OBJECTIVE SPACE NORMALIZATION IN EVOLUTIONARY MULTI-OBJECTIVE OPTIMIZATION

Objective space normalization is crucial for handling real-world multiobjective optimization problems (MOPs) with differently-scaled objectives, as it allows multiobjective evolutionary algorithms (MOEAs) to obtain uniformly-distributed and well-converged solutions.

This thesis addresses five research questions related to objective space normalization in multiobjective optimization and makes several key contributions to the field of multiobjective optimization. First, we provide a comprehensive survey of objective space normalization methods, offering insights into the current state of the art. Second, we explore the theoretical and empirical connections between normalization and weight vector scaling, and their impact on decomposition-based MOEAs. Third, we investigate the effectiveness of different normalization methods in MOEA/D and propose two new strategies for enhancing existing normalization methods. Fourth, we analyze the limitations of existing evaluation methods and introduce a new metric for investigating normalization methods, allowing for a more comprehensive and accurate comparison of different techniques. Finally, we propose a preference-based nonlinear normalization method that aligns the solution set with the desired distribution, demonstrating its flexibility and effectiveness in various MOEAs.

Overall, this thesis contributes to a deeper understanding of the role of normalization in MOEAs and offers practical guidance for researchers and practitioners in the field of multiobjective optimization. The insights and advancements provided in this thesis enhance the performance and applicability of MOEAs in solving real-world problems with differently-scaled objectives.

[1] Q. Zhang, H. Li, D. Maringer, and E. Tsang, “MOEA/D with NBI-styleTchebycheff approach for portfolio management”, in Proceedings of the IEEECongress on Evolutionary Computation, Barcelona, Spain, Jul. 2010, pp. 1–8.
[2] H. Jain and K. Deb, “An evolutionary many-objective optimization algorithmusing reference-point based nondominated sorting approach, part II: Handlingconstraints and extending to an adaptive approach.” IEEE Transactions onEvolutionary Computation, vol. 18, no. 4, pp. 602–622, Aug. 2014.
[3] R. Tanabe and H. Ishibuchi, “An easy-to-use real-world multi-objectiveoptimization problem suite”, Applied Soft Computing, vol. 89, Apr. 2020.
[4] S. Huband, P. Hingston, L. Barone, and L. While, “A review of multiobjectivetest problems and a scalable test problem toolkit”, IEEE Transactions onEvolutionary Computation, vol. 10, no. 5, pp. 477–506, Oct. 2006.
[5] R. Cheng, M. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “Abenchmark test suite for evolutionary many-objective optimization”, Complex& Intelligent Systems, vol. 3, no. 1, pp. 67–81, Mar. 2017.
[6] H. Li, K. Deb, Q. Zhang, P. Suganthan, and L. Chen, “Comparison betweenMOEA/D and NSGA-III on a set of novel many and multi-objectivebenchmark problems with challenging difficulties”, Swarm and EvolutionaryComputation, vol. 46, pp. 104–117, May 2019.
[7] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable multi-objectiveoptimization test problems”, in Proceedings of the Congress on EvolutionaryComputation, IEEE, vol. 1, Honolulu, USA, May 2002, pp. 825–830.
[8] H. Ishibuchi, K. Doi, and Y. Nojima, “On the effect of normalization inMOEA/D for multi-objective and many-objective optimization”, Complex &Intelligent Systems, vol. 3, no. 4, pp. 279–294, Dec. 2017.
[9] L. Paquete and T. Stützle, “A two-phase local search for the biobjectivetraveling salesman problem”, in Proceedings of Evolutionary Multi-CriterionOptimization, Faro, Portugal, Apr. 2003, pp. 479–493.
[10] K. Deb and K. Miettinen, “A review of nadir point estimation proceduresusing evolutionary approaches: A tale of dimensionality reduction”, en, inProceedings of the International Conference on Multiple Criteria DecisionMaking, Auckland, New Zealand, Jan. 2009, pp. 1–14.
[11] K. Deb, K. Miettinen, and S. Chaudhuri, “Toward an estimation of nadirobjective vector using a hybrid of evolutionary and local search approaches”,IEEE Transactions on Evolutionary Computation, vol. 14, no. 6, pp. 821–841,Dec. 2010.
[12] K. Miettinen, M. M. Mäkelä, and K. Kaario, “Experiments with classificationbasedscalarizing functions in interactive multiobjective optimization”, EuropeanJournal of Operational Research, vol. 175, no. 2, pp. 931–947, Dec.2006.
[13] Q. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithmbased on decomposition”, IEEE Transactions on Evolutionary Computation,vol. 11, no. 6, pp. 712–731, Dec. 2007.
[14] K. Deb and H. Jain, “An evolutionary many-objective optimization algorithmusing reference-point-based nondominated sorting approach, part i:Solving problems with box constraints.” IEEE Transactions on EvolutionaryComputation, vol. 18, no. 4, pp. 577–601, Aug. 2014.
[15] S. Liu, Q. Lin, K.-C. Wong, C. A. C. Coello, J. Li, Z. Ming, and J. Zhang, “Aself-guided reference vector strategy for many-objective optimization”, IEEETransactions on Cybernetics, vol. 52, no. 2, pp. 1164–1178, Feb. 2020.
[16] J. Blank and K. Deb, “Pymoo: Multi-objective optimization in Python”,IEEE Access, vol. 8, pp. 89 497–89 509, Apr. 2020.
[17] Y.-H. Zhang, Y.-J. Gong, T.-L. Gu, H.-Q. Yuan, W. Zhang, S. Kwong, andJ. Zhang, “DECAL: Decomposition-based coevolutionary algorithm for manyobjectiveoptimization”, IEEE Transactions on Cybernetics, vol. 49, no. 1,pp. 27–41, Jan. 2019.
