My problem is like, three objective equation \sum_{i=1,6;j=1,3} Ci,j x-2i and six constraints like \sum_{i=1,6; j=1,6} Di,j x-2i ==ai,j. Both the case j refers number of equations. How maximize the three objective equation simultaneously?
Why did you mention simplex method? Do you consider linearisation by replacing x-2i with some new variables yi?
And weighted sum always works (from the technical point of view). If you create a weighted sum of objectives, multiplication and addition don't affect the validity of the resulting formula. The problem is if the weighted sum represents your preferences about the nature of the compromise you want to achieve.
According to my modest experience, if you can reformulate your problem to be MIP or even MIQP, it is better.
Thank you Mr.Kowalik, Here the one more constraints are x-2i > 0 for all i. I agree, i have to use weighted sum method. i found a good program paper i.e Ser.Math.Inform.26(2011),49-63. Thank you Mr.Richard for your information.It contains a good introduction, about my problem. This is physics problem. So i don't know whether i get fixed no of pareto optimal solution or not. Because my problem has six solution for each objective equation.
Selvaganapathy Jaganathan, if -2 in your problem stands for the exponent -2, then constraints x-2i > 0 are redundant. For any xi except 0 those formulas are positive and for 0 undefined (or, if it is useful to consider so, equal to plus infinity).
Thank you for the question and replies so far. We have been studying multi-objective optimization (MOO) based on stochastic methods (metaheuristics) such as genetic algorithms and differential evolution, and their applications to chemical engineering problems involving many variables and constraints. We have two Excel-based MOO programs available free to charge to interested researchers. Please visit my group website: http://cheed.nus.edu.sg/stf/chegpr/Software.html and contact me ([email protected]) if you would like to have these programs.
There are several possibilities. The simpler one would be to minimize the normalized Tchebyshev distance from the ideal solution to the existence space using any numerical Non lineal Programming method. Did you understood me?
Best wishes,
Kowalik sir, x-2i > 0 is the important constraint for my problem. So that xi is not equal to zero.It can be +ve or -ve. The feasible solutions of the objective can have -ve and +ve value. Can we calculate how many feasible solutions are possible for given no.of objective,constraints and variable? Right now i am not considering about pareto optimal solution. Because this objective values are related to the physical coupling strength of the particles when they are exchanging electromagnetic force carriers(Photon) at high energy.
Jose sir, I am sorry. I couldn't understand your point. Can you send reference about minimization of the normalized Tchebyshev distance?
Hello,
As far as I understand, your variables xi are continuous and only their square roots appear in the expression of your constraints and objectives. By replacing the square roots of xi by fresh new variables, you actually have a linear multiobjective problem.
A parametric simplex method or a weighted sum approach will solve your problem directly. Check thiese slides for a good introduction these methods:
http://www.lamsade.dauphine.fr/~gold/COST_slides/HanLecture2_ME.pdf
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