Advertisements
Advertisements
Question
Solve the following linear programming problems by graphical method.
Maximize Z = 40x1 + 50x2 subject to constraints 3x1 + x2 ≤ 9; x1 + 2x2 ≤ 8 and x1, x2 ≥ 0.
Advertisements
Solution
Given that 3x1 + x2 ≤ 9
Let 3x1 + x2 = 9
| x1 | 0 | 3 |
| x2 | 9 | 0 |
Also given that x1 + 2x2 ≤ 8]
Let x1 + 2x2 = 8
| x1 | 0 | 8 |
| x2 | 4 | 0 |
3x1 + x2 = 9 ………(1)
x1 + 2x2 = 8 ……..(2)
6x1 + 2x2 = 18 ……..(3) [Multiply by 2 for eq. (1)]
− 5x1 = − 10
x1 = 2
x1 = 2 substitute in (1)
3(2) + x2 = 9
x2 = 3
The feasible region satisfying all the conditions is OABC.
The co-ordinates of the corner points are O(0, 0), A(3, 0), B(2, 3), C(0, 4)
| Corner points | Z = 40x1 + 50x2 |
| O(0, 0) | 0 |
| A(3, 0) | 120 |
| B(2, 3) | 40 × 2 + 50 × 3 = 80 + 150 = 230 |
| C(0, 4) | 200 |
The maximum value of Z occurs at B(2, 3).
∴ The optimal solution is x1 = 2, x2 = 3 and Zmax = 230
APPEARS IN
RELATED QUESTIONS
A furniture dealer deals in tables and chairs. He has ₹ 1,50,000 to invest and a space to store at most 60 pieces. A table costs him ₹ 1500 and a chair ₹ 750. Construct the inequations and find the feasible solution.
Solve the following LPP by graphical method:
Maximize z = 11x + 8y, subject to x ≤ 4, y ≤ 6, x + y ≤ 6, x ≥ 0, y ≥ 0
The maximum value of z = 10x + 6y subject to the constraints 3x + y ≤ 12, 2x + 5y ≤ 34, x, ≥ 0, y ≥ 0 is ______.
Solve the following LPP:
Maximize z = 2x + 3y subject to x - y ≥ 3, x ≥ 0, y ≥ 0.
Solve the following LPP:
Maximize z = 4x1 + 3x2 subject to
3x1 + x2 ≤ 15, 3x1 + 4x2 ≤ 24, x1 ≥ 0, x2 ≥ 0.
Solve the following LPP:
Minimize z = 4x + 2y
Subject to 3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0
Solve the following L.P.P. by graphical method:
Maximize: Z = 4x + 6y
Subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.
State whether the following is True or False :
The point (1, 2) is not a vertex of the feasible region bounded by 2x + 3y ≤ 6, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.
Which value of x is in the solution set of inequality − 2X + Y ≥ 17
The point which provides the solution of the linear programming problem, Max.(45x + 55y) subject to constraints x, y ≥ 0, 6x + 4y ≤ 120, 3x + 10y ≤ 180, is ______