[18] Y. Yuan, H. Xu, B. Wang, B. Zhang, and X. Yao, “Balancing convergenceand diversity in decomposition-based many-objective optimizers”, IEEETransactions on Evolutionary Computation, vol. 20, no. 2, pp. 180–198, Apr.2015.
[19] Y. Yuan, H. Xu, B. Wang, and X. Yao, “A new dominance relation-basedevolutionary algorithm for many-objective optimization”, IEEE Transactionson Evolutionary Computation, vol. 20, no. 1, pp. 16–37, Feb. 2016.
[20] M. Elarbi, S. Bechikh, A. Gupta, L. B. Said, and Y.-S. Ong, “A newdecomposition-based nsga-ii for many-objective optimization”, IEEE Transactionson Systems, Man, and Cybernetics: Systems, vol. 48, no. 7, pp. 1191–1210, Jul. 2017.
[21] R. Wang, Z. Zhou, H. Ishibuchi, T. Liao, and T. Zhang, “Localized weightedsum method for many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 22, no. 1, pp. 3–18, Feb. 2016.
[22] S. Jiang and S. Yang, “A strength Pareto evolutionary algorithm basedon reference direction for multiobjective and many-objective optimization”,IEEE Transactions on Evolutionary Computation, vol. 21, no. 3, pp. 329–346,Jun. 2017.
[23] H.-L. Liu, L. Chen, Q. Zhang, and K. Deb, “Adaptively allocating search effortin challenging many-objective optimization problems”, IEEE Transactionson Evolutionary Computation, vol. 22, no. 3, pp. 433–448, Jun. 2018.
[24] S. Yang, M. Li, X. Liu, and J. Zheng, “A grid-based evolutionary algorithmfor many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 17, no. 5, pp. 721–736, Oct. 2013.
[25] X. Zhang, Y. Tian, and Y. Jin, “A knee point-driven evolutionary algorithmfor many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 19, no. 6, pp. 761–776, Dec. 2015.
[26] Y. Liu, D. Gong, J. Sun, and Y. Jin, “A many-objective evolutionary algorithmusing a one-by-one selection strategy”, IEEE Transactions on Cybernetics,vol. 47, no. 9, pp. 2689–2702, Sep. 2017.
[27] Y. Liu, H. Ishibuchi, Y. Nojima, N. Masuyama, and K. Shang, “Improving1by1EA to handle various shapes of Pareto fronts”, in Proceedings of theInternational Conference on Parallel Problem Solving from Nature, Coimbra,Portugal, Sep. 2018, pp. 311–322.
[28] A. Jaszkiewicz, “Evaluation of multiple objective metaheuristics”, in Metaheuristicsfor Multiobjective Optimisation, vol. 535, Berlin, Heidelberg: Springer,2004, pp. 65–89.
[29] L. He, Y. Nan, K. Shang, and H. Ishibuchi, “A study of the naïve objectivespace normalization method in MOEA/D”, in IEEE Symposium Series onComputational Intelligence, Xiamen, China, Dec. 2019, pp. 1834–1840.
[30] H. Ishibuchi, R. Imada, Y. Setoguchi, and Y. Nojima, “Performance comparisonof NSGA-II and NSGA-III on various many-objective test problems”, inProceedings of the IEEE Congress on Evolutionary Computation, Vancouver,BC, Canada, Jul. 2016, pp. 3045–3052.
[31] K. Li, K. Deb, Q. Zhang, and S. Kwong, “An evolutionary many-objectiveoptimization algorithm based on dominance and decomposition”, IEEETransactions on Evolutionary Computation, vol. 19, no. 5, pp. 694–716, Oct.2014.
[32] H. Ishibuchi, Y. Setoguchi, H. Masuda, and Y. Nojima, “Performance ofdecomposition-based many-objective algorithms strongly depends on paretofront shapes”, IEEE Transactions on Evolutionary Computation, vol. 21,no. 2, pp. 169–190, Apr. 2017.
[33] Y. Zhou, Y. Xiang, Z. Chen, J. He, and J. Wang, “A scalar projectionand angle-based evolutionary algorithm for many-objective optimizationproblems”, IEEE Transactions on Cybernetics, vol. 49, no. 6, pp. 2073–2084,Jun. 2019.
[34] L. He, H. Ishibuchi, A. Trivedi, H. Wang, Y. Nan, and D. Srinivasan, “Asurvey of normalization methods in multiobjective evolutionary algorithms”,IEEE Transactions on Evolutionary Computation, vol. 25, no. 6, pp. 1028–1048, Dec. 2021.
[35] L. He, K. Shang, Y. Nan, H. Ishibuchi, and D. Srinivasan, “Relation betweenobjective space normalization and weight vector scaling in decompositionbasedmulti-objective evolutionary algorithms”, IEEE Transactions on EvolutionaryComputation, 2022 (Early Access).
[36] L. He, H. Ishibuchi, A. Trivedi, and D. Srinivasan, “Dynamic normalizationin MOEA/D for multiobjective optimization”, in Proceedings of the IEEECongress on Evolutionary Computation, Glasgow, Scotland, United Kingdom,Jul. 2020.
[37] L. He, K. Shang, and H. Ishibuchi, “Simultaneous use of two normalizationmethods in decomposition-based multi-objective evolutionary algorithms”,Applied Soft Computing, vol. 92, Jul. 2020.
[38] L. He, H. Ishibuchi, and D. Srinivasan, “Metric for evaluating normalizationmethods in multiobjective optimization”, in Proceedings of the Genetic andEvolutionary Computation Conference, 2021, pp. 403–411.
[39] L. He, Y. Nan, H. Ishibuchi, and D. Srinivasan, “Preference-based nonlinearnormalization for multiobjective optimization”, in Proceedings of EvolutionaryMulti-Criterion Optimization, Leiden, The Netherlands, Mar. 2023, pp. 563–577.
