A company that produces and distributes electronic equipment in the Northeast, currently partitions the region into two markets. Each market is served by a single warehouse $($i$.$e$.,$ each customer is assigned to a single market and receives deliveries from the warehouse in that region$).$

The following information is the historical weekly demand at each region$/$market for a typical product: $$Product X$($assume $50$ weeks per year$).$ Note that the ordering cost is $$300$ per order, lead time is always $4$ weeks, carrying cost is $$30$ per unit per year, the desired service level is $95\%,$ and they are utilizing a continuous review inventory system. The company has decided to serve both markets$/$regions by utilizing a single centralized warehouse.

Table $1-$ Weekly Demand Data

Week

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

$9$

$10$

Market $1$

$75$

$62$

$70$

$65$

$58$

$74$

$53$

$77$

$70$

$76$

Market $2$

$78$

$93$

$71$

$98$

$65$

$79$

$66$

$95$

$90$

$80$

Calculate the average demand at the centralized warehouse. $(2$ points$)$

Calculate the standard deviation of the demand at the centralized warehouse. $(4$ points$)$

Calculate the average inventory for $$Product X$$ at the centralized warehouse. Please show all your work! $(8$ points$)$

The company would like to choose one of the two existing regional warehouses as the centralized warehouse. The inbound and outbound transportation costs under the current distribution system are provided in Table $2.$ Finally, Table $3$ provides information about inbound and outbound transportation costs between the existing regional warehouses and the other market areas if a particular regional warehouse were to become the centralized warehouse.