Advertisements
Advertisements
Question
Solve the following linear programming problem graphically.
Maximise Z = 4x1 + x2 subject to the constraints x1 + x2 ≤ 50; 3x1 + x2 ≤ 90 and x1 ≥ 0, x2 ≥ 0.
Solution
Maximise Z = 4x1 + x2
Subject to the constraints
x1 + x2 ≤ 50
3x1 + x2 ≤ 90
x1, x2 ≥ 0
Since the decision variables, x1 and x2 are non-negative, the solution lies in the I quadrant of the plane.
Consider the equations
x1 + x2 = 50
| x1 | 0 | 50 |
| x2 | 50 | 0 |
3x1 + x2 = 90
| x1 | 0 | 30 |
| x2 | 90 | 0 |
The feasible region is OABC and its co-ordinates are O(0, 0) A(30, 0) C(0, 50) and B is the point of intersection of the lines
x1 + x2 = 50 ..........(1)
and 3x1 + x2 = 90 .........(2)
Verification of B:
x1 + x2 = 50 ..........(1)
3x1 + x2 = 90 .........(2)
− − −
− 2x1 = − 40
x1 = 20
From (1), 20 + x2 = 50
x2 = 30
∴ B is (20, 30)
| Corner points | Z = 4x1 + x2 |
| O(0, 0) | 0 |
| A(30, 0) | 120 |
| B(20, 30) | 110 |
| C(0, 50) | 50 |
Maximum value occurs at A(30, 0)
Hence the solution is x1 = 30, x2 = 0 and Zmax = 120.
APPEARS IN
RELATED QUESTIONS
Find the feasible solution of the following inequation:
x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0.
Solve the following LPP by graphical method:
Maximize z = 11x + 8y, subject to x ≤ 4, y ≤ 6, x + y ≤ 6, x ≥ 0, y ≥ 0
Solve each of the following inequations graphically using XY-plane:
4x - 18 ≥ 0
Sketch the graph of the following inequation in XOY co-ordinate system:
|x + 5| ≤ y
Solve the following LPP:
Maximize z = 4x1 + 3x2 subject to
3x1 + x2 ≤ 15, 3x1 + 4x2 ≤ 24, x1 ≥ 0, x2 ≥ 0.
A manufacturer produces bulbs and tubes. Each of these must be processed through two machines M1 and M2. A package of bulbs requires 1 hour of work on Machine M1 and 3 hours of work on M2. A package of tubes requires 2 hours on Machine M1 and 4 hours on Machine M2. He earns a profit of ₹ 13.5 per package of bulbs and ₹ 55 per package of tubes. If maximum availability of Machine M1 is 10 hours and that of Machine M2 is 12 hours, then formulate the L.P.P. to maximize the profit.
Choose the correct alternative :
The corner points of the feasible region are (0, 0), (2, 0), `(12/7, 3/7)` and (0,1) then the point of maximum z = 7x + y
Minimize z = 7x + y subjected to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0
A firm manufactures two products A and B on which the profits earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M1 and M2. Product A requires one minute of processing time on M1 and two minutes on M2, While B requires one minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs 30 minutes while M2 is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.
The LPP to maximize Z = x + y, subject to x + y ≤ 1, 2x + 2y ≥ 6, x ≥ 0, y ≥ 0 has ________.
