Skip to main content
SpringerLink
Log in

Adjust weight vectors in MOEA/D for bi-objective optimization problems with discontinuous Pareto fronts

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) is a recently proposed algorithm which is a research focus in the field of multi-objective evolutionary optimization. It decomposes a multi-objective problem into subproblems by mathematic programming methods and applies evolutionary algorithms to optimize the subproblems simultaneously. MOEA/D is good at finding Pareto solutions which are evenly distributed. However, it can be improved for problems with discontinuous Pareto fronts (PF). Many solutions will assemble in breakpoints in this situation. A method for adjusting weight vectors for bi-objective optimization problems with discontinuous PF is proposed. Firstly, this method detects the weight vectors which need to be adjusted using a property of MOEA/D. Secondly, the reserved vectors are divided into several subsets. Thirdly, after calculating the ideal number of vectors in each subset, vectors are adjusted evenly. Lastly, the corresponding solutions are updated by a linear interpolation. Numerical experiment shows the proposed method obtains good diversity and convergence on approached PF.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  • Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76

    Article  Google Scholar 

  • Carvalho RD, Saldanha RR, Gomes B, Lisboa AC, Martins A (2012) A multi-objective evolutionary algorithm based on decomposition for optimal design of Yagi–Uda antennas. IEEE Trans Magn 48(2):803–806

    Article  Google Scholar 

  • Cheng R, Jin Y, Olhofer M, Sendhoff B (2016) A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 20(5):773–791

    Article  Google Scholar 

  • Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657

    Article  MathSciNet  MATH  Google Scholar 

  • Deb K, Jain H (2014) An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans Evol Comput 18(4):577–601

    Article  Google Scholar 

  • Deb K, Mohan M, Mishra S (2005) Evaluating the \(\varepsilon \)-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evol Comput 13(4):501–525

    Article  Google Scholar 

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  • Deb K, Pratap A, Meyarivan T (2001) Constrained test problems for multi-objective evolutionary optimization. Paper presented at the evolutionary multi-criterion optimization

  • Dipama J, Teyssedou A, Aubé F, Lizon-A-Lugrin L (2010) A grid based multi-objective evolutionary algorithm for the optimization of power plants. Appl Therm Eng 30(8):807–816

    Article  Google Scholar 

  • García S, Molina D, Lozano M, Herrera F (2009) A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J Heuristics 15(6):617–644

    Article  MATH  Google Scholar 

  • Gu F-Q, Liu H-L (2010) A novel weight design in multi-objective evolutionary algorithm. Paper presented at the 2010 international conference on computational intelligence and security (CIS)

  • Hillermeier C (2001) Nonlinear multiobjective optimization: a generalized homotopy approach, vol 135. Springer, Berlin

    Book  MATH  Google Scholar 

  • Horn J, Nafpliotis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. Paper presented at proceedings of the first IEEE conference on evolutionary computation, 1994. IEEE world congress on computational intelligence. Orlando, FL

  • Jiang S, Cai Z, Zhang J, Ong Y-S (2011) Multiobjective optimization by decomposition with Pareto-adaptive weight vectors. Paper presented at 2011 seventh international conference on natural computation (ICNC)

  • Konstantinidis A, Yang K (2011) Multi-objective energy-efficient dense deployment in Wireless Sensor Networks using a hybrid problem-specific MOEA/D. Appl Soft Comput 11(6):4117–4134. doi:10.1016/j.asoc.2011.02.031

    Article  Google Scholar 

  • Kukkonen S, Deb K (2006) A fast and effective method for pruning of non-dominated solutions in many-objective problems Parallel Problem Solving from Nature-PPSN IX. Springer, Berlin, pp 553–562

    Google Scholar 

  • Li H, Landa-Silva D (2011) An adaptive evolutionary multi-objective approach based on simulated annealing. Evol Comput 19(4):561–595

    Article  Google Scholar 

  • Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302

    Article  Google Scholar 

  • Liu Y, Gong D, Sun X, Zhang Y (2017) Many-objective evolutionary optimization based on reference points. Appl Soft Comput 50:344–355

