute in Dept. II. One unit of product y requires four (4) minutes in Dept.I and two (2) minutes in Dept.II. Profit contribution is P5/unit of x and P8/unit of y. It is required that at least 25 units of x be made to maximize profit if Dept.I and II have 150 and 160 minutes available respectively. What is the maximum profit?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 D. Solve the following problems using simplex method: 2 Maximize: Z = 5x + 8y subject to: time in dept. 1 3x+ 4y ≤ 150 time in dept .2 1x+ 2y ≤ 160 capacity x ≥ 25 x,y ≥ 0 Z = 5x + 8y + 0S1 + 0S2 + 0S3 + 10A3 3X+ 4Y + 1S1 + 0S2 + 0S3 +0 A3 = 150 1X+ 2Y + 0S1 + 1S2 + 0S3 + 0A3 = 160 1X+ 0Y + 0S1 + 0S2 - 1S3 + 1A3 = 25

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Initial Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

0

S1

150

3

4

1

0

0

0

0

S2

160

1

2

0

1

0

0

10

A3

25

1

0

0

0

-1

1

Zj

250

10

0

0

0

-10

10

Cj - Zj

-250

-5

8

0

0

10

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Initial Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

ETR

0

S1

150

3

4

1

0

0

0

50

0

S2

160

1

2

0

1

0

0

160

10

A3

25

1

0

0

0

-1

1

25

Zj

250

10

0

0

0

-10

10

Cj - Zj

-250

-5

8

0

0

10

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Second (Final) Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

0

S1

75

0

4

1

0

-3

-3

0

S2

135

0

2

0

1

1

-1

5

X

25

1

0

0

0

-1

1

Zj

125

5

0

0

0

-5

5

Cj - Zj

-125

0

8

0

0

5

5

The solution is optimal: X = 25 Y= 0

Max Z = 5X + 8Y = 125

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D3 D. Solve the following problems using simplex method: 3. A manufacturer of commercial chemical has an order for a certain mixture containing of three ingredients x1, x2, x3, which costs P8, P7, and P4 per kilo, respectively. The following are specifications: a. b. c. d.

It cannot contain more than 35 kilos of x. It must contain at least 15 kilos of x2. It cannot contain more than 40 kilos of X3. The weight of the mixture must be 120 kilos.

Find a mixture of the three ingredients, which satisfies his customer’s requirements and still yields the minimum total cost of raw materials.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 D. Solve the following problems using simplex method: 3. Minimize: Z = 8x1 + 7x2 + 4x3 subject to: content of X x1+ x2 + x3 ≤ 35 content of X2 x2 ≥ 15 content of X3 x3 ≤ 40 weight x1+ x2 + x3 = 120 x1,x2, x3≥ 0 Z = 8x1 + 7x2 + 4x3 + 0S1 + 0S2 + 0S3 + 10A2 + 10A4 1x1+ 1x2 + 1x3 + 1S1 + 0S2 + 0S3 + 0A2 +0A4 = 35 0x1+ 1x2 + 0x3 + 0S1 - 1S2 + 0S3 +1A2 +0A4= 15 0x1+ 0x2 + 1x3 + 0S1 + 0S2 + 1S3 +0A2 + 0A4 = 40 1x1+ 1x2 + 1x3 + 0S1 + 0S2 + 0S3 +0A2 + 1A4 = 120

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Initial Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

0

S1

35

1

1

1

1

0

0

0

0

10

A2

15

0

1

0

0

-1

0

1

0

0

S3

40

0

0

1

0

0

1

0

0

10

A4

120

1

1

1

0

0

0

0

1

Zj

1350

10

20

10

0

-10

0

10

10

Cj - Zj

-1350

-2

-13

-6

0

10

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Initial Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

ETR

0

S1

35

1

1

1

1

0

0

0

0

35

10

A2

15

0

1

0

0

-1

0

1

0

15

0

S3

40

0

0

1

0

0

1

0

0

0

10

A4

120

1

1

1

0

0

0

0

1

120

Zj

1350

10

20

10

0

-10

0

10

10

Cj - Zj

-1350

-2

-13

-6

0

10

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Second Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

0

S1

20

1

0

1

1

2

0

-1

0

7

X2

15

0

1

0

0

-1

0

1

0

0

S3

40

0

0

1

0

0

1

0

0

10

A4

105

1

0

1

0

1

0

-1

1

Zj

1155

10

7

10

0

3

0

-3

10

Cj - Zj

-1155

-2

0

-6

0

-3

0

7

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Second Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

ETR

0

S1

20

1

0

1

1

2

0

-1

0

20

7

X2

15

0

1

0

0

-1

0

1

0

0

0

S3

40

0

0

1

0

0

1

0

0

40

10

A4

105

1

0

1

0

1

0

-1

1

105

Zj

1155

10

7

10

0

3

0

-3

10

Cj - Zj

-1155

-2

0

-6

0

-3

0

7

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Third (Final) Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

4

X3

20

1

0

1

1

2

0

-1

0

7

X2

15

0

1

0

0

-1

0

1

0

0

S3

20

1

0

0

-1

-2

1

1

0

10

A4

85

0

0

0

-1

-1

0

0

1

Zj

1035

4

7

4

-6

-9

0

3

10

Cj - Zj

-1035

4

0

0

6

9

0

7

0

The solution is optimal: X3 = 20 X2= 15

ETR

X1 = 0 Min Z = 8X1 + 7X2 + 4X3 + 10A4 A4 = 85 = 1035

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D4 D. Solve the following problems using simplex method: 4. The Jay Gee Bull Ranch has plans to sell cattle to a large meat packaging plant. The manager of the ranch believes that each cattle should receive a minimum of 240 oz. of nutritional ingredients A and a minimum of 150 oz. of nutritional ingredients B each week. There are two grains available which contain both types of nutritional ingredients. One bag of grain 1 contains 12 oz of ingredients A and 25 oz of ingredients B and costs P140/bag; one bag of grain 2 contains 20 oz of ingredients A and five (5) oz. of ingredients B and costs P120/bag. Determine the optimum mixture of grains and the associated minimum cost.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 D. Solve the following problems using simplex method: 4. Minimize: Z = 140x + 120y subject to: Ingredients A 12X + 20Y ≥ 240 Ingredients B 25X + 5Y ≥ 150 x,y ≥ 0

Let X = grain 1 Y = grain 2

Z = 140x + 120y + 0S1 + 0S2 + 1000A1 + 1000A2 12x + 20y -1S1 + 0S2 + 1A1 + 0A2 = 35 25x + 5y + 0S1 - 1S2 + 0A1 + 1A2 = 35

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Initial Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

1000

A1

240

12

20

-1

0

1

0

1000

A2

150

25

5

0

-1

0

1

Zj

390000

37000

25000

-1000

-1000

1000

1000

Cj - Zj

-390000

-36860 -24880

1000

1000

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Initial Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

1000

A1

240

12

20

-1

0

1

0

20

1000

A2

150

25

5

0

-1

0

1

6

Zj

390000

37000

25000

-1000

-1000

1000

1000

Cj - Zj

-390000

-36860 -24880

1000

1000

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Second Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

1000

A1

168

0

88/5

-1

12/25

1

-12/25

140

X

6

1

1/5

0

-1/25

0

1/25

Zj

168840

140

18300

-1000

4794.4

1000

-4794.4

Cj - Zj

-168840

0

-18180

1000

-4794.4

0

5794.4

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Second Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

1000

A1

168

0

88/5

-1

12/25

1

-12/25

840/88

140

X

6

1

1/5

0

-1/25

0

1/25

30

Zj

168840

140

18300

-1000

4794.4

1000

-4794.4

Cj - Zj

-168840

0

-18180

1000

-4794.4

0

5794.4

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Third Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

105/11

0

1

-5/88

3/110

5/88

-3/110

140

X

45/11

1

0

1/88

-1/22

-1/88

1/22

Zj

1145.45

140

120

5.23

-3.09

5.23

-4794.4

Cj - Zj

-1145.55

0

0

-5.23

3.09

994.8

6.36

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Third Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

120

Y

105/11

0

1

-5/88

3/110

5/88

-3/110

-168

140

X

45/11

1

0

1/88

-1/22

-1/88

1/22

360

Zj

1145.45

140

120

5.23

-3.09

5.23

-4794.4

Cj - Zj

-1145.55

0

0

-5.23

3.09

994.8

6.36

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fourth Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

30

5

1

0

-22/110

0

22/110

0

S2

360

88

0

1

-4

-1

4

Zj

3,600

600

120

0

-24

0

24

Cj - Zj

-3,600

-460

0

0

24

1000

976

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fourth Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

120

Y

30

5

1

0

-22/110

0

22/110

0

0

S2

360

88

0

1

-4

-1

4

4.09

Zj

3,600

600

120

0

-24

0

24

Cj - Zj

-3,600

-460

0

0

24

1000

976

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fifth (Final) Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

105/11

0

1

-5/88

3/110

-5/88

-3/110

140

X

45/11

1

0

1/88

-4/88

1/88

4/88

Zj

1718.18

140

120

-5.23

-3.09

5.23

3.09

Cj - Zj

-1718.18

0

0

5.23

3.09

994.77

996.91

The solution is optimal: Y = 105/11 Min Z = 140X + 120Y X= 45/11 = 1718.18

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D5 D. Solve the following problems using simplex method: 5. A manufacturer makes two products; picnic tables and benches, which must be processed through two machine centers. MC1 has up to 60 hours available. MC2 can handle up 48 hours of work. Each picnic table requires four (4) hours in MC1 and two (2) hours in MC2. Each bench takes two (2) hours in MC1 and four(4) hours in MC2. If profit is P80/picnic table and P60/bench, determine the best possible contribution of picnic tables and benches to produce and sell in order to realize the maximum profit.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 D. Solve the following problems using simplex method: 5. Maximize: Z = 80x + 60y subject to: MC 1 4x+ 2y ≤ 60 MC 2 2x+ 4y ≤ 48 Z = 80x + 60y + 0S1 + 0S2 4X+ 2Y + 1S1 + 0S2 = 60 2X+ 4Y + 0S1 + 1S2 = 48

