$1-F(R)=\frac{Qh}{p},R_0={}^{**}z+,n^{**}={}^{**}L(z)$

$n(R)={\displaystyle \underset{R}{\overset{}{}}}(x-R)f(x)dx,Q=\sqrt[\phantom{2}]{\frac{2[K+pn(R)]}{h}}$

Chester is a printing company, which sells printers and cartridges to offices. The cost of one cartridge is $$50$ and requires $3$ months of lead time to be in stock. The company expects to sell $450$ cartridges during the lead time but there is a deviation from three$-$month period to the next, which is $30$ units of cartridge. The interest rate is calculated as $20$ per cent annually for holding costs. Total cost for one order is $$300.$ The shortage penalty is estimated at $$32$ per unit. Assuming that the demand is distributed as normal, find the optimal $(Q,R)$ values.