Question $2.$ Sheldon Town is divided into $2$ districts: People Hill and Everythingelse Heights.

From the census data, we know,

IEOR $162$

Homework $10$

Linear Programming & Network Flows $-$ Page $2$

Recently, the mayor and the city board of directors are considering to place public clinics

throughout the city. After extensive land survey and many long$-$hour meetings, it is decided to

limit the potential locations of clinics to $4$ sites. The distance from the center of each district

to each site is listed:

To ensure fairness, every district should be assigned to exactly one clinic.

$($a$)$ Formulate an integer programming problem to decide which sites to be used for construc$-$

tions of clinics and what districts are assigned to which clinics so that the distance traveled

to the district farthest from its assigned clinics is minimized.

$($b$)$ Suppose we impose a requirement: either site $1$ and $2$ or site $3$ and $4$ must be used. Modify

the previous formulation to add this requirement.

$($c$)$ Suppose it costs $j**s_j$ units of money to build a clinic at site $j,$ where $j=1,2,3,4$ to service

$s_j$ people, where $s_j$ is the total population serviced by a clinic built at site $j.$ The city has

allocated $100$ dollars for our clinic$-$building project. Modify the previous formulation to add

this requirement $($there might not be a solution, but by solving our integer programming

formulation, we can see that easily$).$