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A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm

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Abstract

This article is concerned with the numerical solution for a class of interval multi-objective mixed-integer optimal control problems (IMOMIOCPs). The IMOMIOCPs under investigation are typical NP-hard problems involve unknown-but-bounded interval parameters, multiple objectives, and mixed-integer dynamic controls. Accordingly, a new numerical method based on quantum heuristic algorithm is designed, which has the following modules: (i) Control vector parameterization and the fourth order Runge–Kutta method for model discretization, (ii) interval programming based on interval credibility for addressing interval parameters, (iii) coevolution of Quantum Annealing and Quantum Krill Herd for searching the optimal mixed-integer decisions, and (iv) the multiple populations for multi-objective technology for establishing the Pareto optimal front. The analyses on convergence and computational complexity of the proposed optimization mechanism are given. Moreover, simulation results on benchmark functions and a practical engineering IMOMIOCP verify that the proposed numerical method is more excel at achieving promising results than some classic algorithms and state-of-the-art algorithms.

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References

  • Abdelaziz, F. B., & Masri, H. (2010). A compromise solution for the multiobjective stochastic linear programming under partial uncertainty. European Journal of Operational Research, 202(1), 55–59.

    Article  Google Scholar 

  • Abdelaziz, F. B., Masri, H., & Alaya, H. (2017). A recourse goal programming approach for airport bus routing problem. Annals of Operations Research, 251(1–2), 1–14.

    Google Scholar 

  • Abel, E., Mikhailov, L., & Keane, J. (2018). Inconsistency reduction in decision making via multi-objective optimization. European Journal of Operational Research, 267(1), 212–226.

    Article  Google Scholar 

  • Ali, M., & Pant, M. (2011). Improving the performance of differential evolution algorithm using Cauchy mutation. Soft Computing, 15(5), 991–1007.

    Article  Google Scholar 

  • Allahverdi, A., & Aydilek, H. (2014). Total completion time with makespan constraint in no-wait flowshops with setup times. European Journal of Operational Research, 238(3), 724–734.

    Article  Google Scholar 

  • Arjmandzadeh, Z., Nazemi, A., & Safi, M. (2018). Solving multiobjective random interval programming problems by a capable neural network framework. Applied Intelligence, 49(4), 1566–1579.

    Article  Google Scholar 

  • Artigues, C., & Feillet, D. (2008). A branch and bound method for the job-shop problem with sequence-dependent setup times. Annals of Operations Research, 159, 135–159.

    Article  Google Scholar 

  • Bahri, O., & Talbi, E. G. (2018). Dealing with epistemic uncertainty in multi-objective optimization: A survey information processing and management of uncertainty in knowledge-based systems. Berlin: Springer.

    Book  Google Scholar 

  • Biegler, L. T., & Grossmann, I. E. (2004). Retrospective on optimization. Computer and Chemical Engineering, 28(8), 1169–1192.

    Article  Google Scholar 

  • Bonami, P., Kilinc, M., & Linderoth, J. (2012). Algorithms and software for convex mixed integer nonlinear programs. Mixed Integer Nonlinear Programming, 154, 1–39.

    Article  Google Scholar 

  • Bonami, P., & Lejeune, M. A. (2009). An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research, 57(3), 650–670.

    Article  Google Scholar 

  • Boukouvala, F., Misener, R., & Floudas, C. A. (2016). Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO. European Journal of Operational Research, 252(3), 701–727.

    Article  Google Scholar 

  • Bryson, A. E., Ho, Y. C., & Siouris, G. M. (1979). Applied optimal control: Optimization, estimation, and control. IEEE Transactions on Systems Man and Cybernetics, 9(6), 366–367.

    Article  Google Scholar 

  • Celia, M. A., Russell, T. F., & Herrera, I. (1990). An Eulerian–Lagrangian localized adjoint method for the advection–diffusion equation. Advances in Water Resources, 13, 187–206.

