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.