CMOEA/D-TAP, which incorporates the threshold adaptive penalty (TAP) function into MOEA/D, has shown competitive performance on the CTP and CF benchmark suites. However, its effectiveness diminishes on more challenging CMOPs, such as DASCMOP, LIRCMOP, and MW, where a considerable number of function evaluations are consumed during the early optimization phase due to population stagnation and slow convergence toward the Pareto front. This behavior is primarily attributed to severe constraints, which prevent many solutions from reaching the constrained Pareto front (CPF) within the available evaluation budget. To overcome this limitation, CMOEA/D-TAP is enhanced by incorporating a push and pull search strategy, referred to as PPSTAP. The proposed algorithm operates in two distinct phases: a push phase that drives the population toward the unconstrained Pareto front, followed by a pull phase that guides solutions toward the CPF using an adaptive penalty function method. Extensive experiments on the CF, DASCMOP, LIRCMOP, and MW benchmark suites include sensitivity analysis against PPS and CMOEA/D-TAP and comparison of the best-performing variant, PPSTAP ($ p\, $ = 20), with BiCo, C3M, and CCMO. Performance is evaluated using IGD, HV, and IGDp with statistical validation. The results show that all PPSTAP variants outperform their parent algorithms on average, while PPSTAP ($ p\, $ = 20) achieves the best overall performance across constrained multiobjective optimization problems.
Citation: Muhammad Sagheer, Muhammad Asif Jan, Akhtar Munir Khan, Emel Khan, Farman Shah. Augmenting push and pull search to improve threshold adaptive penalty function–based MOEA/D[J]. AIMS Mathematics, 2026, 11(9): 27585-27632. doi: 10.3934/math.20261104
CMOEA/D-TAP, which incorporates the threshold adaptive penalty (TAP) function into MOEA/D, has shown competitive performance on the CTP and CF benchmark suites. However, its effectiveness diminishes on more challenging CMOPs, such as DASCMOP, LIRCMOP, and MW, where a considerable number of function evaluations are consumed during the early optimization phase due to population stagnation and slow convergence toward the Pareto front. This behavior is primarily attributed to severe constraints, which prevent many solutions from reaching the constrained Pareto front (CPF) within the available evaluation budget. To overcome this limitation, CMOEA/D-TAP is enhanced by incorporating a push and pull search strategy, referred to as PPSTAP. The proposed algorithm operates in two distinct phases: a push phase that drives the population toward the unconstrained Pareto front, followed by a pull phase that guides solutions toward the CPF using an adaptive penalty function method. Extensive experiments on the CF, DASCMOP, LIRCMOP, and MW benchmark suites include sensitivity analysis against PPS and CMOEA/D-TAP and comparison of the best-performing variant, PPSTAP ($ p\, $ = 20), with BiCo, C3M, and CCMO. Performance is evaluated using IGD, HV, and IGDp with statistical validation. The results show that all PPSTAP variants outperform their parent algorithms on average, while PPSTAP ($ p\, $ = 20) achieves the best overall performance across constrained multiobjective optimization problems.
