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The Simplex Method

The Simplex Method. Standard Form. Furniture Manufacturing Problem. x y u v w M. Carp. Finish. Uph. Selecting the Pivot Column:. x y u v w M. Carp. Finish. Uph. Objective Function: 80x + 70y = M

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The Simplex Method

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  1. The Simplex Method Standard Form

  2. Furniture Manufacturing Problem x y u v w M Carp. Finish. Uph.

  3. Selecting the Pivot Column: x y u v w M Carp. Finish. Uph. Objective Function: 80x + 70y = M Which would we want to make first chairs or sofas?

  4. Selecting the Pivot Column: x y u v w M Carp. Finish. Uph. PIVOT COLUMN “Largest Negative” in the bottom row

  5. Selecting the Pivot Row: x y u v w M Carp. Finish. Uph. First a bit of Trial and Error…

  6. x y u v w M x y u v w M C F U C F U C F U C F U C F U C F U

  7. x y u v w M C F U x = 16 y = 0 u = 0 v = 2 w = 40 M = 1280 C F U x = 18 y = 0 u = -12 v = 0 w = 36 M = 1440 C F U x = 36 y = 0 u = -120 v = -18 w = 0 M = 2880

  8. Selecting the Pivot Row: x y u v w M RATIOS Carp. Finish. Uph.

  9. Selecting the Pivot Row: x y u v w M RATIOS Carp. Finish. Uph. PIVOT ROW Smallest Non-Negative Ratio

  10. Putting it all together: x y u v w M Carp. Finish. Uph.

  11. After Completing the Pivot:Are We Done Yet??? x y u v w M Carp. Finish. Uph.

  12. Pivot Column: x y u v w M Carp. Finish. Uph. PIVOT COLUMN “Largest Negative” in the bottom row

  13. Pivot Row: x y u v w M Carp. Finish. Uph. PIVOT ROW Smallest Non-Negative Ratio

  14. After Completing the Pivot:Are We Done Yet??? x y u v w M Carp. Finish. Uph.

  15. FINAL ANSWER: x y u v w M Carp. Finish. Uph. 14 chairs, 4 sofas Profit of $1400

  16. Maximize 10x + 12y + 10z subject to:

  17. System of Equations: x - 2y + u = 6 3x + z +v = 9 y + 3z +w = 12 -10x - 12y - 10z +M = 0

  18. Simplex Tableau: x y z u v w M

  19. Simplex Tableau: x y z u v w M Ratios PIVOT ROW PIVOT COLUMN

  20. New Simplex Tableau: x y z u v w M x = 0, y = 12, z = 0 M = 144

  21. New Simplex Tableau: x y z u v w M PIVOT ROW PIVOT COLUMN

  22. New Simplex Tableau: x y z u v w M x = 3, y = 12, z = 0 M = 174

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