Abstract
This paper proposes a new stopping criterion for decomposition-based multi-objective evolutionary algorithms (MOEA/Ds) to reduce the unnecessary usage of computational resource. In MOEA/D, a multi-objective problem is decomposed into a number of single-objective subproblems using a Tchebycheff decomposition approach. Then, optimal Pareto front (PF) is obtained by optimizing the Tchebycheff objective of all the subproblems. The proposed stopping criterion monitors the variations of Tchebycheff objective at every generation using maximum Tchebycheff objective error (MTOE) of all the subproblems and stops the algorithm, when there is no significant improvement in MTOE. \(\chi ^{2}\) test is used for statistically verifying the significant changes of MTOE for every \(\gamma \) generations. The proposed stopping criterion is implemented in a recently constrained MOEA/D variant, namely CMOEA/D-CDP, and a simulation study is conducted with the constrained test instances for choosing a suitable tolerance value for the MTOE stopping criterion. A comparison with the recent stopping methods demonstrates that the proposed MTOE stopping criterion is simple and has minimum computational complexity. Moreover, the MTOE stopping criterion is tested on real-world application, namely multi-objective \(\hbox {H}_{\infty }\) loop shaping PID controller design. Simulation results revealed that the MTOE stopping criterion reduces the unnecessary usage of computational resource significantly when solving the constrained test instances and multi-objective \(\hbox {H}_{\infty }\) loop shaping PID controller design problems.
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Abbreviations
- CDP:
-
Constraint-domination principle
- DM:
-
Decision maker
- EAs:
-
Evolutionary algorithms
- FE:
-
Function evaluation
- GD:
-
Generational distance
- HV:
-
Hypervolume
- MFE:
-
Maximum number of function evaluation
- MOEA:
-
Multi-objective evolutionary algorithm
- MOP:
-
Multi-objective problems
- PF:
-
Pareto front
- MIMO:
-
Multi-input multi-output
- LSSC:
-
Least square stopping criterion
- maxCD:
-
Maximum crowding distance
- IAE:
-
Integral absolute error
- IGD:
-
Inverted generational distance
- MTOE:
-
Maximum Tchebycheff objective error
- SISO:
-
Single-input single-output
- PIs:
-
Performance indicators
- MDR:
-
Mutual domination rate
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Acknowledgments
The authors gratefully thank the Principal and Management of Thiagarajar College of Engineering, Madurai, India, for their support and continuous encouragement in this research. This research was supported by Indian Government via University Grants Commission’s Maulana Azad National Fellowship scheme (File No.: MANF-MUS-TAM-3045). Also, authors thank to Tobias Wagner, Heike Trautmann and Luis Marti for sharing their MATLAB code of taxonomy of stopping methods for MOEA algorithms.
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This research is funded by University Grants Commission of India via Maulana Azad National Fellowship scheme (File No.: MANF-MUS-TAM-3045).
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Authors declared no conflict of interest.
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Communicated by V. Loia.
Appendix
Appendix
1.1 Calculation of Euclidean distance
In MOEA/D, the subproblems are represented as weight vectors. The Euclidean distance between the two subproblems or weight vectors (for example \(\lambda ^{1}\) and \(\lambda ^{2}\)) is calculated as follows:
where m is the number of objectives in the MOP. In this way, the Euclidean distances between a subproblem (\(\lambda ^{\mathrm{j}}\)) to all subproblems are calculated. Then, T numbers of neighborhood subproblems for a jth subproblem (\(\lambda ^{\mathrm{j}}\)) are selected based on their minimum Euclidean distance (see Tables 8, 9).
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Abdul Kadhar, K.M., Baskar, S. A stopping criterion for decomposition-based multi-objective evolutionary algorithms. Soft Comput 22, 253–272 (2018). https://doi.org/10.1007/s00500-016-2331-7
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DOI: https://doi.org/10.1007/s00500-016-2331-7