White & Becker, a manufacturer of hand$-$held vacuum cleaners for the consumer market has a problem. Its two biggest customers are an Automotive Store and a family$-$owned store selling household appliances, both of which are fiercely competitive. They both want exclusive products so that the consumers cannot make price comparisons of their products. For example, White & Becker sells the same exact car$-$vacuum to each customer but must use packaging customized for each retailer.

Assume the weekly demand for the vacuum cleaner to each retailer is normal with mean $5,000$ and standard deviation $1,200.$ White & Becker's cost of doing a setup to produce a new lot is $$5,000,$ and the cost of holding a vacuum cleaner in stock for a year is $$52$; each lot takes a week to produce.

Assume the company fills retailer orders from stock. As the two retailers are quite important to White & Becker, it sets a target service level $($type I$)$ of $99\mathrm{.}7\%.$

$($a$)$ What is White & Becker's average stock of each of the two versions of this vacuum cleaner?

$(10$ marks$)$

$($b$)$ A consultant for the company has suggested that they stock these vacuum cleaners without putting them into their specialized packaging, which distinguishes the product for the two customers. Once orders are received from the two retailers, the company can fulfill these orders from the same stock, since it does not take much time to package each vacuum cleaner. Demand at the two retailers is quite independent as they operate in different geographical markets. By how much $($in percentage$)$ would this new system reduce White & Becker's stock of vacuum cleaners?

$(10$ marks$)$

$($Total $20$ marks$)$