Manhattan College MATH 103: Simplex Method

⎧ x1 + 2x2 ≤ 8 ​ ​




x1 + x2 ≤ 6
Maximize Z = 3x1 + 4x2 , subject to ⎨
​ ​




.
x2 ≥ 0
​ ​ ​ ​




​




⎩
x1 ≥ 0 ​




SOLUTION
The problem in the canonical form can be written as follows:


Z = 3x1 + 4x2 → max ​ ​




⎧ x1 + 2x2 ≤ 8 ​ ​




⎨ x1 + x2 ≤ 6
​
​ ​
​




⎩
x1 , x2 ≥ 0 ​ ​




Add variables (slack or surplus) to turn all the inequalities into equalities:


Z = 3x1 + 4x2 → max ​ ​




⎧ x1 + 2x2 + S1 = 8 ​ ​ ​




⎨ x1 + x2 + S 2 = 6
​
​ ​ ​
​




⎩
x1 , x2 , S 1 , S 2 ≥ 0
​ ​ ​ ​




Write down the simplex tableau:

Basic x1 ​ x2 ​ S1 ​ S2 ​
Solution

Z −3 −4 0 0 0
S1 ​
1 2 1 0 8
S2 ​ 1 1 0 1 6

The entering variable is x2 , because it has the most negative coefficient −4 in the Z-row.
​




Basic x1 ​ x2 ​ S1 ​ S2 ​
Solution Ratio

Z −3 −4 0 0 0
8
S1 ​ 1 2 1 0 8 2
​
=4
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