In this problem, we will consider the $(Q,R)$ model. Specifically, we will be looking at the version of the model with total expected cost minimization $($using the approximate expected cost expression derived in class$).$ For each of the following problem instances in parts $($a$),($c$),$ and $($d$),$ use a programming language of your choice to solve for the optimal $Q$ and $R$ using the iterative technique demonstrated in class. Perform as many iterations as needed until the absolute change in $Q$ between consecutive iterations is less than $0\mathrm{.}1$ units. Along with your final answer, report your value of $Q$ and $R$ at the end of each iteration.

a$.)$ The lead time is one week. The weekly demand is distributed normally with a mean of $200$ units and a standard deviation of $40$ units. The holding cost is $h=$$$0\mathrm{.}05$ per unit per week, the backorder cost is $b=$$$20$ per unit, and the fixed order cost is $K=$$$50$ per order.

b$.)$ What is the Type I service level associated with your answer to part $($a$),$ as a percentage?

c$.)$ The lead time is one week. The weekly demand is distributed normally with a mean of $300$ units and a standard deviation of $85$ units. The holding cost is $h=$$$0\mathrm{.}08$ per unit per week, the backorder cost is $b=$$$25$ per unit, and the fixed order cost is $K=$$$75$ per order.

d$.)$ All parameters identical to those in part $($c$),$ except that $b=15.$

e$.)$ Based on your answers to $($c$)$ and $($d$),$ what change do you observe in $R$ as the per$-$unit backorder cost decreases? Why do you think that this is the case?