Mixed integer linear programming is a class of problems rather than an algorithm. It consists of all the problems, which boil down to maximizing the cost function, which is linear and has integer values. These assumptions make it easier to create algorithms that solve this very specific problem, and I think this is what you are talking about in the MILP Approach. Implementation can vary greatly, as optimization for specific tasks can be applied on top of a good overall solution.
it is more difficult to determine, because all algorithms related to graph theory do not make it clear that they use a graph, but some nontrivial graph theorems may be required to prove correctness or optimality.
Metaheuristics is a class of algorithms that intend to expand heuristics. Heuristics is a “practical” approach to a problem that does not guarantee optimality, but this is sufficient for the immediate purpose. Metaheuristics raises the level of abstraction one step higher: instead of directly discussing the problem, you will build solutions to the problem (for example, people in GA) and talk about them (i.e., Developing your population in GA).
Route optimization can fall into any of the three categories, you need to be more precise before I can answer correctly, but here are a few examples:
Problems with stream maximization: linear programming.
ex: each route of your network can be used by no more than k trucks, and you want to take your sand from point A to point B in the minimum time, how many trucks can you send along each route? One path can be split into two more limited paths or shrink and allow your trucks to get stuck in the middle of the path or even merge, etc. (Note how it is still based on the graph)
The shortest path, the longest path, number of paths: Pure Graph.
First, a search by depth and width (I suppose you already know this) can solve a huge number of graph-based problems without the need for a more complex approach. A *, for example, is an "only" extended version of DFS. When optimizing a route, you most likely have a route network, presented in a graph, so this can be a good starting point.
Problem with the seller: Meta heuristic
TSP basically finds a path that visits all cities exactly once. It is much more complicated than it looks (NP-complete if it rings). Metaheuristics are very effective here, since an effective solution is not known. Genetic algorithms, Ant-colonial optimization and simulated annealing all give good results if implemented correctly. Iterative local search, as you pointed out, can be used, for example, for local optimizations based on individual sides for each round of global optimization, which gives better results.
I'm sorry that my three examples fall into graphics-related problems, but also show that graphics can help solve an incredible amount of problems, even if the graph-graph is not explicitly stated in the problem statement.
All three are also problems of route optimization, it all depends on what kind of optimization you are looking for. Your problem can be solved using one of these three methods, or maybe by combining the two (local LP optimization for metaheuristics, for example).