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A company manufactures bicycles and t...

A company manufactures bicycles and tricycles, each of which must be processed through two machines A and B. Machine A has maximum of 120 hours available and machine B has a maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.
If profits are ₹ 180 for a bicycle and ₹ 220 for a tricycle, determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

Text Solution

Verified by Experts

Let x bicycles and y tricycles are to be manufactured. Then the total profit is ` z = ₹ (180 x + 220y ) `
This is a linear function which is to be maximized. Hence, it is the objective function. The constraints are as per the following table :

From the table, the constraints are
`6x + 4y le 120, 3x + 10 y le 180 `
Also, the number of bicycles and tricycles cannot be negative
` therefore x ge 0, y ge 0 `.
Hence, the mathematical formulation of given LPP is :
Maximize ` z = 180 x + 220 y `, subject to
` 6x + 4y le 120 , 3 x + 10 y le 180, x ge 0 , y ge 0 ` .
First we draw the lines AB and CD whose equations are ` 6x + 4y = 120 and 3 x + 10 y = 180 ` respectively.

The feasible region is OAPDO which is shaded in the figure.
The vertices of s the feasible region are ` O ( 0, 0 ) , A ( 20, 0), P and D ( 0, 18).`
P is the point of intersection of the lines
`3x + 10 y = 180 " " `...(1)
and `6x + 4y = 120 " " `... (2)
Multiplying equation (1) by 2, we get,
` 6x + 20 y = 360 `
Subtracting equation (2) from this equation, we get,
` 16 y = 240 " " therefore y = 15 `
` therefore ` from ` (1) , 3x + 10 ( 15 ) = 180 `
` therefore 3 x = 30" " therefore x = 10 `
` therefore P = ( 10, 15 ) `
The values of the objective function ` z = 180 x + 220 y ` at these vertices are
` z (O ) = 180 ( 0) + 220 ( 0 ) = 0 `
` z(A) = 180 ( 20 ) + 220 ( 0 ) = 3600 `
` z(P) = 180 (10) + 220 ( 15 ) = 1800 + 3300 = 5100 `
`z ( D) = 180 (0 ) + 220 (18) = 3960 `
` therefore ` the maximum value of ` z ` is 5100 at the point ` (10, 15) `.
Hence, 10 bicycles and 15 tricycles should be manufactured in order to have the maximum profit of ₹ 5100.
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