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A new distribution metric for comparing Pareto optimal solutions

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Abstract

Evolutionary multi-objective optimization has established itself a core field of research and application, with a proliferation of algorithms derived. During the multi-objective optimization processes, the discovered ideal solutions should be diversely distributed at the Pareto front. In order to measure and compare the performances of different multi-objective evolutionary algorithms, or provide a guidance for the search or a stopping criterion, various performance metrics are defined and used. In this paper, two of the most commonly used metrics, the spacing metric and the overall Pareto spread metric, which evaluate the uniformity and the range of the Pareto solutions’ distribution are studied, respectively. A new distribution metric which potentially can combine these two metrics and resolve their deficiencies for comparing Pareto optimal solutions is then proposed. Five typical Pareto fronts and a real practical example are used to demonstrate the effectiveness of the proposed metric by comparing with the subject matter experts’ ratings.

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Abbreviations

EMO:

Evolutionary multi-objective optimization

MOEA:

Multi-objective evolutionary algorithm

SP:

Spacing metric

OS:

Overall Pareto spread metric

DM:

Distribution metric

SME:

Subject matter expert

H :

The number of objectives

S :

The obtained non-dominated solutions

|S|:

The number of the obtained non-dominated solutions

f h (s):

The function values of the h th objective at current solutions s

d i :

The distance measure in the SP

\( \overline{d} \) :

The mean value of d i

P B :

The Utopia design points

P G :

The Nadir design points

σ h :

The standard deviation of all distances obtained in the DM corresponding to the h th objective

μ h :

The mean of all distances obtained in the DM corresponding to the h th objective

R h :

The range of function values corresponding to the h th objective

d′ h e :

The distance value of the eth interval between two adjacent solutions corresponding to the h th objective in the DM

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 51475288, 51275293), “ShuGuang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (12SG14), US Ford Motor Company and ESTECO North America.

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Authors and Affiliations

  1. Institute of Knowledge Based Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, People’s Republic of China

    Kai Zheng & Jie Hu

  2. Research and Advanced Engineering, Ford Motor Company, Dearborn, MI, 48124, USA

    Ren-Jye Yang & Hongyi Xu

Authors
  1. Kai Zheng
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  2. Ren-Jye Yang
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  3. Hongyi Xu
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  4. Jie Hu
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Corresponding author

Correspondence to Jie Hu.

Appendix

Appendix

(Table 8).

Table 7 Detailed SMEs’ ratings on Pareto front 1
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Table 8 Detailed SMEs’ ratings on Pareto front 2
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Table 9 Detailed SMEs’ ratings on Pareto front 3
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Table 10 Detailed SMEs’ ratings on Pareto front 4
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Table 11 Detailed SMEs’ ratings on Pareto front 5
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Table 12 Detailed SMEs’ ratings on Pareto front of side impact problem
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Zheng, K., Yang, RJ., Xu, H. et al. A new distribution metric for comparing Pareto optimal solutions. Struct Multidisc Optim 55, 53–62 (2017). https://doi.org/10.1007/s00158-016-1469-3

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