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Chapter 4

Chapter 4. The Simplex Method. Outline. 4.1 Slack Variables and the Simplex Tableau 4.2 The Simplex Method I: Maximum Problems 4.3 The Simplex Method II: Minimum Problems 4.4 Sensitivity Analysis and Matrix Formulations of Linear Programming Problems 4.5 Duality.

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Chapter 4

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  1. Chapter 4 The Simplex Method Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  2. Outline • 4.1 Slack Variables and the Simplex Tableau • 4.2 The Simplex Method I: Maximum Problems • 4.3 The Simplex Method II: Minimum Problems • 4.4 Sensitivity Analysis and Matrix Formulations of Linear Programming Problems • 4.5 Duality Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  3. 4.1 Slack Variables and the Simplex Tableau • Standard Form • Slack Variable • Solutions Using Slack Variables • Group I and Group II Variables • Simplex Tableau • Solution Corresponding to Simplex Tableau Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  4. Standard Form • A linear programming problem is in standard form if: • The objective function is to be maximized; • Each variable is constrained to be greater than or equal to 0; • All other constraints are of the form • [linear polynomial] < [nonnegative constant]. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  5. Example Standard Form • The following is in standard form: Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  6. Slack Variable • A nonnegative variable that “takes up the slack” between the left-hand side of an inequality and the right-hand side is called a slack variable. The slack variable changes an inequality into an equation. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  7. Example Slack Variable • Use slack variables to change the following into a system of equations. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  8. Example Slack Variable - Answer • For u, v, w> 0 and M as large as possible: • 6x + 3y< 96 becomes 6x + 3y + u = 96, • x + y< 18 becomes x + y + v = 18, • 2x + 6y< 72 becomes 2x + 6y + w = 72, • and • Maximize 80x + 70y becomes -80x - 70y + M = 0 for M as large as possible. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  9. Infinite Number of Solutions • The system of equations using the slack variables have an infinite number of solutions since there are more variables than equations. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  10. Example Infinite Number of Solutions • Find two solutions to the following system of equations with u, v, and w> 0. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  11. Example Solutions - Answer (1) • Write the augmented matrix for the system If x = y = 0, then a solution is u = 96, v = 18, w = 72 and M = 0. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  12. Example Solutions - Answer (2) • If we pivot about a11, we have If y = u = 0, then another solution is x = 16, v = 2, w = 40 and M = 1280. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  13. Group I and Group II Variables • In the preceding solutions, two of the variables were chosen to be 0 and the other 4 were solved for using the equations. The variables chosen to be 0 are Group I variables. The variables solved for once the Group I variables were chosen are called Group II variables. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  14. Example Group I & II Variables • In the two solutions from the previous example, identify the Group I and Group II variables. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  15. Simplex Tableau • You will notice that both matrices from which a solution was obtained in the previous example had the columns of the identity matrix present in some order. Such a matrix is called a simplex tableau. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  16. Example Simplex Tableau • Find the columns of the identity matrix in the following two simplex tableaux. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  17. Solution Corresponding to Simplex Tableau • The solution corresponding to a given simplex tableau is obtained by choosing those variables whose columns correspond to the columns of the identity matrix as Group II variables. The other variables are Group I variables. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  18. Example Solution from Simplex Tableau • Find the solution corresponding to Suppose the columns correspond to x, y, u, v, w and M. Choose u = w = 0, then x = 8, y = 9, v = 0and M = 7. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  19. Summary Section 4.1 - Part 1 • A linear programming problem is in standardform if the linear objective function is to be maximized, every variable is constrained to be nonnegative, and all other constraints are of the form [linear polynomial] < [nonnegative constant]. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  20. Summary Section 4.1 - Part 2 • To form the initial simplex tableau corresponding to a linear programming problem in standard form: • Step 1: For each constraint of the form [linear polynomial] < [nonnegative constant], introduce a slack variable and write the constraint as an equation. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  21. Summary Section 4.1 - Part 3 • Step 2: Introduce a variable M to represent the quantity to be maximized, and form the equation -[objective function] + M = 0. • Form the augmented matrix corresponding to the system of linear equations from steps 1 and 2, with the equation from step 2 at the bottom. This matrix is the initial simplex tableau. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  22. 4.2 The Simplex Method I: Maximum Problems • The Simplex Method for Problems in Standard Form Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  23. The Simplex Method I: Maximum Problems (1) • The Simplex Method for Problems in Standard Form: • Introduce slack variables and state the problem in terms of a system of linear equations. • Construct the simplex tableau corresponding to the system. