$A$ manufacturer produces nuts and bolts. It takes $1\, hour$ of work on machine $A$ and $3\, hours$ on machine $B$ to produce a package of nuts. It takes $3\, hours$ on machine $A$ and $1\, hour$ on machine $B$ to produce a package of bolts. He earns a profit of $Rs.\,17.50$ per package on nuts and $Rs.\,7$ per package on bolts. How many packages of each should be produced each day so as to maximise his profit,if he operates his machines for at the most $12\, hours$ a day?
(B) Let the manufacturer produce $x$ packages of nuts and $y$ packages of bolts.
Therefore,$x \geq 0$ and $y \geq 0$.
The given information can be compiled in a table as follows:
The profit on a package of nuts is $Rs.\,17.50$ and on a package of bolts is $Rs.\,7$. Therefore,the constraints are:
$x + 3y \leq 12$
$3x + y \leq 12$
Total profit,$Z = 17.5x + 7y$.
The mathematical formulation of the given problem is:
Maximise $Z = 17.5x + 7y$ subject to the constraints:
$x + 3y \leq 12$
$3x + y \leq 12$
$x, y \geq 0$
The feasible region determined by the system of constraints has corner points $O(0,0)$,$A(4,0)$,$B(3,3)$,and $C(0,4)$.
The values of $Z$ at these corner points are as follows:
The maximum value of $Z$ is $Rs.\,73.50$ at $(3,3)$.
Thus,$3$ packages of nuts and $3$ packages of bolts should be produced each day to get the maximum profit of $Rs.\,73.50$.
Therefore,$x \geq 0$ and $y \geq 0$.
The given information can be compiled in a table as follows:
| Machine | Nuts | Bolts | Availability |
| Machine $A$ $(h)$ | $1$ | $3$ | $12$ |
| Machine $B$ $(h)$ | $3$ | $1$ | $12$ |
The profit on a package of nuts is $Rs.\,17.50$ and on a package of bolts is $Rs.\,7$. Therefore,the constraints are:
$x + 3y \leq 12$
$3x + y \leq 12$
Total profit,$Z = 17.5x + 7y$.
The mathematical formulation of the given problem is:
Maximise $Z = 17.5x + 7y$ subject to the constraints:
$x + 3y \leq 12$
$3x + y \leq 12$
$x, y \geq 0$
The feasible region determined by the system of constraints has corner points $O(0,0)$,$A(4,0)$,$B(3,3)$,and $C(0,4)$.
The values of $Z$ at these corner points are as follows:
| Corner point | $Z = 17.5x + 7y$ |
| $O(0,0)$ | $0$ |
| $A(4,0)$ | $70$ |
| $B(3,3)$ | $73.5$ (Maximum) |
| $C(0,4)$ | $28$ |
The maximum value of $Z$ is $Rs.\,73.50$ at $(3,3)$.
Thus,$3$ packages of nuts and $3$ packages of bolts should be produced each day to get the maximum profit of $Rs.\,73.50$.
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$A$ manufacturer has three machines $I, II$ and $III$ installed in his factory. Machines $I$ and $II$ are capable of being operated for at most $12 \, hours$ whereas machine $III$ must be operated for at least $5 \, hours$ a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines. The number of hours required for producing $1$ unit of each of $M$ and $N$ on the three machines are given in the following table:
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