Problem $1(55$ marks$)$: Find the optimal ordering policy for the stochastic single$-$period

model with a setup cost. $In$ this model, demand $is$ fulfilled $at$ the beginning $of$ the period, and

$it$ follows the probability density function

$f(D)=\{\begin{array}{ccc}\frac{1}{20}& \text{f}\text{o}\text{r}& 0D20\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

The costs are

Holding cost $=$$$1$ per item,

Shortage cost $=$$$3$ per item,

Purchase price $=$$$2$ per order,

Setup cost $=$$$1\mathrm{.}5$ per item.