I'm studying a simple max flow problem:
Each type of object $a_1, a_2...$ can be stored in some of several stores $b_1,b_2...$. This is described by this graph:
There are $|a_i|$ objects of the type $a_i$. Store $b_j$ can contain at most $|b_j|$ objects. This shows as capacity constraints on the graph. The other edges have no capacity constraint and just mean "this type of object can be stored in this store". We want to maximize the total number $N$ of objects being stored. Max flow on this graph answers to this question.
Now we add a priority idea. Each type of object $a_i$ has an integer priority $p(a_i)$ say from $1$ to $2$ for simplicity. Instead of maximizing the total number of objects, we want to maximize $(N_1,N_2)$ in lexicographical order where $N_p$ stands for the number of objects with priority $p$ being stored.
Do you see a way to formalize this problem as a flow problem (possibly with a broader meaning than just simple max flow)? Or a way to use max flow algorithms to solve it?