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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
29. Unbounded solution example (using `Z_j-C_j` method)
(Previous example)		31. Maximization example-1 (using `C_j-Z_j`method)
(Next example)

30. Algorithm (using `C_j-Z_j`method)


Simplex Method Steps (Rule)
Step-1: Formulate the Problem

a. Formulate the mathematical model of the given linear programming problem.

b. If the objective function is minimization type then change it into maximization type. Min z = - Max (-z)

c. All the `X_(B_i) > 0`. So if any `X_(B_i) < 0` then multiply the corresponding constraint by -1 to make `X_(B_i) > 0`. So sign `<=` changed to `>=` and vice varsa

d. Transform every `<=` constraint into an `=` constraint by adding a slack variable to every constraint and assign a 0 cost coefficient in the objective function.
Step-2: Find out the Initial basic solution

Find the initial basic feasible solution by setting zero value to the decision variables. That shown in the example-1
Step-3: Test for Optimality

a. Calculate the values of `c_j - z_j` in the last row of simplex table.

b. If all `c_j - z_j <= 0` , the current basic feasible solution is the optimal solution.

c. In `c_j - z_j > 0`, then select the variable that has Most Positive `c_j - z_j` and enter this variable into the new table. This column is called key column (pivot column).
Step-4: Test for Feasibility (variable to leave the basis)

a. Find the ratio by dividing the values of `X_B` column by the positive values of key column (say `a_(ij)>0`)

b. Find the minimum ratio and this row is called key row (pivot row) and corresponding variable will leave the solution.

c. The intersection element of key row and key column is called key element (pivot element).
Step-5: Determine the new solution

a. The new values of key row can be obtained by dividing the key row elements by the pivot element.

eg. `R_1`(new)`=``R_1`(old) `-:3`

b. The numbers in the remaining rows can be computed by utilizing the following formula:

Row(new) = Row(old) - (value of key column and Row(old)) `xx` KeyRow(new)

eg. `R_2`(new)`=``R_2`(old) `- 2` `R_1`(new)
Step-6: Repeat the procedure

Goto step 3 and repeat the procedure until all the values of `c_j - z_j<=0`.



This material is intended as a summary. Use your textbook for detail explanation.
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29. Unbounded solution example (using `Z_j-C_j` method)
(Previous example)		31. Maximization example-1 (using `C_j-Z_j`method)
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