Irregular Types of LP Models
Dr. Sania Bhatti
Irregular Types of LP Models
• Theorem:
Every linear program either:
1. is infeasible,
2. is unbounded,
3. has a unique optimal solution value (single
optimal point or multiple optima)
Infeasible LP Model
• Definition A linear program is infeasible if it
has no feasible solutions, i.e. the feasible
region is empty.
• Example:
Max 8X1 + 5X2
s.t. 2X1 + 1X2 ≤ 1000
3X1 + 4X2 ≤ 2400
X1 - X2 ≤ 350
X1
≥ 800
X1, X2 ≥ 0
X2
1000
900
Models With No Solutions
Infeasibility
.
Max
s.t.
800
700
600
8X1 + 5X2
2X1 + 1X2 ≤ 1000
3X1 + 4X2 ≤ 2400
X1 - X2 ≤ 350
X1
≥ 800
X1, X2 ≥ 0
No points in common.
No points satisfy all constraints
simultaneously.
500
400
No Solutions!
Problem is
300
INFEASIBLE.
200
100
100
200
300
400
500
600
700
800
X1
Infeasibility
• A problem is infeasible when there are no solutions that
satisfy all the constraints.
• Infeasibility can occur from
– Input Error
– Misformulation
– Simply an inconsistent set of constraints
Models With An
“Unbounded” Solution
X2
1000
Max
s.t.
900
800
8X1 + 5X2
X1 - X2 ≤ 350
X1
≥ 200
X2 ≥ 300
700
600
Unbounded
Feasible Region
500
Can increase
indefinitely
400
300
200
100
Unbounded Solution
100
200
300
400
500
600
700
800
X1
X2
1000
Models With An Unbounded
Feasible Region – Optimal Solution
Min
s.t.
900
800
8X1 + 5X2
X1 - X2 ≤ 350
X1
≥ 200
X2 ≥ 200
700
600
Unbounded
Feasible Region
500
400
300
200
OPTIMAL POINT
100
100
200
300
400
500
600
700
800
X1
Unboundedness
• An unbounded feasible region extends to infinity in some
direction.
• If the problem is unbounded, the feasible region must be
unbounded.
• If the feasible region is unbounded, the problem may or
may not be unbounded.
• An unbounded solution means you left out some
constraints – you cannot make an “infinite” profit.
Multiple Optimal Solutions
• When an objective function line is parallel to a constraint
the problem can have multiple optimal solutions.
• A problem can have multiple optima(alternative optima)
but a single optimal value of Z.
Example:
MAX 8X1 + 4X2
s.t.
2X1 + 1X2 ≤ 1000
3X1 + 4X2 ≤ 2400
1X1 - 1X2 ≤ 350
X1, X2 ≥ 0
Multiple Optimal Solutions
X2
1000
MAX 8X1 + 4X2
s.t.
2X1 + 1X2 ≤ 1000
3X1 + 4X2 ≤ 2400
1X1 - 1X2 ≤ 350
All points on the
X1, X2 ≥ 0
boundary between
900
800
700
600
Optimal
Extreme
Point
optimal extreme points
are also optimal
500
Optimal
Extreme
Point
400
300
200
100
100
200
300
400
500
600
700
800
X1
Class Exercise
• Solve graphically the following LP models and
categorize them as Infeasible, unbounded or a
problem with single/multiple optima.
1. Max z = 2x1 + 6x2
s.t.
4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
Class Exercise
2.
Max z = 3x1 + 4x2
s.t.
x1 + x2 > 5
3x1 + x2 > 8
x1, x2 > 0
Class Exercise: Solution
• There are no points that satisfy both constraints, hence this
problem has no
x2 feasible region, and no optimal solution.
2x1 + x2 > 8
8
4x1 + 3x2 < 12
4
x1
3
4
Class Exercise: Solution
• The feasible region is unbounded and the objective function line
can be moved parallel to itself without bound so that z can be
x2
increased infinitely.
3x1 + x2 > 8
8
5
x1 + x2 > 5
Max 3x1 + 4x2
2.67
5
x1