Linear Programming Lecture Notes - Penn State Personal Web Server

5. Identifying Unboundedness
We have already identified a theorem for detecting unboundedness. Recall Theorem 5.7:
In a maximization problem, if aji < 0 for all i = 1, . . . , m, and zj − cj < 0, then the linear
programming problem is unbounded.
This condition occurs when a variable xj should enter the basis because ∂z/∂xj > 0
and there is no blocking basis variable. That is, we can arbitrarily increase the value of xj
without causing any variable to become negative. We give an example:
Example 5.11. Consider the Linear programming problem from Example 2.9:
⎧
max z(x1, x2) = 2x1 − x2
⎪⎨ s.t. x1 − x2 ≤ 1
⎪⎩
2x1 + x2 ≥ 6
x1, x2 ≥ 0
We can convert this problem into standard form by adding a slack variable s1 and a surplus
variable s2:
⎧
⎪⎨
⎪⎩
max z(x1, x2) = 2x1 − x2
s.t. x1 − x2 + s1 = 1
2x1 + x2 − s2 = 6
x1, x2, s1, s2 ≥ 0
This yields the matrices:
⎡ ⎤ ⎡ ⎤
2 x1
⎢
c = ⎢−1⎥
⎢x2⎥
⎥
⎣ 0 ⎦ x = ⎢ ⎥
⎣s1⎦
0
s2
A =
1 −1 1
0
2 1 0 −1
We have both slack and surplus variables, so the case when x1 = x2 = 0 is not a valid initial
solution. We can chose a valid solution based on our knowledge of the problem. Assume
that s1 = s2 = 0 and so we have:
B =
1 −1
2 1
In this case we have:
This yields:
x1
xB =
x2
B −1 b =
7/3
4/3
N =
s1
xN =
s2
1 0
0 −1
B −1 N =
cB =
We also have the cost information:
cBB −1 b = 10
3 cBB −1 N = 4
3
2
−1
cN =
1/3
−1/3
−2/3 −1/3
0
0
b =
1
6
1 − 3 cBB −1 N − cN = 4
3
81
1 − 3