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Home > Operation Research > Simplex method example
2. Simplex method example ( Enter your problem )
( Enter your problem )
  1. Structure of Linear programming problem
  2. Algorithm (using `Z`-row method)
  3. Maximization example-1 (using `Z`-row method)
  4. Maximization example-2 (using `Z`-row method)
  5. Maximization example-3 (using `Z`-row method)
  6. BigM method Algorithm (using `Z`-row method)
  7. Minimization example-1 (using `Z`-row method)
  8. Minimization example-2 (using `Z`-row method)
  9. Minimization example-3 (using `Z`-row method)
  10. Degeneracy example-1 (Tie for leaving basic variable) (using `Z`-row method)
  11. Degeneracy example-2 (Tie first Artificial variable removed) (using `Z`-row method)
  12. Unrestricted variable example (using `Z`-row method)
  13. Multiple optimal solution example (using `Z`-row method)
  14. Infeasible solution example (using `Z`-row method)
  15. Unbounded solution example (using `Z`-row method)
  16. Algorithm (using `Z_j-C_j` method)
  17. Maximization example-1 (using `Z_j-C_j` method)
  18. Maximization example-2 (using `Z_j-C_j` method)
  19. Maximization example-3 (using `Z_j-C_j` method)
  20. BigM method Algorithm (using `Z_j-C_j` method)
  21. Minimization example-1 (using `Z_j-C_j` method)
  22. Minimization example-2 (using `Z_j-C_j` method)
  23. Minimization example-3 (using `Z_j-C_j` method)
  24. Degeneracy example-1 (Tie for leaving basic variable) (using `Z_j-C_j` method)
  25. Degeneracy example-2 (Tie first Artificial variable removed) (using `Z_j-C_j` method)
  26. Unrestricted variable example (using `Z_j-C_j` method)
  27. Multiple optimal solution example (using `Z_j-C_j` method)
  28. Infeasible solution example (using `Z_j-C_j` method)
  29. Unbounded solution example (using `Z_j-C_j` method)
  30. Algorithm (using `C_j-Z_j`method)
  31. Maximization example-1 (using `C_j-Z_j`method)
  32. Maximization example-2 (using `C_j-Z_j`method)
  33. Maximization example-3 (using `C_j-Z_j`method)
  34. BigM method Algorithm (using `C_j-Z_j`method)
  35. Minimization example-1 (using `C_j-Z_j`method)
  36. Minimization example-2 (using `C_j-Z_j`method)
  37. Minimization example-3 (using `C_j-Z_j`method)
  38. Degeneracy example-1 (Tie for leaving basic variable) (using `C_j-Z_j`method)
  39. Degeneracy example-2 (Tie first Artificial variable removed) (using `C_j-Z_j`method)
  40. Unrestricted variable example (using `C_j-Z_j`method)
  41. Multiple optimal solution example (using `C_j-Z_j`method)
  42. Infeasible solution example (using `C_j-Z_j`method)
  43. Unbounded solution example (using `C_j-Z_j`method)
 		Other related methods
  1. Formulate linear programming model
  2. Graphical method
  3. Simplex method (BigM method)
  4. Two-Phase method
  5. Primal to dual conversion
  6. Dual simplex method
  7. Integer simplex method
  8. Branch and Bound method
  9. 0-1 Integer programming problem
  10. Revised Simplex method
2. Graphical method
(Previous method)		2. Algorithm (using `Z`-row method)
(Next example)

1. Structure of Linear programming problem


General structure of Linear programming problem

(Maximize or Minimize) `Z_x=c_1x_1+c_2x_2+c_3x_3+...+c_nx_n` (objective function)
Subject to the constraints
`a_11x_1+a_12x_2+a_13x_3+...+a_(1n)x_n(<=,>=,=)b_1`
`a_21x_1+a_22x_2+a_23x_3+...+a_(2n)x_n(<=,>=,=)b_2`
:
:
`a_(m1)x_1+a_(m2)x_2+a_(m3)x_3+...+a_(mn)x_n(<=,>=,=)b_m` (constraints)
and `x_1,x_2,...,x_n>=0` (non-negative condition)


Here `x_1,x_2,...,x_n` are decision variables

Each constraint may be either
1. `<=` (less than or equal to)
2. `>=` (greater than or equal to)
3. `=` (equal to)

`x_1,x_2,...,x_n>=0` means each `x_i` must be non-negative


A linear problem is said to be in standard form if
  1. It is maximization problem
  2. There are only equalities (=) (by adding slack, surplus or artificial variables)
  3. All variables are restricted to be non-negativity


Any linear program can in fact be transformed into an equivalent linear program in standard form
  1. If the objective function is minimize `Z=2x_1-3x_2` then we can simply maximize it by multiplying -1,
    maximize `Z'=` minimize `-Z=-2x_1+3x_2`
  2. If constraint is `<=` type then we have to add slack variable
    eg. `3x_1+2x_2<=4` then `3x_1+2x_2+S_1=4` (here `S_1` is slack variable)
  3. If constraint is `>=` type then we have to subtract surplus variable and add aritificial variable
    eg. `5x_1+3x_2>=6` then `5x_1+3x_2-S_2+A_1=6` (here `S_2` is surplus variable and `A_1` is aritificial variable)
  4. If constraint is `=` type then we have to add aritificial variable
    eg. `x_1-x_2=1` then `x_1-x_2+A_2=1` (here `A_2` is aritificial variable)
  5. If `x_1` is unrestricted in sign then we can introduce two new decision variables `x_1'` and `x_1''`
    and now `x_1=x_1'-x_1''` where `x_1,x_1',x_1'' >= 0`


For eg.
Min `Z=2x_1-3x_2`
subject to the constraint
`3x_1+2x_2<=4`
`5x_1+3x_2>=6`
`x_1-x_2=1`
and `x_1,x_2>=0`

Then it is converted to
Max `Z'=-2x_1+3x_2`
subject to the constraint
`3x_1+2x_2+S_1=4`
`5x_1+3x_2-S_2+A_1=6`
`x_1-x_2+A_2=1`
and `x_1,x_2,S_1,S_2,A_1,A_2>=0`




This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


2. Graphical method
(Previous method)		2. Algorithm (using `Z`-row method)
(Next example)


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