[40] Y. Tian, R. Cheng, X. Zhang, F. Cheng, and Y. Jin, “An indicator-basedmultiobjective evolutionary algorithm with reference point adaptation forbetter versatility”, IEEE Transactions on Evolutionary Computation, vol. 22,no. 4, pp. 609–622, Aug. 2018.
[41] A. Habib, H. K. Singh, T. Chugh, T. Ray, and K. Miettinen, “A multiplesurrogate assisted decomposition-based evolutionary algorithm for expensivemulti/many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 23, no. 6, pp. 1000–1014, Dec. 2019.
[42] R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, “A reference vector guidedevolutionary algorithm for many-objective optimization”, IEEE Transactionson Evolutionary Computation, vol. 20, no. 5, pp. 773–791, Oct. 2016.
[43] H. Li, J. Sun, Q. Zhang, and Y. Shui, “Adjustment of weight vectors ofpenalty-based boundary intersection method in MOEA/D”, in Proceedingsof Evolutionary Multi-Criterion Optimization, East Lansing, MI, USA, Mar.2019, pp. 91–100.
[44] Y. Sun, G. G. Yen, and Z. Yi, “IGD indicator-based evolutionary algorithm formany-objective optimization problems”, IEEE Transactions on EvolutionaryComputation, vol. 23, no. 2, pp. 173–187, Apr. 2019.
[45] Y. Sun, B. Xue, M. Zhang, and G. G. Yen, “A new two-stage evolutionary algorithmfor many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 23, no. 5, pp. 748–761, Oct. 2019.
[46] K. Ikeda, H. Kita, and S. Kobayashi, “Failure of Pareto-based MOEAs:Does non-dominated really mean near to optimal?” In Proceedings of theIEEE Congress on Evolutionary Computation, Seoul, South Korea, May 2001,pp. 957–962.
[47] R. C. Purshouse and P. J. Fleming, “On the evolutionary optimization of manyconflicting objectives”, IEEE Transactions on Evolutionary Computation,vol. 11, no. 6, pp. 770–784, Dec. 2007.
[48] M. Li, S. Yang, and X. Liu, “Bi-goal evolution for many-objective optimizationproblems”, Artificial Intelligence, vol. 228, pp. 45–65, Nov. 2015.
[49] H. Ishibuchi, T. Fukase, N. Masuyama, and Y. Nojima, “Effects of dominanceresistant solutions on the performance of evolutionary multi-objective andmany-objective algorithms”, in Proceedings of the Genetic and EvolutionaryComputation Conference, Cancún, Mexico, Jun. 2020, pp. 507–515.
[50] Z. Wang, Y.-S. Ong, and H. Ishibuchi, “On scalable multiobjective test problemswith hardly dominated boundaries”, IEEE Transactions on EvolutionaryComputation, vol. 23, no. 2, pp. 217–231, Apr. 2019.
[51] M. Li, L. Liu, and D. Lin, “A fast steady-state ϵ-dominance multi-objectiveevolutionary algorithm”, Computational Optimization and Applications, vol. 48,no. 1, pp. 109–138, Jan. 2011.
[52] T. Kondoh, T. Tatsukawa, A. Oyama, T. Watanabe, and K. Fujii, “Effects ofdiscrete design-variable precision on real-coded genetic algorithm”, in IEEESymposium Series on Computational Intelligence, Athens, Greece, Dec. 2016,pp. 1–8.
[53] M. Laumanns, L. Thiele, K. Deb, and E. Zitzler, “Combining convergenceand diversity in evolutionary multiobjective optimization”, EvolutionaryComputation, vol. 10, no. 3, pp. 263–282, Sep. 2002.
[54] K. S. Bhattacharjee, H. K. Singh, T. Ray, and Q. Zhang, “Decompositionbased evolutionary algorithm with a dual set of reference vectors”, in Proceedingsof the IEEE Congress on Evolutionary Computation, San Sebastian,Spain, Jun. 2017, pp. 105–112.
[55] L. Chen, K. Deb, H.-L. Liu, and Q. Zhang, “Effect of objective normalizationand penalty parameter on penalty boundary intersection decompositionbased evolutionary many-objective optimization algorithms”, EvolutionaryComputation, pp. 1–25, Jun. 2020.
[56] X. Cai, Z. Mei, Z. Fan, and Q. Zhang, “A constrained decomposition approachwith grids for evolutionary multiobjective optimization”, IEEE Transactionson Evolutionary Computation, vol. 22, no. 4, pp. 564–577, Aug. 2018.
[57] Y. Tian, R. Cheng, X. Zhang, Y. Su, and Y. Jin, “A strengthened dominancerelation considering convergence and diversity for evolutionary many-objectiveoptimization”, IEEE Transactions on Evolutionary Computation, vol. 23,no. 2, pp. 331–345, Apr. 2019.
[58] X. He, Y. Zhou, Z. Chen, and Q. Zhang, “Evolutionary many-objectiveoptimization based on dynamical decomposition”, IEEE Transactions onEvolutionary Computation, vol. 23, no. 3, pp. 361–375, Jun. 2018.
[59] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjectivegenetic algorithm: NSGA-II”, IEEE Transactions on EvolutionaryComputation, vol. 6, no. 2, pp. 182–197, Apr. 2002.
[60] J. Blank, K. Deb, and P. C. Roy, “Investigating the normalization procedureof NSGA-III”, in Proceedings of Evolutionary Multi-Criterion Optimization,vol. 11411, East Lansing, MI, USA, Mar. 2019, pp. 229–240.
[61] Z. Fan, W. Li, X. Cai, H. Li, K. Hu, and H. Yin, “An improved ideal pointsetting in multiobjective evolutionary algorithm based on decomposition”, inInternational Conference on Industrial Informatics - Computing Technology,Intelligent Technology, Industrial Information Integration, Wuhan, China,Dec. 2015, pp. 63–70.
