When modeling a multi-objective problem, is there a simple way of choosing to fully minimize one function, then to go on and minimize the second?

Question

I am modelling a problem where I have two objectives. My goal is to fully minimize the first objective function, then choose among the solutions that fully minimized the first objective function to minimize the second objective function as much as possible.

Currently I normalize the two functions and weight the first one by 100 and the second one by 1, so that it puts priority on the first function, then minimizes the second one as much as possible. Is there a more elegant/correct way of doing this?

Answer 1

One other popular approach is a lexicographic search:

1. Solve with first objective (e.g. $z_1 = \min f_1(x)$)
2. Add a constraint to the model, so the first objective can not deteriorate (e.g. $f_1(x) \le z_1^*$)
3. Solve with second objective

See for instance here.