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Basic Variable

-1	1	1	0	0	11
1	1	0	1	0	27
2	5	0	0	0	90
-4	-6	0	0	0	0
					↑
					Current z-value

For this initial simplex tableau, the basic variables are , , and , and the nonbasic variables are , and . Note that the basic variables are labeled to the right of the simplex tableau next to the appropriate rows. This technique is important as you proceed through the simplex method. It helps keep track of the changing basic variables, as shown in Example 1.
, and are the nonbasic variables in this initial tableau, so they have an initial value of zero, yielding a current z-value of zero. From the columns that are farthest to the right, the basic variables have initial values of , , and . So the current solution is

This solution is a basic feasible solution and is often written as

The entry in the lower right corner of the simplex tableau is the current value of z. Note that the bottom-row entries under , and are the negatives of the coefficients of , and in the objective function

To perform an optimality check for a solution represented by a simplex tableau, look at the entries in the bottom row of the tableau. If any of these entries are negative (as above), then the current solution is not optimal.
Pivoting
After you have set up the initial simplex tableau for a linear programming problem, the simplex method consists of checking for optimality and then, when the current solution is not optimal, improving the current solution. (An improved solution is one that has a larger z-value than the current solution.) To improve the current solution, bring a new basic variable into the solution, the entering variable. This implies that one of the current basic variables (the departing variable) must leave, otherwise you would have too many variables for a basic solution. Choose the entering and departing variables as listed below [3].
1. The entering variable corresponds to the least (the most negative) entry in the bottom row of the tableau, excluding the “b-column.”
2. The departing variable corresponds to the least nonnegative ratio in the column determined by the entering variable, when .
3. The entry in the simplex tableau in the entering variable’s column and the departing variable’s row is the pivot.
Finally, to form the improved solution, apply Gauss-Jordan elimination to the column that contains the pivot, as illustrated in Example 1 ([3]). (This process is called pivoting.)