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4.2 The Simplex Method I: Maximum Problems

4.2 The Simplex Method I: Maximum Problems. The Simplex Method for Problems in Standard Form. 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.

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4.2 The Simplex Method I: Maximum Problems

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  1. 4.2 The Simplex Method I: Maximum Problems • The Simplex Method for Problems in Standard Form

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. Example Simplex Method I

  8. Example Simplex Method I - Step 1 • Let u, v and w be the slack variables. The corresponding linear system is

  9. Example Simplex Method I - Step 2 • Set up the initial simplex tableau. x y u v w M u v w M

  10. Example Simplex Method I - Step 3 • Determine if maximum has been reached. At least one negative entry. Maximum has not been reached.

  11. 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

  12. Example Simplex Method I - Step 4c • Pivot. x y u v w M x v w M Group II variables

  13. 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.

  14. Example Simplex Method I - continued • Choose pivot. 16/(1/2) = 32 2/(1/2) = 4 40/5 = 8 pivot row pivot column

  15. 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

  16. 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.

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