Problem $1.$

A county in Florida has decided to build emergency shelters to protect its citizens in case of disasters. The county has been divided into $15$ areas, each of which needs to be close enough to a shelter. After a long debate, it has been decided that the 'close$-$enough' threshold is $35$ miles, measured from the geometric center of an area to the shelter. Eight sites have been selected as candidates to place a shelter, each having a difference cost.

The county wants to determine in which of the eight sites a shelter must be placed, so that all areas are close enough to a shelter, with minimum total cost.

For each area $(1$ up to $15),$ the distance to each possible shelter location $(A,B,$dots,$H$ is provided, measured from the area's center. Also, given is the cost of placing a shelter for each of the possible sites.

We will formulate this problem as an integer linear program.

Formulate the integer linear program for this problem.

Suppose that after examining the emergency site locations, the city council identified an issue and stated the following additional requirement: an emergency center in site $B$ can be built only if there is an emergency center in site A and there is no emergency center in site $C.$

$($a$)$ Describe how you will modify your model to consider this new restriction by explicitly listing your new linear constraints along with any other changes you will make to the existing ones. If needed, you can define additional parameters or decision variables here.

$($b$)$ Update your Excel implementation to take into account this new restriction, solve it to optimality, and generate the Answer Report. What is the optimal objective value? Which sites are selected to have emergency shelters?

Emergency Shelters

Maximum distance: $$

$($c$)$