I’m looking for a little help on a new problem. I’m in a linear programming class and trying to work on a project exploring methods on nonlinear optimization and I came across the following question in a book:
Minimize the total distance traveled by students from $P$ subdivisions $N_j$ to determine the locations $(a,b)$ and $(c,d)$ of high schools $A$ and $B$
minimize $$\sum_{j=1}^P w_{1j}\left((a-x_j)^2+(b-y_j)^2)^{1/2}+w_{2j}((c-x_j)^2+(d-y_j)^2\right)^{1/2}$$
Subject to $$\sum_{j} w_{ij}\le c_i$$
$$w_{1j}+w_{2j}=r_j$$
where $w_{1,j}$ is the number of students going to school $A$ and $w_{2,j}$ is the number going to school $B$, the capacity of school $A$ is $c_1$ and the capacity of school $B$ is $c_2$, and the total number of students in each subdivision is $r_j$, and the location of each subdivision is a single point $(x_i, y_i)$.
I understand the objective function and I think I could figure out how to solve the problem, but I’m a little confused about the variables. It seems clear to me that $w_{1,j}$, $w_{2,j}$ are the variables, but if we are ultimately solving for the locations of schools $(a,b)$ and $(c,d)$, how will I end up finding $a,b,c,d$ if I’m solving for the number of students from each subdivision? In other words, where do $a,b,c,d$ come from if they are not in the constraints?
I attempted to make up some schools and some numbers, trying to reverse engineer it a little but I'm stuck. The reason I'm interested in this problem is because the school district in my home town is going through the process of drawing new boundaries and I have access to some relevant data and I thought it would be interesting. Thanks in advance.