2. SIMPLEX METHOD
The general problem of optimizing a linear
function of several variables subject to a set of
linear constraints:
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3. SIMPLEX METHOD
• Consider the following linear programming problem in two
variables:
• A feasible solution to this problem is any point (x, y) that
satisfies all the constraints of the problem
• An optimal solution, a point in the feasible region with the
largest value of the objective function z = 3x + 5y.
• Our task is to find an optimal solution
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4. SIMPLEX METHOD
• To apply the simplex method to a linear
programming problem, it has to be represented
in a special form called the standard form.
• The standard form has the following
requirements:
– It must be a maximization problem.
– All the constraints (except the nonnegativity
constraints) must be in the form of linear equations
with nonnegative right-hand sides.
– All the variables must be required to be nonnegative.
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7. SIMPLEX METHOD – Solved Example
• Constraint is given as an inequality
• Add slack variable to convert it into equivalent
equation.
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8. SIMPLEX METHOD – Solved Example
• Convert the given problem into a the standard
form of linear programming problem in four
variables:
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9. SIMPLEX METHOD – Solved Example
• Now the Problem is of the standard form:
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10. Create the initial Simplex Tableau
• Use the Constraints for the first two rows of the table
• Use the negative coefficient values of the Objective Function for the 3rd row
Objective Row
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13. Steps to be Followed
• Find Entering Variable
– Find the Pivot Column
• Find Departing Variable
– Calculate value
– Find the Pivot Row
• Find Pivot Value (c)
• Pivoting
– Update the Pivot Row with the Pivot Value (row/c)
– Update the other Two Rows
• Check for Convergence
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16. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Least Column Value in the Objective Row
Objective Row
Pivot Column
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17. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Least Column Value in the Objective Row
Objective Row
Pivot Column
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18. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Least Column Value in the Objective Row
Objective Row
Pivot Column
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19. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Objective Row
Pivot Column
Least Value
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20. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Objective Row
Pivot Column
Least Value
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21. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Objective Row
Pivot Column
Least Value
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22. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Objective Row
Pivot Column
Least Value
Pivot Row
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23. Identify Pivot Row
• compute the by dividing the row’s last
entry by the entry in the pivot column.
Objective Row
Pivot Column
Least Value
Pivot Row
Pivot Value
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24. Identify Entering & Departing Variables
Objective Row
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Entering Variable Y
Departing Variable V
25. Identify Entering & Departing Variables
Objective Row
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Entering Variable Y
Departing Variable V
26. Identify Entering & Departing Variables
Objective Row
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Entering Variable Y
Departing Variable V
27. Pivoting
Divide all the entries of the pivot row by the pivot
value
Objective Row
Pivot Row
Pivot Value
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51. Steps to be Followed
• Find Entering Variable
– Find the Pivot Column
• Find Departing Variable
– Calculate value
– Find the Pivot Row
• Find Pivot Value (c)
• Pivoting
– Update the Pivot Row with the Pivot Value (row/c)
– Update the other Two Rows
• Check for Convergence
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