Question

A manufacturer of patent medicines is preparing a production plan on medicines $A$ and $B$. There are sufficient row material available to make $20000$ bottles of $A$ and $40000$ bottles of $B$, but there are $45000$ 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 Rs. $8$ per bottle for $A$ and Rs. $7$ per bottle for $B$. Construct the maximization problem.

Answer 1

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

Since profits on each 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 $\frac{3x}{1000}$ hours and $\frac{y}{1000}$ hours respectively, but only $66$ hours are available, so

$\frac{3x}{1000}+\frac{y}{1000}\le 66$

or, $3x+y\le 66000$.

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,

$x\le 2000$, $y\le 40000$, $x+y\le 45000$ with $x,y\ge 0$.

So mathematical formulation of the given LPP is

Max $Z=8x+7y$

Subject to $3x+y\le 66000$, $x\le 2000$, $y\le 40000$, $x+y\le 45000$ with $x,y\ge 0$.