Let X = picnic table Y = bench

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Initial Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

0

S1

60

4

2

1

0

0

S2

48

2

4

0

1

Zj

0

0

0

0

0

Cj - Zj

-0

80

60

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Initial Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

ETR

0

S1

60

4

2

1

0

15

0

S2

48

2

4

0

1

24

Zj

0

0

0

0

0

Cj - Zj

-0

80

60

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Second Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

80

X

15

1

1/2

1/4

0

0

S2

18

0

3

-1/2

1

Zj

1200

80

40

20

0

Cj - Zj

-1200

0

20

-20

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Second Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

ETR

80

X

15

1

1/2

1/4

0

7.5

0

S2

18

0

3

-1/2

1

6

Zj

1200

80

40

20

0

Cj - Zj

-1200

0

20

-20

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Third Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

80

X

4

1

0

1/3

-1/6

60

Y

8

0

1

-1/6

1/3

Zj

800

80

60

16.67

6.67

Cj - Zj

-800

0

0

-16.67

-6.67

The solution is optimal: X = 4 y=8

Max Z = 80X + 60Y = 800

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-6 D. Solve the following problems using simplex method: 6.

Mr. Winston Pe can produce in his shop any mix of three products x,y, and z. The prices and variable cost per unit are: Product Price/unit Variable cost/unit X P20 P11 Y 12 8 Z 8 6 Winston processes three product in each of the three departments I, II, III. The time requirements per department are as follows:

80 hours per day is available at each of the department. Develop the set of constraints and the objective function to maximize profit. How much is the maximum profit?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 D. Solve the following problems using simplex method: 6. Maximize: Z = 9x + 4y + 2z subject to: Dept. 1 2x+ 4y + 3y ≤ 80 Dept. 2 x+ 6y + 5y ≤ 80 Dept. 3 7x + 2y + 9z ≤ 80 Z = 9x + 4y + 2z + 0S1 + 0S2 + 0S3 2X+ 4Y + 3z + 1S1 + 0S2 + 0S3 = 80 X+ 6Y + 5z + 0S1 + 1S2 + 0S3 = 80 7X+ 2Y + 9z + 0S1 + 0S2 + 1S3 = 80

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Initial Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

80

2

4

3

1

0

0

0

S2

80

1

6

5

0

1

0

0

S2

80

7

2

9

0

0

1

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

9

4

2

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Initial Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

80

2

4

3

1

0

0

40

0

S2

80

1

6

5

0

1

0

80

0

S2

80

7

2

9

0

0

1

11.43

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

9

4

2

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Second Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

400/7

0

24/7

3/7

1

0

-2/7

0

S2

480/7

0

40/7

26/7

0

1

-1/7

9

X

80/7

1

2/7

9/7

0

0

1/7

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Second Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

400/7

0

24/7

3/7

1

0

-2/7

133.3

0

S2

480/7

0

40/7

26/7

0

1

-1/7

18.46

9

X

80/7

1

2/7

9/7

0

0

1/7

8.89

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Third Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

160/3

-1/3

10/3

0

1

0

-1/3

0

S2

4940/63

-26/9

308/63

0

0

1

-35/63

2

Z

80/9

7/9

2/9

1

0

0

1/9

Zj

17.78

1.56

0.44

2

0

0

0.22

Cj - Zj

-17,78

7.44

3.56

0

0

0

-0.22

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Third Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

160/3

-1/3

10/3

0

1

0

-1/3

-160

0

S2

4940/63

-26/9

308/63

0

0

1

-35/63

-27.25

2

Z

80/9

7/9

2/9

1

0

0

1/9

11.43

Zj

17.78

1.56

0.44

2

0

0

0.22

Cj - Zj

-17,78

7.44

3.56

0

0

0

-0.22

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fourth Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

1200/21

0

72/21

3/7

1

0

-6/21

0

S2

40

0

360/63

26/7

0

1

-9/63

9

X

80/7

1

2/7

9/7

0

0

1/7

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fourth Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

1200/21

0

72/21

3/7

1

0

-6/21

21.05

0

S2

40

0

360/63

26/7

0

1

-9/63

7

9

X

80/7

1

2/7

9/7

0

0

1/7

31.5

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fifth (Final) Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

232/7

0

0

-9/5

1

-3/5

-13/35

4

Y

7

0

1

13/20

0

7/40

-1/40

9

X

66/7

1

0

11/7

0

-1/20

19/140

Zj

112.85

9

4

16.74

0

0.25

1.12

Cj - Zj

-112.86

0

0

-14,74

0

-0.25

-1.12

The solution is optimal: Y= 7 X = 66/7 Z=0

Max Z = 9X + 4Y + 2Z = 112.86

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-D7 D. Solve the following problems using simplex method: 7.

The poultry farmer must supplement the vitamins in the feeds he buys. He is considering two supplements, each of which contains the feed required but in different amounts. He must meet or exceed the minimum vitamin requirements.

The vitamin content per gram of the supplements is given in the following table:

Vitamin 1 2 3

Supplement 1 2 2 2

supplement 2 1 9 3

Supplement 1 costs P5 per gram and supplement 2 costs P4 per gram. The feed must contain at least 12 units of vitamin 1, 36 units of vitamin 2, and 24 units of vitamin 3. Determine the combination that has the minimum cost.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 D. Solve the following problems using simplex method: 7. Minimize: Z = 5x + 4y subject to: vita 1 2x+ 1y ≥ 12 vita 2 2x+ 9y ≥ 36 vita 3 2x + 3y ≥ 24

Let X = supplement 1 Y = supplement 2

Z = 5x + 4y + 0S1 + 0S2 + 0S3 +10A1 + 10A2 + 10A3 2X + 1Y - 1S1 + 0S2 + 0S3 +1A1 + 0A2 + 0A3 = 12 2X + 9Y + 0S1 - 1S2 + 0S3 +0A1 + 1A2 + 0A3 = 36 2X + 3Y + 0S1 + 0S2 - 1S3 +0A1 + 0A2 + 1A3 = 24

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Initial Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

10

A1

12

2

1

-1

0

0

1

0

0

12

10

A2

36

2

9

0

-1

0

0

1

0

4

10

A3

24

2

3

0

0

-1

0

0

1

8

Zj

720

60

130

-10

-10

-10

10

10

10

Cj Zj

-0

-55

-126

10

10

10

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Second Tableau Cont. Soln.

5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

Qty. 10

A1

8

16/9

0

-1

1/9

0

1

-1/9

0

4

Y

4

2/9

1

0

-1/9

0

0

1/9

0

10

A3

12

4/3

0

0

1/3

-1

0

-1/3

1

Zj

216

32

4

-10

4

-10

10

-4

10

Cj - Zj

-216

-27

0

10

-4

10

0

14

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Second Tableau 5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

ETR

Co nt.

Sol n.

Qty.

10

A1

8

16/9

0

-1

1/9

0

1

-1/9

0

4.5

4

Y

4

2/9

1

0

-1/9

0

0

1/9

0

18

10

A3

12

4/3

0

0

1/3

-1

0

-1/3

1

9

Zj

216

32

4

-10

4

-10

10

-4

10

Cj Zj

-216

-27

0

10

-4

10

0

14

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Third Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

5

X

9/2

1

0

-9/16

1/16

0

9/16

-1/16

0

4

Y

3

0

1

1/8

-1/8

0

-1/8

1/8

0

10

A3

6

0

0

3/4

1/4

-1

-3/4

-1/4

1

Zj

94.5

5

4

0.188

2.31

-10

-5.19

-2.31

10

Cj Zj

-94.5

0

0

-0.188

-2.31

10

15.19 12.31

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Third Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

5

X

9/2

1

0

-9/16

1/16

0

9/16

-1/16

0

72

4

Y

3

0

1

1/8

-1/8

0

-1/8

1/8

0

-24

10

A3

6

0

0

3/4

1/4

-1

-3/4

-1/4

1

24

Zj

94.5

5

4

0.188

2.31

-10

-5.19

-2.31

10

Cj Zj

-94.5

0

0

-0.188

-2.31

10

15.19 12.31

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Fourth Final Tableau Cont. Soln.

