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Dave Powell, USDA Forest Service, forestryimages Image number 1210054

FORS 8450 • Advanced Forest Planning Lecture 3 Linear Programming - The Simplex Method (Appendix B of Bettinger, Boston, Siry, and Grebner). Dave Powell, USDA Forest Service, www.forestryimages.org Image number 1210054. Create DCM. Linear programming Simplex method. Calc. opportunity

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Dave Powell, USDA Forest Service, forestryimages Image number 1210054

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  1. FORS 8450 • Advanced Forest Planning Lecture 3 Linear Programming - The Simplex Method (Appendix B of Bettinger, Boston, Siry, and Grebner) Dave Powell, USDA Forest Service, www.forestryimages.org Image number 1210054

  2. Create DCM Linear programming Simplex method Calc. opportunity costs Optimal ? Yes Stop No ID Pivot Column Calc. maximum contributions ID Pivot Row ID Pivot Number Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  3. Linear programming Simplex method - Example Maximize 2.5 X + 8 Y Subject to: 15 X + 20 Y  60 X + 5 Y  10 Y X + 5 Y  10 X 15 X + 20 Y  60

  4. Linear programming Simplex method - Example X Y S1 S2 Q Objective function 2.5 8 2.5 X + 8 Y Constraint 1 15 20 1 60 15 X + 20 Y + S1 = 60 Constraint 2 1 5 1 10 X + 5 Y + S2 = 10 Initially, Q = RHS values

  5. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop No X Y S1 S2 Q ID Pivot Column 2.5 8 15 20 1 60 Calc. maximum contributions 1 5 1 10 ID Pivot Row ID Pivot Number Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  6. Linear programming Simplex method - Example Maximize 2.5 X + 8 Y Subject to: 15 X + 20 Y  60 X + 5 Y  10 Where are we? Y X + 5 Y  10 X 15 X + 20 Y  60

  7. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row Variables currently in the solution have an opportunity cost of 0 or less. "Their value" is their value in the objective function. ID Pivot Number Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  8. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row OCx = (Objective function value of variable X) - ((Columnx1 value) ("their value1")) - ((Columnx2 value) ("their value2")) ID Pivot Number Calc. "Ratio" OCx = (2.5) - ((15) (0)) - ((1) (0)) = 2.5 OCy = (8) - ((20) (0)) - ((5) (0)) = 8.0 Transform Pivot Row OCS1 = (0) - ((1) (0)) - ((0) (0)) = 0 Transform Non-Pivot Row(s) OCS2 = (0) - ((0) (0)) - ((1) (0)) = 0

  9. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row 2.5 8.0 0 0 Opportunity costs ID Pivot Number  Not the optimal solution, since some opportunity costs are > 0. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  10. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row 2.5 8.0 0 0 Opportunity costs ID Pivot Number  Pivot Column is the column with the highest opportunity cost  Whichever variable that column refers to is "coming into the solution." Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  11. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row ID Pivot Number  Maximum contributions are the most that the variable represented by the Pivot Column can bring into the solution. Max contribution = (Q row / Pivot Column value row ) Calc. "Ratio" Transform Pivot Row MC constraint #1 = (60 / 20) = 3 MC constraint #2 = (10 / 5) = 2 Transform Non-Pivot Row(s)

  12. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row ID Pivot Number  The most of Y that we can bring into the solution is 2 units (related to constraint row #2). Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  13. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row ID Pivot Number  The Pivot Row is the row related to the maximum contribution of variable Y Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  14. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row ID Pivot Number  The Pivot Number is the value of the cell at the intersection of the Pivot Row and the Pivot Column. Here, the Pivot Number = 5 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  15. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 1 5 1 10 S2 0 ID Pivot Row ID Pivot Number  The "Ratio" for this simple 2-constraint problem is the value in the cell of the non-Pivot Row / Pivot Column (20) divided by the Pivot Number. Here, the "ratio" = (20 / 5) = 4 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  16. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 0.2 1 0.2 2 S2 0 ID Pivot Row ID Pivot Number  Transform the Pivot Row by dividing all values in that row by the Pivot Number. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  17. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  This indicates that we are bringing the Y variable into the solution. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  18. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Transform the Non-Pivot Row (the other constraint row). New cell value = (old cell value) - (old Pivot Row value for the new cell's column x "ratio") Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  19. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 15 20 1 60 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Example: X variable column, 1st constraint row: Old cell value = 15 Old Pivot Row value for the new cell's column = 1 "ratio" = 4 New cell value = 15 - (1 x 4) = 11 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  20. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  21. Linear programming Simplex method - Example Maximize 2.5 X + 8 Y Subject to: 15 X + 20 Y  60 X + 5 Y  10 Where are we? Y X + 5 Y  10 X 15 X + 20 Y  60

