$(30$ points$)We$ are given the following information for a product:

Order cost $=$$$50$

Annual demand $=N(960,55\mathrm{.}43)$

Annual holding cost $=$$$\frac{6}{?}$ item $\frac{?}{?}$ year

Shortage cost $=$$$80$ per unit

Lead time $=$ one month

Sales price $=$$$40$ per unit

Product cost $=$$$30$ per unit

$(a)(8$ points$)$ Determine the order quantity and the reorder point under the assumption that all

demands are backordered.

$(b)(8$ points$)$ Determine the order quantity and reorder point under the lost sales assumption.

$(c)(7$ points$)$ What $is$ the maximum $f(x)$ value for the lead time demand? For which $xf(x)=\frac{e^{-\frac{(x-{)}^2}{2{}^2}}}{\sqrt[\phantom{2}]{2{}^2}}is$ the probability density function $(pdf)of$ Normal

distribution.$SL{M}_2$ value for the solution you obtained for backorder case. For the

stockout probability $of$ a cycle, show which area under $pdf$ and which point $oncdf$

corresponds $to,$ and explain the relation between them.