Question $2.(10$ points$)$ Before the Springfield Motors factory in Springfield began

manufacturing the Homer, they were shipped in from Ogdenville. While the cars are

now assembled locally, some key components still make their journey from Ogdenville,

but with a twist$$they arrive via the famed Monorail system. This innovative transport

method shortens the travel time to just $3$ days. Particularly, the crucial battery

component, essential for the car's operation, is sourced from Ogdenville and delivered

right to Springfield's doorstep by the Monorail.

Each Homer requires a single battery pack, which is composed of $192$ cells. The cost

for a single cell$$s materials is $$6,$ with an associated carrying cost of $35\%$ per year.

When placing an order for battery cells from Ogdenville, the cost is estimated at

$$11,500,$ which is the delivery fee of the monorail. $($Yes$,$ delivery by monorail is that

expensive $$ have you ever had delivery by monorail before? Didn$$t think so$).$

Now, unlike question $1,$ assume that demand for the next year is somewhat uncertain.

In other words, assume that the demand for Homers is thought to follow a normal

distribution and that you can use the past $12$ months of sales data to estimate the

average and variability in demand. For the sake of simplicity, assume that each of the

months has the same number of days$30$ of them$$thus making a year consist of $360$

days.

Springfield Motors aims for a $95\%$ service level for their assembly plant to prevent any

disruptions due to missing parts, as this would bring production to a standstill.

Based on this information, please answer the questions below.

$($a$)$ What is the optimal order quantity for battery cells from Ogdenville?

$($b$)$ What is the approximate time between orders from Ogdenville $($in days$)?$

$($c$)$ What is the safety stock of battery cell inventory that Springfield Motors should

be aiming for?

$($d$)$ What is the total annual inventory cost for Springfield Motors? $($Expressed as the

sum of the purchasing costs, the holding costs, and the ordering costs.$)$

$($e$)$ What is the reorder point? In other words, when the number of battery cells at

Springfield Motors falls below this number, it makes a new order from

Ogdenville.

$($f$)$ Now, assume that the unit price of a battery cell can be decreased from its

original price of $$6$ to $$5\mathrm{.}80$ if at least $200,000$ units are ordered within a single

order, and it can be even further reduced to $$5\mathrm{.}65$ if at least $300,000$ units are

ordered. This price reduction applies to all the units being purchased in a single

order. Is there a new optimal order quantity? Please show all the details of your

calculation.