Please help me with the attached question on operations research.
Let's call X1, X2, ..., X12 the decision variables. These variables will be binary and interpreted as follows. If Xi = 1, it would mean that there is a fire station in neighborhood i. If Xi = 0, it would mean that there is no fire station in neighborhood i. Clearly, then, the objective of this problem is to minimize the sum of all Xi's, as this would mean we are minimizing the total number of fire stations.
Now, let's call C1, C2, ..., C12 to the 'coverage' of the neighborhoods. Specifically, Ci represents the number of fire stations that can reach neighborhood i. The 'coverage' for each neighborhood will be the sum of the fire stations in that neighborhood plus in all adjacent neighborhoods. For example, we can see that:
C1 = X1 + X2 + X4 + X5 + X6
The solution includes a full setup of the linear programming problem (with objective function and constraints) that minimizes the number of fire stations to build.