| [1] | K. Miettinen, Nonlinear multiobjective optimization, Berlin: Springer, 1999. |
| [2] | K. Deb, Multi-Objective Optimisation Using Evolutionary Algorithms: An Introduction, Berlin: Springer, 2011. |
| [3] |
Q. Zhang, H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Trans. Evol. Comput., 11 (2007), 712–731. https://doi.org/10.1109/TEVC.2007.892759 doi: 10.1109/TEVC.2007.892759
|
| [4] |
J. Liang, X. Ban, K. Yu, B. Qu, K. Qiao, C. Yue, et al., A survey on evolutionary constrained multiobjective optimization, IEEE Trans. Evol. Comput., 27 (2023), 201–221. https://doi.org/10.1109/TEVC.2022.3155533 doi: 10.1109/TEVC.2022.3155533
|
| [5] |
R. Saravanan, S. Ramabalan, N. G. R. Ebenezer, C. Dharmaraja, Evolutionary multi criteria design optimization of robot grippers, Appl. Soft Comput., 9 (2009), 159–172. https://doi.org/10.1016/j.asoc.2008.04.001 doi: 10.1016/j.asoc.2008.04.001
|
| [6] |
M. Basu, Economic environmental dispatch using multi-objective differential evolution, Appl. Soft Comput., 11 (2011), 2845–2853. https://doi.org/10.1016/j.asoc.2010.11.014 doi: 10.1016/j.asoc.2010.11.014
|
| [7] |
X. Zuo, C. Chen, W. Tan, M. Zhou, Vehicle scheduling of an urban bus line via an improved multiobjective genetic algorithm, IEEE Trans. Intell. Transport. Syst., 16 (2014), 1030–1041. https://doi.org/10.1109/TITS.2014.2352599 doi: 10.1109/TITS.2014.2352599
|
| [8] |
Y. Cui, Z. Geng, Q. Zhu, Y. Han, Multi-objective optimization methods and application in energy saving, Energy, 125 (2017), 681–704. https://doi.org/10.1016/j.energy.2017.02.174 doi: 10.1016/j.energy.2017.02.174
|
| [9] |
M. Premkumar, P. Jangir, B. S. Kumar, R. Sowmya, H. H. Alhelou, L. Abualigah, et al., A new arithmetic optimization algorithm for solving real-world multiobjective cec-2021 constrained optimization problems: Diversity analysis and validations, IEEE Access, 9 (2021), 84263–84295. https://doi.org/10.1109/ACCESS.2021.3085529 doi: 10.1109/ACCESS.2021.3085529
|
| [10] |
M. Niu, C. Wan, Z. Xu, A review on applications of heuristic optimization algorithms for optimal power flow in modern power systems, J. Modern Power Syst. Clean Energy, 2 (2014), 289–297. https://doi.org/10.1007/s40565-014-0089-4 doi: 10.1007/s40565-014-0089-4
|
| [11] |
N. Srinivas, K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evol. Comput., 2 (1994), 221–248. https://doi.org/10.1162/evco.1994.2.3.221 doi: 10.1162/evco.1994.2.3.221
|
| [12] |
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Trans. Evol. Comput., 6 (2002), 182–197. https://doi.org/10.1109/4235.996017 doi: 10.1109/4235.996017
|
| [13] |
K. Deb, H. Jain, 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 (2014), 577–601. https://doi.org/10.1109/TEVC.2013.2281535 doi: 10.1109/TEVC.2013.2281535
|
| [14] |
T. Murata, H. Ishibuchi, MOGA: Multi-objective genetic algorithms, IEEE Int. Confer. Evol. Comput., 1 (1995), 289–294. https://doi.org/10.1109/ICEC.1995.489161 doi: 10.1109/ICEC.1995.489161
|
| [15] | J. Horn, N. Nafpliotis, D. E. Goldberg, A niched Pareto genetic algorithm for multiobjective optimization, In: Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence, Orlando: IEEE, 1994, 82–87. https://doi.org/10.1109/ICEC.1994.350037 |
| [16] |
E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Trans. Evol. Comput., 3 (1999), 257–271. https://doi.org/10.1109/4235.797969 doi: 10.1109/4235.797969
|
