For instance, if i formulate mathematical equations to solve multi-objective optimization problem (1 value should decrease while other value should increase at the same time) then what algorithm can be the best algorithm to dealt with.
Exact methods: those of mathematical programming, depending on the nature of the equations and variables, i.e. Simplex, Branch & Bound, Non-linear methods, etc.
The multiobjective problem is transformed into a single-objective problem using diverse methods. The most used are weights method and epsilon-constraint method. I recommend you to read:
Ehrgott, M. (2005). Multicriteria optimization (Vol. 491). Springer Science & Business Media.
Hi, do you have a classical (single objective) problem to solve (constrained or not) or do you want to solve a multi-objective problem?
Richard Epenoy As mentioned in description of question.Its multi-objective optimization problem (1 value should decrease while other value should increase at the same time) .
Hi, you will find below a interesting link to some codes dedicated to multi-objective optimization:
plato.asu.edu/sub/multiobj.html
Best
Why not use a ranking approach: Suppose the first objective is F_1, and the other is F_2. Parameterize them through a non-negative convex combination like this:
P * F_1 + (1-P)* F_2,
with P in the interval [0,1]. For simplicity, suppose you use these values of P:
P = {0, 0.1, 0.2, ... , 0.8, 0.9, 1.0}. Looking at the corresponding objective values in R^2 you have the responsibility to choose what you prefer.
Quite often the process does not end here, but there might be a further step, when a decision-maker (you, or someone else) looks at these 11 points, and simply decide that one of these are better, for whatever reason.
Bi-criterion and multi-objective problems ARE difficult, because of the need to involve a decision-maker.
Exact methods: those of mathematical programming, depending on the nature of the equations and variables, i.e. Simplex, Branch & Bound, Non-linear methods, etc.
The multiobjective problem is transformed into a single-objective problem using diverse methods. The most used are weights method and epsilon-constraint method. I recommend you to read:
Ehrgott, M. (2005). Multicriteria optimization (Vol. 491). Springer Science & Business Media.
Distinguished, Muhammad Ali, I am not sure I have understood your question, but following the recommendations of the other colleagues, it occurs to me that you can try treating (transform) the problem as a meta programming problem (Goal programming problem).
I hope my suggestion can be of some use.
Many successes in your research.
José Hernández.
In all practical optimization problems, whether multi-objective or not, you need to distinguish two aspects/phases:
1) You need to define, as exactly as possible WHAT you want to optimize and
2) then you have to decide HOW to do the optimization which normally reduces to finding a maximum (or minimum) of some objective function.
The difficult part is step (1) because that is NOT MATHEMATICAL. What you want is up to you! In multi-objective problems, for example, you might want to reach too many objectives which are in conflict with each other. How much weight you assign to one objective with respect to the others, and/or to an objective being met only partially, that is totally up to you, obviously. There is no "exact" approach to it, at least not if by "exact" you mean "objective" (please, clarify).
The simple part is step (2) which is basically a matter of choosing between many available algorithms (there are many books about this). When the solution is unique, the algorithms should all lead to the same result, so that the comparison regards only execution times. When there are multiple solutions, it becomes a bit more complex since different algorithms can lead to different "solutions" but then all such solutions are valid and, in a sense, "exact".
You can use goal programming approach to solve this kind of optimization problem. This can be done using WinQsB solver
I think the problem you are having is how to construct your objective function in such a way that as the variable changes ( for instance increases or decreases) their values, there will be corresponding change in the dependent variables in a manner that an increase in one will give corresponding decrease in the other dependent variables.
Having said that, there seem not to be a generic form of modelling such. It all depend on the problem you are trying to solve.
You may provide the problem, then providing the the associated model will be lot more easy.
Cheers.
Mbachu Victor M.
Use NSGA - II method. i have used it to solve multi objective design optimiztion problems using NSGA-II
Please see the following book:
Ehrgott, M.: Multicriteria Optimization. Springer, Berlin 2005.
Best Regards
The fixation on the topic of multiple objectives in this forum is truly puzzling, and I'd like to know why. The initial question didn't refer to multiple objectives, but degenerated into one very quickly ...
Dear Muhammad
I don't think that you can talk of optimality in a multi objectives problem, probably because an optimal solution does not exist due to conflicting criteria, such as minimizing costs while maximizing benefits.
You can look for a compromise solution but not for optimization
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