Problem $3$: In Problem $2,$ suppose now that Amazon wants to build $2$ new distribution

centers. The objective is to minimize the total distance that its shipments travel from the

distribution centers to each of the demand points.

$($a$)$ Formulate an NLP model for this problem? $(10$ points$)$

Hint: the objective function should minimize the total distance the shipments travel

to each demand location from the closest distribution center. But how do we find the

closest distribution center? Let$$s think of an example. Miami needs $13$ shipments.

Suppose Miami is $200$ miles from DC $1$ and $1000$ miles from DC $2.$ To calculate the

total distance that the shipments to Miami will travel, we can use the min function as

follows: min$\{200,1000\}\backslash$times $13=2600.$ In other words: min$\{$distance from Miami to DC

$1,$ distance from Miami to DC $2\}\backslash$times number of shipments to Miami. Thus, your objective

function should be in the following form:

min $($min$\{$distance from NY to DC $1,$ distance from NY to DC $2\}\backslash$times number of shipments to NY$)+$

$($min$\{$distance from Boston to DC $1,$ distance from Boston to DC $2\}\backslash$times number of shipments to Boston$)+$

$...$

$($min$\{$distance from Seattle to DC $1,$ distance from Seattle to DC $2\}\backslash$times number of shipments to Seattle$)$