I work on a multi-objective problem and I am preparing a Pareto front. Among the solutions I want to select the best solution. I know there are many methods such as min-max or fuzzy methods. Does anyone know a better method to find the best solution? Or does anyone know any report or paper that compares the different methods in this context?
I would suggest you using Evolutionary Algorithms. Please, check the next paper.
Regards.
http://ai2-s2-pdfs.s3.amazonaws.com/34b4/adafb5b55140fc03b95806b6d4ae195ea9b3.pdf
My teacher (in case study: optimization methods) said Pareto GA is the best method. It has a multi-objective function for the fitness function. Of course, several evolutionary algorithms are good. Search tne new methods in the optimization journals (mathematic journals are good). Their papers give new ideas for your work and you can compare the new algorithms with the prior ones.
I suggest to use NSGA-II first. There are many multi-objective evolutionary algorithms (e.g. MOGA, MOPSO,...). The success of the algorithm depends heavily on the nature of the problem. So you should try using more than one.
For decision making , you can use level diagram(LD)tool :
http://www.mathworks.com/matlabcentral/fileexchange/24042-leveldiagrams-for-multiobjective-decision-making
we used NSGA-II & LD in this paper:
MOHARAM, A., EL-HOSSEINI, M. A., ALI, H. A., Design of Optimal PID Controller Using NSGA-II Algorithm and Level Diagram, Studies Informatics and Control
In this paper; NSGA-II is compared against four optimization methods (i.e. Box method , Random search algorithm, MO genetic algorithm, MO Dragonfly algorithm(MODA))
http://www.mathworks.com/matlabcentral/fileexchange/24042-leveldiagrams-for-multiobjective-decision-making
Thank you very much for your answers. It seems that Evolutionary Algorithms are the best methods for finding the best solution. Among different Evolutionary Algorithms such as NSGA-II, I will check which one suits my problem best.
Dear Dr. Shafie-khah,
I used IGA (a modified version of GA algorithm) to find Pareto optimal front in a multi-objective problem.
In order to select a final solution among the solutions which belong to the optimal front, I used the fuzzy satisfying approach in which a membership function is assigned to each solution. The fuzzy satisfying approach is a widely-used and simple method to select the final solution (Ref: A. Soroudi. Hybrid Immune-Genetic algorithm method for benefit maximization of distribution network operators and DG owners in a deregulated environment).
To the best of my knowledge, the attached book is a good reference on Pareto optimal front .
Best Regards.
Dear Miadreza Shafie-khah,
Multi-objective evolutionary algorithms are used in order to find the Pareto-optimal set. But if you just want to choose a few solution from Pareto-optimal set, I recommend used a multicriteria decision approach.
I recommend the following book (chapter 9) that summarizes the approaches of multicriteria decision-making
Coello, C. A. C., Lamont, G. B., & Van Veldhuizen, D. A. (2007). Evolutionary algorithms for solving multi-objective problems (Vol. 5). New York: Springer.
ISO 690
And the following article for an application, where they select the best compromise solution from the Pareto-optimal set.
Abido, M. A. (2003). A novel multiobjective evolutionary algorithm for environmental/economic power dispatch. Electric power systems research, 65(1), 71-81.
NSGA-II is the best for arriving pareto front and to find the best compromising solution among the available pareto solutions use methods like TOPSIS, AHP, PROMOTHEE etc
To get a compromise solution among the available pareto solutions user can choose the trade-off of the objectives taken.
I believe that one of the simplest methods wasn't mentioned in this thread. So there you go: the utopian point method. Check this paper here for understanding.
Article Multi-objective optimization based on the utopian point meth...
By definition, Pareto efficient solutions are all equally good, in a mathematical sense. Your question is about how to pick the best solution from this set. We have developed an approach to ordering solutions in the Pareto efficient set that does not require any additional preference. Perhaps it is applicable to your problem? Check out: Preprint An Approach to Ordering Objectives and Pareto Efficient Solutions
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