Chapter 9 – Simplex Method
S. Neuburger
Linear Programming – Simplex Method
The simplex method can be used for problems with more than 2 variables. This method iteratively examines corner
points of feasible region. This algebraic method can easily be automated by software.
Convert each constraint into an equation:
Original Format
Variables Introduced In Conversion
≤ constraint
Slack variable
≥ constraint
Surplus and Artificial variables
= constraint
Artificial variable
Incorporate all variables in all equations. If they don’t appear assign a coefficient of 0.
Variable Type
Slack Variable
Represents
Measures unused resources
Coefficient in objective function
0
Used in
≤ constraint
Surplus Variable
How much solution exceeds the
constraint resource
Computational tool, ensures nonnegativity of surplus variables
0
≥ constraint
M (some huge cost)
≥ constraint and
= constraint
Artificial Variable
The simplex method:
 A basic feasible solution to a system of n equations is found by setting all but n variables equal to 0 and
solving for the other variables.
 The method considers only feasible solutions and will only touch the corner points of the feasible region.
 The numbers in the body of the simplex tableau can be thought of as substitution rates for the variables in
the solution mix.
 Any variable that appears in the solution mix column must have the number 1 occupying one cell in its
column and 0’s in every other place in that column.
 The Zj value for the Quantity column provides the total contribution to objective function (frequently gross
profit) of the given solution.
 The Zj values for the other columns (variables) represent the gross profit given up by adding one unit of this
variable into the current solution.
 The Cj – Zj number in each column represents the net profit that will result from introducing 1 unit of each
product or variable into the solution,
 i.e., the profit gained minus the profit given up.
 It is not calculated for the quantity column.
 A negative number in the Cj - Zj row would tell us that profits would decrease if the
corresponding variable were added to the solution mix while a positive number indicates that the
profits would increase.
 An optimal solution is reached in the simplex method when the Cj - Zj row contains no positive numbers
for a maximization problem or no negative numbers for a minimization problem.
 After the initial tableau is completed, proceed through a series of five steps to compute all the numbers
needed in the next tableau.
Chapter 9 – Simplex Method
S. Neuburger
The 5 Steps for maximization [minimization]
1. Choose the variable with the greatest positive [negative] Cj - Zj to enter the solution.
2. Determine the row to be replaced by selecting that one with the smallest [non-negative] quantity-to-pivotcolumn ratio.
3. Calculate the new values for the pivot row: new # = old # / pivot #
4. Calculate the new values for the other row(s): (new #) = (old #) – (# in pivot column)(new # in pivot row)
Gauss-Jordan elimination
5. Calculate the Cj and Cj - Zj values for this tableau.
o If there are any Cj - Zj values greater than [less than] zero, return to step 1.
o Otherwise, the final tableau has been created.
Chapter 9 – Simplex Method
S. Neuburger
Special case
Infeasibility
General Description
Contradictory constraints
Indication in Simplex Method
Artificial variable included in the solution mix
Unboundedness
Solution approaches infinity
Ratios of pivot column entries are all ≤ 0
Degeneracy
3 constraints pass through 1 point
Alternate optimal solutions
Several optimal solutions exist
More than one smallest ratio in pivot column;
can cause cycling
Cj – Zj = 0 for variable not in the solution mix
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Sensitivity Analysis shows how the optimal solution and the corresponding value of objective function
change, given changes in various inputs to the problem.
Computer programs handling LP problems of all sizes provide sensitivity analysis as an important output
feature.
Those programs often use the information provided in the final simplex tableau to compute ranges for the
objective function coefficients and ranges for the RHS values.
They also provide “shadow prices,” the value of one additional unit of a scarce resource.
The range over which shadow prices remain valid is called right-hand-side ranging which is computed by
dividing the quantity by the indicated value of the variable.