5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

Qty. 5

X

3

1

0

-3/4

0

1/4

3/4

0

-1/4

4

Y

6

0

1

1/2

0

-1/2

-1/2

0

1/2

0

S2

24

0

0

3

1

-4

-3

-1

4

Zj

39

5

4

-1.75

0

-0.75

1.75

0

0.75

Cj - Zj

-39

0

0

1.75

0

0.75

8.25

0

9.25

The solution is optimal: Y= 6 X=3 S2 = 24

Min Z = 5X + 4Y = 39

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D8 D. Solve the following problems using simplex method: 8. The Ajax Manufacturing Company makes three products, x1, x2, and x3. The profit per unit for each is as follows: x1, P2; x2, P4; and x3, P3. The three products pass through three manufacturing centers as part of the manufacturing process. Product x1 requires 3 hours in center 1, two (2) hours in center 2, and one (1) hour in center 3; and product x2 requires 4 hours in center 1, 1 hour in center 2, and 3 hours in center 3; and product x3 requires 2 hours in each of the 3 centers. Each center has time available as follows; center 1, 60 hours; center 2, 40 hours; and center 3, 80 hours. Determine the optimum product mix for next week production schedule.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 D. Solve the following problems using simplex method: 8. Maximize: Z = 2x1 + 4x2 + 3x3 subject to: Center 1 3x1+ 4x2 + 2x3 ≤ 60 Center 2 2x1+ 1x2 + 2x3 ≤ 40 Center 3 1x1+ 3x2 + 2x3 ≤ 80 x1,x2, x3≥ 0 Z = 2x1 + 4x2 + 3x3 + 0S1 + 0S2 + 0S3 3x1+ 4x2 + 2x3 + 1S1 + 0S2 + 0S3 = 60 2x1+ 1x2 + 2x3 + 0S1 + 1S2 + 0S3 = 40 1x1+ 3x2 + 2x3 + 0S1 + 0S2 + 1S3 = 80

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Initial Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

0

S1

60

3

4

2

1

0

0

0

S2

40

2

1

2

0

1

0

0

S3

80

1

3

2

0

0

1

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

2

4

3

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Initial Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

ETR

0

S1

60

3

4

2

1

0

0

15

0

S2

40

2

1

2

0

1

0

40

0

S3

80

1

3

2

0

0

1

26.67

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

2

4

3

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Second Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

4

X2

15

3/4

1

2/4

1/4

0

0

0

S2

25

1/4

0

2/4

-1/4

1

0

0

S3

35

-5/4

0

2/4

-3/4

0

1

Zj

60

3

4

2

1

0

0

Cj - Zj

-60

-1

0

1

-1

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Second Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

ETR

4

X2

15

3/4

1

2/4

1/4

0

0

30

0

S2

25

1/4

0

2/4

-1/4

1

0

50

0

S3

35

-5/4

0

2/4

-3/4

0

1

70

Zj

60

3

4

2

1

0

0

Cj - Zj

-60

-1

0

1

-1

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Third (Final) Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

3

X3

30

3/2

2

1

1/2

0

0

0

S2

10

-1/2

-1

0

-1/2

1

0

0

S3

20

-2

-1

0

-1

0

1

Zj

90

4.5

6

3

1.5

0

0

Cj - Zj

-90

-2.5

-2

0

-1.5

0

0

The solution is optimal: X3= 30 S2 = 10 S3 = 20

ETR

Max Z = 2X1 + 4X2 + 3X3 = 90

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D9 D. Solve the following problems using simplex method: 9. The dean of the university, College of B.A, plans the course offerings for the second semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of P2,500,000 in faculty wages, and each graduate course costs P3,000,000. How many undergraduate and graduate courses should be taught in the second semester so that total faculty salaries are kept to a minimum?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 D. Solve the following problems using simplex method: 9. Minimize: Z = 2.5Mx + 3My subject to: Course offerings x ≥ 30 y ≥ 20 x + y ≥ 60

Let X = undergraduate Y = graduate M = million pesos

Z = 2.5Mx + 3My + 0S1 + 0S2 + 0S3 +10MA1 + 10MA2 + 10MA3 1X + 0Y - 1S1 + 0S2 + 0S3 +1A1 + 0A2 + 0A3 = 30 0X + 1Y + 0S1 - 1S2 + 0S3 +0A1 + 1A2 + 0A3 = 20 1X + 1Y + 0S1 + 0S2 - 1S3 +0A1 + 0A2 + 1A3 = 60

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Initial Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

10M

A1

30

1

0

-1

0

0

1

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

10M

A3

60

1

1

0

0

-1

0

0

1

Zj

1100M

20M

20M

-10M

-10M

-10M

10M

10M

10M

Cj Zj

-1100M

-17.5M

-17M

10M

10M

10M

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Initial Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

10M

A1

30

1

0

-1

0

0

1

0

0

30

10M

A2

20

0

1

0

-1

0

0

1

0

0

10M

A3

60

1

1

0

0

-1

0

0

1

60

Zj

1100M

20M

20M

-10M

-10M

-10M

10M

10M

10M

Cj Zj

-1100M

-17.5M

-17M

10M

10M

10M

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Second Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

30

1

0

-1

0

0

1

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

10M

A3

30

0

1

1

0

-1

-1

0

1

Zj

575M

2.5M

20M

7.5M

-10M

-10M

-7.5M

10M

10M

Cj Zj

-575M

0

-17M -7.5M

10M

10M

2.5M

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Second Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

2.5M

X

30

1

0

-1

0

0

1

0

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

20

10M

A3

30

0

1

1

0

-1

-1

0

1

30

Zj

575M

2.5M

20M

7.5M

-10M

-10M

-7.5M

10M

10M

Cj Zj

-575M

0

-17M -7.5M

10M

10M

2.5M

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Third Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

30

1

0

-1

0

0

1

0

0

3M

Y

20

0

1

0

-1

0

0

1

0

10M

A3

10

0

0

1

1

-1

-1

-1

1

Zj

235M

2.5M

3M

7.5M

7M

-10M

-7.5M

-7M

10M

Cj Zj

-235M

0

0M

-7.5M

-7M

10M

17.5M

17M

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Third Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

2.5M

X

30

1

0

-1

0

0

1

0

0

-2.5

3M

Y

20

0

1

0

-1

0

0

1

0

0

10M

A3

10

0

0

1

1

-1

-1

-1

1

10

Zj

235M

2.5M

3M

7.5M

7M

-10M

-7.5M

-7M

10M

Cj Zj

-235M

0

0M

-7.5M

-7M

10M

17.5M

17M

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Fourth Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

40

1

0

0

1

-1

0

-1

1

3M

Y

20

0

1

0

-1

0

0

1

0

0

S1

10

0

0

1

1

-1

-1

-1

1

Zj

160M

2.5M

3M

0

-0.5M -2.5M

0

0.5M

2.5M

Cj Zj

-160M

0

0

0

0.5M

10M

9.5M

7.5M

The solution is optimal: X= 40 Y = 20 S1 = 10

2.5M

Min Z = 2.5MX + 3MY = 160M Where: M= million pesos

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D10 D. Solve the following problems using simplex method: 10. Use the final table to answer the following questions: a. What is the best product mix? What is the highest profit? b. What are the shadow prices for the two constraints? c. Perform RHS ranging for constraint 1. d. Find the range of optimality for the profit of x.

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Solution to Exercise 9-D10 10. a. The best product mix: x = 12 and y = 6 The highest profit: P132 b. The shadow prices: x = 8 and y = 6 c. Perform RHS ranging for constraint 1. d. Find the range of optimality for the profit of x.

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Linear Programming Simplex Method

Exercise 9 Solving Linear Programming Problems

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______1. An iterative process is a procedure, which is repeated over and over following a random pattern. ______ 2. An inequality relationship may be converted into an equality by the addition of a slack variable. ______ 3. An infeasible solution is characterized as one where at least one constraint is violated. ______ 4. The Cj column in a maximization simplex table indicates the profit per unit for only those variables in the product mix. ______ 5. The significance of the quantity column in the simplex table is that it always shows the right hand side values of the constraints. ______ 6. Artificial, slack, and surplus variables are added to inequality constraints in order to convert them to equalities. Artificial variables are also added to constraints which were equalities to begin with.

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______ 7. There are always more variables in a simplex problem than there are constraints. ______ 8. If an artificial variable appears in the product mix of a table, which is optimal, it is likely that somebody has defined two or more incompatible constraints. ______ 9. Shadow prices are identified with variables, which are included in the product mix. ______ 10. For any basic variable in the product mix of a simplex table, the value of Cj – Zj in the column headed by the basic variables will always be negative in a maximization problem and positive in a minimization problem.

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. (Answer is in red letter) ______1. An iterative process is a procedure, which is repeated over and over following a random pattern. True ______ 2. An inequality relationship may be converted into an equality by the addition of a slack variable. True ______ 3. An infeasible solution is characterized as one where at least one constraint is violated. True ______ 4. The Cj column in a maximization simplex table indicates the profit per unit for only those variables in the product mix. True ______ 5. The significance of the quantity column in the simplex table is that it always shows the right hand side values of the constraints. True ______ 6. Artificial, slack, and surplus variables are added to inequality constraints in order to convert them to equalities. Artificial variables are also added to constraints which were equalities to begin with. True

Chapter 3

Linear Programming Simplex Method

Exercise 9-A True/False Questions. Write True on the blank if the statement is correct and False if it is an incorrect statement. ______ 7. There are always more variables in a simplex problem than there are constraints. False ______ 8. If an artificial variable appears in the product mix of a table, which is optimal, it is likely that somebody has defined two or more incompatible constraints. False ______ 9. Shadow prices are identified with variables, which are included in the product mix. True ______ 10. For any basic variable in the product mix of a simplex table, the value of Cj – Zj in the column headed by the basic variables will always be negative in a maximization problem and positive in a minimization problem. True

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 1. Which of the following is not true for the simplex procedures? a. Each iteration results a new solution, which is as good as or better than the previous solution. b. The procedure indicates when the optimum solution has been reached. c. The procedure ensures that the optimum solution is reached in the minimum number of iteration. d. All of the above choices are correct. ______ 2. If for a given solution, a slack variable is equal to zero then, a. the solution is not feasible b. the solution is optimal c. the entire amount of resource has been used d. all of the above choices are correct