  22. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row OCx = (Objective function value of variable X) - ((Columnx1 value) ("their value1")) - ((Columnx2 value) ("their value2")) ID Pivot Number Calc. "Ratio" OCx = (2.5) - ((11) (0)) - ((0.2) (8)) = 0.9 OCy = (8) - ((0) (0)) - ((1) (8)) = 0 Transform Pivot Row OCS1 = (0) - ((1) (0)) - ((0) (8)) = 0 Transform Non-Pivot Row(s) OCS2 = (0) - ((-4) (0)) - ((0.2) (8)) = -1.6

  23. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row 0.9 0 0 -1.6 Opportunity costs ID Pivot Number  Not the optimal solution, since some opportunity costs are > 0. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  24. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row 0.9 0 0 -1.6 Opportunity costs ID Pivot Number  Pivot Column is the column with the highest opportunity cost  Whichever variable that column refers to is "coming into the solution." Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  25. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Maximum contributions are the most that the variable represented by the Pivot Column can bring into the solution. Max contribution = (Q row / Pivot Column value row ) Calc. "Ratio" Transform Pivot Row MC constraint #1 = (20 / 11) = 1.8182 MC constraint #2 = (2 / 0.2) = 10 Transform Non-Pivot Row(s)

  26. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  The most of X that we can bring into the solution is 1.8182 units (related to constraint row #1). Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  27. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  The Pivot Row is the row related to the maximum contribution of variable X Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  28. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  The Pivot Number is the value of the cell at the intersection of the Pivot Row and the Pivot Column. Here, the Pivot Number = 11 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  29. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 11 0 1 -4 20 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  The "Ratio" for this simple 2-constraint problem is the value in the cell of the non-Pivot Row / Pivot Column (0.2) divided by the Pivot Number. Here, the "ratio" = (0.2 / 11) = 0.018182 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  30. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 S1 0 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Transform the Pivot Row by dividing all values in that row by the Pivot Number. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  31. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  This indicates that we are bringing the X variable into the solution. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  32. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Transform the Non-Pivot Row (the other constraint row). New cell value = (old cell value) - (old Pivot Row value for the new cell's column x "ratio") Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  33. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0.2 1 0.2 2 Y 8 ID Pivot Row ID Pivot Number  Example: X variable column, 2nd constraint row: Old cell value = 0.2 Old Pivot Row value for the new cell's column = 11 "ratio" = 0.018182 New cell value = 0.2 - (11 x 0.018182) = 0 Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  34. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0 1 -0.018 0.273 1.636 Y 8 ID Pivot Row ID Pivot Number  This indicates that we are bringing the X variable into the solution. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  35. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0 1 -0.018 0.273 1.636 Y 8 ID Pivot Row OCx = (Objective function value of variable X) - ((Columnx1 value) ("their value1")) - ((Columnx2 value) ("their value2")) ID Pivot Number Calc. "Ratio" OCx = (2.5) - ((1) (2.5)) - ((0) (8)) = 0 OCy = (8) - ((0) (2.5)) - ((1) (8)) = 0 Transform Pivot Row OCS1 = (0) - ((0.09) (2.5)) - ((-.018) (8)) = -0.081 Transform Non-Pivot Row(s) OCS2 = (0) - ((-0.364) (2.5)) - ((0.273) (8)) = -1.27

  36. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0 1 -0.018 0.273 1.636 Y 8 ID Pivot Row 0 0 -0.081 -1.27 Opportunity costs ID Pivot Number  The optimal solution, since no opportunity costs are > 0. Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  37. Create DCM Linear programming Simplex method - Example Calc. opportunity costs Optimal ? Yes Stop Variables in the solution No X Y S1 S2 Q Their value ID Pivot Column 2.5 8 1 0 0.09 -0.364 1.818 X 2.5 Calc. maximum contributions 0 1 -0.018 0.273 1.636 Y 8 ID Pivot Row Optimal solution: X = 1.818 Y = 1.636 Solution value = 1.818 (2.5) + 1.636 (8) = 17.633 ID Pivot Number Calc. "Ratio" Transform Pivot Row Transform Non-Pivot Row(s)

  38. Linear programming Simplex method - Example Maximize 2.5 X + 8 Y Subject to: 15 X + 20 Y  60 X + 5 Y  10 Where are we? Y X + 5 Y  10 X 15 X + 20 Y  60

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