Q13 of 118 Page 31

A firm manufacturing two type of electric items, A and B, can make a profit of 20 per unit of A and ₹ 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programming problem for maximum profit and solve it graphically.

Let x units of item A and y units of item B were manufactured.


Numbers of items cannot be negative. Therefore,


x, y 0


The given information can be tabulated as follows:



Further, it is given that type B is an export model, whose supply is restricted to 65 units per month.


Therefore, the constraints are


3x + 2y 210


4x + 4y 300


y 65


A and B can make profit of Rs 20 and Rs 30 per unit respectively.


Therefore, profit gained from x units of item A and y units of item B is Rs 20x and 30y respectively.


Total Profit = Z = 20x + 30y which according to question is to be maximised.


Thus the mathematical formulation of the given LPP is,


Max Z = 20x + 30y


Subject to constraints


3x + 2y 210


4x + 4y 300


y 65


x, y 0


Region represented by 3x + 2y 210: The line 3x + 2y = 210 meets the axes at A(70,0), B(0,105) respectively.


Region containing the origin represents 3x + 2y 210 as origin satisfies 3x + 2y 210.


Region represented by 4x + 4y 300: The line 4x + 4y = 300 meets the axes at C(75,0), D(0,75) respectively.


Region containing the origin represents 4x + 4y 300 as origin satisfies 4x + 4y 300


y = 65 is the line passing through the point E(0,65) and is parallel to X - axis.


Region x,y 0: it represents the first quadrant.


13.jpg


The corner points are O(0,0), E(0,65), G(10,65), F(60,15) and A(70,0).


The values of Z at these corner points are as follows:



The maximum value of Z is 2150 which is attained at G(10,65).


Thus, the maximum profit is Rs. 2150 obtained when 10 units of item A and 65 units of item B are manufactured.


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