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 3. A variable, which is not included in the product mix column for a given solution is: a. always equal to zero b. never equal to zero c. called a basic variable d. none of the above choices are correct ______ 4. With simplex, the initial solution contains only: a. slack variable in the product mix b. artificial variables in the product mix c. slack and artificial variables in the product mix d. real variables in the product mix ______ 5. The procedure for solving a minimization problem with simplex is exactly the same as solving a maximization problem but with one exception, which is: a. the identification of the replaced row b. the identification of the optimum column c. the computation of the Cj – Zj values d. the computation of the replacing row

Chapter 3

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. _____ 6. In solving a problem where 100 different products (100 real variables) are produced using just two resources (2 slack variables), then the combination that will yield the optimal profit is: a. produce all 100 products if both resources are used to full capacity b. produce, at most 2 products c. produce only one product d. none of the above choices are correct ______ 7. In converting > constraint to equality one will a. add an artificial variable b. add a slack variable c. subtract a surplus variable and add an artificial variable d. subtract a surplus variable ______ 8. In solving maximization problem using simplex method, the optimum column is obtained by getting the column with a. the largest positive value in the Cj – Zj row b. the largest negative value in the Cj – Zj row c. zero value in the Cj – Zj row d. smallest positive value in the Cj – Zj rowariables

Chapter 3

Linear Programming Simplex Method

Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ______ 9. Which of the following in a simplex table indicates that an optimal solution for maximization problem has been formed? a. All the Cj – Zj values are negative or zero. b. All the Cj – Zj values are positive or zero. c. There are no more slack variables in the product mix. d. All the Cj – Zj values are zeros. ______ 10. Which of the following must equal to zero? a. Basic variables b. Solution mix variables c. Non-basic variables d. Objective function coefficients for artificial variables

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __b____ 1. Which of the following is not true for the simplex procedures? a. Each iteration results a new solution, which is as good as or better than the previous solution. b. The procedure indicates when the optimum solution has been reached. c. The procedure ensures that the optimum solution is reached in the minimum number of iteration. d. All of the above choices are correct. __c____ 2. If for a given solution, a slack variable is equal to zero then, a. the solution is not feasible b. the solution is optimal c. the entire amount of resource has been used d. all of the above choices are correct

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __d____ 3. A variable, which is not included in the product mix column for a given solution is: a. always equal to zero b. never equal to zero c. called a basic variable d. none of the above choices are correct __c____ 4. With simplex, the initial solution contains only: a. slack variable in the product mix b. artificial variables in the product mix c. slack and artificial variables in the product mix d. real variables in the product mix __d____ 5. The procedure for solving a minimization problem with simplex is exactly the same as solving a maximization problem but with one exception, which is: a. the identification of the replaced row b. the identification of the optimum column c. the computation of the Cj – Zj values d. the computation of the replacing row

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. __d___ 6. In solving a problem where 100 different products (100 real variables) are produced using just two resources (2 slack variables), then the combination that will yield the optimal profit is: a. produce all 100 products if both resources are used to full capacity b. produce, at most 2 products c. produce only one product d. none of the above choices are correct ___C___ 7. In converting > constraint to equality one will a. add an artificial variable b. add a slack variable c. subtract a surplus variable and add an artificial variable d. subtract a surplus variable __b____ 8. In solving maximization problem using simplex method, the optimum column is obtained by getting the column with a. the largest positive value in the Cj – Zj row b. the largest negative value in the Cj – Zj row c. zero value in the Cj – Zj row d. smallest positive value in the Cj – Zj rowariables

Chapter 3

Linear Programming Simplex Method

Answer to Exercise 9-B Multiple Choice Questions. Write the letter of the correct answer on the blank. ___a___ 9. Which of the following in a simplex table indicates that an optimal solution for maximization problem has been formed? a. All the Cj – Zj values are negative or zero. b. All the Cj – Zj values are positive or zero. c. There are no more slack variables in the product mix. d. All the Cj – Zj values are zeros. ___d___ 10. Which of the following must equal to zero? a. Basic variables b. Solution mix variables c. Non-basic variables d. Objective function coefficients for artificial variables

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 1.

Maximize: Z = 4x1 + 5x2 subject to: 3x1+ 6x2 ≤ 36 4x1 + 2x2 ≤ 36 x1 + x2 ≤ 8 x1,x2 ≥ 0

2.

Maximize: Z = 6x + 3y subject to: 2x+ 4y ≤ 30 4x + 2y ≤ 30 2x + 3y ≥ 18 x1,x2 ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C1 C. Adding the appropriate slack, surplus, and artificial variables. 1.

Maximize: Z = 4x1 + 5x2 subject to: 3x1+ 6x2 ≤ 36 4x1 + 2x2 ≤ 36 x1 + x2 ≤ 8 x1,x2 ≥ 0

Z = 4x1 + 5x2 + 0S1 + 0S2 + 0S3 3x1+ 6x2 + 1S1 + 0S2 + 0S3= 36 4x1 + 2x2 + 0S1 + 1S2 + 0S3 = 36 x1 + x2 + 0S1 + 0S2 + 1S3 = 8

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C2 C. Adding the appropriate slack, surplus, and artificial variables.

2. Maximize: Z = 6x + 3y ubject to: 2x+ 4y ≤ 30 4x + 2y ≤ 30 2x + 3y ≥ 18 x1,x2 ≥ 0

Z = 6x + 3y + 1S1 + 0S2 + 0S3 + 10A 2x+ 4y + 1S1 + 0S2 + 0S3 + 0 A3 =30 4x + 2y + 0S1 + 1S2 + 0S3 + OA3 = 30 2x + 3y + 0S1 + 0S2 - 1S3 +1A3 = 18

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C3 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 3.

Minimize: C = 8x + 4y subject to: 10x+ 25y ≥ 100 y≤4 x,y ≥ 0

4.

Minimize: C = 4x + 5y subject to: x+ 6y = 12 x≤4 y≤ 8 y≥5 x,Y ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C3 C. Adding the appropriate slack, surplus, and artificial variables. 3. Minimize: C = 8x + 4y subject to: 10x+ 25y ≥ 100 y≤4 x,y ≥ 0

C = 8x + 4y + 0S1 + 0S2 + 10 A1 10x+ 25y + 1S1 + 0S2 + 1A1 = 100 0x + y - 0S1 + 1S2 + 0A1 = 4

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C4 C. Adding the appropriate slack, surplus, and artificial variables.

4. Minimize: C = 4x + 5y subject to: x+ 6y = 12 x≤4 y≤ 8 y≥5 x,Y ≥ 0

C = 4x + 5y + 0S2 + 0S3 + 0S4 +10A1 +10A4 x+ 6y + 0S2 + 0S3 + 0S4 +1A1 + 0A4 = 12 x + 0y + 1S2 + 0S3 + 0S4 +0A1 + 0A4 = 4 0x + y + 0S2 + 1S3 + 0S4 +0A1 + 0A4 = 8 0x + y + 0S2 + 0S3 - 1S4 +0A1 + 1A4 = 5

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C5 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 5.

Minimize: C = 0.05x + 0.06y subject to: 5x+ 25y ≥ 50 25x + 10y ≥ 100 10x + 10y ≥ 60 36x + 20y ≥ 180 x,y ≥ 0

6.

Maximize: Z = 4A + 5B subject to: 3A+ 4B ≤ 150 A+2B ≤ 60 A ≥ 25 A,B ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C6 C. Adding the appropriate slack, surplus, and artificial variables. 5. Minimize: C = 0.05x + 0.06y subject to: 5x+ 25y ≥ 50 25x + 10y ≥ 100 10x + 10y ≥ 60 36x + 20y ≥ 180 x,y ≥ 0 C = 0.05x + 0.06y + 0S1 +0S2 + 0S3 + 0S4 +10A1 + 10A2 + 10A3 +10A4 5x+ 25y - 1S1 +0S2 + 0S3 + 0S4 +1A1 + 0A2 + 0A3 +0A4 = 50 25x + 10y + 0S1 -1S2 + 0S3 + 0S4 +0A1 + 1A2 + 0A3 +0A4 = 100 10x + 10y + 0S1 +0S2 - 1S3 + 0S4 +0A1 + 0A2 + 1A3 +0A4 = 60 36x + 20y + 0S1 +0S2 + 0S3 - 1S4 +0A1 + 0A2 + 0A3 +1A4 = 180

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C6 C. Adding the appropriate slack, surplus, and artificial variables. 6.

Maximize: Z = 4A + 5B subject to: 3A+ 4B ≤ 150 A+2B ≤ 60 A ≥ 25 A,B ≥ 0

Z = 4A + 5B + 0S1 +0S2 + 0S3 + 10A 3A+ 4B + 1S1 +0S2 + 0S3 + 0A = 150 A+ 2B + 0S1 + 1S2 + 0S3 + 0A = 60 A + 0B + 0S1 +0S2 - 1S3 + 1A = 25

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C7 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 7.

Minimize: C = 2x1 + 5x2 subject to: 3x1+ 2x2 = 30 x1 ≤ 5 x2 ≥ 10 x1,x2 ≥ 0

8.

Minimize: C = 2x1 + 5x2 + 9x3 subject to: 2x1+ 2x2 + x3 = 20 2x1+ 4x2 + 5x3 ≤ 20 x1 + 2x2 ≥ 6 x2 ≥ 10 x1,x2, x3 ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C7 C. Adding the appropriate slack, surplus, and artificial variables. 7.

