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in Linear Programming by (25.9k points)
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A chemical company produces a chemical containing three basic elements A, B, C so that it has at least 16 liters of A, 24 liters of B and 18 liters of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 liters of A, 12 liters of B, 2 liters of C. Each unit of compound II has 2 liters of A, 2 liters of B and 6 liters of C. The cost per unit of compound I is Rs 800/- and that of compound II is Rs 640/-. Formulate the problem as L.P.P. and solve it to minimize the cost.

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Best answer

Let the company buy x units of compound I and y units of compound II. 

Then the total cost is z = Rs (800x + 640y). 

This is the objective function which is to be minimized. 

The constraints are as per the following table :

From the table, the constraints are 4x + 2y ≥ 16, 12x + 2y ≥ 24, 2x + 6y ≥ 18. 

Also, the number of units of compound I and compound II cannot be negative. 

∴ x ≥ 0, y ≥ 0. 

∴ the mathematical formulation of given LPP is 

Minimize z = 800x + 640y, subject to 4x + 2y ≥ 16, 12x + 2y ≥ 24, 2x + 6y ≥ 18, x ≥ 0, y ≥ 0. 

First we draw the lines AB, CD and EF whose equations are 4x + 2y = 16, 12x + 2y = 24 and 2x + 6y = 18

The feasible region is shaded in the graph. The vertices of the feasible region are E(9, 0), P, Q, and D(0, 12). 

P is the point of intersection of the lines 

2x + 6y = 18 … (1) 

and 4x + 2y = 16 … (2) 

Multiplying equation (1) by 2, we get 

4x + 12y = 36 

Subtracting equation (2) from this equation, we get

10y = 20 

∴ y = 2 

∴ from (1), 2x + 6(2) = 18 

∴ 2x = 6 

∴ x = 3 

∴ P = (3, 2) 

Q is the point of intersection of the lines 

12x + 2y = 24 … (3) 

and 4x + 2y = 16 … (2) 

On subtracting, we get 

8x = 8 

∴ x = 1 

∴ from (2), 4(1) + 2y = 16 

∴ 2y = 12 

∴ y = 6 

∴ Q = (1, 6) 

The values of the objective function z = 800x + 640y at these vertices are 

z(E) = 800(9)+ 640(0) =7200 + 0 = 7200 

z(P) = 800(3) + 640(2) = 2400 + 1280 = 3680 

z(Q) = 800(1) + 640(6) =800 + 3840 = 4640 

z(D) = 800(0) + 640(12) = 0 + 7680 = 7680 

∴ the minimum value of z is 3680 at the point (3, 2).

Hence, the company should buy 3 units of compound I and 2 units of compound II to have the minimum cost of Rs 3680.

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