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 \(\frac{3x}{1000}\) hours and \(\frac{y}{1000}\) hours respectively, bout only 66 hours are available, so,
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,
[Since production of bottles can not be negative]
Hence mathematical formulation of the given LPP is,
Max Z = 8x + 7y
Subject to constraints,
Region 3x + y ≤ 66000: 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.
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.