[62] H. Ishibuchi, K. Doi, and Y. Nojima, “Reference point specification inMOEA/D for multi-objective and many-objective problems”, in Proceedingsof the International Conference on Systems, Man and Cybernetics, Budapest,Hungary, Oct. 2016, pp. 4015–4020.
[63] R. Wang, J. Xiong, H. Ishibuchi, G. Wu, and T. Zhang, “On the effect ofreference point in MOEA/D for multi-objective optimization”, Applied SoftComputing, vol. 58, pp. 25–34, Sep. 2017.
[64] K. Deb, S. Chaudhuri, and K. Miettinen, “Towards estimating nadir objectivevector using evolutionary approaches”, in Proceedings of the Genetic andEvolutionary Computation Conference, Seattle, Washington, USA, Jul. 2006,pp. 643–650.
[65] H. Fukumoto and A. Oyama, “Impact of estimation method of ideal/nadirpoints on practically-constrained multi-objective optimization problems fordecomposition-based multi-objective evolutionary algorithm”, in IEEE SymposiumSeries on Computational Intelligence, Xiamen, China, Dec. 2019,pp. 2138–2145.
[66] H. Wang, L. Jiao, and X. Yao, “Two_Arch2: An improved two-archive algorithmfor many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 19, no. 4, pp. 524–541, Aug. 2015.
[67] Y. Xiang, Y. Zhou, M. Li, and Z. Chen, “A vector angle-based evolutionary algorithmfor unconstrained many-objective optimization”, IEEE Transactionson Evolutionary Computation, vol. 21, no. 1, pp. 131–152, Feb. 2017.
[68] L. Chen, K. Deb, and H.-L. Liu, “Explicit control of implicit parallelism indecomposition-based evolutionary many-objective optimization algorithms”,IEEE Computational Intelligence Magazine, vol. 14, no. 4, pp. 52–64, Nov.2019.
[69] E. Zitzler and S. Künzli, “Indicator-based selection in multiobjective search”,in Proceedings of the International Conference on Parallel Problem Solvingfrom Nature, vol. 3242, Birmingham, United Kingdom, Sep. 2004, pp. 832–842.
[70] N. Chen, W.-N. Chen, Y.-J. Gong, Z.-H. Zhan, J. Zhang, Y. Li, and Y.-S.Tan, “An evolutionary algorithm with double-level archives for multiobjectiveoptimization”, IEEE Transactions on Cybernetics, vol. 45, no. 9, pp. 1851–1863, Sep. 2015.
[71] K. Li, S. Kwong, Q. Zhang, and K. Deb, “Interrelationship-based selectionfor decomposition multiobjective optimization”, IEEE Transactions onCybernetics, vol. 45, no. 10, pp. 2076–2088, Oct. 2015.
[72] Q. Lin, G. Jin, Y. Ma, K.-C. Wong, C. A. C. Coello, J. Li, J. Chen, and J.Zhang, “A diversity-enhanced resource allocation strategy for decompositionbasedmultiobjective evolutionary algorithm”, IEEE Transactions on Cybernetics,vol. 48, no. 8, pp. 2388–2401, Aug. 2018.
[73] X. Cai, Z. Mei, and Z. Fan, “A decomposition-based many-objective evolutionaryalgorithm with two types of adjustments for direction vectors”, IEEETransactions on Cybernetics, vol. 48, no. 8, pp. 2335–2348, Aug. 2018.
[74] Y. Qi, Q. Zhang, X. Ma, Y. Quan, and Q. Miao, “Utopian point baseddecomposition for multi-objective optimization problems with complicatedPareto fronts”, Applied Soft Computing, vol. 61, pp. 844–859, Dec. 2017.
[75] R. Tanabe and H. Ishibuchi, “A decomposition-based evolutionary algorithmfor multi-modal multi-objective optimization”, in Proceedings of theInternational Conference on Parallel Problem Solving from Nature, Coimbra,Portugal, Sep. 2018, pp. 249–261.
[76] D. Sharma, S. Z. Basha, and S. A. Kumar, “Diversity over dominance approachfor many-objective optimization on reference-points-based framework”, inProceedings of Evolutionary Multi-Criterion Optimization, East Lansing, MI,USA, Mar. 2019, pp. 278–290.
[77] K. Shang and H. Ishibuchi, “A new hypervolume-based evolutionary algorithmfor many-objective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 24, no. 5, pp. 839–852, 2020.
[78] A. B. Ruiz, R. Saborido, and M. Luque, “A preference-based evolutionaryalgorithm for multiobjective optimization: The weighting achievement scalarizingfunction genetic algorithm”, Journal of Global Optimization, vol. 62,no. 1, pp. 101–129, May 2015.
[79] J. Luo, X. Sun, Y. Qi, and J. Xie, “Approximating the irregularly shapedPareto front of multi-objective reservoir flood control operation problem”,Applied Mathematical Modelling, vol. 54, pp. 502–516, Feb. 2018.
[80] D. Sharma and P. K. Shukla, “Line-prioritized environmental selection andnormalization scheme for many-objective optimization using reference-linesbasedframework”, Swarm and Evolutionary Computation, vol. 51, Dec. 2019.
[81] H. Seada and K. Deb, “A unified evolutionary optimization procedure forsingle, multiple, and many objectives”, IEEE Transactions on EvolutionaryComputation, vol. 20, no. 3, pp. 358–369, Jun. 2016.
[82] R. Wang, H. Ishibuchi, Y. Zhang, X. Zheng, and T. Zhang, “On the effectof localized PBI method in MOEA/D for multi-objective optimization”, inIEEE Symposium Series on Computational Intelligence, Athens, Greece, Dec.2016, pp. 1–8.
[83] S. Zapotecas Martínez, V. A. Sosa Hernández, H. Aguirre, K. Tanaka, andC. A. Coello Coello, “Using a family of curves to approximate the Pareto frontof a multi-objective optimization problem”, in Proceedings of the InternationalConference on Parallel Problem Solving from Nature, vol. 8672, Ljubljana,Slovenia, Sep. 2014, pp. 682–691.
