Objective function is Z = 3x + 4 y ……(1).
The given constraints are : x + y ≤ 4, x ≥ 0, y ≥ 0.
The corner points obtained by constructing the line x+ y= 4, are (0,0),(0,4) and (4,0).
Corner points | Z = 3x +4y |
O ( 0 ,0 ) | Z = 3(0)+4(0) = 0 |
A ( 4 , 0 ) | Z = 3(4) + 4 (0) = 12 |
B ( 0 , 4 ) | Z = 3(0) + 4 ( 4) = 16 …( Max. ) |
therefore Z = 16 is maximum at ( 0 , 4 ) .
Objective function is Z = - 3x + 4 y ……………………(1).
The given constraints are : x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.
The corner points obtained by constructing the line x+2y=8 and 3x +2y = 12 are (0,0),(0,2),(3,0) and (20/19,45/19)
Corner points | Z = 5x + 3 y |
O(0 , 0 ) | 0 |
B ( 2 , 0 ) | 10 |
C( 0 , 3 ) | 9 |
D ( 20/19 , 45/19 ) | 235/19 ……………….(Max.) |
Here , Z = -12 is minimum at C ( 4 , 0 ) .
Objective function is Z = 3x + 5 y ……………………(1).
The given constraints are : x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0 .
The corner points obtained by drawing the lines x+3y=3 and x+y=2 are (0,0), (3,0),(0,2) and (3/2,1/2)
Corner points | Z = 3x + 5 y |
A(3 , 0 ) | 9 |
B ( 0 ,2 ) | 10 |
C( 3/2 , 1/2 ) | 7………………..( Min. ) |
Here Z = 7 is minimum at ( 3/2 , ½ ) .
Objective function is Z = 5x + 3 y ……………………(1).
The given constraints are : 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
The corner points obtained by drawing the lines 3x+5y=15 and 5x+2y=10 are (0,0),(0,3), (2,0) and (20/19,45/19)
Corner points | Z = 5x + 3 y |
O(0 , 0 ) | 0 |
B ( 2 , 0 ) | 10 |
C( 0 , 3 ) | 9 |
D ( 20/19 , 45/19 ) | 235/19 ……………….(Max.) |
Here , Z = 235/19 is maximum at ( 20/19 , 45/19 ) .
Objective function is Z = 3x + 2 y ……………………(1).
The given constraints are : x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0. The corner points obtained by drawing the lines 3x+y=15 and x+2y=10 graphically are (0,0),(0,5), (5,0) and (4,3).
Corner points | Z = 3x + 2y |
O(0 ,0 ) | 0 |
A(5,0) | 15 |
B(0,5) | 10 |
C(4,3) | 18……………………..(Max.) |
Here , Z = 18 is maximum at ( 4, 3 )
Objective function is Z = x + 2 y ……………………(1).
The given constraints are : 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0 .
Corner points | Z = x + 2y |
A(0 ,3 ) | 6…………………..(Minimum) |
B(6,0) | 6………………………(Minimum) |
Here , Z = 18 is minimum at ( 0, 3 ) and ( 6 , 0 ) .
Minimum Z = 6 at all the points on the line segment joining the points (6, 0) and (0, 3).
Objective function is Z = 5x + 10 y ……………………(1).
The given constraints are : x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0 .
The corner points are obtained by drawing the lines x+2y =120, x+y = 60 and x-2y = 0. The points so obtained are (60,30),(120,0), (60,0) and (40,20)
Corner points | Z = 5x + 10y |
D(60 ,30 ) | 600 |
A(120,0) | 600 |
B(60,0) | 300……………………..(Min.) |
C(40,20) | 400 |
Here , Z = 300 is minimum at ( 60, 0 ).
Objective function is Z = x + 2 y ……………………(1).
The given constraints are : x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.
The corner points are obtained by drawing the lines x+2y=100, 2x-y=0 and 2x+y=200.
The points so obtained are (0,50),(20,40), (50,100) and (0,200)
Corner points | Z = x + 2y |
D(0 ,50 ) | 100……………..(Min.) |
A(20,40) | 100……………………..(Min.) |
B(50,100) | 250 |
C(0,200) | 400 |
Here , Z = 100 is minimum at ( 0, 50) and ( 20 ,40).
Minimum Z = 100 at all the points on the line segment joining the points (0, 50) and (20, 40).
Let number of food type P = x
And number of units of food type Q = y
Therefore , the above L.P.P. is given as :
Minimise , Z = 60x +80y , subject to the constraints : 3 x + 4y ≥ 8, 5x + 2y ≥ 11, x,y ≥ 0.
The corner points can be obtained by drawing the lines 3x+4y=8 and 5x+2y=11 graphically.
The points so obtained are (8/3,0), (2,1/2), (0,11/2)
Corner points | Z = - x + 2y |
D(8/3,0 ) | 160………………….(Min.) |
A(2,1/2) | 160………………(Min.) |
B(0,11/2) | 440 |
Here Z = 160 is minimum. i.e. Minimum cost = Rs 160 at all points lying on segment joining .