Dear Achour,

For more information about this question i suggest to you look at links and attached files in topics.

- Topics in Theoretical Computer Science: The First IFIP WG 1.8 ...

https://books.google.dz/books?isbn=3319286781

- Evolutionary Multi-Criterion Optimization: 8th International ...

https://books.google.dz/books?isbn=3319158929

- Modelling Foundations and Applications: 8th European Conference, ...

https://books.google.dz/books?isbn=3642314910

- Proceedings - Page 48 - Résultats Google Recherche de Livres

https://books.google.dz/books?isbn=2875870157

- Artificial Evolution: 7th International Conference, Evolution ...

https://books.google.dz/books?isbn=3540335900

Best regards

Conference Paper Approximating the Pareto Front of Multi-criteria Optimization Problems

Book Multiobjective optimization. Interactive and evolutionary approaches

Conference Paper Genetic algorithm with pareto front selection for multi-crit...

Article Pareto front-based multi-objective real-time traffic signal ...

Article Combined Pareto Front Visualization Method in Multi-Criteria...

Yes. You are ok, the functions are not antagonist. If you maximize f1, f2 is maximized; if you minimize f1, f2 is minimized. You can find antagonist functions in 

http://dx.doi.org/10.1016/j.cej.2011.09.007

I guess you are minimizing. My impression is, first of all, that you have dominated (inferior) solutions, so it is nor really a PAreto front, but a set of solutions. For a Pareto front, you should delete all dominated solutions.

Second, although the fron looks straight, my guess is that it is not, and, as usual, it is convex. You are just finding the vertices of the feasible region in the objective space.  

If is were straight, there would be no way to find so many solutions...

I hope it helps.

Cheers,

Vlad

Thank you all for the responses

Mr Vladimir Marianov, How can I delete the dominated solutions ? 

thank you

If at least some of your best solutions are in the Pareto-front, that means one of them must be a maximization function, and the other one is a minimization one. If they were both min or max functions, the graph would like a curve from top-left corner to bottom right corner, indicating that you can't max/min them together.

In this case, you CAN maximize or minimize them together, and this is not what you want (or else there would be only one point in the Pareto-front), so f1 and f2 are min/max. To remove visually the dominated solutions, you should first discover which one is min and max first, then remove the points that have strictly better solutions, that is, one function is improved while the other one is at least the same.

Hope it helps

Felipe

Solutions dominated are those that have all values of the objective functions worse than or equal than any another (better) solution. This is particularly pertinent and very easy to perform when populations of solutions are evaluated in each generation using an evolutionary algorithm, for example.  For obtaining a Pareto Front (after every dominated solutions have been removed) the objective functions must be contradictory. For example with two functions, function f1 represents a cost (represented in horizontal x axis) that need to be minimized and f2 represents a grade (represented in vertical axis y) that needs to be minimized too, but which decreases when the cost increases. This explains the frequent behavior of the curve in a two-objective optimization. Multi-objective optimization links of Wikipedia, for example, are very simple and explain clearly the most important concepts that need to be fully understood.

Dear António Manuel Abreu Freire Diogo, Felipe de Oliveira Mota,Vladimir Marianov.

As I understood, ones I got the Pareto front with its dominated and non dominated solutions, I have to eliminate the non dominated ones . can you give me an idea about how to eliminate them from the Front Pareto . 

thank you 

best regards

If you use a regular mathematical programming formulation, it will by itself find only the non-dominated solutions. Judging by the figure, I guess you are using a heuristic and finding both dominated and non-dominated solutions. The "brut force" method would be to take the solutions one by one, say  s_1, s_2, s_3... s_q and compare the corresponding objective Z_1 and Z_2 values with every other solution. Suppose you are comparing s_1 and s_4. If Z_1(s_1)

Dear Vladimir Marianov

I'm using a heuristic method, and I think that the Pareto front I got every time contains the both dominated and non dominated solutions, What do you think about creating a Matlab code that separate the non dominated solutions from the overall solutions. ?

do you know other efficient method that I can use in Matlab? 

Use NSGA-II for multi-objective optimization. Its code is available in MATLAB File Exchange

No idea what the hell is medical news

It depends. I suppose you have only a Pareto front, then all solutions must be non-dominated, but it's not like that. Review:

Approximates values...

1.05 3.53

1.075 3.59

1.077 3.599

1.078 3.63

1.08 3.61 (Here there is a problem)

Then, I concluded you have a wrong Pareto Front