[84] R. Hernández Gómez, C. A. Coello Coello, and E. Alba, “A parallel versionof SMS-EMOA for many-objective optimization problems”, in Proceedingsof the International Conference on Parallel Problem Solving from Nature,Coimbra, Portugal, Sep. 2016, pp. 568–577.
[85] R. Hernández Gómez, C. A. Coello Coello, and E. Alba Torres, “A multiobjectiveevolutionary algorithm based on parallel coordinates”, in Proceedingsof the Genetic and Evolutionary Computation Conference, Denver, Colorado,USA, Jul. 2016, pp. 565–572.
[86] I. Das, “Nonlinear multicriteria optimization and robust optimality”, PhDDissertation, Rice University, Houston Texas, Apr. 1997.
[87] I. Das and J. E. Dennis, “Normal-boundary intersection: A new methodfor generating the Pareto surface in nonlinear multicriteria optimizationproblems”, SIAM Journal on Optimization, vol. 8, no. 3, pp. 631–657, Jul.1998.
[88] M. Asafuddoula, T. Ray, and R. Sarker, “A decomposition-based evolutionaryalgorithm for many objective optimization”, IEEE Transactions onEvolutionary Computation, vol. 19, no. 3, pp. 445–460, Jun. 2015.
[89] M. Elarbi, S. Bechikh, C. A. C. Coello, M. Makhlouf, and L. B. Said, “Approximatingcomplex Pareto fronts with pre-defined normal-boundary intersectiondirections”, IEEE Transactions on Evolutionary Computation, vol. 24, no. 5,pp. 809–823, 2019.
[90] R. Tanabe, H. Ishibuchi, and A. Oyama, “Benchmarking multi- and manyobjectiveevolutionary algorithms under two optimization scenarios”, IEEEAccess, vol. 5, pp. 19 597–19 619, Sep. 2017.
[91] H. K. Singh, K. S. Bhattacharjee, and T. Ray, “Distance-based subset selectionfor benchmarking in evolutionary multi/many-objective optimization”, IEEETransactions on Evolutionary Computation, vol. 23, no. 5, pp. 904–912, Oct.2019.
[92] A. Panichella, “An adaptive evolutionary algorithm based on non-euclideangeometry for many-objective optimization”, in Proceedings of the Genetic andEvolutionary Computation Conference, Prague, Czech Republic, Jul. 2019,pp. 595–603.
[93] Y. Xiang, Y. Zhou, X. Yang, and H. Huang, “A many-objective evolutionaryalgorithm with Pareto-adaptive reference points”, IEEE Transactions onEvolutionary Computation, vol. 24, no. 1, pp. 99–113, Feb. 2020.
[94] M. Asafuddoula, T. Ray, and R. Sarker, “A decomposition based evolutionaryalgorithm for many objective optimization with systematic sampling andadaptive epsilon control”, in Proceedings of Evolutionary Multi-CriterionOptimization, Sheffield, United Kingdom, Mar. 2013, pp. 413–427.
[95] S. Jiang and S. Yang, “Convergence versus diversity in multiobjective optimization”,in Proceedings of the International Conference on Parallel ProblemSolving from Nature, Coimbra, Portugal, Sep. 2016, pp. 984–993.
[96] S. Zapotecas-Martinez, C. A. Coello Coello, H. E. Aguirre, and K. Tanaka,“A review of features and limitations of existing scalable multiobjective testsuites”, IEEE Transactions on Evolutionary Computation, vol. 23, no. 1,pp. 130–142, Feb. 2019.
[97] S. Ying, L. Li, Z. Wang, W. Li, and W. Wang, “An improved decompositionbasedmultiobjective evolutionary algorithm with a better balance of convergenceand diversity”, Applied Soft Computing, vol. 57, pp. 627–641, Aug.2017.
[98] H. Seada, M. Abouhawwash, and K. Deb, “Multiphase balance of diversityand convergence in multiobjective optimization”, IEEE Transactions onEvolutionary Computation, vol. 23, no. 3, pp. 503–513, Jun. 2019.
[99] H. K. Singh, A. Isaacs, and T. Ray, “A Pareto corner search evolutionaryalgorithm and dimensionality reduction in many-objective optimizationproblems”, IEEE Transactions on Evolutionary Computation, vol. 15, no. 4,pp. 539–556, Aug. 2011.
[100] Y. Tian, R. Cheng, X. Zhang, and Y. Jin, “PlatEMO: A MATLAB platform forevolutionary multi-objective optimization”, IEEE Computational IntelligenceMagazine, vol. 12, no. 4, pp. 73–87, Nov. 2017.
[101] H. Wang and X. Yao, “Corner sort for Pareto-based many-objective optimization”,IEEE Transactions on Cybernetics, vol. 44, no. 1, pp. 92–102, Jan.2014.
[102] M. Asafuddoula, H. K. Singh, and T. Ray, “An enhanced decomposition-basedevolutionary algorithm with adaptive reference vectors”, IEEE Transactionson Cybernetics, vol. 48, no. 8, pp. 2321–2334, Aug. 2018.
[103] G. Lei, J. G. Zhu, Y. G. Guo, J. F. Hu, W. Xu, and K. R. Shao, “Robustdesign optimization of PM-SMC motors for six sigma quality manufacturing”,IEEE Transactions on Magnetics, vol. 49, no. 7, pp. 3953–3956, Jul. 2013.
[104] M. Asafuddoula, H. K. Singh, and T. Ray, “Six-sigma robust design optimizationusing a many-objective decomposition-based evolutionary algorithm”,IEEE Transactions on Evolutionary Computation, vol. 19, no. 4, pp. 490–507,Aug. 2015.
[105] R. Hernández Gómez and C. A. Coello Coello, “Improved metaheuristicbased on the R2 indicator for many-objective optimization”, in Proceedingsof the Genetic and Evolutionary Computation Conference, Madrid, Spain,Jul. 2015, pp. 679–686.