Minimize: C = 2x1 + 5x2 subject to: 3x1+ 2x2 = 30 x1 ≤ 5 x2 ≥ 10 x1,x2 ≥ 0 C = 2x1 + 5x2 3x1+ 2x2 -1S1 +0S2 + 0S3 + 1A1 + 0A3 = 30 x1 + 0x2 +0S1 +1S2 + 0S3 + 0A1 + 0A3 = 5 0x1 + x2 +0S1 +0S2 - 1S3 + 0A1 + 1A3 = 10

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C8 C. Adding the appropriate slack, surplus, and artificial variables. 8. Minimize: C = 2x1 + 5x2 + 9x3 subject to: 2x1+ 2x2 + x3 = 20 2x1+ 4x2 + 5x3 ≤ 20 x1 + 2x2 ≥ 6 x2 ≥ 10 x1,x2, x3 ≥ 0 C = 2x1 + 5x2 + 9x3 + 0S2 + 0S3 + 0S4 + 10A1 + 10A3 + 10A4 2x1+ 2x2 + x3 + 0S2 + 0S3 + 0S4 + 1A1 + 0A3 + 0A4 = 20 2x1+ 4x2 + 5x3 + 1S2 + 0S3 + 0S4 + 0A1 + 0A3 + 0A4 = 20 x1 + 2x2 +0x3 + 0S2 - 1S3 + 0S4 + 0A1 + 1A3 + 0A4 = 6 0x1 + x2 +0x3 + 0S2 + 0S3 - 1S4 + 0A1 + 0A3 + 1A4 = 10

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C9 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 9.

Maximize: Z = 8x1 + 6x2 + 2x3 subject to: 3x1+ 2x2 + 6 ≤ 24 4x1+ 3x2 + 6x3 = 48 x1,x2, x3 ≥ 0

10.

Minimize: C = 4x + 5y + 3z subject to: 4x+ 2y + 3z = 100 x ≥ 20 y≤6 z ≥8 x,y, z ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C9 C. Adding the appropriate slack, surplus, and artificial variables. 9. Maximize: Z = 8x1 + 6x2 + 2x3 subject to: 3x1+ 2x2 + 6 ≤ 24 4x1+ 3x2 + 6x3 = 48 x1,x2, x3 ≥ 0 Z = 8x1 + 6x2 + 2x3 + 0S1 + 10A2 3x1+ 2x2 + 6 + 1S1 + 0A2 = 24 4x1+ 3x2 + 6x3 + 0S1 + 1A2 = 48

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C10 C. Adding the appropriate slack, surplus, and artificial variables. 10. Minimize: C = 4x + 5y + 3z subject to: 4x+ 2y + 3z = 100 x ≥ 20 y≤6 z ≥8 x,y, z ≥ 0

C = 4x + 5y + 3z + 0S2 + 0S3 + 0S4 + 10A1 +10A2 +10A4 4x + 2y + 3z + 0S2 + 0S3 + 0S4 + 1A1 + 0A2 + 0A4 = 100 1x + 0y + 0z - 1S2 + 0S3 + 0S4 + 0A1 +1A2 +0A4 = 20 0x + 1y + 0z + 0S2 + 1S3 + 0S4 + 0A1 +0A2 +0A4 = 6 0x + 0y + 1z + 0S2 + 0S3 - 1S4 + 0A1 +0A2 +1A4 = 8

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C11-C12 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 11.

Minimize: C = 6x1 + 16x2 subject to: x1+ x2 = 400 x1= 150 x2 ≥ 200 x1,x2, ≥ 0

12. Minimize: C = 1000x + 1500y subject to: 20x+ 20y ≥ 160 30x+ 60y ≥ 300 x≥ 2 x,y, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C11 C. Adding the appropriate slack, surplus, and artificial variables. 11. Minimize: C = 6x1 + 16x2 subject to: x1+ x2 = 400 x1= 150 x2 ≥ 200 x1,x2, ≥ 0 C = 6x1 + 16x2 + 0S3 + 10A1 + 10A2 + 10A3 1x1+ 1x2 + 0S3 + 1A1 + 0A2 + 0A3 = 400 1x1+ 0x2 + 0S3 + 0A1 + 1A2 + 0A3 = 150 0x1+ 1x2 - 1S3 + 0A1 + 0A2 + 1A3 = 200

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C12 C. Adding the appropriate slack, surplus, and artificial variables. 12. Minimize: C = 1000x + 1500y subject to: 20x+ 20y ≥ 160 30x+ 60y ≥ 300 x≥ 2 x,y, ≥ 0 C = 1000x + 1500y + 0S1 + 0S2 + 0S3 + 10A1 +10A2 +10A3 20x + 20y - 1S1 + 0S2 + 0S3 + 1A1 +0A2 +0A3 = 160 30x + 60y + 0S1 - 1S2 + 0S3 + 0A1 +1A2 +0A3 = 300 1x + 0y + 0S1 + 0S2 - 1S3 + 0A1 +0A2 +1A3 = 2

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-C12-C14 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 13. Minimize: C = 10x1 + 20x2 subject to: x1+ x2 ≥ 10 3x1 + x2 ≤ 12 x1,x2, ≥ 0

14. Minimize: C = 120x1 + 100x2 subject to: 2x1+ x2 ≤ 18 5x1+ 4x2 ≥ 60 x1,x2, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C13 C. Adding the appropriate slack, surplus, and artificial variables. 13. Minimize: C = 10x1 + 20x2 subject to: x1+ x2 ≥ 10 3x1 + x2 ≤ 12 x1,x2, ≥ 0

C = 10x1 + 20x2 + 0S1 + 0S2 + 10A1 1x1+ 1x2 - 1S1 + 0S2 + 1A1 = 10 3x1+ 1x2 + 0S1 + 1S2 + 0A1 = 12

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C14 C. Adding the appropriate slack, surplus, and artificial variables. 14. Minimize: C = 120x1 + 100x2 subject to: 2x1+ x2 ≤ 18 5x1+ 4x2 ≥ 60 x1,x2, ≥ 0

C = 120x1 + 100x2 + 0S1 + 0S2 + 10A2 1x1+ 1x2 + 1S1 + 0S2 + 0A2 = 18 3x1+ 1x2 + 0S1 - 1S2 + 1A2 = 60

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-C15 C. Add the appropriate slack, surplus, and artificial variables. Use the value of 10 for the objective function coefficient of artificial variables. Do not solve. 15. Minimize: Z = 2x1 + 1x2 subject to: 2x1+ 3x2 ≥ 12 3x1+ 4x2 ≤ 21 6x1+ 5x2 ≤ 30 x1,x2, ≥ 0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-C15 C. Adding the appropriate slack, surplus, and artificial variables. 15. Minimize: Z = 2x1 + 1x2 subject to: 2x1+ 3x2 ≥ 12 3x1+ 4x2 ≤ 21 6x1+ 5x2 ≤ 30 x1,x2, ≥ 0 Z = 2x1 + 1x2 + 0S1 + 0S2 + 0S3 + 10A1 1x1+ 1x2 - 1S1 + 0S2 + 0S3 + 1A1 = 12 1x1+ 0x2 + 0S1 + 1S2 + 0S3 + 0A1 = 21 0x1+ 1x2 + 0S1 + 0S2 + 1S3 + 0A1 = 30

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D D. Solve the following problems using simplex method: 1. Ignacio Furniture Co. makes two types of playhouses: a standard model and a deluxe model. The playhouse are sold to the independent dealers at a profit of P200/standard and P300/ deluxe. A standard requires 30 man-hours for assembly, and 20 man-hours for painting and finishing and 10 man-hours for inspecting. A deluxe requires 75 man-hours for assembly, 25 manhours for painting and finishing, and five (5) man-hours for inspecting. A production run generally has 15000 man-hours available for assembly, 6500 man-hours available for painting and finishing and 2500 man-hours for inspecting. Determine the maximum profit and optimal values of the decisions and slack variables.

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Solution to Exercise 9-D1 D. Solve the following problems using simplex method: 1 Maximize: Z = 200x1 + 300x2 subject to: Assembly 30x1+ 75x2 ≤ 15,000 Painting/finishing 20x1+ 25x2 ≤ 6,500 inspecting 10x1+ 5x2 ≤ 2,500 x1,x2, ≥ 0

Let x1 = standard model x2 = deluxe model

Z = 200x1 + 300x2 + 0S1 + 0S2 + 0S3 30x1+ 75x2 + 1S1 + 0S2 + 0S3 = 15,000 20x1+ 25x2 + 0S1 + 1S2 + 0S3 = 6,500 10x1+ 5x2 + 0S1 + 0S2 + 1S3 = 2,500

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Initial Tableau Cont.

Soln.

Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

0

S1

15,000

30

75

1

0

0

0

S2

6,500

20

25

0

1

0

0

S3

2,500

10

5

0

0

0

Zj

0

0

0

0

0

0

200

300

0

0

0

Cj - Zj

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Initial Tableau Cont.

Soln.

Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

ETR

0

S1

15,000

30

75

1

0

0

200

0

S2

6,500

20

25

0

1

0

260

0

S3

2,500

10

5

0

0

0

500

Zj

0

0

0

0

0

0

200

300

0

0

0

Cj - Zj

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Second Tableau 300

0

0

0

X1

X2

S1

S2

S3

Cont.

Soln.

300

X1

200

2/5

1

1/75

0

0

0

S2

1,500

10

0

-1/3

1

0

0

S3

1,500

8

0

-1/15

0

0

Zj

60,000

120

300

4

0

0

80

o

-4

0

0

Cj - Zj

Qty.

