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A tri-objective preference-based uniform weight design method using Delaunay triangulation

Soft Computing volume 25, pages 9703–9729 (2021)Cite this article

Abstract

User-preference based multi-objective evolutionary algorithms (MOEAs) have attracted much attention recently because it helps save computational cost, make better use of the knowledge offered by the decision-maker, and offer more insight into solutions in the region of interest (ROI). Weight vectors based MOEAs can be converted to their user-preference based versions by offering a set of evenly distributed weight vectors located in ROI. Yet existing weight design methods can only generate weight vectors in the whole unit plane in the weight space. To generate an arbitrary number of weight vectors in ROI, this paper proposes a tri-objective user-preference based uniform weight design method using Delaunay Triangulation (PUWD-DT), so that weight vectors can be fine-tuned to uniformity in ROI. Furthermore, the proposed PUWD-DT based preference method with the achievement scalarizing function is assembled into MOEA/D to convert it into its user-preference based version (MOEA/D+PUWD-DT) and the convergence of population in ROI for optimization problems with irregular shaped Pareto front is also promoted. Finally, the MOEA/D+PUWD-DT is applied to the reservoir flood control operation problem, and our experimental results indicate that the proposed preference-based MOEA method performs better than the state-of-the-art.

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References

  • Adra SF, Griffin I, Fleming PJ (2007) A comparative study of progressive preference articulation techniques for multiobjective optimisation. International conference on evolutionary multi-criterion optimization. Springer, Berlin Heidelberg, Berlin, Heidelberg, pp 908–921

  • Antonio M, Jeremy O, Ludovic H, Paolo RG, Franck M (2018) An application-based characterization of dynamical distance geometry problems. Optimization Letters pp 1–15

  • Bern M, Eppstein D (1970) Mesh generation and optimal triangulation, World Scientific Publishing Co. Pte. Ltd., pp 23–90

  • 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 (1996) Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. Siam J Optim 8(3):631–657

    MathSciNet  Article  Google Scholar 

  • Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley-Interscience series in systems and optimization, John Wiley & Sons, Ltd., Chichester, with a foreword by David E. Goldberg

  • 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 Computat 18(4):577–601

    Article  Google Scholar 

  • Deb K, Kumar A (2007) Light beam search based multi-objective optimization using evolutionary algorithms. In: 2007 IEEE Congress on Evolutionary Computation, pp 2125–2132, 10.1109/CEC.2007.4424735

  • Fang KT (1980) Uniform design-the application of number theoretic method in experiment design. Acta Mathematicae Applicatae Sinica pp 363–371

  • Field DA (1988) Laplacian smoothing and delaunay triangulations. Commun Appl Num Methods 4(6):709–712

    Article  Google Scholar 

  • Fortune S (1997) Voronoi diagrams and delaunay triangulations. Handbook of Discrete and Computational Geometry pp 377–388

  • Ho ESL, Komura T, Tai CL (2010) Spatial relationship preserving character motion adaptation p 1

  • 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

    Article  Google Scholar 

  • Jaszkiewicz A, Slowinski R (1999) The ‘light beam search’ approach - an overview of methodology and applications. Eur J Op Res 113(2):300–314

    Article  Google Scholar 

  • Kaisa M (1998) Nolinear multiobjective optimization. International Series in Operation Research & Management Science, Springer, US

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Li K, Deb K, Yao X (2018) R-metric: evaluating the performance of preference-based evolutionary multiobjective optimization using reference points. IEEE Trans Evol Comput 22(6):821–835

    Article  Google Scholar 

  • Ma X, Qi Y, Li L, Liu F, Jiao L, Wu J (2014) Moea/d with uniform decomposition measurement for many-objective problems. Soft Comput A Fusion Found Methodol Appl 18(12):2541–2564

    Google Scholar 

  • Ma X, Liu F, Qi Y, Li L, Jiao L, Deng X, Wang X, Dong B, Hou Z, Zhang Y (2016) Moea/d with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem. Soft Comput A Fusion Found Methodol Appl 20(12):4999–5023

    Google Scholar 

  • Mohammadi A, Omidvar MN, Li X (2012) Reference point based multi-objective optimization through decomposition. In: 2012 IEEE Congress on Evolutionary Computation, pp 1–8

  • Mohammadi A, Omidvar MN, Li X, Deb K (2014) Integrating user preferences and decomposition methods for many-objective optimization. In: 2014 IEEE Congress on Evolutionary Computation (CEC), pp 421–428

  • Qi Y, Ma X, Yin M, Liu F, Wei J (2014) Moea/d with a delaunay triangulation based weight adjustment. In: Proceedings of the Companion Publication of the 2014 annual conference on genetic and evolutionary computation, ACM, New York, NY, USA, GECCO Comp ’14, pp 93–94, 10.1145/2598394.2598416

  • Qi Y, Yin M, Li X (2016) A delaunay triangulation based density measurement for evolutionary multi-objective optimization. In: Australasian conference on artificial life & computational intelligence, Springer International Publishing, pp 183–192

  • Qi Y, Li X, Yu J, Miao Q (2019) User-preference based decomposition in moea/d without using an ideal point. Swarm Evol Comput 44:597–611

  • Tan YY, Jiao Yc, Li H, Wang Xk (2013) Moea/d + uniform design: a new version of moea/d for optimization problems with many objectives. Comput Op Res 40(6):1648–1660

    MathSciNet  Article  Google Scholar 

  • Wang R, Zhang Q, Zhang T (2016) Decomposition-based algorithms using pareto adaptive scalarizing methods. IEEE Trans Evol Comput 20(6):821–837

    Article  Google Scholar 

  • Wierzbicki AP (1980) The use of reference objectives in multiobjective optimization. Springer, Berlin Heidelberg, Berlin, Heidelberg

    Book  Google Scholar 

  • Wilcoxon F (1945) Individual comparisons by ranking methods. Biomet Bull 1(6):80–83

    Article  Google Scholar 

  • Zapotecas-Martínez S, Coello Coello CA, Aguirre HE, Tanaka K (2019) A review of features and limitations of existing scalable multiobjective test suites. IEEE Trans Evol Comput 23(1):130–142. https://doi.org/10.1109/TEVC.2018.2836912

    Article  Google Scholar 

  • 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 

  • Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithm – a comparative case study. Int Conf Parallel Problem Solving from Nature (PPSN-V), 1998 1498(3):292–301

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 61772392.

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Authors and Affiliations

  1. School of Computer Science and Technology, Xidian University, Xi’an, 710071, China

    Dazhuang Liu, Yutao Qi, Rui Yang, Yining Quan & Qiguang Miao

  2. School of Computer Science and IT, RMIT University, Melbourne, Australia

    Xiaodong Li

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  1. Dazhuang Liu
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  2. Yutao Qi
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  3. Rui Yang
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Correspondence to Yutao Qi.

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Author Dazhuang Liu declares that he has no conflict of interest. Author Yutao Qi declares that he has no conflict of interest. Author Rui Yang declares that he has no conflict of interest. Author Yining Quan declares that he has no conflict of interest. Author Xiaodong Li declares that he has no conflict of interest. Author Qiguang miao declares that he has no conflict of interest.

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Liu, D., Qi, Y., Yang, R. et al. A tri-objective preference-based uniform weight design method using Delaunay triangulation. Soft Comput 25, 9703–9729 (2021). https://doi.org/10.1007/s00500-021-05868-1

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Keywords

  • Decomposition based multi-objective evolutionary optimization
  • Delaunay triangulation
  • Reservoir flood control operation
  • User-preference based evolutionary multi-objective optimization
  • Weight vectors