[106] J. G. Falcón-Cardona and C. A. Coello Coello, “A new indicator-basedmany-objective ant colony optimizer for continuous search spaces”, SwarmIntelligence, vol. 11, no. 1, pp. 71–100, Mar. 2017.
[107] Y. Qi, J. Yu, X. Li, Y. Quan, and Q. Miao, “Enhancing robustness of theinverted PBI scalarizing method in MOEA/D”, Applied Soft Computing,vol. 71, pp. 1117–1132, Oct. 2018.
[108] H. Sato, “Analysis of inverted PBI and comparison with other scalarizingfunctions in decomposition based MOEAs”, Journal of Heuristics, vol. 21,no. 6, pp. 819–849, 2015.
[109] D. K. Saxena and S. Kapoor, “On timing the nadir-point estimation and/ortermination of reference-based multi- and many-objective evolutionary algorithms”,en, in Proceedings of Evolutionary Multi-Criterion Optimization,vol. 11411, East Lansing, MI, USA, Mar. 2019, pp. 191–202.
[110] D. K. Saxena, S. Mittal, S. Kapoor, and K. Deb, “A localized high fidelitydominance based many-objective evolutionary algorithm”, COIN Lab, MichiganState University, COIN Report No. 2021002, Jan. 2021.
[111] H.-L. Liu, F. Gu, and Q. Zhang, “Decomposition of a multiobjective optimizationproblem into a number of simple multiobjective subproblems”, IEEETransactions on Evolutionary Computation, vol. 18, no. 3, pp. 450–455, Jun.2014.
[112] T. Kohira, H. Kemmotsu, O. Akira, and T. Tatsukawa, “Proposal of benchmarkproblem based on real-world car structure design optimization”, inProceedings of the Genetic and Evolutionary Computation Conference, Kyoto,Japan, Jul. 2018, pp. 183–184.
[113] H. Ishibuchi, L. He, and K. Shang, “Regular Pareto front shape is not realistic”,in Proceedings of the Congress on Evolutionary Computation, Wellington,New Zealand, Jun. 2019, pp. 2034–2041.
[114] R. E. Steuer and R. Steuer, “Multiple Criteria Optimization: Theory, Computation,and Application”, Wiley New York, 1986.
[115] R. Cheng and M. Gen, “Evolution program for resource constrained projectscheduling problem”, in Proceedings of the IEEE Congress on EvolutionaryComputation, vol. 2, Orlando, FL, USA, Jun. 1994, pp. 736–741.
[116] R. Cheng and M. Gen, “Genetic Algorithms and Engineering Design”, JohnWiley, 1997.
[117] Z.-Z. Liu and Y. Wang, “Handling constrained multiobjective optimizationproblems with constraints in both the decision and objective spaces”, IEEETransactions on Evolutionary Computation, vol. 23, no. 5, pp. 870–884, Oct.2019.
[118] K. Li, R. Chen, G. Fu, and X. Yao, “Two-archive evolutionary algorithm forconstrained multiobjective optimization”, IEEE Transactions on EvolutionaryComputation, vol. 23, no. 2, pp. 303–315, Apr. 2019.
[119] E. Zitzler, L. Thiele, M. Laumanns, C. Fonseca, and V. da Fonseca, “Performanceassessment of multiobjective optimizers: An analysis and review”,IEEE Transactions on Evolutionary Computation, vol. 7, no. 2, pp. 117–132,Apr. 2003.
[120] L. M. Pang, H. Ishibuchi, and K. Shang, “Decomposition-based multiobjectiveevolutionary algorithm design under two algorithm frameworks”,IEEE Access, vol. 8, pp. 163 197–163 208, 2020.
[121] I. Giagkiozis, R. C. Purshouse, and P. J. Fleming, “Towards understandingthe cost of adaptation in decomposition-based optimization algorithms”, inProceedings of the International Conference on Systems, Man and Cybernetics,Manchester, UK, Oct. 2013, pp. 615–620.
[122] B. Li, J. Li, K. Tang, and X. Yao, “Many-objective evolutionary algorithms:A survey”, ACM Computing Surveys, vol. 48, no. 1, pp. 1–35, Sep. 2015.
[123] A. Trivedi, D. Srinivasan, K. Sanyal, and A. Ghosh, “A survey of multiobjectiveevolutionary algorithms based on decomposition”, IEEE Transactionson Evolutionary Computation, vol. 21, no. 3, pp. 440–462, 2016.
[124] Y. Liu, H. Ishibuchi, G. G. Yen, Y. Nojima, N. Masuyama, and Y. Han, “Onthe normalization in evolutionary multi-modal multi-objective optimization”,in Proceedings of the IEEE Congress on Evolutionary Computation, Glasgow,United Kingdom, Jul. 2020, pp. 1–8.
[125] Y. Liu, H. Ishibuchi, N. Masuyama, and Y. Nojima, “Adapting referencevectors and scalarizing functions by growing neural gas to handle irregularPareto fronts”, IEEE Transactions on Evolutionary Computation, vol. 24,no. 3, pp. 439–453, Jun. 2020.
[126] M. Li and X. Yao, “What weights work for you? Adapting weights forany Pareto front shape in decomposition-based evolutionary multi-objectiveoptimisation”, Evolutionary Computation, vol. 28, no. 2, pp. 227–253, Jun.2020.
[127] X. Ma, Y. Yu, X. Li, Y. Qi, and Z. Zhu, “A survey of weight vector adjustmentmethods for decomposition-based multiobjective evolutionary algorithms”,IEEE Transactions on Evolutionary Computation, vol. 24, no. 4, pp. 634–649,2020.
[128] K. Li, Q. Zhang, S. Kwong, M. Li, and R. Wang, “Stable matching-basedselection in evolutionary multiobjective optimization”, IEEE Transactionson Evolutionary Computation, vol. 18, no. 6, pp. 909–923, Dec. 2014.