| [17] | E. Zitzler, M. Laumanns, L. Thiele, SPEA2: Improving the strength Pareto evolutionary algorithm, TIK Report, 2001. https://doi.org/10.3929/ethz-a-004284029 |
| [18] | D. W. Corne, J. D. Knowles, M. J. Oates, The Pareto envelope-based selection algorithm for multiobjective optimization, In: International Conference on Parallel Problem Solving from Nature, Berlin: Springer, 2000,839–848. https://doi.org/10.1007/3-540-45356-3_82 |
| [19] |
Y. Chen, W. H. Chan, E. L. M. Su, Q. Diao, Multi-objective optimization for smart cities: A systematic review of algorithms, challenges, and future directions, Peer J Comput. Sci., 11 (2025), e3042. https://doi.org/10.7717/peerj-cs.3042 doi: 10.7717/peerj-cs.3042
|
| [20] |
M. G. de Alcantara, A. F. da Silva, F. A. S. Marins, R. de Carvalho Miranda, Systematic literature review on multi-objective simulation optimization, Complex Syst. Model. Simul., 5 (2025), 190–220. https://doi.org/10.23919/CSMS.2024.0037 doi: 10.23919/CSMS.2024.0037
|
| [21] | E. Zitzler, S. Künzli, Indicator-based selection in multiobjective search, In: International Conference on Parallel Problem Solving from Nature, Berlin: Springer, 2004,832–842. https://doi.org/10.1007/978-3-540-30217-9_84 |
| [22] |
J. Bader, E. Zitzler, Hype: An algorithm for fast hypervolume-based many-objective optimization, Evol. Comput., 19 (2011), 45–76. https://doi.org/10.1162/EVCO_a_00009 doi: 10.1162/EVCO_a_00009
|
| [23] | D. H. Phan, J. Suzuki, R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization, In: 2013 IEEE Congress on Evolutionary Computation, Cancun: IEEE, 2013. https://doi.org/10.1109/CEC.2013.6557783 |
| [24] | T. Zheng, J. Liu, S. Tan, H. Wang, An indicator based multiobjective evolutionary algorithm utilizing inverse modeling, In: 2024 $43^{rd}$ Chinese Control Conference (CCC), Kunming: IEEE, 2024, 2088–2093. https://doi.org/10.23919/CCC63176.2024.10661744 |
| [25] |
H. Li, Q. Zhang, Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-Ⅱ, IEEE Trans. Evol. Comput., 13 (2008), 284–302. https://doi.org/10.1109/TEVC.2008.925798 doi: 10.1109/TEVC.2008.925798
|
| [26] | Q. Zhang, W. Liu, H. Li, The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances, In: 2009 IEEE Congress on Evolutionary Computation, Trondheim: IEEE, 2009,203–208. https://doi.org/10.1109/CEC.2009.4982949 |
| [27] |
K. Li, K. Deb, Q. Zhang, S. Kwong, An evolutionary many-objective optimization algorithm based on dominance and decomposition, IEEE Trans. Evol. Comput., 19 (2014), 694–716. https://doi.org/10.1109/TEVC.2014.2373386 doi: 10.1109/TEVC.2014.2373386
|
| [28] |
Y. Qi, X. Ma, F. Liu, L. Jiao, J. Sun, J. Wu, MOEA/D with adaptive weight adjustment, Evol. Comput., 22 (2014), 231–264. https://doi.org/10.1162/EVCO_a_00109 doi: 10.1162/EVCO_a_00109
|
| [29] |
L. R. de Farias, A. F. Araújo, A decomposition-based many-objective evolutionary algorithm updating weights when required, Swarm Evol. Comput., 68 (2022), 100980. https://doi.org/10.1016/j.swevo.2021.100980 doi: 10.1016/j.swevo.2021.100980
|
| [30] |
J. Qiao, H. Zhou, C. Yang, S. Yang, A decomposition-based multiobjective evolutionary algorithm with angle-based adaptive penalty, Appl. Soft Comput., 74 (2019), 190–205. https://doi.org/10.1016/j.asoc.2018.10.028 doi: 10.1016/j.asoc.2018.10.028
|
| [31] |
Y. Zhu, Y. Qin, D. Yang, H. Xu, H. Zhou, An enhanced decomposition-based multi-objective evolutionary algorithm with a self-organizing collaborative scheme, Expert Syst. Appl., 213 (2023), 118915. https://doi.org/10.1016/j.eswa.2022.118915 doi: 10.1016/j.eswa.2022.118915