200

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Second Tableau 300

0

0

0

X1

X2

S1

S2

S3

ETR

Cont.

Soln.

300

X1

200

2/5

1

1/75

0

0

500

0

S2

1,500

10

0

-1/3

1

0

150

0

S3

1,500

8

0

-1/15

0

0

187.5

Zj

60,000

120

300

4

0

0

80

o

-4

0

0

Cj - Zj

Qty.

200

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D1 Third (Final) Tableau Qty.

200

300

0

0

0

X1

X2

S1

S2

S3

Cont.

Soln.

300

X1

140

0

1

11/75

-1/25

0

200

X1

150

1

0

-1/30

1/10

0

0

S3

300

0

0

13/5

-1/80

0

Zj

72,000

200

300

112/3

8

0

Cj - Zj

-72,000

0

0

-112/3

-8

0

The solution is optimal: X2 = 140 X1= 150

Max Z = 200x1 + 300x2 = 72,000

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D2 D. Solve the following problems using simplex method: 2. To make one unit of product x requires three (3) minutes in Dept. I and one (1) minute in Dept. II. One unit of product y requires four (4) minutes in Dept.I and two (2) minutes in Dept.II. Profit contribution is P5/unit of x and P8/unit of y. It is required that at least 25 units of x be made to maximize profit if Dept.I and II have 150 and 160 minutes available respectively. What is the maximum profit?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 D. Solve the following problems using simplex method: 2 Maximize: Z = 5x + 8y subject to: time in dept. 1 3x+ 4y ≤ 150 time in dept .2 1x+ 2y ≤ 160 capacity x ≥ 25 x,y ≥ 0 Z = 5x + 8y + 0S1 + 0S2 + 0S3 + 10A3 3X+ 4Y + 1S1 + 0S2 + 0S3 +0 A3 = 150 1X+ 2Y + 0S1 + 1S2 + 0S3 + 0A3 = 160 1X+ 0Y + 0S1 + 0S2 - 1S3 + 1A3 = 25

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Initial Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

0

S1

150

3

4

1

0

0

0

0

S2

160

1

2

0

1

0

0

10

A3

25

1

0

0

0

-1

1

Zj

250

10

0

0

0

-10

10

Cj - Zj

-250

-5

8

0

0

10

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Initial Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

ETR

0

S1

150

3

4

1

0

0

0

50

0

S2

160

1

2

0

1

0

0

160

10

A3

25

1

0

0

0

-1

1

25

Zj

250

10

0

0

0

-10

10

Cj - Zj

-250

-5

8

0

0

10

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D2 Second (Final) Tableau Cont. Soln.

Qty.

5

8

0

0

0

10

X

Y

S1

S2

S3

A3

0

S1

75

0

4

1

0

-3

-3

0

S2

135

0

2

0

1

1

-1

5

X

25

1

0

0

0

-1

1

Zj

125

5

0

0

0

-5

5

Cj - Zj

-125

0

8

0

0

5

5

The solution is optimal: X = 25 Y= 0

Max Z = 5X + 8Y = 125

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D3 D. Solve the following problems using simplex method: 3. A manufacturer of commercial chemical has an order for a certain mixture containing of three ingredients x1, x2, x3, which costs P8, P7, and P4 per kilo, respectively. The following are specifications: a. b. c. d.

It cannot contain more than 35 kilos of x. It must contain at least 15 kilos of x2. It cannot contain more than 40 kilos of X3. The weight of the mixture must be 120 kilos.

Find a mixture of the three ingredients, which satisfies his customer’s requirements and still yields the minimum total cost of raw materials.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 D. Solve the following problems using simplex method: 3. Minimize: Z = 8x1 + 7x2 + 4x3 subject to: content of X x1+ x2 + x3 ≤ 35 content of X2 x2 ≥ 15 content of X3 x3 ≤ 40 weight x1+ x2 + x3 = 120 x1,x2, x3≥ 0 Z = 8x1 + 7x2 + 4x3 + 0S1 + 0S2 + 0S3 + 10A2 + 10A4 1x1+ 1x2 + 1x3 + 1S1 + 0S2 + 0S3 + 0A2 +0A4 = 35 0x1+ 1x2 + 0x3 + 0S1 - 1S2 + 0S3 +1A2 +0A4= 15 0x1+ 0x2 + 1x3 + 0S1 + 0S2 + 1S3 +0A2 + 0A4 = 40 1x1+ 1x2 + 1x3 + 0S1 + 0S2 + 0S3 +0A2 + 1A4 = 120

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Initial Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

0

S1

35

1

1

1

1

0

0

0

0

10

A2

15

0

1

0

0

-1

0

1

0

0

S3

40

0

0

1

0

0

1

0

0

10

A4

120

1

1

1

0

0

0

0

1

Zj

1350

10

20

10

0

-10

0

10

10

Cj - Zj

-1350

-2

-13

-6

0

10

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Initial Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

ETR

0

S1

35

1

1

1

1

0

0

0

0

35

10

A2

15

0

1

0

0

-1

0

1

0

15

0

S3

40

0

0

1

0

0

1

0

0

0

10

A4

120

1

1

1

0

0

0

0

1

120

Zj

1350

10

20

10

0

-10

0

10

10

Cj - Zj

-1350

-2

-13

-6

0

10

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Second Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

0

S1

20

1

0

1

1

2

0

-1

0

7

X2

15

0

1

0

0

-1

0

1

0

0

S3

40

0

0

1

0

0

1

0

0

10

A4

105

1

0

1

0

1

0

-1

1

Zj

1155

10

7

10

0

3

0

-3

10

Cj - Zj

-1155

-2

0

-6

0

-3

0

7

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Second Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

ETR

0

S1

20

1

0

1

1

2

0

-1

0

20

7

X2

15

0

1

0

0

-1

0

1

0

0

0

S3

40

0

0

1

0

0

1

0

0

40

10

A4

105

1

0

1

0

1

0

-1

1

105

Zj

1155

10

7

10

0

3

0

-3

10

Cj - Zj

-1155

-2

0

-6

0

-3

0

7

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D3 Third (Final) Tableau Cont.

Soln.

Qty.

8

7

4

0

0

0

10

10

X1

X2

X3

S1

S2

S3

A2

A4

4

X3

20

1

0

1

1

2

0

-1

0

7

X2

15

0

1

0

0

-1

0

1

0

0

S3

20

1

0

0

-1

-2

1

1

0

10

A4

85

0

0

0

-1

-1

0

0

1

Zj

1035

4

7

4

-6

-9

0

3

10

Cj - Zj

-1035

4

0

0

6

9

0

7

0

The solution is optimal: X3 = 20 X2= 15

ETR

X1 = 0 Min Z = 8X1 + 7X2 + 4X3 + 10A4 A4 = 85 = 1035

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D4 D. Solve the following problems using simplex method: 4. The Jay Gee Bull Ranch has plans to sell cattle to a large meat packaging plant. The manager of the ranch believes that each cattle should receive a minimum of 240 oz. of nutritional ingredients A and a minimum of 150 oz. of nutritional ingredients B each week. There are two grains available which contain both types of nutritional ingredients. One bag of grain 1 contains 12 oz of ingredients A and 25 oz of ingredients B and costs P140/bag; one bag of grain 2 contains 20 oz of ingredients A and five (5) oz. of ingredients B and costs P120/bag. Determine the optimum mixture of grains and the associated minimum cost.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 D. Solve the following problems using simplex method: 4. Minimize: Z = 140x + 120y subject to: Ingredients A 12X + 20Y ≥ 240 Ingredients B 25X + 5Y ≥ 150 x,y ≥ 0

Let X = grain 1 Y = grain 2

Z = 140x + 120y + 0S1 + 0S2 + 1000A1 + 1000A2 12x + 20y -1S1 + 0S2 + 1A1 + 0A2 = 35 25x + 5y + 0S1 - 1S2 + 0A1 + 1A2 = 35

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Initial Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

1000

A1

240

12

20

-1

0

1

0

1000

A2

150

25

5

0

-1

0

1

Zj

390000

37000

25000

-1000

-1000

1000

1000

Cj - Zj

-390000

-36860 -24880

1000

1000

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Initial Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

1000

A1

240

12

20

-1

0

1

0

20

1000

A2

150

25

5

0

-1

0

1

6

Zj

390000

37000

25000

-1000

-1000

1000

1000

Cj - Zj

-390000

-36860 -24880

1000

1000

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Second Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

1000

A1

168

0

88/5

-1

12/25

1

-12/25

140

X

6

1

1/5

0

-1/25

0

1/25

Zj

168840

140

18300

-1000

4794.4

1000

-4794.4

Cj - Zj

-168840

0

-18180

1000

-4794.4

0

5794.4

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Second Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

1000

A1

168

0

88/5

-1

12/25

1

-12/25

840/88

140

X

6

1

1/5

0

-1/25

0

1/25

30

Zj

168840

140

18300

-1000

4794.4

1000

-4794.4

Cj - Zj

-168840

0

-18180

1000

-4794.4

0

5794.4

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Third Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

105/11

0

1

-5/88

3/110

5/88

-3/110

140

X

45/11

1

0

1/88

-1/22

-1/88

1/22

Zj

1145.45

140

120

5.23

-3.09

5.23

-4794.4

Cj - Zj

-1145.55

0

0

-5.23

3.09

994.8

6.36

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Third Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

120

Y

105/11

0

1

-5/88

3/110

5/88

-3/110

-168

140

X

45/11

1

0

1/88

-1/22

-1/88

1/22

360

Zj

1145.45

140

120

5.23

-3.09

5.23

-4794.4

Cj - Zj

-1145.55

0

0

-5.23

3.09

994.8

6.36

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fourth Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

30

5

1

0

-22/110

0

22/110

0

S2

360

88

0

1

-4

-1

4

Zj

3,600

600

120

0

-24

0

24

Cj - Zj

-3,600

-460

0

0

24

1000

976

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fourth Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

ETR

120

Y

30

5

1

0

-22/110

0

22/110

0

0

S2

360

88

0

1

-4

-1

4

4.09

Zj

3,600

600

120

0

-24

0

24

Cj - Zj

-3,600

-460

0

0

24

1000

976

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D4 Fifth (Final) Tableau Cont. Soln.