[129] Z. Wang, J. Deng, Q. Zhang, and Q. Yang, “On the parameter setting of thepenalty-based boundary intersection method in MOEA/D.” In Proceedingsof Evolutionary Multi-Criterion Optimization, Shenzhen, China, Mar. 2021,pp. 413–423.
[130] E. Zitzler, D. Brockhoff, and L. Thiele, “The hypervolume indicator revisited:On the design of Pareto-compliant indicators via weighted integration”,in Proceedings of Evolutionary Multi-Criterion Optimization, Matsushima,Japan, Mar. 2007, pp. 862–876.
[131] H. Ishibuchi, R. Imada, Y. Setoguchi, and Y. Nojima, “How to specify areference point in hypervolume calculation for fair performance comparison”,Evolutionary Computation, vol. 26, no. 3, pp. 411–440, Sep. 2018.
[132] H. Ishibuchi, N. Akedo, and Y. Nojima, “A study on the specification of ascalarizing function in MOEA/D for many-objective knapsack problems”, inInternational Conference on Learning and Intelligent Optimization, vol. 7997,Catania, Italy, Jan. 2013, pp. 231–246.
[133] H. Ishibuchi, N. Tsukamoto, and Y. Nojima, “Evolutionary many-objectiveoptimization: A short review”, in Proceedings of the IEEE Congress onEvolutionary Computation, IEEE, Hong Kong, China, Jun. 2008, pp. 2419–2426.
[134] H. Ishibuchi, N. Akedo, and Y. Nojima, “Behavior of multiobjective evolutionaryalgorithms on many-objective knapsack problems”, IEEE Transactionson Evolutionary Computation, vol. 19, no. 2, pp. 264–283, Apr. 2015.
[135] E. Zitzler and L. Thiele, “Multiobjective optimization using evolutionaryalgorithms—A comparative case study”, in Proceedings of the InternationalConference on Parallel Problem Solving from Nature, Amsterdam, The Netherlands,Sep. 1998, pp. 292–301.
[136] K. Musselman and J. Talavage, “A tradeoff cut approach to multiple objectiveoptimization”, European Journal of Operational Research, vol. 28, no. 6,pp. 1424–1435, 1980.
[137] H. Ishibuchi, R. Imada, Y. Setoguchi, and Y. Nojima, “Hypervolume subsetselection for triangular and inverted triangular Pareto fronts of three-objectiveproblems”, in Proceedings of the 14th ACM/SIGEVO Conference on Foundationsof Genetic Algorithms, ACM, 2017, pp. 95–110.
[138] H. Ishibuchi, R. Imada, Y. Setoguchi, and Y. Nojima, “Reference point specificationin hypervolume calculation for fair comparison and efficient search”,in Proceedings of the Genetic and Evolutionary Computation Conference,ACM, Berlin, Germany, Jul. 2017, pp. 585–592.
[139] L. While, L. Bradstreet, and L. Barone, “A fast way of calculating exacthypervolumes”, IEEE Transactions on Evolutionary Computation, vol. 16,no. 1, pp. 86–95, 2012.
[140] J. D. Knowles, “Local-search and hybrid evolutionary algorithms for Paretooptimization”, Ph.D. dissertation, Department of Computer Science, Universityof Reading, Reading, U.K., Jan. 2002.
[141] E. Zitzler, J. Knowles, and L. Thiele, “Quality assessment of Pareto setapproximations”, in Multiobjective Optimization, vol. 5252, Berlin, Heidelberg:Springer, 2008, pp. 373–404.
[142] E. Zitzler, “Evolutionary algorithms for multiobjective optimization: Methodsand applications”, en, PhD Dissertation, Swiss Federal Institute of Technology,Zurich, Switzerland, Nov. 1999.
[143] J. Knowles and D. Corne, “On metrics for comparing nondominated sets”, inProceedings of the IEEE Congress on Evolutionary Computation, Honolulu,HI, USA, 2002, pp. 711–716.
[144] N. Drechsler, R. Drechsler, and B. Becker, “Multi-objective optimisationbased on relation favour”, in Proceedings of Evolutionary Multi-CriterionOptimization, vol. 1993, Zurich, Switzerland, Mar. 2001, pp. 154–166.
[145] N. Beume, B. Naujoks, and M. Emmerich, “SMS-EMOA: Multiobjectiveselection based on dominated hypervolume”, European Journal of OperationalResearch, vol. 181, no. 3, pp. 1653–1669, Sep. 2007.
[146] J. Bader and E. Zitzler, “HypE: An algorithm for fast hypervolume-basedmany-objective optimization”, Evolutionary Computation, vol. 19, no. 1,pp. 45–76, Mar. 2011.
[147] S. Jiang, J. Zhang, Y.-S. Ong, A. N. Zhang, and P. S. Tan, “A simple and fasthypervolume indicator-based multiobjective evolutionary algorithm”, IEEETransactions on Cybernetics, vol. 45, no. 10, pp. 2202–2213, Oct. 2015.
[148] E. Zitzler, M. Laumanns, and L. Thiele, “SPEA2: Improving the strengthPareto evolutionary algorithm”, TIK-Report, vol. 103, 2001.
[149] A. J. Keane, “Statistical improvement criteria for use in multiobjective designoptimization”, AIAA Journal, vol. 44, no. 4, pp. 879–891, Apr. 2006.
[150] K. Shang, H. Ishibuchi, and X. Ni, “R2-based hypervolume contributionapproximation”, IEEE Transactions on Evolutionary Computation, vol. 24,no. 1, pp. 185–192, Apr. 2019.
[151] Z. Fan, W. Li, X. Cai, H. Li, C. Wei, Q. Zhang, K. Deb, and E. Goodman,“Push and pull search for solving constrained multi-objective optimizationproblems”, Swarm and Evolutionary Computation, vol. 44, pp. 665–679, Feb.2019.
