# A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items. A fan costs Rs. 360 and a sewing machine cost Rs. 240. He can sell a fan at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise the profit? Translate the problem as an LPP and solve it graphically.

Toolbox:
• Let $R$ be the feasible region for a linear programming problem and let $z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
Let the no of fans purchased by the dealer be $x$
Let the no of sewing machines purchased by the dealer be $y$
Since the dealer has space for at most 20 item
$\therefore x+y\leq 20$
A fan costs Rs.360 and a sewing machine costs Rs.240
Total cost of $x$ fans and $y$ sewing machines is Rs $(360x+240y)$
But the dealer has only Rs.5760 to invest.
$\therefore 360x+240y\leq 5760$
Since the dealer can sell all the items that he can buy and the profit on a fan is of Rs.22 and on a sewing machine the profit is of Rs18
$\therefore$ Total profit on selling $x$ fans and $y$ sewing machines is of Rs.$(22x+18y)$
Step 2:
Let $Z$ denote the total profit
$\therefore Z=22x+18y$
Clearly $x,y\geq 0$
Thus the mathematical formulation of the given problem is maximize $Z=22x+18y$ subject to :
$x+y \leq 20$
$360x+240y\leq 5760$ and $x\geq 0,y\geq 0$
Step 3:
Let us solve this LPP graphically
Let us draw the line.
$360x+240y=5760$
$x+y=20$
The corner points of the feasible region $OA_2,PB$ are $O(0,0),A_2(16,0),P(8,12)$ and $B_1(0,20)$
Step 4:
Let us obtain the values of the objective function $Z$ at corner points of the feasible region
At the points $(x,y)$ the value of the objective function subjected to $Z=22x+18y$
At $O(0,0)$ the value of the objective function $Z=0$
At $A_2(16,0)$ the value of the objective function $Z=22\times 16+18\times 0=352$
At $P(8,12)$ the value of the objective function $Z=22\times 8+18\times 12=392$
At $A_2(0,20)$ the value of the objective function $Z=22\times 0+20\times 18=360$
Clearly $Z$ is maximum at $x=8$ and $y=12$
The maximum value of $Z$ is 392
Hence the dealer should purchase 8fans and 12 sewing machines to obtain the maximum profit.