Objective function is Z = - x + 2 y ……………………(1).
The given constraints are : x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.
Corner points | Z = - x + 2y |
D(6,0 ) | -6 |
A(4,1) | -2 |
B(3,2) | 1 |
Here , the open half plane has points in common with the feasible region .
Therefore , Z has no maximum value.
Let number of cakes of first type = x
And number of cakes of second type = y
Therefore , the above L.P.P. is given as :
Minimise , Z = x +y , subject to the constraints : 200x +100y ≤ 5000 and. 25x +50y ≤ 1000, i.e. 2x + y ≤ 50 and x +2y ≤ 40 x, y ≥ 0.
The corner points can be obtained by constructing the lines x+2y=40 , 2x+y= 50 and x+2y = 40.
The points so obtained are (0,0),(25,0), (20,10), and (0,20).
Corner points | Z = x + y |
O( 0 , 0 ) | 0 |
D(25,0 ) | 25 |
A(20,10) | 30……………..(Max.) |
B(0,20) | 20 |
Here Z = 30 is maximum.
i.e Maximum number of cakes = 30 , 20 of kind one and 10 cakes of another kind .
Let number of rackets made = x
And number of bats made = y
Therefore , the above L.P.P. is given as :
Maximise , Z = x +y , subject to the constraints : 1.5x +3y ≤ 42 and. 3x +y ≤ 24, i.e.0.5x + y ≤ 14 i.e. x +2y ≤ 28 and 3x +y ≤ 24 , x, y ≥ 0.
Corner points | Z = x + y |
O( 0 , 0 ) | 0 |
D(0,14 ) | 14 |
A(8,0) | 8 |
B(4,12) | 16…………………(Max.) |
Here Z = 16 is maximum. i.e Maximum number of rackets = 4 and number of bats = 12.
Let number of rackets made = x
And number of bats made = y
Therefore , the above L.P.P. is given as :
Maximise , Z = x +y , subject to the constraints : 1.5x +3y ≤ 42 and. 3x +y ≤ 24, i.e.0.5x + y ≤ 14 i.e. x +2y ≤ 28 and 3x +y ≤ 24 , x, y ≥ 0.
Corner points | Z = x + y |
O( 0 , 0 ) | 0 |
D(0,14 ) | 14 |
A(8,0) | 8 |
B(4,12) | 16…………………(Max.) |
Here Z = 16 is maximum. i.e Maximum number of rackets = 4 and number of bats = 12.
Here , profit function is P = 20x + 10y
Profit is maximum at x = 4 and y = 12 .
Therefore , maximum profit = 20(4) + 10 ( 12) = 200.i.e. Rs.200.
Let number of packages of nuts produced = x
And number of packages of bolts produced = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 17.50x +7y , subject to the constraints : x +3y ≤ 12 and. 3x +y ≤ 12, x, y ≥ 0.
Corner points | Z =17.50 x +7 y |
O( 0 , 0 ) | 0 |
D(4,0 ) | 70 |
A(0,4) | 28 |
B(3,3) | 73.50…………………(Max.) |
Here Z = 73.50 is maximum.
i.e 3 packages of nuts and 3 packages of bolts;
Maximum profit = Rs 73.50.
Let number of packages of screws A produced = x
And number of packages of screws B produced = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 7x +10y , subject to the constraints : 4x +6y ≤ 240 and. 6x +3y ≤ 240 i.e. 2x +3y ≤ 120 and 2x +y ≤ 80 , x, y ≥ 0.
Corner points | Z =7 x +10 y |
O( 0 , 0 ) | 0 |
D(40,0 ) | 280 |
A(0,40) | 400 |
B(30,20) | 410…………………(Max.) |
Here Z = 410 is maximum.
i.e 30 packages of screws A and 20 packages of screws B; Maximum profit = Rs 410.
Let number of pedestal lamps manufactured = x
And number of wooden shades manufactured = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 5x +3y , subject to the constraints : 2x +y ≤ 12 and. 3x +2y ≤ 20 , x, y ≥ 0.
Corner points | Z =5x +3 y |
O( 0 , 0 ) | 0 |
D(6,0 ) | 30 |
A(0,10) | 30 |
B(4,4) | 32…………………(Max.) |
Here Z = 32 is maximum.
i.e 30 packages of screws A and 20 packages of screws B; Maximum profit = Rs 410.
i.e. 4 Pedestal lamps and 4 wooden shades; Maximum profit = Rs 32 .
Let number of souvenirs of type A = x
And number of souvenirs of type B = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 5x +6y , subject to the constraints : 5x +8y ≤ 200 and. 10x +8y ≤ 240 , x, y ≥ 0.
Corner points | Z =5x +6 y |
O( 0 , 0 ) | 0 |
D(0,25 ) | 150 |
A(24,0) | 120 |
B(8,20) | 160…………………(Max.) |
Here Z = 160 is maximum.
i.e. 8 Souvenir of types A and 20 of Souvenir of type B; Maximum profit = Rs 160.
Let number of desktop model computers = x
And number of portable model computers = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 4500x +5000y , subject to the constraints : x +y ≤ 250 and 25000x +40000y ≤ 700000 .i.e. x +y ≤ 250 and 5x +8y ≤ 1400 , x, y ≥ 0.
Corner points | Z =4500x +5000 y |
O( 0 , 0 ) | 0 |
D(250,0 ) | 1125000 |
A(0,175) | 87500 |
B(200,50) | 1150000…………………(Max.) |
Here Z = 1150000 is maximum.
i.e. 200 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150000 .
Let number of units of food F1 = x
And number of units of food F2 = y
Therefore , the above L.P.P. is given as :
Minimise , Z = 4x +6y , subject to the constraints : 3 x + 6y ≥ 80, 4x + 3y ≥ 100, x,y ≥ 0.,
Corner points | Z =4x +6 y |
B(80/3 , 0 ) | 320/3 |
D(24,4/3 ) | 104…………………(Min.) |
A(0,100/3) | 200 |
Corner points Z =4x +6 y B(80/3 , 0 ) 320/3 D(24,4/3 ) 104…………………(Min.) A(0,100/3) 200 Here Z = 104 is minimum. i.e. Minimum cost = Rs 104.