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  24. The Simplex Method I: Maximum Problems (2) • Determine if the left part of the bottom row contains negative entries. If none are present, the solution corresponding to the tableau yields a maximum and the problem is solved. • If the left part of the bottom row contains negative entries, construct a new simplex tableau. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  25. The Simplex Method I: Maximum Problems (3) • Choose the pivot column by inspecting the entries of the last row of the current tableau, excluding the right-hand entry. The pivot column is the one containing the most-negative of these entries. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  26. The Simplex Method I: Maximum Problems (4) • Choose the pivot element by computing ratios associated with the positive entries of the pivot column. The pivot element is the one corresponding to the smallest nonnegative ratio. • Construct the new simplex tableau by pivoting around the selected element. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  27. The Simplex Method I: Maximum Problems (5) • Return to step 3. Steps 3 and 4 are repeated as many times as necessary to find a maximum. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  28. Example Simplex Method I Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  29. Example Simplex Method I - Step 1 • Let u, v and w be the slack variables. The corresponding linear system is Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  30. Example Simplex Method I - Step 2 • Set up the initial simplex tableau. x y u v w M u v w M Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  31. Example Simplex Method I - Step 3 • Determine if maximum has been reached. At least one negative entry. Maximum has not been reached. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  32. Example Simplex Method I - Steps 4a,b • Choose the pivot element 96/6 = 16 18/1 = 18 72/2 = 36 Smallest positive ratio Most negative entry Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  33. Example Simplex Method I - Step 4c • Pivot. x y u v w M x v w M Group II variables Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  34. Example Simplex Method I - Step 5 • Determine if maximum has been reached. x y u v w M x v w M Group II variables At least one negative entry. Maximum has not been reached. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  35. Example Simplex Method I - continued • Choose pivot. 16/(1/2) = 32 2/(1/2) = 4 40/5 = 8 pivot row pivot column Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  36. Example Simplex Method I - continued • New tableau: x y u v w M x y w M Group II variables No negative entries Solution: x = 14, y = 4 and Maximum = 1400 Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  37. Summary Section 4.2 • The simplex method entails pivoting around entries in the simplex tableau until the bottom row contains no negative entries except perhaps the entry in the last column. The solution can be read off the final tableau by letting the variables heading columns with 0 entries in every row but the ith row take on the value in the ith row of the right-most column, and setting the other variables equal to 0. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  38. 4.3 The Simplex Method II: Minimum Problems • Simplex Method for Problems in Nonstandard Form • Negative Constants • Minimization Problem • Further Comments Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  39. Simplex Method for Problems in Nonstandard Form (1) • Simplex Method for Problems in Nonstandard Form : • If necessary, convert all inequalities (except the nonnegative inequalities) into the form [linear polynomial] < [constant] where the constant can be negative. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  40. Simplex Method for Problems in Nonstandard Form (2) • If a negative number appears in the upper part of the last column of the simplex tableau, remove it by pivoting. • Select one of the negative entries in its row. The column containing the entry will be the pivot column. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  41. Simplex Method for Problems in Nonstandard Form (3) • Select the pivot element by determining the least of the positive ratios associated with entries in the pivot column (except the bottom entry). • Pivot. • Repeat step 2 until there are no negative entries in the upper part of the right-hand column of the simplex tableau. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  42. Simplex Method for Problems in Nonstandard Form (4) • Proceed to apply the simplex method for tableaux in standard form. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  43. Example Nonstandard Constraints • Maximize the objective function 5x + 10y subject to Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  44. Example Nonstandard Constraints (2) • The second inequality is in nonstandard form. Multiply by -1 to reverse the inequality. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  45. Example Nonstandard Constraints (3) • Form the tableau. 20/1 = 20 -10/(-2) = 5 Negative entry in selected row Negative entry Pivot column Pivot about –2. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  46. Example Nonstandard Constraints (4) • Most negative entry 15/(3/2) = 10 Pivot about 3/2. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  47. Example Nonstandard Constraints (5) x y u v M yx M Solution: y = 10, x = 10, Maximum = 150 Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  48. Example Minimum Problem • Minimize the objective function 3x + 2y subject to Multiply the objective function by -1 to change it to a maximum problem: Maximize the objective function -3x - 2y. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  49. Example Minimum Problem (2) • The first inequality is in nonstandard form. Multiply by -1 to reverse the inequality. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

  50. Example Minimum Problem (3) • Form the tableau. -10/(-1) = 10 Negative entries in selected row Negative entry We’ll choose the 2nd column as Pivot column. Pivot about –1. Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e

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