|
| [32] |
H. Cheng, L. Li, L. You, A weight vector adjustment method for decomposition-based multi-objective evolutionary algorithms, IEEE Access, 11 (2023), 42324–42330. https://doi.org/10.1109/ACCESS.2023.3270806 doi: 10.1109/ACCESS.2023.3270806
|
| [33] |
Y. Su, Q. Lin, Z. Ming, K. C. Tan, Adapting decomposed directions for evolutionary multiobjective optimization, IEEE Trans. Cyber., 53 (2022), 6289–6302. https://doi.org/10.1109/TCYB.2022.3165557 doi: 10.1109/TCYB.2022.3165557
|
| [34] |
J. Wang, Y. Su, Q. Lin, L. Ma, D. Gong, J. Li, et al., A survey of decomposition approaches in multiobjective evolutionary algorithms, Neurocomputing, 408 (2020), 308–330. https://doi.org/10.1016/j.neucom.2020.01.114 doi: 10.1016/j.neucom.2020.01.114
|
| [35] |
Y. Su, J. Wang, L. Ma, X. Wang, Q. Lin, C. A. C. Coello, et al., A hybridized angle-encouragement-based decomposition approach for many-objective optimization problems, Appl. Soft Comput., 78 (2019), 355–372. https://doi.org/10.1016/j.asoc.2019.02.026 doi: 10.1016/j.asoc.2019.02.026
|
| [36] |
Y. Su, Q. Lin, J. Wang, J. Li, J. Chen, Z. Ming, A constrained solution update strategy for multiobjective evolutionary algorithm based on decomposition, Complexity, 2019 (2019), 3251349. https://doi.org/10.1155/2019/3251349 doi: 10.1155/2019/3251349
|
| [37] |
E. Mezura-Montes, C. A. C. Coello, Constraint-handling in nature-inspired numerical optimization: Past, present and future, Swarm Evol. Comput., 1 (2011), 173–194. https://doi.org/10.1016/j.swevo.2011.10.001 doi: 10.1016/j.swevo.2011.10.001
|
| [38] | T. Runarsson, X. Yao, Evolutionary search and constraint violations, In: The 2003 Congress on Evolutionary Computation, Canberra: IEEE, 2003. https://doi.org/10.1109/CEC.2003.1299836 |
| [39] |
M. A. Jan, M. Sagheer, H. U. Khan, M. I. Uddin, R. A. Khanum, M. Mahmoud, et al., Hybrid stochastic ranking for constrained optimization, IEEE Access, 8 (2020), 227270–227287. https://doi.org/10.1109/ACCESS.2020.3044439 doi: 10.1109/ACCESS.2020.3044439
|
| [40] |
C. Wang, R. Xu, An angle based evolutionary algorithm with infeasibility information for constrained many-objective optimization, Appl. Soft Comput., 86 (2020), 105911. https://doi.org/10.1016/j.asoc.2019.105911 doi: 10.1016/j.asoc.2019.105911
|
| [41] |
Y. Hao, C. Zhao, Y. Zhang, Y. Cao, Z. Li, Constrained multi-objective optimization problems: Methodologies, algorithms and applications, Knowledge-Based Syst., 299 (2024), 111998. https://doi.org/10.1016/j.knosys.2024.111998 doi: 10.1016/j.knosys.2024.111998
|
| [42] |
R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1–23. https://doi.org/10.1090/S0002-9904-1943-07818-4 doi: 10.1090/S0002-9904-1943-07818-4
|
| [43] |
C. W. Carroll, The created response surface technique for optimizing nonlinear, restrained systems, Oper. Res., 9 (1961), 169–184. https://doi.org/10.1287/opre.9.2.169 doi: 10.1287/opre.9.2.169
|
| [44] |
A. V. Fiacco, G. P. McCormick, Extensions of sumt for nonlinear programming: Equality constraints and extrapolation, Manag. Sci., 12 (1966), 816–828. https://doi.org/10.1287/mnsc.12.11.816 doi: 10.1287/mnsc.12.11.816
|
| [45] | M. A. Jan, Q. Zhang, MOEA/D for constrained multiobjective optimization: Some preliminary experimental results, In: 2010 UK Workshop on Computational Intelligence (UKCI), Colchester: IEEE, 2010. https://doi.org/10.1109/UKCI.2010.5625585 |
