max z = 5x + 3y
Constraint
4x + 5y ≤ 1000
5x + 2y ≤ 1000
3x + 8y ≤ 1200
x,y ≥ 0
intersection point of line (1) and (2)
5 x equation (2) - 2 equation(1) ⇒ 25x + 16y - 8x - 16y = 5000 - 2000
⇒ 17x = 3000 ⇒ x = \(\frac{3000}{17}\)
5\(\big(\frac{3000}{17}\big)\) + 2y = 1000
⇒ 2y = 1000 - \(\frac{15000}{17}\) = \(\frac{2000}{17}\)
⇒ y = \(\frac{1000}{17}\)
\(\big(\frac{3000}{17},\frac{1000}{17}\big)\)
intersection point of line (2) and (3)
4 x equation (2) - equation (3)
⇒ 20x + 8y - 3x - 8y = 4000 - 1200
⇒ 17x = 2800 ⇒ x = \(\frac{2800}{17}\)
2y = 1000 - 5\(\big(\frac{2800}{17}\big)\) = \(\frac{17000-14000}{17}\) = \(\frac{3000}{17}\)
y = \(\frac{1500}{17}\)
\(\big(\frac{2800}{17},\frac{1500}{17}\big)\)
intersection point of (1) and (3)
4 x equation(3) - 3 x equation(1) ⇒ 12x + 32y - 12x - 15y = 4800 - 3000
⇒ 17y = 1800
⇒ y = \(\frac{1800}{17}\)
4x = 1000 - 5y = 1000 - 5\(\big(\frac{1800}{17}\big)\) = \(\frac{17000-9000}{17}\)
x = \(\frac{2000}{17}\)
\(\big(\frac{2000}{17},\frac{1800}{17}\big)\)
Clearly by graph corner points are
(0,0), (0,150), (\(\big(\frac{2000}{17},\frac{1800}{17}\big)\),\(\big(\frac{3000}{17},\frac{1000}{17}\big)\) and (200,0)
Points |
z = 5x + 3y |
(0,0) |
0 |
(0,150) |
450 |
\(\big(\frac{2000}{17},\frac{1800}{17}\big)\) |
\(\frac{15400}{17}\) = 905.88 |
\(\big(\frac{3000}{17},\frac{1000}{17}\big)\) |
\(\frac{18000}{17}\) = 1058.82 (maximum) |
(200,0) |
1000 |
Therefore, maximum profit = 1058.82 at point \(\big(\frac{3000}{17},\frac{1000}{17}\big)\).