Qty.

140

120

0

0

1000

1000

X

Y

S1

S2

A1

A2

120

Y

105/11

0

1

-5/88

3/110

-5/88

-3/110

140

X

45/11

1

0

1/88

-4/88

1/88

4/88

Zj

1718.18

140

120

-5.23

-3.09

5.23

3.09

Cj - Zj

-1718.18

0

0

5.23

3.09

994.77

996.91

The solution is optimal: Y = 105/11 Min Z = 140X + 120Y X= 45/11 = 1718.18

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D5 D. Solve the following problems using simplex method: 5. A manufacturer makes two products; picnic tables and benches, which must be processed through two machine centers. MC1 has up to 60 hours available. MC2 can handle up 48 hours of work. Each picnic table requires four (4) hours in MC1 and two (2) hours in MC2. Each bench takes two (2) hours in MC1 and four(4) hours in MC2. If profit is P80/picnic table and P60/bench, determine the best possible contribution of picnic tables and benches to produce and sell in order to realize the maximum profit.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 D. Solve the following problems using simplex method: 5. Maximize: Z = 80x + 60y subject to: MC 1 4x+ 2y ≤ 60 MC 2 2x+ 4y ≤ 48 Z = 80x + 60y + 0S1 + 0S2 4X+ 2Y + 1S1 + 0S2 = 60 2X+ 4Y + 0S1 + 1S2 = 48

Let X = picnic table Y = bench

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Initial Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

0

S1

60

4

2

1

0

0

S2

48

2

4

0

1

Zj

0

0

0

0

0

Cj - Zj

-0

80

60

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Initial Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

ETR

0

S1

60

4

2

1

0

15

0

S2

48

2

4

0

1

24

Zj

0

0

0

0

0

Cj - Zj

-0

80

60

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Second Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

80

X

15

1

1/2

1/4

0

0

S2

18

0

3

-1/2

1

Zj

1200

80

40

20

0

Cj - Zj

-1200

0

20

-20

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Second Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

ETR

80

X

15

1

1/2

1/4

0

7.5

0

S2

18

0

3

-1/2

1

6

Zj

1200

80

40

20

0

Cj - Zj

-1200

0

20

-20

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D5 Third Tableau Cont. Soln.

Qty.

80

60

0

0

X

Y

S1

S2

80

X

4

1

0

1/3

-1/6

60

Y

8

0

1

-1/6

1/3

Zj

800

80

60

16.67

6.67

Cj - Zj

-800

0

0

-16.67

-6.67

The solution is optimal: X = 4 y=8

Max Z = 80X + 60Y = 800

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-6 D. Solve the following problems using simplex method: 6.

Mr. Winston Pe can produce in his shop any mix of three products x,y, and z. The prices and variable cost per unit are: Product Price/unit Variable cost/unit X P20 P11 Y 12 8 Z 8 6 Winston processes three product in each of the three departments I, II, III. The time requirements per department are as follows:

80 hours per day is available at each of the department. Develop the set of constraints and the objective function to maximize profit. How much is the maximum profit?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 D. Solve the following problems using simplex method: 6. Maximize: Z = 9x + 4y + 2z subject to: Dept. 1 2x+ 4y + 3y ≤ 80 Dept. 2 x+ 6y + 5y ≤ 80 Dept. 3 7x + 2y + 9z ≤ 80 Z = 9x + 4y + 2z + 0S1 + 0S2 + 0S3 2X+ 4Y + 3z + 1S1 + 0S2 + 0S3 = 80 X+ 6Y + 5z + 0S1 + 1S2 + 0S3 = 80 7X+ 2Y + 9z + 0S1 + 0S2 + 1S3 = 80

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Initial Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

80

2

4

3

1

0

0

0

S2

80

1

6

5

0

1

0

0

S2

80

7

2

9

0

0

1

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

9

4

2

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Initial Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

80

2

4

3

1

0

0

40

0

S2

80

1

6

5

0

1

0

80

0

S2

80

7

2

9

0

0

1

11.43

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

9

4

2

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Second Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

400/7

0

24/7

3/7

1

0

-2/7

0

S2

480/7

0

40/7

26/7

0

1

-1/7

9

X

80/7

1

2/7

9/7

0

0

1/7

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Second Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

400/7

0

24/7

3/7

1

0

-2/7

133.3

0

S2

480/7

0

40/7

26/7

0

1

-1/7

18.46

9

X

80/7

1

2/7

9/7

0

0

1/7

8.89

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Third Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

160/3

-1/3

10/3

0

1

0

-1/3

0

S2

4940/63

-26/9

308/63

0

0

1

-35/63

2

Z

80/9

7/9

2/9

1

0

0

1/9

Zj

17.78

1.56

0.44

2

0

0

0.22

Cj - Zj

-17,78

7.44

3.56

0

0

0

-0.22

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Third Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

160/3

-1/3

10/3

0

1

0

-1/3

-160

0

S2

4940/63

-26/9

308/63

0

0

1

-35/63

-27.25

2

Z

80/9

7/9

2/9

1

0

0

1/9

11.43

Zj

17.78

1.56

0.44

2

0

0

0.22

Cj - Zj

-17,78

7.44

3.56

0

0

0

-0.22

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fourth Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

1200/21

0

72/21

3/7

1

0

-6/21

0

S2

40

0

360/63

26/7

0

1

-9/63

9

X

80/7

1

2/7

9/7

0

0

1/7

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fourth Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

ETR

0

S1

1200/21

0

72/21

3/7

1

0

-6/21

21.05

0

S2

40

0

360/63

26/7

0

1

-9/63

7

9

X

80/7

1

2/7

9/7

0

0

1/7

31.5

Zj

102.86

9

2.57

11.57

0

0

1.29

Cj - Zj

-102.86

0

1.43

-9.57

0

0

-1.29

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D6 Fifth (Final) Tableau Cont Soln. .

9

4

2

0

0

0

Qty.

X

Y

Z

S1

S2

S3

0

S1

232/7

0

0

-9/5

1

-3/5

-13/35

4

Y

7

0

1

13/20

0

7/40

-1/40

9

X

66/7

1

0

11/7

0

-1/20

19/140

Zj

112.85

9

4

16.74

0

0.25

1.12

Cj - Zj

-112.86

0

0

-14,74

0

-0.25

-1.12

The solution is optimal: Y= 7 X = 66/7 Z=0

Max Z = 9X + 4Y + 2Z = 112.86

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Exercise 9-D7 D. Solve the following problems using simplex method: 7.

The poultry farmer must supplement the vitamins in the feeds he buys. He is considering two supplements, each of which contains the feed required but in different amounts. He must meet or exceed the minimum vitamin requirements.

The vitamin content per gram of the supplements is given in the following table:

Vitamin 1 2 3

Supplement 1 2 2 2

supplement 2 1 9 3

Supplement 1 costs P5 per gram and supplement 2 costs P4 per gram. The feed must contain at least 12 units of vitamin 1, 36 units of vitamin 2, and 24 units of vitamin 3. Determine the combination that has the minimum cost.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 D. Solve the following problems using simplex method: 7. Minimize: Z = 5x + 4y subject to: vita 1 2x+ 1y ≥ 12 vita 2 2x+ 9y ≥ 36 vita 3 2x + 3y ≥ 24

Let X = supplement 1 Y = supplement 2

Z = 5x + 4y + 0S1 + 0S2 + 0S3 +10A1 + 10A2 + 10A3 2X + 1Y - 1S1 + 0S2 + 0S3 +1A1 + 0A2 + 0A3 = 12 2X + 9Y + 0S1 - 1S2 + 0S3 +0A1 + 1A2 + 0A3 = 36 2X + 3Y + 0S1 + 0S2 - 1S3 +0A1 + 0A2 + 1A3 = 24

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Initial Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

10

A1

12

2

1

-1

0

0

1

0

0

12

10

A2

36

2

9

0

-1

0

0

1

0

4

10

A3

24

2

3

0

0

-1

0

0

1

8

Zj

720

60

130

-10

-10

-10

10

10

10

Cj Zj

-0

-55

-126

10

10

10

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Second Tableau Cont. Soln.

5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

Qty. 10

A1

8

16/9

0

-1

1/9

0

1

-1/9

0

4

Y

4

2/9

1

0

-1/9

0

0

1/9

0

10

A3

12

4/3

0

0

1/3

-1

0

-1/3

1

Zj

216

32

4

-10

4

-10

10

-4

10

Cj - Zj

-216

-27

0

10

-4

10

0

14

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Second Tableau 5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

ETR

Co nt.

Sol n.

Qty.

