A manufacturer of a line of patent medicines is preparing a production plane on medicines A and B. There are sufficient ingredients available to make 20,000 bottles of A and 40, 000 bottles of B but there are only 45, 000 bottles into which either of the medicines can be put. Further more, it takes 3 hours to prepare enough material to fill 100 bottles of A, it takes on hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. Te profit is Rs. 8 per bottle for A and Rs. 7 per bottle for B. Formulate this problem as a linear programming problem.

Explanation

Let production of every of `A and B` are `x and y `respectively.

Since profits on every bottle of A and B are `Rs. 7` per bottle respectively.
So profit on `x` bottles of `A` and `y` bottles 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 `(3x)/1000 hours`and `y/1000 hours` respectively
but only 66 hours are available, so
`(3x)/1000+y/1000≤66`
or, `3x+y≤66000`.

Since raw material 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≤2000, y≤40000, x+y≤45000 `with `x,y≥0`.

So mathematical formulation of the given LPP is

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