First of all, you should be verified the optimization model used with others public examples in order to show its performance with other previous methods. Then , you can use it directly with specific case study.

Hi,

I suggest you to see links and attached files on topic.

https://www.springer.com/gp/book/9780387332543

https://www.springer.com/gp/book/9783540729631

Article Numerical results for the multiobjective trust region algorithm MHT

Conference Paper A Scalable Multi-objective Test Problem Toolkit

Chapter Solving Hard Multiobjective Optimization Problems Using ε-Co...

Best regards

Dear Sina

If you solve a problem using different MCDM methods it is normal to obtain different results, that in my opinion, are due to subjective issues.

Now, you want to compare results using different criteria?

Reasonably, results will be probably different. In my opinion you should use the same set of criteria, with the same atributes, for both methods. There are many methods to choose from, and their result may or may not coincide with yours.

I don't understand why you use a genetic algorithm when you have many different MCDM methods available.

Your problem appears to be rather simple, however, if you address two objectives, I doubt that you can speak about optimization

Regarding validation, you can't validate because for that you have to have a yardstick, something that you don't have.

There are different ways to check the correctness and accuracy of the implemented code. The following approaches to check the optimization can be addressed briefly as follows:

A) Comparison with classical approaches: In some problems, it can be possible to run both the classical approach and multi-variate GA to make sure the results are correct and accurate. However, in most cases, because of mathematical complexity using a classical approach for comparison is not possible.

B) Using Well-known Benchmarks: The correctness of the code can be also proven though existing benchmarks. The used algorithm can be validated via well-known benchmarks [1-5].

C) Using Expert Idea: An expert of the system can check the results and their rationality. They should all agree that the final results are logical.

D) Convergence: The convergence of the implemented multi-objective genetic algorithm can be easily observed through the plot of cost vs. iterations. There are also other factors such as roundoff errors and local extrema that have the same perspective.

E) Domination Factor: The NSGA-II optimization algorithm and other recent versions have the function to separate dominant results from non-dominant one. So, all final results should be dominant and with a proper initial population, it is very rare to be trapped by a local minimum. For more information regarding the domination please check [6].

[1] Yao, X., Liu, Y., & Lin, G. (1999). Evolutionary programming made faster. IEEE Transactions on Evolutionary Computation, 3(2), 82-102.

[2] Digalakis, J. G., & Margaritis, K. G. (2001). On benchmarking functions for genetic algorithms. International journal of computer mathematics, 77(4), 481-506.

[3] Molga, M., & Smutnicki, C. (2005). Test functions for optimization needs. Test functions for optimization needs, 101.

[4] Yang, X. S. (2010). Test problems in optimization. arXiv preprint arXiv:1008.0549.

[5] Mirjalili, S., & Lewis, A. (2013). S-shaped versus V-shaped transfer functions for binary particle swarm optimization. Swarm and Evolutionary Computation, 9, 1-14.

[6] Elarbi, M., Bechikh, S., Gupta, A., Said, L. B., & Ong, Y. S. (2017). A new decomposition-based NSGA-II for many-objective optimization. IEEE transactions on systems, man, and cybernetics: systems, 48(7), 1191-1210.

Koorosh Aslansefat has given a comprehensive answer to the query.

Dear Koorosh

NM-With due respect to your opinions and comments, I disagree

KA - There are different ways to check the correctness and accuracy of the implemented code. The following approaches to check the optimization can be addressed briefly as follows:

NM. You can’t check the correctness of the solution simply because you don’t have any reference to compare it with.

In addition, you can’t check the optimization because there is no method that can optimize in multiple criteria, you only aim at reaching to a compromise solution.

KA - A) Comparison with classical approaches: In some problems, it can be possible to run both the classical approach and multi-variate GA to make sure the results are correct and accurate. However, in most cases, because of mathematical complexity using a classical approach for comparison is not possible.

NM. You can use other methods for the same problem, but what do you get? Simply, different solutions. Which is the best to compare with? You don’t know it, therefore, the maximum you can get is that your solution is similar to other and dissimilar to another.

KA - B) Using Well-known Benchmarks: The correctness of the code can be also proven though existing benchmarks. The used algorithm can be validated via well-known benchmarks [1-5].

NM – And which are those benchmarks?

KA - C) Using Expert Idea: An expert of the system can check the results and their rationality. They should all agree that the final results are logical.

NM – That makes sense, but it does not mean accuracy. You only can claim accuracy, when you know which the solution is

KA - D) Convergence: The convergence of the implemented multi-objective genetic algorithm can be easily observed through the plot of cost vs. iterations. There are also other factors such as roundoff errors and local extrema that have the same perspective.

NM – Could you elaborate a little more please? What do you mean with your first sentence?

KA- E) Domination Factor: The NSGA-II optimization algorithm and other recent versions have the function to separate dominant results from non-dominant one. So, all final results should be dominant and with a proper initial population, it is very rare to be trapped by a local minimum. For more information regarding the domination please check [6].

NM – Again, it does not assure that you bet the ‘true’ solution.

There are many benchmark function which you can compare your algorithm. But if your algorithm validated with the benchmark function, It may don't answer for your application. In fact, In this type of optimization you cant grantee the answer is the most optimized point, but you can just say the best answer of my problem with this GA parameters is obtained. May be by changing GA parameters you can find better answer. you should change GA parameters several times and compare the answers to find the best optimized point.

Dear Mohsen

I agree with you however, I don't understand you reference to the benchmark function. This is new for me.

I will be glad if you can explain it to me.

Thank you