A manufacturer produces two products *A *and *B*. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product *A* requires 3 hours on both machines and each unit of product *B* requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹7 profit and that of *B *at a profit of ₹4. Find the production level per day for maximum profit graphically.

#### Solution

Let *x* units of product A and *y* units of product B be manufactured by the manufacturer per day.

It is given that one unit of product A requires 3 hours of processing time on first machine, while one unit of product B requires 2 hours of processing time on first machine. It is also given that first machine is available for 12 hours per day.

∴ 3*x* + 2*y* ≤ 12

Also, one unit of product A requires 3 hours of processing time on second machine, while one unit of product B requires 1 hour of processing time on second machine. It is also given that second machine is available for 9 hours per day.

∴ 3*x* + *y* ≤ 9

The profits on one unit each of product A and product B is ₹ 7 and ₹ 4, respectively.

So, the objective function is given by *Z* = ₹ (7*x* + 4*y*).

Therefore, the mathematical formulation of the given linear programming problem can be stated as:**Maximize** *Z* = 7*x* + 4*y *

Subject to the constraints

3*x* + 2*y* ≤ 12 .....(1)

3*x* + *y* ≤ 9 .....(2)*x* ≥ 0, *y* ≥ 0 .....(3)

The feasible region determined by constraints (1) to (3) is graphically represented as:

Here, it is seen that OABCO is the feasible region and it is bounded. The values of *Z *at the corner points of the feasible region are represented in tabular form as:

Corner Point |
Z = 7x + 4y |

O(0, 0) | Z = 7 × 0 + 4 × 0 = 0 |

A(3, 0) | Z = 7 × 3 + 4 × 0 = 21 |

B(2, 3) | Z = 7 × 2 + 4 × 3 = 26 |

C(0, 6) | Z = 7 × 0 + 4 × 6 = 24 |

The maximum value of *Z* is 26, which is obtained at *x* = 2 and *y* = 3.

Thus, **2 units of product A** and **3 units of product B** should be manufactured by the manufacturer per day in order to maximize the profit.

Also, the maximum daily profit of the manufacturer is **₹ 26**.