Risk Pooling $(20$ points$)$

A mail$-$order firm has four regional warehouses. Weekly demand at each warehouse

is normally distributed with a mean of $10,000$ units and a standard deviation of $2,000$

units. The company purchases each unit of product at $$10.$ Annual holding cost of

one unit of product is $25\%$ of its value. Each order incurs an ordering cost of $$1,000$

$($primarily from fixed transportation costs$),$ and lead time is $1$ week. The company

wants the probability of stocking out during the lead time at each warehouse to be

no more than $5\%.$ Assume $50$ working weeks in a year.

$($a$)(2$ points$)$ What is the optimal order quantity for one single warehouse?

$($