In my bi-objective model the range of solution for the first objective is between 1300000 to 1800000, while the second objective is in range of 0.02 to 0.09. I decide to obtain a single solution by weighted sum approach. How can I do that?
One of the simplest (and the best at the same time) approaches is to optimize each of the objectives individually first. Then divide each objective by those optimum values and then sum up all normalized terms as one objective. The new objective will be dimensionless.
You can also follow the "goal programming" approach. However, it requires a similar trick to make the terms dimensionless.
We should note that there is no single solution for a MOO problem and the solution is a Pareto front. If you are using the weighted sum approach, you should vary the weights to obtain the front.
The weighted sum approach works well only when the Pareto front is convex. If parts of Pareto front is concave, Pareto based approaches like MOPSO, NGSA-II, etc. are preferred.
Please have a look at first chapter of this book which has a good discussion on normalization of objective functions
Haftka, R. T., & Gurdal, Z. (1992). Elements of Structural Optimization: Springer Netherlands.
Another approach is to introduce some utility function for comparing alternative decisions and maximize it.
If the Phi(P) is the first objective function and Psi(P) is the second one you can use the weighted sum defined by {[Phi(P)/Phi(P0)] + [Psi(P)/Psi(P0)]} where P0 is an initial parameter value.
You can also replace P by P/P0 to have a dimensionless form of parameters.
You might want to take a look at this great survey of approaches by Marler and Arora.
http://web.cse.ohio-state.edu/~parent/classes/788/Au10/OptimizationPapers/MultiObjective/journal_survey.pdf
The global minimum of the weighted sum is on a convex area of the Pareto front, but you will need either a global optimization algorithm to effectively reach it or to be sure that a specific local algorithm can reach the global minimum of your particular problem. There are two different issues there that can lead to sub optimal solutions : the limitations of the optimization algorithms and the issues of the scalarization methods (weighted sum or others). This is very confusing.
One of the simplest (and the best at the same time) approaches is to optimize each of the objectives individually first. Then divide each objective by those optimum values and then sum up all normalized terms as one objective. The new objective will be dimensionless.
You can also follow the "goal programming" approach. However, it requires a similar trick to make the terms dimensionless.
what can we call this approach?
A very good reference for weighted sum approach is the article by Prof. Arora. You can get the full text from RG.
In his book (introduction to optimum design, there is a chapter on multi-objective optimization. I use the book as the main reference in a graduate course I teach. I certainly recommend the book.
Marler, R. T., & Arora, J. S. (2010). The weighted sum method for multi-objective optimization: new insights. Structural and multidisciplinary optimization, 41(6), 853-862.
http://link.springer.com/article/10.1007/s00158-009-0460-7
Hi,
I suggest that you have a look on the following paper for problems related to the normalization approaches:
*/ A novel heuristic for multi-objective optimization of analog circuit
performances
Analog Integr Circ Sig Process (2009) 61:47–64
Dear Ashkan: you can use the so called Compromise Programming method, which not only calculates a relative value and puts a wheight on it,(like Michele does above), but also includes exponentiating each term, and then taking a root of the whole aditive term. I would recommend you to look for a paper of Prof. Serafim Opricovic who developed this method or of Prof. Lucien Duckstein who taught it to me. The formula for Compromise Programming can be found in my paper: Multiobjective Decision Making Techniques for Reservoir Operation, Water Resources Bulletin, February 1992. Best regards, Ricardo
To normalize multi-objective functions, in order to obtain an undimenssional numerical form, it is necessary in a fist time to divise each objective function by the norm of initial or corresponding experimental values. Computation of relative functionsby division with the corresponding function value get at initial state or initial measurement can be also used (relative least squares errors - see wikipedia).
Best regards,
Adinel GAVRUS
INSA Rennes
Dear shkan,
Since you know the varying range of your OF, one way is to normalize each function and bring it to the range [0,1]: Feqi=(f-fmin)/(fmax-fmin), see for instance
A high performances CMOS CCII and high frequency
applications
BenSalem eta., Analog Integr Circ Sig Process (2006) 49:71–78
However, some drawbacks emerge here, you can found details in:
A novel heuristic for multi-objective optimization of analog circuit performances
Fakhfakh et al. October 2009, Volume 61, Issue 1, pp 47-64.
BR
I think that you want optimization 2 or 3 variables together. so you should find sum of variables but order of variable not Same !!!
you can use Penalti method that each variable has a effective factor and change order of variables.
for example: A and B
50000
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