    Article  Google Scholar 

  • Ma X, Liu F, Qi Y, Li L, Jiao L, Deng X et al. (2015) MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem. Soft Comput. doi:10.1007/s00500-015-1789-z

  • Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscip Optim 25(2):86–98

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, Boston

    MATH  Google Scholar 

  • Phan DH, Suzuki J (2013) R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization. Paper presented at 2013 IEEE congress on evolutionary computation (CEC)

  • Qi Y, Ma X, Liu F, Jiao L, Sun J, Wu J (2014) MOEA/D with adaptive weight adjustment. Evol Comput 22(2):231–264. doi:10.1162/EVCO_a_00109

    Article  Google Scholar 

  • Schaffer JD (1985) Some experiments in machine learning using vector evaluated genetic algorithms. Vanderbilt University, Nashville

    Google Scholar 

  • Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248

    Article  Google Scholar 

  • Takahama T, Sakai S (2006) Constrained optimization by the \(\varepsilon \) constrained differential evolution with gradient-based mutation and feasible elites. Paper presented at the 2006 IEEE congress on evolutionary computation (CEC), Vancouver, BC

  • Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 17(5):721–736

    Article  Google Scholar 

  • Zhang CJ, Lin Q, Gao L (2015) A novel adaptive \(\varepsilon \)-constrained method for constrained problem. Paper presented at the proceedings of the 18th Asia Pacific symposium on intelligent and evolutionary systems, vol 1. Singapore

  • Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731

    Article  Google Scholar 

  • Zhang Q, Liu W, Li H (2009) The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances. Paper presented at the IEEE congress on evolutionary computation, Trondheim

  • Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2008) Multiobjective optimization test instances for the CEC 2009 special session and competition. technical report. University of Essex, Colchester, UK and Nanyang technological University, Singapore

  • Zhang Y, Yang R, Zuo J, Jing X (2015) Enhancing MOEA/D with uniform population initialization, weight vector design and adjustment using uniform design. J Syst Eng Electron 26(5):1010–1022

    Article  Google Scholar 

  • Zhu Y, Wang J, Qu B (2014) Multi-objective economic emission dispatch considering wind power using evolutionary algorithm based on decomposition. Int J Electr Power Energy Syst 63:434–445. doi:10.1016/j.ijepes.2014.06.027

    Article  Google Scholar 

  • Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. Paper presented at the parallel problem solving from nature-PPSN VIII

  • Zitzler E, Laumanns M, Thiele L (2001) SPEA2: Improving the strength Pareto evolutionary algorithm. Technical Report. Zürich, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH), Institut für Technische Informatik und Kommunikationsnetze (TIK)

  • Zitzler E, Thiele L (1998a) An evolutionary algorithm for multiobjective optimization: The strength pareto approach. Zürich, Switzerland: Technical Report 43, Eidgenössische Technische Hochschule Zürich (ETH), Institut für Technische Informatik und Kommunikationsnetze (TIK)

  • Zitzler E, Thiele L (1998b) Multiobjective optimization using evolutionary algorithms—a comparative case study. Paper presented at the international conference on parallel problem solving from nature

  • Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132

    Article  Google Scholar 

Download references

Acknowledgements

This research work is supported by the National Key Technology Support Program under Grant No. 2015BAF01B04, and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51421062 and 61232008, and China Scholarship Council (CSC).

Author information

Authors and Affiliations

  1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Chunjiang Zhang & Liang Gao

  2. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Chunjiang Zhang

  3. Department of Computer Science, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

    Kay Chen Tan

  4. Department of Industrial and Systems Engineering, National University of Singapore, Singapore, 117576, Singapore

    Loo Hay Lee

Authors
  1. Chunjiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kay Chen Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Loo Hay Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Liang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liang Gao.

Ethics declarations

Conflict of interest

All author declares that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by V. Loia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, C., Tan, K.C., Lee, L.H. et al. Adjust weight vectors in MOEA/D for bi-objective optimization problems with discontinuous Pareto fronts. Soft Comput 22, 3997–4012 (2018). https://doi.org/10.1007/s00500-017-2609-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-017-2609-4

Keywords

Search

Navigation