3/9/2017
Happens when there is no feasible region. No feasible solution
Happens when there exists lines that do not contribute to the feasible region.
More than one optimal solution
Special Cases in LP Graph
Redundancy
Happens when there is more than one point that give the same Z value.
Unbounded
Happens when the problem is a maximization problem, but the feasible region is open.
1
3/9/2017
Maximize: Z = 10X1 + 15X2 Subject to: 6X1 + 5X2 ≥ 30 2X1 + 4X2 ≤ 12 10X1 + 8X2 ≤ 40 x1, X2 ≥ 0 Line that conflicts with other lines which causes the nonexistence of feasible region. Since there is no feasible region, no optimal solution can be found.
Maximize: Z = 5X1 + 9X2 Subject to: 2X1 + 8X2 ≥ 16 6X1 + 3X2 ≥ 24 7X1 + 5X2 ≥ 35 x1, X2 ≥ 0
The problem is a maximization problem but the feasible region is unbounded outward (open). When unboundedness happens, no optimal solution can be found.
2
3/9/2017
Maximize: Z = 4X1 + 3X2 Subject to: 8X1 + 6X2 ≤ 48 7X1 + 2X2 ≤ 28 x1, X2 ≥ 0
Points
Z = 4X1 + 3X2
(0, 0)
0
(0, 8)
24
(2.77, 4.31)
24
(4, 0)
16
Optimal solution: X1 = 0, X2 = 8 X1 = 2.77, X2 = 4.31 Z = 24 There are two points that give the same maximum Z value. Therefore, both points are considered as optimal solutions.
Maximize: Z = 4X1 + 8X2 Subject to: 6X1 + 3X2 ≤ 18 2X1 + 5X2 ≤ 10 3X1 + 7X2 ≤ 21 x1, X2 ≥ 0
Points
Z = 4X1 + 8X2
(0, 0)
0
(0, 2)
16
(2.5, 1)
18
(3, 0)
12
Optimal solution: X1 = 2.5, X2 = 1, Z = 18 Line that does not contribute to the feasible region but does not conflict with other lines. This line is called redundant line. When redundancy happens, optimal solution is still available.
3