| [46] | M. A. Jan, R. A. Khanum, N. M. Tairan, W. K. Mashwani, Performance of a constrained version of MOEA/D on CTP-series test instances, Int. J. Adv. Comput. Sci. Appl., 7 (2016), 196–505. |
| [47] |
H. Jain, K. Deb, An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part ii: Handling constraints and extending to an adaptive approach, IEEE Trans. Evol. Comput., 18 (2014), 602–622. https://doi.org/10.1109/TEVC.2013.2281534 doi: 10.1109/TEVC.2013.2281534
|
| [48] |
K. Deb, An efficient constraint handling method for genetic algorithms, Comput. Meth. Appl. Mech. Eng., 186 (2000), 311–338. https://doi.org/10.1016/S0045-7825(99)00389-8 doi: 10.1016/S0045-7825(99)00389-8
|
| [49] |
Z. Z. Liu, B. C. Wang, K. Tang, Handling constrained multiobjective optimization problems via bidirectional coevolution, IEEE Trans. Cyber., 52 (2022), 10163–10176. https://doi.org/10.1109/TCYB.2021.3056176 doi: 10.1109/TCYB.2021.3056176
|
| [50] |
J. Zhou, J. Zou, S. Yang, J. Zheng, D. Gong, T. Pei, Niche-based and angle-based selection strategies for many-objective evolutionary optimization, Inform. Sci., 571 (2021), 133–153. https://doi.org/10.1016/j.ins.2021.04.050 doi: 10.1016/j.ins.2021.04.050
|
| [51] | M. Asafuddoula, T. Ray, R. Sarker, K. Alam, An adaptive constraint handling approach embedded MOEA/D, In: 2012 IEEE Congress on Evolutionary Computation, Brisbane: IEEE, 2012. https://doi.org/10.1109/CEC.2012.6252868 |
| [52] |
J. Zou, R. Sun, S. Yang, J. Zheng, A dual-population algorithm based on alternative evolution and degeneration for solving constrained multi-objective optimization problems, Inform. Sci., 579 (2021), 89–102. https://doi.org/10.1016/j.ins.2021.07.078 doi: 10.1016/j.ins.2021.07.078
|
| [53] |
Y. Tian, T. Zhang, J. Xiao, X. Zhang, Y. Jin, A coevolutionary framework for constrained multiobjective optimization problems, IEEE Trans. Evol. Comput., 25 (2020), 102–116. https://doi.org/10.1109/TEVC.2020.3004012 doi: 10.1109/TEVC.2020.3004012
|
| [54] |
Y. Tian, Y. Zhang, Y. Su, X. Zhang, K. C. Tan, Y. Jin, Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization, IEEE Trans. Cyber., 52 (2022), 9559–9572. https://doi.org/10.1109/TCYB.2020.3021138 doi: 10.1109/TCYB.2020.3021138
|
| [55] |
Z. Z. Liu, Y. Wang, Handling constrained multiobjective optimization problems with constraints in both the decision and objective spaces, IEEE Trans. Evol. Comput., 23 (2019), 870–884. https://doi.org/10.1109/TEVC.2019.2894743 doi: 10.1109/TEVC.2019.2894743
|
| [56] |
R. Sun, J. Zou, Y. Liu, S. Yang, J. Zheng, A multistage algorithm for solving multiobjective optimization problems with multiconstraints, IEEE Trans. Evol. Comput., 27 (2023), 1207–1219. https://doi.org/10.1109/TEVC.2022.3224600 doi: 10.1109/TEVC.2022.3224600
|
| [57] |
Z. Fan, W. Li, X. Cai, H. Li, C. Wei, Q. Zhang, et al., Push and pull search for solving constrained multi-objective optimization problems, Swarm Evol. Comput., 44 (2019), 665–679. https://doi.org/10.1016/j.swevo.2018.08.017 doi: 10.1016/j.swevo.2018.08.017
|
| [58] |
A. M. Khan, M. A. Jan, M. Sagheer, R. A. Khanum, M. I. Uddin, S. Ahmad, Near feasibility driven adaptive penalty functions embedded MOEA/D, IEEE Access, 11 (2023), 105182–105213. https://doi.org/10.1109/ACCESS.2023.3317818 doi: 10.1109/ACCESS.2023.3317818
|
| [59] |