[152] H. Wang, S. He, and X. Yao, “Nadir point estimation for many-objectiveoptimization problems based on emphasized critical regions”, Soft Computing,vol. 21, no. 9, pp. 2283–2295, May 2017.
[153] Y. Tian, X. Xiang, X. Zhang, R. Cheng, and Y. Jin, “Sampling reference pointson the pareto fronts of benchmark multi-objective optimization problems”,in Proceedings of the IEEE Congress on Evolutionary Computation, Rio deJaneiro, Brazil, Jul. 2018.
[154] A. Kumar, G. Wu, M. Z. Ali, Q. Luo, R. Mallipeddi, P. N. Suganthan,and S. Das, “A benchmark-suite of real-world constrained multi-objectiveoptimization problems and some baseline results”, Swarm and EvolutionaryComputation, vol. 67, p. 100 961, Dec. 2021.
[155] F. Englmaier, A. Schmöller, and T. Stowasser, “Price discontinuities in anonline market for used cars”, Manage Sci, vol. 64, no. 6, pp. 2754–2766, Dec.2018.
[156] S. Vosper and J.-F. Mercure, “Assessing the effectiveness of South Africa’semissions-based purchase tax for private passenger vehicles: A consumerchoice modelling approach”, Journal of Energy in Southern Africa, vol. 27,no. 4, pp. 25–37, Dec. 2016.
[157] E. Fernandez, E. Lopez, F. Lopez, and C. A. C. Coello, “Increasing selectivepressure towards the best compromise in evolutionary multiobjective optimization:The extended NOSGA method”, Information Sciences, vol. 181,no. 1, pp. 44–56, Jan. 2011.
[158] J. Branke, T. Kaußler, and H. Schmeck, “Guidance in evolutionary multiobjectiveoptimization”, Advances in Engineering Software, vol. 32, no. 6,pp. 499–507, Jun. 2001.
[159] L. Rachmawati and D. Srinivasan, “Incorporating the notion of relativeimportance of objectives in evolutionary multiobjective optimization”, IEEETransactions on Evolutionary Computation, vol. 14, no. 4, pp. 530–546, Aug.2010.
[160] K. Deb, A. Sinha, P. J. Korhonen, and J. Wallenius, “An interactive evolutionarymultiobjective optimization method based on progressively approximatedvalue functions”, IEEE Transactions on Evolutionary Computation, vol. 14,no. 5, pp. 723–739, Oct. 2010.
[161] M. Köksalan and I. Karahan, “An interactive territory defining evolutionaryalgorithm: iTDEA”, IEEE Transactions on Evolutionary Computation, vol. 14,no. 5, pp. 702–722, Oct. 2010.
[162] R. Battiti and A. Passerini, “Brain–computer evolutionary multiobjectiveoptimization: A genetic algorithm adapting to the decision maker”, IEEETransactions on Evolutionary Computation, vol. 14, no. 5, pp. 671–687, Oct.2010.
[163] J. Branke, S. Greco, R. Słowiński, and P. Zielniewicz, “Interactive evolutionarymultiobjective optimization driven by robust ordinal regression”, Bulletin ofthe Polish Academy of Sciences. Technical Sciences, vol. 58, no. 3, pp. 347–358, 2010.
[164] K. Deb and J. Sundar, “Reference point based multi-objective optimizationusing evolutionary algorithms”, in Proceedings of the Genetic and EvolutionaryComputation Conference, Seattle Washington, USA, Jul. 2006, pp. 635–642.
[165] K. Narukawa, Y. Setoguchi, Y. Tanigaki, M. Olhofer, B. Sendhoff, and H.Ishibuchi, “Preference representation using Gaussian functions on a hyperplanein evolutionary multi-objective optimization”, Soft Computing, vol. 20,no. 7, pp. 2733–2757, Jul. 2016.
[166] J. Yi, J. Bai, H. He, J. Peng, and D. Tang, “Ar-MOEA: A novel preferencebaseddominance relation for evolutionary multiobjective optimization”, IEEETransactions on Evolutionary Computation, vol. 23, no. 5, pp. 788–802, Oct.2019.
[167] L. Thiele, K. Miettinen, P. J. Korhonen, and J. Molina, “A preferencebasedevolutionary algorithm for multi-objective optimization”, EvolutionaryComputation, vol. 17, no. 3, pp. 411–436, 2009.
[168] T. Wagner and H. Trautmann, “Integration of preferences in hypervolumebasedmultiobjective evolutionary algorithms by means of desirability functions”,IEEE Transactions on Evolutionary Computation, vol. 14, no. 5,pp. 688–701, Oct. 2010.
[169] A. K. M. E. S. Vijay K. Rohatgi, “An introduction to probability andstatistics”, John Wiley & Sons, 2015.
[170] H. G. Tucker, “A generalization of the Glivenko-Cantelli theorem”, TheAnnals of Mathematical Statistics, vol. 30, no. 3, pp. 828–830, Sep. 1959.
[171] K. Li, K. Deb, and X. Yao, “R-metric: Evaluating the performance ofpreference-based evolutionary multiobjective optimization using referencepoints”, IEEE Transactions on Evolutionary Computation, vol. 22, no. 6,pp. 821–835, Dec. 2018.
[172] H. Wang, Y. Jin, and X. Yao, “Diversity assessment in many-objectiveoptimization”, IEEE Transactions on Cybernetics, vol. 47, no. 6, pp. 1510–1522, Jun. 2017.
[173] E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective evolutionaryalgorithms: Empirical results”, Evolutionary Computation, vol. 8, no. 2,pp. 173–195, Jun. 2000.
[174] A. Desai, “100,000 UK used car data set, Version 3, https://www.kaggle.com/datasets/adityadesai13/used-car-dataset-ford-and-mercedes.Last accessed 9 Apr 2023”.