Advertisements
Advertisements
प्रश्न
Solve the following linear programming problems by graphical method.
Minimize Z = 20x1 + 40x2 subject to the constraints 36x1 + 6x2 ≥ 108; 3x1 + 12x2 ≥ 36; 20x1 + 10x2 ≥ 100 and x1, x2 ≥ 0.
Advertisements
उत्तर
Given that 36x1 + 6x2 ≥ 108
Let 36x1 + 6x2 = 108
6x1 + x2 = 18
| x1 | 0 | 3 | 2 |
| x2 | 18 | 0 | 6 |
Also given that 3x1 + 12x2 ≥ 36
Let 3x1 + 12x2 = 36
x1 + 4x2 = 12
| x1 | 0 | 12 | 4 |
| x2 | 3 | 0 | 2 |
Also given that 20x1 + 10x2 ≥ 100
Let 20x1 + 10x2 = 100
2x1 + x2 = 10
| x1 | 0 | 5 | 4 |
| x2 | 10 | 0 | 2 |
The feasible region satisfying all the conditions is ABCD.
The coordinates of the comer points are A(12, 0), B(4, 2), C(2, 6) and D(0, 18).
| Corner points | Z = 20x1 + 40x2 |
| A(12, 0) | 240 |
| B(4, 2) | 80 + 80 = 160 |
| C(2, 6) | 40 + 240 = 280 |
| D(0, 18) | 720 |
The minimum value of Z occurs at B(4, 2)
∴ The optimal solution is x1 = 4, x2 = 2 and Zmin = 160
APPEARS IN
संबंधित प्रश्न
Solve the following L.P.P. by graphical method:
Minimize: z = 8x + 10y
Subject to: 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.
Select the appropriate alternatives for each of the following question:
The value of objective function is maximum under linear constraints
The maximum value of z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x, y ≥ 0 is ______.
Of all the points of the feasible region, the optimal value of z obtained at the point lies ______.
Solve each of the following inequations graphically using XY-plane:
4x - 18 ≥ 0
A company manufactures two types of fertilizers F1 and F2. Each type of fertilizer requires two raw materials A and B. The number of units of A and B required to manufacture one unit of fertilizer F1 and F2 and availability of the raw materials A and B per day are given in the table below:
| Raw Material\Fertilizers | F1 | F2 | Availability |
| A | 2 | 3 | 40 |
| B | 1 | 4 | 70 |
By selling one unit of F1 and one unit of F2, company gets a profit of ₹ 500 and ₹ 750 respectively. Formulate the problem as L.P.P. to maximize the profit.
Solve the following linear programming problem graphically.
Maximize Z = 3x1 + 5x2 subject to the constraints: x1 + x2 ≤ 6, x1 ≤ 4; x2 ≤ 5, and x1, x2 ≥ 0.
The set of feasible solutions of LPP is a ______.
Solve the following problems by graphical method:
Maximize z = 4x + 2y subject to 3x + y ≥ 27, x + y ≥ 21, x ≥ 0 y ≥ 0
Solve the following LPP by graphical method:
Maximize: z = 3x + 5y Subject to: x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0
