Q21 of 118 Page 31

A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B, but there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is ₹ 8 per bottle for A and ₹ 7 per bottle for B. How should the manufacturer schedule his production in order to maximize his profit?

Let production of each bottle of A and B are x and y respectively.


Since profits on each bottle of A and B are Rs 8 and Rs 7 per bottle respectively. So, profit on x bottles of A and y bottles of of B are 8x and 7y respectively. Let Z be total profit on bottles so,


Z = 8x + 7y


Since, it takes 3 hours and 1 hour to prepare enough material to fill 1000 bottles of Type A and Type B respectively, so x bottles of A and y bottles of B are preparing is hours and hours respectively, bout only 66 hours are available, so,



3x + y 66000


Since raw materials available to make 2000 bottles of A and 4000 bottles of B but there are 45000 bottles in which either of these medicines can be put so,


x 20000


y 40000


x + y 45000


x,y 0. [ Since production of bottles can not be negative]


Hence mathematical formulation of the given LPP is,


Max Z = 8x + 7y


Subject to constraints,


3x + y 66000


x 20000


y 40000


x + y 45000


x,y 0


Region 3x + y66000: line 3x + y = 66000 meets the axes at A(22000,0), B(0,66000) respectively.


Region containing origin represents 3x + y 10000 as (0,0) satisfy 3x + y 66000


Region x + y 45000: line x + y = 45000 meets the axes at C(45000,0), D(0,45000) respectively.


Region towards the origin will satisfy the inequation as (0,00 satisfies the inequation


Region represented by x 20000,


x = 20000 is the line passes through (20000, 0) and is parallel to the Y - axis. The region towards the origin will satisfy the inequation.


Region represented by y 40000,


y = 40000 is the line passes through (0,40000) and is parallel to the X - axis. The region towards the origin will satisfy the inequation.


Region x,y 0: it represents first quadrant.


21.jpg


The corner points are O(0,0), B(0,40000), G(10500,34500), H(20000,6000), A(20000,0).


The values of Z at these corner points are,



The maximum value of Z is 325500 which is attained at G(10500, 34500).


Thus the maximum profit is Rs 325500 obtained when 10500 bottles of A and 34500 bottles of B are manufactured.


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