10

A1

8

16/9

0

-1

1/9

0

1

-1/9

0

4.5

4

Y

4

2/9

1

0

-1/9

0

0

1/9

0

18

10

A3

12

4/3

0

0

1/3

-1

0

-1/3

1

9

Zj

216

32

4

-10

4

-10

10

-4

10

Cj Zj

-216

-27

0

10

-4

10

0

14

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Third Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

5

X

9/2

1

0

-9/16

1/16

0

9/16

-1/16

0

4

Y

3

0

1

1/8

-1/8

0

-1/8

1/8

0

10

A3

6

0

0

3/4

1/4

-1

-3/4

-1/4

1

Zj

94.5

5

4

0.188

2.31

-10

-5.19

-2.31

10

Cj Zj

-94.5

0

0

-0.188

-2.31

10

15.19 12.31

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Third Tableau 5

4

0

0

0

10

10

10

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

5

X

9/2

1

0

-9/16

1/16

0

9/16

-1/16

0

72

4

Y

3

0

1

1/8

-1/8

0

-1/8

1/8

0

-24

10

A3

6

0

0

3/4

1/4

-1

-3/4

-1/4

1

24

Zj

94.5

5

4

0.188

2.31

-10

-5.19

-2.31

10

Cj Zj

-94.5

0

0

-0.188

-2.31

10

15.19 12.31

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D7 Fourth Final Tableau Cont. Soln.

5

4

0

0

0

10

10

10

X

Y

S1

S2

S3

A1

A2

A3

Qty. 5

X

3

1

0

-3/4

0

1/4

3/4

0

-1/4

4

Y

6

0

1

1/2

0

-1/2

-1/2

0

1/2

0

S2

24

0

0

3

1

-4

-3

-1

4

Zj

39

5

4

-1.75

0

-0.75

1.75

0

0.75

Cj - Zj

-39

0

0

1.75

0

0.75

8.25

0

9.25

The solution is optimal: Y= 6 X=3 S2 = 24

Min Z = 5X + 4Y = 39

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D8 D. Solve the following problems using simplex method: 8. The Ajax Manufacturing Company makes three products, x1, x2, and x3. The profit per unit for each is as follows: x1, P2; x2, P4; and x3, P3. The three products pass through three manufacturing centers as part of the manufacturing process. Product x1 requires 3 hours in center 1, two (2) hours in center 2, and one (1) hour in center 3; and product x2 requires 4 hours in center 1, 1 hour in center 2, and 3 hours in center 3; and product x3 requires 2 hours in each of the 3 centers. Each center has time available as follows; center 1, 60 hours; center 2, 40 hours; and center 3, 80 hours. Determine the optimum product mix for next week production schedule.

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 D. Solve the following problems using simplex method: 8. Maximize: Z = 2x1 + 4x2 + 3x3 subject to: Center 1 3x1+ 4x2 + 2x3 ≤ 60 Center 2 2x1+ 1x2 + 2x3 ≤ 40 Center 3 1x1+ 3x2 + 2x3 ≤ 80 x1,x2, x3≥ 0 Z = 2x1 + 4x2 + 3x3 + 0S1 + 0S2 + 0S3 3x1+ 4x2 + 2x3 + 1S1 + 0S2 + 0S3 = 60 2x1+ 1x2 + 2x3 + 0S1 + 1S2 + 0S3 = 40 1x1+ 3x2 + 2x3 + 0S1 + 0S2 + 1S3 = 80

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Initial Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

0

S1

60

3

4

2

1

0

0

0

S2

40

2

1

2

0

1

0

0

S3

80

1

3

2

0

0

1

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

2

4

3

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Initial Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

ETR

0

S1

60

3

4

2

1

0

0

15

0

S2

40

2

1

2

0

1

0

40

0

S3

80

1

3

2

0

0

1

26.67

Zj

0

0

0

0

0

0

0

Cj - Zj

-0

2

4

3

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Second Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

4

X2

15

3/4

1

2/4

1/4

0

0

0

S2

25

1/4

0

2/4

-1/4

1

0

0

S3

35

-5/4

0

2/4

-3/4

0

1

Zj

60

3

4

2

1

0

0

Cj - Zj

-60

-1

0

1

-1

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Second Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

ETR

4

X2

15

3/4

1

2/4

1/4

0

0

30

0

S2

25

1/4

0

2/4

-1/4

1

0

50

0

S3

35

-5/4

0

2/4

-3/4

0

1

70

Zj

60

3

4

2

1

0

0

Cj - Zj

-60

-1

0

1

-1

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D8 Third (Final) Tableau Cont.

Soln.

Qty.

2

4

3

0

0

0

X1

X2

X3

S1

S2

S3

3

X3

30

3/2

2

1

1/2

0

0

0

S2

10

-1/2

-1

0

-1/2

1

0

0

S3

20

-2

-1

0

-1

0

1

Zj

90

4.5

6

3

1.5

0

0

Cj - Zj

-90

-2.5

-2

0

-1.5

0

0

The solution is optimal: X3= 30 S2 = 10 S3 = 20

ETR

Max Z = 2X1 + 4X2 + 3X3 = 90

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D9 D. Solve the following problems using simplex method: 9. The dean of the university, College of B.A, plans the course offerings for the second semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of P2,500,000 in faculty wages, and each graduate course costs P3,000,000. How many undergraduate and graduate courses should be taught in the second semester so that total faculty salaries are kept to a minimum?

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 D. Solve the following problems using simplex method: 9. Minimize: Z = 2.5Mx + 3My subject to: Course offerings x ≥ 30 y ≥ 20 x + y ≥ 60

Let X = undergraduate Y = graduate M = million pesos

Z = 2.5Mx + 3My + 0S1 + 0S2 + 0S3 +10MA1 + 10MA2 + 10MA3 1X + 0Y - 1S1 + 0S2 + 0S3 +1A1 + 0A2 + 0A3 = 30 0X + 1Y + 0S1 - 1S2 + 0S3 +0A1 + 1A2 + 0A3 = 20 1X + 1Y + 0S1 + 0S2 - 1S3 +0A1 + 0A2 + 1A3 = 60

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Initial Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

10M

A1

30

1

0

-1

0

0

1

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

10M

A3

60

1

1

0

0

-1

0

0

1

Zj

1100M

20M

20M

-10M

-10M

-10M

10M

10M

10M

Cj Zj

-1100M

-17.5M

-17M

10M

10M

10M

0

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Initial Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

10M

A1

30

1

0

-1

0

0

1

0

0

30

10M

A2

20

0

1

0

-1

0

0

1

0

0

10M

A3

60

1

1

0

0

-1

0

0

1

60

Zj

1100M

20M

20M

-10M

-10M

-10M

10M

10M

10M

Cj Zj

-1100M

-17.5M

-17M

10M

10M

10M

0

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Second Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

30

1

0

-1

0

0

1

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

10M

A3

30

0

1

1

0

-1

-1

0

1

Zj

575M

2.5M

20M

7.5M

-10M

-10M

-7.5M

10M

10M

Cj Zj

-575M

0

-17M -7.5M

10M

10M

2.5M

0

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Second Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

2.5M

X

30

1

0

-1

0

0

1

0

0

0

10M

A2

20

0

1

0

-1

0

0

1

0

20

10M

A3

30

0

1

1

0

-1

-1

0

1

30

Zj

575M

2.5M

20M

7.5M

-10M

-10M

-7.5M

10M

10M

Cj Zj

-575M

0

-17M -7.5M

10M

10M

2.5M

0

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Third Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

30

1

0

-1

0

0

1

0

0

3M

Y

20

0

1

0

-1

0

0

1

0

10M

A3

10

0

0

1

1

-1

-1

-1

1

Zj

235M

2.5M

3M

7.5M

7M

-10M

-7.5M

-7M

10M

Cj Zj

-235M

0

0M

-7.5M

-7M

10M

17.5M

17M

0

ETR

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Third Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

ETR

2.5M

X

30

1

0

-1

0

0

1

0

0

-2.5

3M

Y

20

0

1

0

-1

0

0

1

0

0

10M

A3

10

0

0

1

1

-1

-1

-1

1

10

Zj

235M

2.5M

3M

7.5M

7M

-10M

-7.5M

-7M

10M

Cj Zj

-235M

0

0M

-7.5M

-7M

10M

17.5M

17M

0

Linear Programming Simplex Method

Chapter 3

Solving Linear Programming Problems

Solution to Exercise 9-D9 Fourth Tableau 2.5M

3M

0

0

0

10M

10M

10M

Con t.

Soln.

Qty.

X

Y

S1

S2

S3

A1

A2

A3

2.5M

X

40

1

0

0

1

-1

0

-1

1

3M

Y

20

0

1

0

-1

0

0

1

0

0

S1

10

0

0

1

1

-1

-1

-1

1

Zj

160M

2.5M

3M

0

-0.5M -2.5M

0

0.5M

2.5M

Cj Zj

-160M

0

0

0

0.5M

10M

9.5M

7.5M

The solution is optimal: X= 40 Y = 20 S1 = 10

2.5M

Min Z = 2.5MX + 3MY = 160M Where: M= million pesos

ETR

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Exercise 9-D10 D. Solve the following problems using simplex method: 10. Use the final table to answer the following questions: a. What is the best product mix? What is the highest profit? b. What are the shadow prices for the two constraints? c. Perform RHS ranging for constraint 1. d. Find the range of optimality for the profit of x.

Chapter 3

Linear Programming Simplex Method

Solving Linear Programming Problems

Solution to Exercise 9-D10 10. a. The best product mix: x = 12 and y = 6 The highest profit: P132 b. The shadow prices: x = 8 and y = 6 c. Perform RHS ranging for constraint 1. d. Find the range of optimality for the profit of x.

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