I have a bi-objective MILP (Mixed Integer Linear Programming) problem, where I call the two objective functions as
I would like to simultaneously maximize both of them via the
Augmented Epsilon-Constraint Method.
It will generate a set of Pareto-optimal solutions (called "Pareto Front"), where each of them represents a trade-off between the two objectives.
My question is:
Obj2 (which are conflicting) represent a special case where, for example:
Obj1 = x1 + x2 + x3 Obj2 = - (x1 + x2 + x3) + (x4 + x5 + x6)
Obj2 is equal to the reversed version of
Obj1, plus some other terms,
Obj2 represent even a more special case where
Obj2 is equal to
Obj1 multiplied by
-1, such as:
Obj1 = x1 + x2 + x3 Obj2 = - (x1 + x2 + x3)
So, in both cases, the two objectives are linearly dependent.
Are these special cases not recommended or problematic (for some reasons which come from Operational Research theory) to be solved as a bi-objective MILP via the
Augmented Epsilon-Constraint Method, or are they ok?
PS: In the examples above, for a sake of simplicity, I used two objectives made up of the sum of few variables, while in my case each of them is made up by the sum of many (thousands) terms.