Prob$\mathrm{.}3.$ Solve the following problems with related to the safety stock. $(30$ points$)$

$(1)$ Now we have the stock of $88$ton. The demand follows the normal distribution and the average demand per

month is ton$/$month$]$ and the standard deviation is ton$/$month$].$ Find the probability that the amount of

stock becomes below zero within four months.

$(2)$ There are two regions $($a$)$ and $($b$),$ and the warehouse A and B cover retailers in each reason respectively.

Products are carried to the warehouses once a month. The monthly demand of region $($a$)$ follows the normal

distribution $N(10,2^2)$ and that of $($b$)$ follows $N(10,3^2).N($mean$,$ variance$)$ The demand of region $($a$)$ and $($b$)$

are not independent and the correlation coefficient between them is $0\mathrm{.}50.$ We will make a new

warehouse that covers both region A and $B.$ How much should the new warehouse have safety

stock for the monthly demand? They expect that the risk of stockout is less than $5\mathrm{.}0\%.$

$(3)$ Discuss the advantage of centralized distribution system from the view points of cost and service level. $(10)$