**Solve the following problem :**

A company manufactures bicyles and tricycles, each of which must be processed through two machines A and B Maximum availability of machine A and B is respectively 120 and 180 hours. 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 on a tricycle, determine the number of bicycles and tricycles that should be manufacturing in order to maximize the profit.

#### Solution

Let x number of bicycles and y number of tricycles be manufactured by the company.

∴ Total profit Z = 180x + 220y

This is the objective function to be maximized.

The given information can be tabulated as shown below:

Bicycles (x) |
Tricycles (y) |
Maximum availability of time (hrs) | |

Machine A | 6 | 4 | 120 |

Machine B | 3 | 10 | 180 |

∴ The constraints are 6x + 4y ≤ 120, 3x + 10y ≤ 180, x ≥ 0, y ≥ 0

∴ Given problem can be formulated as

Maximize Z = 180x + 220y

Subject to, 6x + 4y ≤ 120, 3x + 10y ≤ 180 , x ≥ 0, y ≥ 0.

To draw the feasible region, construct the table as follows:

Inequality | 6x + 4y ≤ 120 | 3x + 10y ≤ 180 |

Corresponding equation (of line) | 6x + 4y = 120 | 3x + 10y = 180 |

Intersection of line with X-axis | (20, 0) | (60, 0) |

Intersection of line with Y-axis | (0, 30) | (0, 18) |

Region | Origin side | Origin side |

Shaded portion OABC is the feasible region,

whose vertices are O ≡ (0, 0), A ≡ (20, 0), B and C ≡ (0, 18)

B is the point of intersection of the lines 3x + 10y = 180 and 6x + 4y = 120.

Solving the above equations, we get

B ≡ (10, 15)

Here the objective function is,

Z = 180x + 220y

∴ Z at O(0, 0) = 180(0) + 220(0) = 0

Z at A(20, 0) = 180(20) + 220(0) = 3600

Z at B(10, 15) = 180(10) + 220(15) = 5100

Z at C(0, 18) = 180(0) + 220(18) = 3960

∴ Z has maximum value 5100 at B(10, 15)

∴ Z is maximum when x = 10, y = 15

Thus, the company should manufacture 10 bicycles and 15 tricycles to gain maximum profit of ₹ 5100.