Y. Tian, R. Cheng, X. Zhang, Y. Jin, PlatEMO: A MATLAB platform for evolutionary multi-objective optimization [educational forum], IEEE Comput. Intell. Magaz., 12 (2017), 73–87. https://doi.org/10.1109/MCI.2017.2742868 doi: 10.1109/MCI.2017.2742868
|
| [60] | Q. Zhang, A. Zhou, S. Zhao, P. N. Suganthan, W. Liu, S. Tiwari, Multiobjective optimization test instances for the cec 2009 special session and competition, Technical report, In: University of Essex, Colchester, UK and Nanyang Technological University, Singapore, Special Session on Performance Assessment of Multi-Objective Optimization Algorithms, 264 (2008), 1–30. |
| [61] |
Z. Fan, W. Li, X. Cai, H. Huang, Y. Fang, Y. You, et al., An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions, Soft Comput., 23 (2019), 12491–12510. https://doi.org/10.1007/s00500-019-03794-x doi: 10.1007/s00500-019-03794-x
|
| [62] |
Z. Fan, W. Li, X. Cai, H. Li, C. Wei, Q. Zhang, et al., Difficulty adjustable and scalable constrained multiobjective test problem toolkit, Evol. Comput., 28 (2020), 339–378. https://doi.org/10.1162/evco_a_00259 doi: 10.1162/evco_a_00259
|
| [63] |
Z. Ma, Y. Wang, Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons, IEEE Trans. Evol. Comput., 23 (2019), 972–986. https://doi.org/10.1109/TEVC.2019.2896967 doi: 10.1109/TEVC.2019.2896967
|
| [64] | A. Mohammadi, M. N. Omidvar, X. Li, A new performance metric for user-preference based multi-objective evolutionary algorithms, In: 2013 IEEE Congress on Evolutionary Computation, Cancun: IEEE, 2013, 2825–2832. https://doi.org/10.1109/CEC.2013.6557912 |
| [65] |
C. A. C. Coello, N. C. Cortés, Solving multiobjective optimization problems using an artificial immune system, Genet. Program Evolvable Mach., 6 (2005), 163–190. https://doi.org/10.1007/s10710-005-6164-x doi: 10.1007/s10710-005-6164-x
|
| [66] | D. A. Van Veldhuizen, Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations, Dayton: Air Force Institute of Technology, 1999. |
| [67] | E. Zitzler, L. Thiele, Multiobjective optimization using evolutionary algorithms–a comparative case study, In: International Conference on Parallel Problem Solving from Nature, Berlin: Springer, 1998,292–301. https://doi.org/10.1007/BFb0056872 |
| [68] | H. Ishibuchi, H. Masuda, Y. Tanigaki, Y. Nojima, Modified distance calculation in generational distance and inverted generational distance, In: International Conference on Evolutionary Multi-Criterion Optimization, 2015,110–125. https://doi.org/10.1007/978-3-319-15892-1_8 |
| [69] | H. Ishibuchi, R. Imada, N. Masuyama, Y. Nojima, Comparison of hypervolume, IGD and IGD+ from the viewpoint of optimal distributions of solutions, In: International Conference on Evolutionary Multi-Criterion Optimization, 2019,332–345. https://doi.org/10.1007/978-3-030-12598-1_27 |
| [70] | E. Zitzler, J. Knowles, L. Thiele, Quality assessment of pareto set approximations, In: Multiobjective Optimization: Interactive and Evolutionary Approaches, Berlin: Springer, 2008,373–404. https://doi.org/10.1007/978-3-540-88908-3_14 |
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Muhammad Sagheer, Muhammad Asif Jan, Akhtar Munir Khan, Emel Khan, Farman Shah. Augmenting push and pull search to improve threshold adaptive penalty function–based MOEA/D[J]. AIMS Mathematics, 2026, 11(9): 27585-27632. doi: 10.3934/math.20261104
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