1. Max Z = x1 + x2 2 x1 + 3 x2  6 (1) x2  1.5 (2) x1 - x2  2 (3)
Graphical solution
Max Z = x1 + x2
2 x1 + 3 x2  6 (1)
x2  (2)
x1 - x2  (3)
x1 , x2  0

2. Graphical solution Max Z = x1 + x2 2 x1 + 3 x2  6 (1) x2  1.5 (2)
OABCD represents the feasible space x2
Max Z = x1 + x2
2 x1 + 3 x2  6 (1)
x2  (2)
x1 - x2  (3)
x1 , x2  0
(3) A B
(2) C x1 O D
(1)

3. Graphical solution Max Z = x1 + x2 2 x1 + 3 x2  6 (1) x2  1.5 (2)
(3) A B
(2)
Z = 1
Optimal Point
2 x*1 + 3 x*2 = 6
x*1 – x*2 = 2 x2 * C
Z = 0 x1 * x1 O D
(1)
x*1 = 2.4, x*2 = 0.4 Z* = 2.8
Direction of the maximization

4. Graphical solution: 2nd method
Theorem 1: If an optimal solution of a LP exists, then at least one of the corner points is optimal. x2
Corner points
(3)
O (0, 0) Z = 0
A (0, 1.5) Z = 1.5
B (0.75, 1.5 ) Z = 2.25
C (2.4, 0.4) Z = 2.8
D (2, 0) Z = 2 A B
(2) C x1 O D
(1)

5. Graphical solution: example 2
Max Z = x1 + 2 x2
2 x1 + 3 x2  6 (1)
x2  (2)
x1 - x2  (3)
x1 , x2  0 x2
Optimal Point
(3) A B
(2)
Z = 1
O (0, 0) Z = 0
A (0, 1.5) Z = 3
B (0.75, 1.5 ) Z = 3.75
C (2.4, 0.4) Z = 3.2
D (2, 0) Z = 2 C Z*
Z = 0 x1 O D
(1)

6. Graphical solution: example 3
Theorem 2: If a LP has more than one optimal solution, then there is an infinite number of optimal solutions. x2
Max Z = 4 x1 + 6 x2
2 x1 + 3 x2  6 (1)
x2  (2)
x1 - x2  (3)
x1 , x2  0
All the points of [BC] are optimal
(3) A B
(2)
Z = 4
O (0, 0) Z = 0
A (0, 1.5) Z = 9
B (0.75, 1.5 ) Z = 12
C (2.4, 0.4) Z = 12
D (2, 0) Z = 8 C Z*
Z = 0 x1 O D
(1)

7. Graphical solution: example 4
(2)
Max Z = x1 + 3 x2
x1 + 2 x2  (1)
- x1 + 2 x2  4 (2)
x1 , x2  0 x2
Infeasible Linear Program:
There is no solution that satisfies all the constraints x1 O
(1)

8. Graphical solution: example 5
Unbounded feasible space
Max Z = x1 + x2
-2 x1 + 3 x2  6 (1)
x2  (2)
x1 , x2  0
(1) x2
Unbounded Linear Program:
The objective function can always increase (maximization) without violating any constraint.
Z = 1
(2)
Z = 0
No optimal solution x1 O