Each week, a computer store sells an average of $100$ sets of audio speakers. Weekly demand is normally distributed with a standard deviation of $4$ sets. The store orders speakers from a national distributor. Each order is filled in two weeks. The cost of placing each order is $$16,$ and the weekly cost of holding one set of speaker in inventory is $$2.$ The per$-$unit stockout cost is assumed to be $$4.$ The store is willing to assume that all demand is back$-$logged.

$($a$)(4)$

Determine a good size of order.

$($b$)(8)$

Determine a good reorder point.

$($c$)(3)$ What is the safety stock level?

$($d$)(4)$ If the new order is not always filled in one week, instead, the time for the new order to arrive has a mean of $1$ week and standard deviation of $1$ week. Determine a good size of order.

$($e$)(8)$ Continue part $($d$),$ determine a good reorder point.

$($You have to use the following critical point for standard normal distribution: $z_{0\mathrm{.}5}=0\mathrm{.}0,z_{0\mathrm{.}55}=0\mathrm{.}1,z_{0\mathrm{.}6}=0\mathrm{.}2,z_{0\mathrm{.}65}=0\mathrm{.}3,z_{0\mathrm{.}7}=0\mathrm{.}5,z_{0\mathrm{.}75}=0\mathrm{.}6,z_{0\mathrm{.}8}=0\mathrm{.}8,z_{0\mathrm{.}85}=1\mathrm{.}0,z_{0\mathrm{.}9}=2\mathrm{.}0,z_{0\mathrm{.}95}=3\mathrm{.}0,z_{0\mathrm{.}99}=4\mathrm{.}0,z_{0\mathrm{.}999}=5\mathrm{.}0.$ If you can not find the exact number, please use the closest number as an approximation.$)$