Abstract
In multi-objective and many-objective optimization, weight vectors are particularly crucial to the performance of decomposition-based optimization algorithms. The uniform weight vectors are not suitable for complex Pareto fronts (PFs), so it is necessary to improve the distribution of weight vectors. Besides, the balance between convergence and diversity is a difficult issue as well in multi-objective optimization, and it becomes increasingly important with the augment of the number of objectives. To address these issues, a self-adaptive weight vector adjustment strategy for decomposition-based multi-objective differential evolution algorithm (AWDMODE) is proposed. In order to ensure that the guidance of weight vectors becomes accurate and effective, the adaptive adjustment strategy is introduced. This strategy distinguishes the shapes and adjusts weight vectors dynamically, which can ensure that the guidance of weight vectors becomes accurate and effective. In addition, a self-learning strategy is adopted to produce more non-dominated solutions and balance the convergence and diversity. The experimental results indicate that AWDMODE outperforms the compared algorithms on WFG suites test instances, and shows a great potential when handling the problems whose PFs are scaled with different ranges in each objective.
This is a preview of subscription content, log in via an institution to check access.
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
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. https://doi.org/10.1109/TEVC.2016.2519378
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. https://doi.org/10.1109/TEVC.2013.2281535
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. https://doi.org/10.1109/4235.996017
Fan R, Wei L, Li X, Hu Z (2018) A novel multi-objective PSO algorithm based on completion-checking. J Intell Fuzzy Syst 34(1):321–333. https://doi.org/10.3233/JIFS-171291
Fan R, Wei L, Sun H, Hu Z (2019) An enhanced reference vectors-based multi-objective evolutionary algorithm with neighborhood-based adaptive adjustment. Neural Comput Appl. https://doi.org/10.1007/s00521-019-04660-5
Gee SB, Arokiasami WA, Jiang J, Tan KC (2016) Decomposition-based multi-objective evolutionary algorithm for vehicle routing problem with stochastic demands. Soft Comput 20(9):3443–3453. https://doi.org/10.1007/s00500-015-1830-2
Hu Z, Yang J, Sun H, Wei L, Zhao Z (2017) An improved multi-objective evolutionary algorithm based on environmental and history information. Neurocomputing 222(C):170–182. https://doi.org/10.1016/j.neucom.2016.10.014
Hu Z, Wei Z, Sun H, Yang J, Wei L (2019a) Optimization of metal rolling control using soft computing approaches: a review. Arch Comput Methods Eng 10:1–17. https://doi.org/10.1007/s11831-019-09380-6
Hu Z, Yang J, Cui H, Wei L, Fan R (2019b) MOEA3D: a MOEA based on dominance and decomposition with probability distribution model. Soft Comput 23(4):1219–1237. https://doi.org/10.1007/s00500-017-2840-z
Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506. https://doi.org/10.1109/TEVC.2005.861417
Ishibuchi H, Setoguchi Y, Masuda H, Nojima Y (2017) Performance of decomposition-based many-objective algorithms strongly depends on pareto front shapes. IEEE Trans Evol Comput 21(2):169–190. https://doi.org/10.1109/TEVC.2016.2587749
Jiang S, Yang S (2016) An improved multiobjective optimization evolutionary algorithm based on decomposition for complex pareto fronts. IEEE Trans Cybern 46(2):421–437. https://doi.org/10.1109/TCYB.2015.2403131
Jiang S, Cai Z, Zhang J, Ong YS (2011) Multiobjective optimization by decomposition with pareto-adaptive weight vectors. In: 2011 seventh international conference on natural computation (ICNC). IEEE, vol 3, pp 1260–1264. https://doi.org/10.1109/ICNC.2011.6022367
Lee LH, Chew EP, Yu Q, Li H, Liu Y (2014) A study on multi-objective particle swarm optimization with weighted scalarizing functions. In: Proceedings of the 2014 winter simulation conference. IEEE Press, pp 3718–3729.https://doi.org/10.1109/WSC.2014.7020200
