Tutorial 7 on LP MEMO
Question 1
Consider the feasible region given below:
70
60
50
40
30 C
B
20
10
A D
0
0 10 20 F 30 E 40 50 60 70
The objective function is given by: 𝑐 = 2𝑥 + 3𝑦
Let 𝑐 = 3 ∴ 3𝑦 = 3 − 2𝑥
2𝑥
𝑦 =1−
3
a. Find the coordinates of the points that will give a maximum and minimum
value.
Coordinates: Point C (15;30) Max value (green dotted line)
Point A (10;6.67) Min value (purple dotted line)
b. Use the extreme point method (table below) to find the coordinates of the
points and value of the objectve function. Identify the extreme values.
Point Coordinates Value of objective function Extreme value
A (10;6.67) 40.01 Min
B (10;30) 110
C (15;30) 120 Max
D (40;5) 95
E (40;0) 80
F (30;0) 60
, Question 2
The constraints for a LP-problem are:
𝑥 + 2𝑦 ≤ 8
4𝑥 + 2𝑦 ≤ 8
0.5𝑥 + 𝑦 ≤ 3
𝑥, 𝑦 ≥ 0
The objective function for the LP-problem is:
Maximise 4𝑥 + 8𝑦
a. Draw the inequalities and objective function (dotted line) and shade the
feasible region.
A
B
b. Find the coordinates of the point that will maximise the profit.
Let 𝑝 = 8 ∴ 4𝑥 + 8𝑦 = 𝑝
8𝑦 = 𝑝 − 4𝑥
𝑥
𝑦=1−
2
The profit is maximised on the line segment ̅̅̅̅
𝐴𝐵 ((𝑥, 𝑦) = (0,3) up to
(0.67, 2.67))
Use Cramer to find the point B (intersection):
4 2
| |=3
0.5 1
8 2
| | 2
𝑥= 3 1 = = 0.67
3 3
4 8
| | 8
𝑦= 0.5 3 = = 2.67
3 3
Question 1
Consider the feasible region given below:
70
60
50
40
30 C
B
20
10
A D
0
0 10 20 F 30 E 40 50 60 70
The objective function is given by: 𝑐 = 2𝑥 + 3𝑦
Let 𝑐 = 3 ∴ 3𝑦 = 3 − 2𝑥
2𝑥
𝑦 =1−
3
a. Find the coordinates of the points that will give a maximum and minimum
value.
Coordinates: Point C (15;30) Max value (green dotted line)
Point A (10;6.67) Min value (purple dotted line)
b. Use the extreme point method (table below) to find the coordinates of the
points and value of the objectve function. Identify the extreme values.
Point Coordinates Value of objective function Extreme value
A (10;6.67) 40.01 Min
B (10;30) 110
C (15;30) 120 Max
D (40;5) 95
E (40;0) 80
F (30;0) 60
, Question 2
The constraints for a LP-problem are:
𝑥 + 2𝑦 ≤ 8
4𝑥 + 2𝑦 ≤ 8
0.5𝑥 + 𝑦 ≤ 3
𝑥, 𝑦 ≥ 0
The objective function for the LP-problem is:
Maximise 4𝑥 + 8𝑦
a. Draw the inequalities and objective function (dotted line) and shade the
feasible region.
A
B
b. Find the coordinates of the point that will maximise the profit.
Let 𝑝 = 8 ∴ 4𝑥 + 8𝑦 = 𝑝
8𝑦 = 𝑝 − 4𝑥
𝑥
𝑦=1−
2
The profit is maximised on the line segment ̅̅̅̅
𝐴𝐵 ((𝑥, 𝑦) = (0,3) up to
(0.67, 2.67))
Use Cramer to find the point B (intersection):
4 2
| |=3
0.5 1
8 2
| | 2
𝑥= 3 1 = = 0.67
3 3
4 8
| | 8
𝑦= 0.5 3 = = 2.67
3 3
