$5\mathrm{.}13$ ABC Tools has divided its northeast U$.$S$.$ market into three customer regions. The weekly demands at the regions are normally distributed with mean and standard deviation as shown in Table $5\mathrm{.}18.$ Assume that the regional demands are independent. Currently, the company has three dedicated DCs$,$ one for each demand region and maintains an $80\%$ service level to its customers. The company is studying a new option to have just one central DC to serve all three demand regions.

TABLE $5\mathrm{.}18$

Weekly Demands $($Exercise $5\mathrm{.}13)$

$\backslash$table$[[$Region$,$Mean,$\backslash$table$[[$Standard$],[$Deviation$]]],[1,2000,100],[2,3000,50],[3,1500,50]]$

Given the replenishment lead time from the plant to any DC is $1$ week, answer the following questions:

$($a$)$ Determine the current level of safety stock in the supply chain under the present $($three DC$)$ system.

$($b$)$ Compute the new level of safety stock in the supply chain under the new option $($one central DC$),$ if the company continues to maintain $80\%$ service level to its customers.

$($c$)$ Suppose the company wants to maintain the current level of safety stock found in part $($a$)$ for the new $($one DC$)$ option also. What will be the new Service Level to its customers in that case?