Role of Examples and Interpretation of Results in Developing Multi-Objective Optimization Techniques

The paper evaluates the suitability of examples used in developing averaging techniques of multi-objective optimization (MOO). Most of the examples used for proposing these techniques were not suitable. The results of these examples have also not been interpreted correctly. An appropriate example has also been solved with existing and improved averaging techniques of multi-objective optimization.


Introduction
The importance of examples in understanding the mathematical theories or mathematical interpretations is very well recognised. Examples are the principle devices used to illustrate and communicate concepts to the learner. Examples are quite relevant for making any mathematical theory or concept more realistic and acceptable. The present study evaluates the suitability of the examples used in the development of new averaging MOO techniques. After Sen's MOO technique [1], several averaging MOO techniques [2]- [11] have been proposed during last three decades. Many examples have been used for testing the applicability of these techniques. Seven examples used in these MOO techniques have been selected for the present analysis. The presence of conflicts amongst objectives is the main characteristic of an appropriate example. The results of these examples and their interpretations have also been reviewed. The achievement of the objec-How to cite this paper: Sen, C. (2020) Role of Examples and Interpretation of Results in Developing Multi-Objective Optimiza-tives using MOO techniques has been compared with the results of individual optimization. The results of the existing averaging MOO techniques using these examples have not been interpreted correctly. An appropriate example has also been solved using existing and improved averaging MOO techniques [12] for comparison.

Multi-Objective Optimization Techniques
The mathematical forms of Sen where, j θ is the optimal value of j th objective function. where, Min. 15 10 20 12 30 18 3 3 3 6 Subject to: Min.

Interpretation of the Results
The solutions of all the above mentioned examples are presented in Table 1.  were declared superior over the MOO technique with lower values of multi-objective functions in the most of the studies, which is not appropriate. An appropriate example has been solved using the existing and improved MOO techniques.

Appropriate Example
Further a new example for testing existing and improved averaging MOO techniques is mentioned below: Example 8: , , , , 0 X X X X X ≥ The example was solved for achieving each objective and the results are presented in Table 2  The existing averaging MOP techniques using mean, geometric mean and harmonic mean have been applied for solving the above example. The example was also solved using improved techniques of mean, geometric mean and harmonic mean and the results are presented in Table 3.
The results of multi-objective optimization with existing averaging techniques  However all the improved averaging techniques have also generated the unique solution but achieved all the objectives simultaneously. The improved averaging techniques have generated the compromised and more acceptable solutions than the existing averaging techniques.

Conclusion
The present analysis reveals that the examples used for testing existing averaging MOO techniques in many studies were not suitable for the purpose. The individual optimization revealed that all the examples were with non conflicting objectives and thus unsuitable in the application of MOO technique. The results have also not been interpreted appropriately. The values of multi-objective functions have been considered as achievements of all the objectives which are not correct. The values of basic objectives should have been considered for any conclusion. The study has been extended by adding an appropriate example and improved MOO techniques. The eighth example was found suitable for the validation of existing and improved averaging MOO techniques. The existing MOO techniques have been found inefficient in solving MOO problems.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.