If you're looking for an approach/method to pick one non-dominated solution out of your Pareto non-dominated trade-off, there is one simple and logical method.

You can calculate a "regret" value for each solution point of your trade-off. Assuming you have n objectives, for each solution point i you can compute:

regret( i ) = sqrt( [((Z1 (i) )^2 - (Z1 (opt))^2]+...+ [((Zn (i))^2 - (Zn (opt))^2] )

where Z1 (i) is the value of the objective function 1 from non-dominated solution point i, Z1 (opt) is the optimum value of objective function 1 obtained by your algorithm; that is the smallest value for Z1 from the Pareto front if objective function 1 must be minimized and the largest value of Z1 if this objective must be maximized.

Finally, when you calculate all regret values for non-dominated solutions, the one with the lowest "regret" can be chosen as choosing it incurs the lowest loss when all objectives are considered.

Surely there are other approaches to do this job, but they are rather more sophisticated. Some examples are

TABOADA H.A., and COIT D.W., "Multi-objective scheduling problems: Determination of pruned Pareto sets", IIE Transactions (2008) 40, 552–564 Giagkiozisa I. and Fleming P.J., "Methods for Multi-Objective Optimization: An Analysis", Information Sciences (2015) 293, 1–16

Veerappa V., and Letier E., "Understanding clusters of optimal solutions in multi-objective decision problems," 2011 IEEE 19th International Requirements Engineering Conference, Trento, 2011, pp. 89-98. doi: 10.1109/RE.2011.6051654

Please feel free to contact me if my explanaition was naot clear or you have any further queries.

Best,

Majed

One of the possible ways to compare between two different non dominated solutions Is to compute the distances between them and the ideal point in the objective space.

The non dominated solutions are compared according to the correspondence of each solution with the preferences system of of the decider. These solutions really constitute a population of solutions that is what requires the decider to be able to choose solutions using not only it own imagination but also simulation and perhaps graphic representation models. For this reason the most appropriate approach consists on generating populations of close to the best commitment solutions, selected preliminarily by the decider, the rectification of its evaluation for complementary models, including the subjective criterion of the decider and the generation from a population of solutions to the higher span system, orderly according to the studied system criteria (that constitutes an element of the higher span system). Ahead could be solved the decision making task of the higher span system.

U should find there mean and variance (statistically analysis) compare u problem with other approaches.

The non dominate set of solutions could be reduced seraching solutions populations close to desirable criteria values in the place of ideal ones. d

Refering to the ideal solution is beter in the abscence of another desirable criteria

Once having multiple solutions, a Multicriteria decision method can be used, as Promethee, Electre, AHP, and so on, I guess. There is a lot of literature on that.(MCDM)

If you're looking for an approach/method to pick one non-dominated solution out of your Pareto non-dominated trade-off, there is one simple and logical method.

You can calculate a "regret" value for each solution point of your trade-off. Assuming you have n objectives, for each solution point i you can compute:

regret( i ) = sqrt( [((Z1 (i) )^2 - (Z1 (opt))^2]+...+ [((Zn (i))^2 - (Zn (opt))^2] )

where Z1 (i) is the value of the objective function 1 from non-dominated solution point i, Z1 (opt) is the optimum value of objective function 1 obtained by your algorithm; that is the smallest value for Z1 from the Pareto front if objective function 1 must be minimized and the largest value of Z1 if this objective must be maximized.

Finally, when you calculate all regret values for non-dominated solutions, the one with the lowest "regret" can be chosen as choosing it incurs the lowest loss when all objectives are considered.

Surely there are other approaches to do this job, but they are rather more sophisticated. Some examples are

TABOADA H.A., and COIT D.W., "Multi-objective scheduling problems: Determination of pruned Pareto sets", IIE Transactions (2008) 40, 552–564 Giagkiozisa I. and Fleming P.J., "Methods for Multi-Objective Optimization: An Analysis", Information Sciences (2015) 293, 1–16

Veerappa V., and Letier E., "Understanding clusters of optimal solutions in multi-objective decision problems," 2011 IEEE 19th International Requirements Engineering Conference, Trento, 2011, pp. 89-98. doi: 10.1109/RE.2011.6051654

Please feel free to contact me if my explanaition was naot clear or you have any further queries.

Best,

Majed

The minimum regret method proposed by Majed Khadem is very good if you have random parameter values as it avoids being too off in case of the "worst" realization of your parameters.

In the same spirit with answers of Vladimir and Majed, you shall use the Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) or VIKOR.

Article Topsis for MODM

Article Compromise solution by MCDM methods: a comparative analysis ...

Non dominated solutions are not comparable by definition: the point of multi objective optimisation is precisely to get multiple equally good solutions, among which no preference criterion has been defined.

Further preference can obvioulsy be expressed afterwards by an informed decision maker on the basis of some additional knowledge that was not employed in the formulation of the multi objective problem.

If you just want one solution, have no other clue or information and your Pareto front has a very steep L shape, you might choose the point closest to the Utopia point. In that case, the Pareto front is telling you that you can get good performance overall without sacrificing too much on any specific performance.

EDIT: what I just wrote is an intuitive view of the minimal regret criterion described by Majed Khadem. Beware that, as formulated, this criterion requires the non dominated solutions to be at different distance with respect to the Utopia point. It's easy to construct examples for which this criterion will not give you a single solution

Please see the Performance assessment section in the following papers:

https://www.researchgate.net/publication/321065586_Quasi-Newton_methods_for_multiobjective_optimization_problems

Article Barzilai and Borwein’s method for multiobjective optimization problems