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. https://doi.org/10.1109/TEVC.2008.925798
Li K, Zhang Q, Kwong S, Li M, Wang R (2014) Stable matching-based selection in evolutionary multiobjective optimization. IEEE Trans Evol Comput 18(6):909–923. https://doi.org/10.1109/TEVC.2013.2293776
Li F, Liu J, Tan S, Yu X (2015a) R2-MOPSO: a multi-objective particle swarm optimizer based on R2-indicator and decomposition. In: 2015 IEEE congress on evolutionary computation (CEC), pp 3148–3155. https://doi.org/10.1109/CEC.2015.7257282
Li K, Deb K, Zhang Q, Kwong S (2015b) An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans Evol Comput 19(5):694–716. https://doi.org/10.1109/TEVC.2014.2373386
Liu HL, Gu F, Cheung Y (2010) T-MOEA/D: MOEA/D with objective transform in multi-objective problems. In: 2010 international conference of information science and management engineering, vol 2, pp 282–285. https://doi.org/10.1109/ISME.2010.274
Liu Y, Gong D, Sun X, Zhang Y (2017) Many-objective evolutionary optimization based on reference points. Appl Soft Comput 50:344–355. https://doi.org/10.1016/j.asoc.2016.11.009
Medina MA, Das S, Coello CAC, Ramírez JM (2014) Decomposition-based modern metaheuristic algorithms for multi-objective optimal power flow—a comparative study. Eng Appl Artif Intell 32:10–20. https://doi.org/10.1016/j.engappai.2014.01.016
Nag K, Pal T, Pal NR (2015) ASMiGA: an archive-based steady-state micro genetic algorithm. IEEE Trans Cybern 45(1):40–52. https://doi.org/10.1109/TCYB.2014.2317693
Purshouse RC, Fleming PJ (2007) On the evolutionary optimization of many conflicting objectives. IEEE Trans Evol Comput 11(6):770–784. https://doi.org/10.1109/TEVC.2007.910138
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
Trivedi A, Srinivasan D, Pal K, Saha C, Reindl T (2015) Enhanced multiobjective evolutionary algorithm based on decomposition for solving the unit commitment problem. IEEE Trans Ind Inf 11(6):1346–1357. https://doi.org/10.1109/TII.2015.2485520
Wang R, Purshouse RC, Fleming PJ (2015) Preference-inspired co-evolutionary algorithms using weight vectors. Eur J Oper Res 243(2):423–441. https://doi.org/10.1016/j.ejor.2014.05.019
Wang R, Xiong J, Ishibuchi H, Wu G, Zhang T (2017a) On the effect of reference point in MOEA/D for multi-objective optimization. Appl Soft Comput 58:25–34. https://doi.org/10.1016/j.asoc.2017.04.002
Wang Z, Zhang Q, Li H, Ishibuchi H, Jiao L (2017b) On the use of two reference points in decomposition based multiobjective evolutionary algorithms. Swarm Evol Comput 34:89–102. https://doi.org/10.1016/j.swevo.2017.01.002
Wei LX, Li X, Fan R, Sun H, Hu ZY (2018) A hybrid multiobjective particle swarm optimization algorithm based on R2 indicator. IEEE Access 6:14710–14721. https://doi.org/10.1109/ACCESS.2018.2812701
Ying S, Li L, Wang Z, Li W, Wang W (2017) An improved decomposition-based multiobjective evolutionary algorithm with a better balance of convergence and diversity. Appl Soft Comput 57:627–641. https://doi.org/10.1016/j.asoc.2017.03.041
Yuan Y, Xu H, Wang B, Zhang B, Yao X (2016) Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans Evol Comput 20(2):180–198. https://doi.org/10.1109/TEVC.2015.2443001
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731. https://doi.org/10.1109/TEVC.2007.892759
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evol Comput 3(4):257–271. https://doi.org/10.1109/4235.797969
Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength pareto evolutionary algorithm. TIK-report 103
Acknowledgements
The authors would like to thank the editor and reviewers for their helpful comments and suggestions to improve the quality of this paper. This work was supported by the Natural Science Foundation of Hebei Province (No. F2016203249).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fan, R., Wei, L., Li, X. et al. Self-adaptive weight vector adjustment strategy for decomposition-based multi-objective differential evolution algorithm. Soft Comput 24, 13179–13195 (2020). https://doi.org/10.1007/s00500-020-04732-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-04732-y