This document discusses the process for determining optimality and selecting entering and leaving variables in the simplex method for linear programming. It states that for a maximization problem, all numbers in the net evaluation row must be greater than or equal to zero for an optimal solution. The entering variable is the one with the largest positive value in the net evaluation row. To select the leaving variable, you divide each basis variable's current objective value (Q) by its sub-rate in the pivot column and choose the one with the smallest positive result.

1. Simplex
Part 3 of 4
Professor Ed Dansereau

2. 3. Test for Optimality
To check to determine if the current extreme point is the optimal solution is a simple
interpretation of the Net Eval row.
● For a Maximization problem
○ All numbers in the Net Eval row must be Greater or Equal than zero
● For a Minimization problem
○ All numbers in the Net Eval row must be Less than or Equal to zero

3. 3. Test for Optimality
Our example we want to maximize profit
For a Max problem the net eval row must be <= 0 for an optimal solution
We do not have an optimal solution.
Net Eval 250 400 0 0 0

4. 4. Identify Entering & Leaving Variables
To move to the next extreme point, we must identify which variable will move into
the basis and which will leave. We start by identifying the Entering Variable by
looking at the Net Evaluation row.
For a Maximization problem (Zmax)
Choose the number that has the largest positive value, the farthest from zero.
For a Minimization problem (Zmin)
Choose the number that has the largest negative value, the farthest from zero.
The column of the entering variable is referred to as the Pivot Column.

5. 4. Identify Entering & Leaving Variables
Initial Tableau
250 400 0 0 0
Basis C X1 X2 S1 S2 S3 Q
S1 0 2.5 3 1 0 0 30
S2 0 8 4 0 1 0 80
S3 0 2 6 0 0 1 48
Z row 0 0 0 0 0 0
Net Eval 250 400 0 0 0
X2 is the entering variable because it is the most positive

6. 4. Identify Entering & Leaving Variables
The number of values in the basis can not change. To add the entering variable, we
must take one away, the leaving variable. To find the leaving variable
For each variable in the basis, divide the Q by the sub.rate of the pivot column. Of
the numbers generate select the smallest positive Q/(sub.rate). Negative and
undefined numbers may not be selected.
An undefined number is any number divided by zero, 48/0 = undefined.

7. 4. Identify Entering & Leaving Variables
The Leaving Variable is S3 because it has the smallest positive Q/(sub.rate)
250 400 0 0 0
Basis C X1 X2 S1 S2 S3 Q Q/(Sub.rate)
S1 0 2.5 3 1 0 0 30 30/3 = 10
S2 0 8 4 0 1 0 80 80/4 = 20
S3 0 2 6 0 0 1 48 48/6 = 8
Z row 0 0 0 0 0 0
Net Eval 250 400 0 0 0
Leaving Variable