$A$ company makes $3$ models of calculators: $A, B$ and $C$ at factory $I$ and factory $II.$ The company has orders for at least $6400$ calculators of model $A, 4000$ calculators of model $B$ and $4800$ calculators of model $C.$ At factory $I, 50$ calculators of model $A, 50$ of model $B$ and $30$ of model $C$ are made every day; at factory $II, 40$ calculators of model $A, 20$ of model $B$ and $40$ of model $C$ are made every day. It costs $Rs. 12000$ and $Rs. 15000$ each day to operate factory $I$ and $II,$ respectively. Find the number of days each factory should operate to minimize the operating costs and still meet the demand.

(B) Let factory $I$ operate for $x$ days and factory $II$ operate for $y$ days.
At factory $I, 50$ calculators of model $A$ and at factory $II, 40$ calculators of model $A$ are made every day. The company has orders for at least $6400$ calculators of model $A$.
$\therefore 50x + 40y \geq 6400 \Rightarrow 5x + 4y \geq 640$
At factory $I, 50$ calculators of model $B$ and at factory $II, 20$ calculators of model $B$ are made every day. The company has orders for at least $4000$ calculators of model $B$.
$\therefore 50x + 20y \geq 4000 \Rightarrow 5x + 2y \geq 400$
At factory $I, 30$ calculators of model $C$ and at factory $II, 40$ calculators of model $C$ are made every day. The company has orders for at least $4800$ calculators of model $C$.
$\therefore 30x + 40y \geq 4800 \Rightarrow 3x + 4y \geq 480$
Also,$x \geq 0, y \geq 0.$
We have to minimize the cost $Z = 12000x + 15000y$ subject to the constraints:
$5x + 4y \geq 640$
$5x + 2y \geq 400$
$3x + 4y \geq 480$
$x, y \geq 0$
The feasible region is unbounded with corner points $A(160, 0), B(80, 60), C(32, 120),$ and $D(0, 200).$

Corner points	Value of $Z = 12000x + 15000y$
$(160, 0)$	$1920000$
$(80, 60)$	$1860000$ (Minimum)
$(32, 120)$	$2184000$
$(0, 200)$	$3000000$

To verify the minimum,we plot $12000x + 15000y < 1860000$ or $4x + 5y < 620.$ As shown in the figure,there are no common points with the feasible region,so the minimum value is $1860000.$
Thus,factory $I$ should operate for $80$ days and factory $II$ should operate for $60$ days.