A city is divided in 6 districts and it is planning the location of firefighter stations such that every district is covered. However, not every district needs its own station as one station in a district can serve other districts as long as it is located less than 10 minutes away.
In the next table you can see the requiered time to get from one district to an other.
For example, district 1 can cover districts 1, 3, and 5.
I want to know if I stated the problem in the right way. I want to minimize the number of stations that I use and I want to cover all the districts.
I am using x1, x2, x3, x4, x5, x6 as binary decision variables
My objective function is: Min x1 + x2 + x3 + x4 + x5 + x6
My restrictions are:
1<= x1 + 18x2 + 9x3 + 13x4 + 5x5 + 26x6 <= 10
1 <= 18x1 + 19x3 + 8x4 + 17x5 + 6x6 + x2 <=10
1 <= 9x1 + 19x2 + x3 + 55x4 + 14x5 + 15x6 <= 10 ...(and so on for the time to reach all the stations)
Guide:
We do not care about the actual distance but rather if each region is cover.
For example, to make sure that district $1$ is covered, $$x_1 + x_3+x_5 \ge 1$$
Repeat this for each district.