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%20Salehizadeh's gravatar image

Reza Salehiz...
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edited 01 Dec '12, 08:28

fbahr's gravatar image

fbahr ♦
4.6k716

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...

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.

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answered 30 Nov '12, 15:02

Ehsan's gravatar image

Ehsan ♦
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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 ♦

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)

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answered 30 Nov '12, 14:17

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gmavrotas
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accept rate: 50%

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Asked: 30 Nov '12, 12:47

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