Weekly demand for DVDs at a retailer is normally distributed with a mean of $1000$ boxes and a standard deviation of $150.$ Currently, the store places orders via paper that is faxed to the supplier. Assume $50$ working weeks in a year and the following data:

$-$Lead time for delivery of an order is $4$ weeks.

$-$Fixed cost $($ordering and transportation$)$ is $$100$ per order

$-$Each box of DVDs costs $$1$

$-$The retailer currently orders $20,000$ DVD boxes when stock on hand reaches $4,200.$

$-$Cost of capital is $25\%$

a$)$ Currently how long, on average does a DVD spend in the store? What is the annual ordering and holding cost under such a policy?

b$)$ Assuming that the retailer wants the probability of stocking out in a cycle to be no more than $5\%,$ recommend an optimal inventory policy $($a policy regarding order quantity and safety stock$).$ Under your recommended policy, how long on average would a box of DVDs spend in the store?

c$)$ Claiming that it will lower lead time to $1$ week, the supplier is trying to push an EDI $($Electronic Data Interchange$)$ system on the retailer. In terms of costs and flow times, what benefits can the retailer expect to realize by adopting the EDI system? Explain why.