Problem $2(10$ points$)$

Consider the following network of $5$ cities A$,$ B$,$ C$,$ D$,$ and E $($see graph$).$ Each city is

associated with a demand of a product indicated by the first number in the box

next to a city. For example, city A has a demand of $100$ units of the product. An

edge between two cities indicate that it is possible to go from each city to the

other directly. For example, it is possible to go from city A to city B directly and

vice versa, but it is not possible to go from city A directly to city $E.$ The number

next to an edge indicates the distance between the two cities at the endpoints of

the edge e$.$g$.$ there is a distance of $14$ between cities A and B$.$

The problem is to locate a warehouse at three cities. A warehouse can meet

$($cover$)$ the demand of the city it is located in as well as cover the demand of all

cities less than or equal to $15$ units of distance from the city. For example, if a

warehouse is located at city A then the warehouse can cover the demand at cities

$A,B,$ and C for a total coverage of $100+250+200=500$ units of demand.

Formulate the problem of locating $3$ warehouses so that demand coverage is

maximized as an integer program. Be sure to clearly define all variables.