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Let us consider a simple but realistic supply chain scenario. ElectroWorks is a company that manufactures and distributes electronic parts to various regions in the United States. To serve the Chicago$-$Milwaukee region, the company has one warehouse in Milwaukee and another in Chicago. Because the company produces thousands of parts, it is considering an alternative strategy of locating a single, centralized warehouse between the two markets$-$say, in Kenosha, Wisconsin$-$that will serve all customer orders. Delivery time, referred to as lead time, from manufacturing to the warehouse$($s$)$ and ultimately to customers is unaffected by the new strategy.

Consider comparing a decentralized versus a centralized inventory system as it ultimately relates to the amount of safety stock $($extra inventory over and above mean demand$)$ held in the system.

Focus on one specific part: SurgeArrester. The lead time for this part from manufacturing to warehouses is one week. Suppose that the CEO of ElectroWorks requires a $99\%$ customer service level. This means that the probability of satisfying customer demand during the lead time is $0\mathrm{.}99.$ Assume that lead time demands for the Milwaukee warehouse, Chicago warehouse, and centralized warehouse are Normally distributed with the following means $$ and standard deviations $$ :

$x=$ Milwaukee warehouse

$Y=$ Chicago warehouse

Correlation between $x$ and $Y$

$x+Y=$ centralized warehouse

${}_x=415$ units $,{}_x=48$ units

${}_Y=2689$ units ${}_Y=272$ units

$=0\mathrm{.}52$

${}_{x+Y}=3104$ units $,{}_{x+Y}=299\mathrm{.}78$ units

$($Use decimal notation. Use Table A$.$ Round your answer to the nearest integer.$)$