Still papers are coming using weighted sum approach for solving bi-objective problems even though Pareto optimal algorithms such as NSGA-II are popular.
how to explain the concept of Pareto front using weighted sum approach. Is it possible to draw Pareto front curves using weighted sum approach.
The Pareto optimal solutions are those that are not worse of any others simultaniously by all the criteria. Using Tchebyshev Program for solving your multiplecriteria optimization problem you can find Pareto front or close to it solving Tchebyshev Program of your particular problem (but not weigted sum of criteria) in a variation cicle of the weigted coefficients. In my reserchagate page your can find works that could be. help your for understanding this idea.
Best regards,
Jose Arzola
Good morning,
You could find some answers to your interesting question in these two papers:
1) Fonseca, Carlos M., and Peter J. Fleming. "An overview of evolutionary algorithms in multiobjective optimization." Evolutionary computation 3.1 (1995): 1-16.
2) Khan, Salman A., and Shafiqur Rehman. "Iterative non-deterministic algorithms in on-shore wind farm design: A brief survey." Renewable and Sustainable Energy Reviews 19 (2013): 370-384.
They underline foure main drawbacks:
1.optimal solution could be not acceptable, either due to the function used excluding aspects of the problem or to an inappropriate setting of the coefficients (Fonseca and Flaming,1995)
2. If the solution is not acceptable, new runs of the optimizer is required;
3.A small change in in weights may results in big changes in the objective vectors (Khan and Rehman, 2013)
4.Significantly different weights may produce nearly similar objective vectors (Khan and Rehman, 2013)
Dear Sabarinath Pandurengan;
I have published a paper solving the optimization problem by NSGA-II. It may help you to answer your question.
http://dx.doi.org/10.1016/j.epsr.2014.09.010
Best,
Jos is right. If the original problem is convex, then the weighted sum method is fine. But if it is non-convex (e.g. because some variables are constrained to be integers), then there may exist non-dominated solutions that cannot be found using weighted sums. See papers by, e.g., Matthias Ehrgott.
Dear Colleague,
I have published a paper solving a complex multi-objective optimization problem by using MOALO. It may help you to answer your question?
DOI 10.1108/COMPEL-05-2018-0208.
Dear Sabarinath Pandurengan
The problem starts with the weights. Take a look at this paragraph from "Introduction to Optimum Design" by Arora:
"1- First, even with the use of some of the methods discussed in the literature for determining weights, a satisfactory a priori selection of weights does not necessarily guarantee that the final solution will be acceptable; one may have to resolve the problem with new weights. In fact, this is true of most weighted methods.
2- The second problem is that it is impossible to obtain points on nonconvex portions of the Pareto optimal set in the criterion space. Although nonconvex Pareto optimal sets are relatively uncommon, some examples are noted in the literature.
3- The final difficulty with the weighted sum method is that varying the weights consistently and continuously may not necessarily result in an even distribution of Pareto optimal points and an accurate, complete representation of the Pareto optimal set."
You can rely on weighted sum method if you are sure that the objective function of your optimization problem is convex in nature. Otherwise it is better to either convert the problem to quasi convex if you really want to use weighted sum method or maybe switch to more advance statistical approaches to solve your problem.
Well, one major drawback, ironically, is right there in its name -- weighted. Weight factors are incorporated into the objective functions based off of relative importance by programmers. This subjects the process to biases and the likelihood of yielding local-optimal results are exceedingly high. We need to understand that multi-objective optimization problems (MOOPs) are in essence tradeoffs between different objective functions. Pareto optimality, where every objective function is optimized to its global optimal value, is merely theoretical in nature. Many solutions to multi-objective engineering problems are therefore given as a Pareto set, i.e., feasible solutions with different tradeoffs of objective functions. Solutions are selected based on technical requirements during actual design.
As for plotting the Pareto front, realize that the weight sum method is a non-Pareto-based technique that produces only a single solution. Ergo, Pareto fronts are not plotted using this method. However, you can circumvent this limitation by methodologically varying the weight values of the objective functions and perform the subsequent optimization task, you should be able to have a rough sketch of the Pareto-front this way.
My advice, save yourself the headache and just use Pareto-based methods.
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