# multi-objective optimization assessment

 2 Dear Friends, We have 3 objective function (f1,f2,f3). I would like to compare the two following approaches by some appropriate measures: 1) considering f1 f2 f3 in multi-objective optimization. 2) considering f1, f2 as the objective functions and f3 as a constraint in the objective function. How can I compare them? Best,Reza asked 30 Nov '12, 12:47 Reza Salehiz... 58●7 accept rate: 0% fbahr ♦ 4.6k●7●16 How are you planning to balance three objectives in the first case (archimedean weights, weighted norm, distance from a utopia point, preemptive priorities, ...)? I think the answer to that might have implications for how to do the comparison. (05 Dec '12, 17:32) Paul Rubin ♦♦ Thank you Paul. We find pareto optimal front for both approaches so the method of normalization is not matter of importance I think. (05 Dec '12, 22:38) Reza Salehiz...

 2 Your second modeling method is similar to $$\epsilon$$-constraint, which is known to give Pareto optimal solutions for different $$\epsilon$$ values. Considering that your solution method is capable of finding Pareto optimal sets for each modeling method (e.g., using an evolutionary algorithm such as NSGA-II), you might compare these modeling methods based on multiple criteria including: Number: number of Pareto optimal solutions obtained by each method. Quality: ratio of number of Pareto optimal solutions obtained by each method to the total number of Pareto optimal solutions obtained by both methods. Spacing: Spacing measures how uniformly each Pareto front is distributed. You might see the following papers for more discussion of the above metrics: Schaffer, J.D. (1985). Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. The first International conference on genetic algorithms, Hillsdale, New Jersy. Srinivas, N. and K. Deb (1994). Multi-Objective Optimization Using Non-Dominated Sorting in Genetic Algorithms." Evolutionary Computation, 2(2): 221-248. Some other criteria are discussed by Deb in his book on multiobjective evolutionary algorithms. answered 30 Nov '12, 15:02 Ehsan ♦ 4.8k●3●12●24 accept rate: 16% Thank you. but if we would like to solve both models via epsilon-constraint, what kind of measures should we use? (30 Nov '12, 15:14) Reza Salehiz... $$\epsilon$$-constraint is designed to work with one objective function as the main one and the other objective functions as constraints. Hence if you want to use $$\epsilon$$-constraint method, you have to change your current models that would result in three models each with one of objective functions $$f_1$$, $$f_2$$, and $$f_3$$ and the other objective functions as constraints. Therefore, I don't think your models could be compared with each other anymore if you use $$\epsilon$$-constraint method. (30 Nov '12, 15:21) Ehsan ♦
 0 Different things. In the first case the Pareto fron is a 3-D surface and in the second case a 2-D curve. The 2-D curve is actually a cut of the 3-D surface in the specific point (the RHS of the f3 constraint) answered 30 Nov '12, 14:17 gmavrotas 26●1 accept rate: 50%
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