Q19 of 118 Page 31

A manufacturer makes two products A and B. Product A sells at 200 each and takes 1/2 hour to make. Product A sells at ₹ 300 each and takes 1 hours to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is ₹ 20 and on product B is Rs 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.

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


Number of units cannot be negative.


Therefore, x,y 0.


According to question, the given information can be tabulated as:



Also, the availability of time is 40 hours and the revenue should be atleast Rs 10000.


Further, it is given that there is a permanent order for 14 units of Product A and 16 units of product B.


Therefore, the constraints are,


200x + 300y 10000,


0.5x + y 40


x 14


y 16.


If the profit on each of product A is Rs 20 and on product B is Rs 30. Therefore, profit gained on x units of product A and y units of product B is Rs 20x and Rs 30y respectively.


Total profit = 20x + 30y which is to be maximized.


Thus, the mathematical formulation of the given LPP is,


Max Z = 20x + 30y


Subject to constraints,


200x + 300y 10000,


0.5x + y 40


x 14


y 16


x,y 0.


Region 200x + 300y 10000: line 200x + 300y = 10000 meets the axes at A(50,0), B(0,) respectively.


Region not containing origin represents 200x + 300y 10000 as (0,0) does not satisfy 200x + 300y 10000.


Region 0.5x + y 40: line 0.5x + y = 40 meets the axes at C(80,0), D(0,40) respectively.


Region containing origin represents 0.5x + y 40 as (0,0) satisfies 0.5x + y 40.


Region represented by x 14,


x = 14 is the line passes through (14,0) and is parallel to the Y - axis. The region to the right of the line x = 14 will satisfy the inequation.


Region represented by y 16,


y = 14 is the line passes through (16,0) and is parallel to the X - axis. The region to the right of the line y = 14 will satisfy the inequation.


Region x,y 0: it represents first quadrant.


19.jpg


The corner points of the feasible region are E(26,16), F(48,16), G(14,33), H(14,24).


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



The maximum value of Z is Rs 1440 which is attained at F(48,16).


Thus, the maximum profit is Rs 1440 obtained when 48 units of product A and 16 units of product B are manufactured.


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