$(30$ points$)$ Consider a product for which the random demand $D_{i}is$ zero, one, $or$ two

per week. Past data indicates that $P(D_i=0)=\frac{1}{5},P(D_i=1)=\frac{3}{5},$ and

$P(D_i=2)=\frac{1}{5}.$ The unit procurement cost $is$ $$80$ and the unit selling price $is$

$$100.$ The unit loss$-of-$goodwill cost $is$ $$15(per$ week $in$ the case $of$ backordering$)$ and

the unit holding cost $is$ $$5$ per week. The replenishment orders can $be$ placed $(for$ any

number $of$ items$)at$ the beginning $of$ each week. The objective $isto$ minimize the

average cost per week over $an$ infinite horizon. Calculate the optimal order$-up-to$ level

$in$ each $of$ the following settings:

$(a)(10$ points$)$ All excess demand $is$ lost and there $isno$ replenishment lead time.

$(b)(10$ points$)$ All excess demand $is$ backordered and there $isno$ replenishment lead

time.

$(c)(10$ points$)$ All excess demand $is$ backordered and the replenishment lead time $is$

one week.