The difficulty when dealing with multiple objectives is to find a simple way to order the solutions found by the algorithm during the optimization process. This is generally done by introducing dominance measures.

Dominance is the central concept of multiobjective optimization. The most used is the Pareto dominance, but many variations can be found as well (epsilon Pareto dominance, lexicographic optimality, Geoffrion dominance...).

Additionally, algorithms refine the dominance measures by considering also the position of a particular with respect to other solutions (density, position on the Pareto front, see the NSGA algorithm)

Hope it helps,

Pierre

@ Pierre, thanks for the info.

You can use scalarization methods to transform multiple objective to single objective. For example weighted-sum scalarization, Tchebycheff Scalarization function, conic scalarization, epsilon constraint scalarization etc. I hope this information useful for you.

Best regards

Ozden 

@ Ozden, thanks for the info.

Dear Rajesh, no need to write algorithm just read and understand some work  from Prof. K Deb and some other articles free for dowload at :

http://www.ijsmdo.org/articles/smdo/abs/2007/01/smdo0703/smdo0703.html

http://www.ijsmdo.org/articles/smdo/abs/2008/02/smdo2108/smdo2108.html

http://www.ijsmdo.org/articles/smdo/abs/2009/04/smdo2009020/smdo2009020.html

Multi objectif should be deal different than mono objectif. and no need to focus on unique solution better front of pareto

@ David, thanks for the info.

Multiobjective optimization has been, to my knowlegde, often investigated in finite and infinite dimensional optimization using NLP methods. If you are interested in the whole Pareto front --- if the dimension is not too high  --- I would recommend to look through the papers of Michael Dellnitz, University of Paderborn, Germany; see http://www2.math.uni-paderborn.de/ags/ag-dellnitz/people/members/prof-dr-michael-dellnitz.html. His works has often impressed me.

If you are interested in a "practical" solution for a specific problem, you can try to adopt the following procedure, which I have once applied for an multiobjective optimal control problem:

1) Compute the optimal solution for each objective.

2) Then decide the priority of the objectives and chosen appropriate scalars for a scalarization of the multiobjective problem by establishing a convex combination of all objectives.

This gives you only ONE point of the Pareto front. By changing of the parameters of the convex combination you will be able to compute a few Pareto points very easily.

Alternatively, you can proceed with another second step:

2) Add to the minimal values of each objective an acceptable margin and formulate inequality constraints which bound each of the objectives to be minimized by their minimal value plus these individual acceptable margins.

3) Add another margin to these bounds and minimize this SINGLE parameter subject to the constraints of the problems and these artificially introduced inequality constraints as described above.

This gives you in some sense the "best" Pareto point from a practical view.

However note that the necessary optimality conditions for both approaches are equivalent. Each set of scalars is associated to a set of margins.

I hope that these ideas can also be applied to genetic algorithms and will help you.

By the way, are you sure that sophisticated optimization methods wouldn't be the better alternative, at least for a postprocessing with GA after knowing good initial guesses?

Hans Josef

@ Hans , thanks for the update.

Dear Rajesh,

sure. Please let me know if this procedures also works in your case. You also can send me an email privately. You can easily google me.

Good luck,

Hans Josef

Try to use Matlab toolbox for mulobj command