cottage firm employed 5 men and 9 women to make two types of articles P and Q. A man takes 2 hours, while a woman takes 3 hours to make one unit of article P and a man takes 1 hour, while a woman takes 2 hours to make one unit of article Q. A man and a woman can work for a maximum of 8 hours a day. The firm makes a profit of Rs 35 and Rs 20 by selling one unit of article P and one unit of article Q respectively. Formulate this problem as a linear programming problem.
To formulate this problem as a linear programming problem, we need to define the decision variables, objective function, and constraints.
Let's define the decision variables:
Let x be the number of units of article P produced, and y be the number of units of article Q produced.
Objective function:
The objective is to maximize the profit, which is given by:
Z = 35x + 20y
Constraints:
1. Time constraint for article P production:
2x + 3y ≤ 8(5 + 9)
This constraint ensures that the total time utilized by men and women to produce article P does not exceed their maximum working hours.
2. Time constraint for article Q production:
x + 2y ≤ 8(5 + 9)
This constraint ensures that the total time utilized by men and women to produce article Q does not exceed their maximum working hours.
3. Non-negativity constraint:
x ≥ 0
y ≥ 0
These constraints ensure that the number of units produced cannot be negative.
Considering all the constraints, the final linear programming problem can be formulated as:
Maximize Z = 35x + 20y
Subject to:
2x + 3y ≤ 8(5 + 9)
x + 2y ≤ 8(5 + 9)
x ≥ 0
y ≥ 0
This problem can now be solved using various linear programming algorithms or software to find the optimal values of x and y that maximize the profit.