Question $4(20$ pts$)$ Consider a hierarchical facility location model. There are three customer sites, i$.$e$.,$

$N=\{1,2,3\}$ and $3$ potential facility locations, i$.$e$.,M=\{4,5,6\}.$ Locations $4$ and $6$ are candidates for a

type $1$ facility, whereas locations $5$ and $6$ are candidates for a type $2$ facility, i$.$e$.,M1=\{4,6\}$ and $M2=$

$\{5,6\}.$ We would like to open one facility for each type, i$.$e$.,p1=p2=1.$ The number of customers at

sites $1,2,3$ for service types $1$ and $2$ are $500($site $1,$ type $1),2000($site $1,$ type $2),400($site $2,$ type $1),$

$200($site $2,$ type $2),750($site $3,$ type $1)$ and $1000($site $3,$ type $2)$ respectively. We would like to decide

which sites of potential facilities to open considering minimization of total distance between customers

and facility sites. Below is the distance matrix between customers and potential facilities.