Hi Hanane. I found myself in this position several times. To tackle this I generally try to provide theoretical model of the problem (possible unsolvable in the general case). Then, I show the (optimal) result for increasingly complex data sets until the point that my computing power becomes insufficient. Then I run the same data sets on the heuristic but I go further in complexity. The result are two curves, one with more data (on the x axis) and another with less. Here you might just find that you could use curve fitting on the theoretical model to predict missing values or just leave it as is. Hope it helps.

Thank you Juan for your answer and help :). I do the same thing for the small instances , cause I have a mathematical model witch gives me the optimal solution for the small instances but as you said, the computing power becomes rapidly insufficient. I want to try your idea, is it possible to give me a title of an article or works where this method is applied ? thank you again so much.

Dear Hanane Krim,

I want to share my thought on your question. I hasten to note that my opinion is not popular in the community of researchers.

Please have a look on my answer from https://www.researchgate.net/post/What_is_the_future_of_OPTIMIZATION

"Dear Ljubomir Jacić,

I would like to argue about your phrase "The future of optimization belongs to metaheuristics algorithms!". From my point of view, in Combinatorial Optimization (CO), all the metaheuristics algorithms (MA) are not any Science. In my view, any MA is only a Craft. Suppose we try to solve a CO model, where OPT(D) is an optimal solution, D is an intitial data. As a rule, we can propose a set of approxinate solutions UB(D) by MA without any problem. A main problem arises: How close this solution UB(D) to the optimal OPT)D) ? None of the known algorithms MA can answer this question. We can consider a metric q = 100% * ( UP(D) - OPT(D) ) / OPT(D) as a measure of closeness UP(D) to OPT(D). Since OPT(D) is unknown, we can consider another metric p = 100% * ( UP(D) - LB(D) ) / LB(D) , where LB(D) is a Lower Bound, LB(D)

Dear Gennady Fedulov,

Thank you for your answer, but as I mentioned in the question, for instance, I tried to find another way to evaluate the solutions returned by the metaheuristic that I adapted to my problem...

Do you have other suggestions?

Thank you

Dear Hanane Krim,

In my view, there is no "another way to evaluate the solutions returned by the metaheuristic". The Lower Bound (LB) is the only real way, there are no other way at all. Otherwise, this method would have been found and proposed. I want to emphasize again: in my personal opinion, the finding the best LBs is a real science, but finding metaheuristic solutions is not science at all.

Dear Hanane,

I once read an article where the authors, facing the same problem, compared their metaheuristics to the best solution obtained after carrying out several runs where 100 000 solutions are evaluated by an iterated local search algorithm.

This will provide you with the lowest known upper bound for your problem.

I hope this helped,

Kind regards,

Imène

 I'd like to suggest another angle for looking at the problem. Typically, you want to compare to the current state of the art. What techniques do people use to solve your problem? If something exists in the literature, you should use that technique as a comparison baseline, hopefully showing that you can do better with respect to some criterion i.e., results closer to optimality, less time needed to generate results, etc.

If nobody ever attempted to solve this problem, or existing solutions are not applicable due to context factors, what people typically do is comparing metaheuristics with the most naive baseline, i.e., random search. In general, I advise doing so anyway, because in my experience there exists a number of problems for which random search works just as well as sophisticated metaheuristics.

hi

i use PSO ,can you send to me your objective problem with constraints if exist in order to try to solve it 

Hi Hanane,

Metaheuristics are usually compared in a “relative” manner, in the sense that it’s usually more important to know how an algorithm performs when compared to another algorithm than the actual (“absolute”) error of the found solution (as Stefano said “you want to compare to the current state of the art”). I´m not saying this is the best way… but is a very common one. This comparison is frequently performed using measures such as the mean, the median, the standard deviation and the minimum and maximum of several runs of a given metaheuristic (usually with a common stopping criteria for all the algorithms being compared). These measures do not necessarily need to be computed from the absolute error, but sometimes they can be directly computed from the objective function value.

However, if you want to have an idea of the error (respect the global optimum) your algorithms are achieving without knowing the actual optimum, then the best approach is to attempt to estimate (as accurately as possible) the fitness of that optimum. The suggestion I could think of have already been suggested:

Hope this helps.

Dear everybody,

More info can be found here:

N. K. T. El-Omari, "Sea Lion Optimization Algorithm for Solving the Maximum Flow Problem", International Journal of Computer Science and Network Security (IJCSNS), e-ISSN: 1738-7906, DOI: 10.22937/IJCSNS.2020.20.08.5, 20(8):30-68, 2020.

Or simply refer to the same paper at the following address:

https://www.researchgate.net/publication/344324283_Sea_Lion_Optimization_Algorithm_for_Solving_the_Maximum_Flow_Problem