Section B

$(1)$ A retail shop sells a type of electronic fan. The unit cost of an electronic fan is $100$ dollars. The monthly demand for the electronic fans follows a normal distribution with mean $800$ units and standard deviation of $200$ units. The lead time for order delivery is $1$ month. The cost of placing an order is $500$ dollars. The annual inventory holding cost rate is $25\%.$

$($Note: conversion between time units $1$ year $=12$ months $=52$ weeks; the $z-$value of a standard normal distribution corresponding to $0\mathrm{.}99$ is $2\mathrm{.}33).$

$($a$)$ The shop is using a $(Q,R)$ policy for inventory management wherein they order every time inventory drops below re$-$order point. Design the $(Q,R)$ policy for the company.

$($b$)$ Suppose the shop wants to switch to a base$-$stock policy, and order once every month, at the beginning of the month. Under the base stock policy, what would be the order$-$up$-$to level required to achieve a service level of $99\%?$