Consider a continuous review inventory model, where the p$.$d$.$f of the demand during lead time follows a uniform distribution over the range of $0$ to $100.$ The annual demand is $600$ units and the cost of each unit is $$20.$ The inventory carrying cost is $$2$ per unit per year and the order cost is $$20$ per order. Based on experience, it is found that a re$-$order level of $80$ units could be optimal. A$)$ What are the optimum order quantity and the implied shortage cost? Assume that shortages are backordered.

b$)$ Solve problem $1$ assuming that shortages result in lost sales and lost sales double the implied shortage cost.

Use the attached equations: the expected shortage quantity per cycle $\frac{?}{\text{b}\text{a}\text{r}}(S)(x)$ is

$\frac{?}{\text{b}\text{a}\text{r}}(S)(x)={\displaystyle \underset{0}{\overset{}{}}}S(x)f(x)dx={\displaystyle \underset{r}{\overset{}{}}}(x-r)f(x)dx$

$Q^{**}=\sqrt[\phantom{2}]{\frac{2D[A+(\frac{?}{\text{b}\text{a}\text{r}}(S))(x)]}{h}}$

${\displaystyle \underset{r^{**}}{\overset{}{}}}f(x)dx=\frac{h{Q}^{**}}{D}$