It is a very sensible definition of a set of points that are natural candidates to choose between in a multi-objective optimization problem.
In brief it is defined as the set of vectors, say x, in your problem for which the property holds that if you were to change the solution vector only so slightly in order for ONE of the objectives to become better, then the value of at least one other objective would be worse. That kind of move is not sensible to make - hence we refer to such a solution as Pareto optimal.
There are algorithms that support the search for such points; whether they are successful depend on certain factors, like the continuity of the functions and of the properties of the set of possible (feasible) choices of the vectors x.
A fairly basic discussion on the topic is found here:
https://en.wikipedia.org/wiki/Pareto_efficiency
It is not to be considered recent or state-of-the-art - on the contrary - but it fleshes out some of the ideas I mentioned, and a few more, in quite a bit of detail.
Good Luck!
Thnaks,
If I have two objective functions then can we solve mathematical model by assigning equal weightage to both objective functions? or Pareto optimal is necessary for it?
On your second question:
Assigning 0.50 weight to both of the objectives would lead to a formulation of a single objective problem and that is not what you would like to do, I suppose. You have to understand the nature of a multi-objective problem, then everything will be clear. Maybe an example would be better for this:
Imagine a beam which is fixed in one end and free in the other end (cantilever). You apply a force at the free end so you have bending and deformation. Assume that you want to minimize both the depth of the beam (or height of the cross section) and the deformation. It is obvious that if you have a low depth (less material) then the deformation would be high, while if you want to end up with a small deformation then the depth has to be high. You cannot have simultaneously both low deformation and low depth, it just cannot be done. This is the notion of "conflicting criteria" which are essential for a multi-objective optimization problem.
If one criterion improves (deformation), then the other deteriorates (depth of beam) and vice versa. Now if you want to get all the pairs of (depth, deformation) that are optimal in the sense that you cannot find another pair that improves both of them simultaneously, that is the Pareto front! So the Pareto front consists of solutions that are optimal in the sense that you cannot find other solutions that improve BOTH the criteria simultaneously, under the given contraints.
It depends on the designer to pick up a single solution from the pareto front curve.
According to the linear weighting method, you can approach the pareto front curve by assigning different weights to the two criteria. Note that even if in theory this can be done, there are problems in this method as the various points may not be well distributed along the pareto front curve. Also, normalization of the two objective functions has to be done first in order to obtain any meaningful results. If deformation =0.001 m and depth of beam=0.50 m, you cannot just assign weights to these two numbers and add them, as 0.50>>0.001.
Vagelis Plevris, Thanks for your kind suggestions, Can you please suggest me software for Pareto ?