    Article  Google Scholar 

  • Chang, N. B., Chen, Y. L., & Wang, S. F. (1997). A fuzzy interval multiobjective mixed integer programming approach for the optimal planning of solid waste management systems. Fuzzy Sets and Systems, 89(1), 35–60.

    Article  Google Scholar 

  • Chen, C. H., Lin, J., & Chick, S. E. (2000). Simulation budget allocation for further enhancing the efficiency of ordinal optimization. Discrete Event Dynamic Systems, 10(3), 251–270.

    Article  Google Scholar 

  • Chen, L., & Wu, Z. (2020). Stochastic optimal control problem in advertising model with delay. Journal of Systems Science and Complexity, 33(3), 968–987.

    Article  Google Scholar 

  • Chen, T. L., Cheng, C. Y., & Chou, Y. H. (2020). Multi-objective genetic algorithm for energy-efficient hybrid flow shop scheduling with lot streaming. Annals of Operations Research, 290, 813–836.

    Article  Google Scholar 

  • Coello, C., Van Veldhuizen, D., & Lamont, G. (2002). Evolutionary algorithms for solving multi-objective problem. Berlin: Springer.

    Book  Google Scholar 

  • Crispin, A., & Syrichas, A. (2013). Quantum annealing algorithm for vehicle scheduling. In Proceedings of the IEEE international conference on systems, man, and cybernetics (pp. 3523–3528).

  • Cuate, O., Derbel, B., Liefooghe, A., et al. (2017). An approach for the local exploration of discrete many objective optimization problems. In Proceedings of international conference on evolutionary multi-criterion optimization (pp. 135–150).

  • Deb, K. (2001). Multi-objective optimization using evolutionary algorithms. Hoboken: Wiley.

    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 Transactions on Evolutionary Computation, 18(4), 577–601.

    Article  Google Scholar 

  • Deb, K., Pratap, A., Agarwal, S., et al. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197.

    Article  Google Scholar 

  • Deep, K., Singh, K. P., Kansal, M. L., & Mohan, C. (2009). A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation, 212, 505–518.

    Article  Google Scholar 

  • Duran, M. A., & Grossmann, I. E. (1986). An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming, 36(3), 307–339.

    Article  Google Scholar 

  • Engin, O., & Guclu, A. (2018). A new hybrid ant colony optimization algorithm for solving the no-wait flow shop scheduling problems. Applied Soft Computing, 72, 166–176.

    Article  Google Scholar 

  • Feng, Z., Zhang, Q., Zhang, Q., et al. (2015). A multiobjective optimization based framework to balance the global exploration and local exploitation in expensive optimization. Journal of Global Optimization, 61(4), 677–694.

    Article  Google Scholar 

  • Floudas, C. A. (1995). Nonlinear and mixed-integer optimization: Fundamentals and applications. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Gandomi, A. H., & Alavi, A. H. (2012). Krill herd: A new bio-inspired optimization algorithm. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4831–4845.

    Article  Google Scholar 

  • Ge, Y., & Li, S. (2013). Computation of reservoir relative permeability curve based on RBF neural network. Journal of Chemical Industry and Engineering (China), 64(12), 4571–4577.

    Google Scholar 

  • Ghawadri, N., Senu, N., Fawzi, F. A., et al. (2019). Diagonally implicit Runge–Kutta type method for directly solving special fourth-order ordinary differential equations with Ill-posed problem of a beam on elastic foundation. Algorithm, 12(1), 10.

    Article  Google Scholar 

  • Giagkiozis, I., & Fleming, P. (2014). Pareto front estimation for decision making. Evolutionary Computation, 22(4), 651–678.

    Article  Google Scholar 

  • Grossmann, I. E. (2002). Review of nonlinear mixed-integer and disjunctive programming techniques. Optimization Engineering, 3(3), 227–252.

    Article  Google Scholar 

  • Guo, Y. N., Cheng, J., Yang, Z., & Wang, C. (2016). Knowledge-inducing MOEA/D for interval multi-objective optimization problems. IEEE Congress on Evolutionary Computation, 2016, 2729–2735.

    Google Scholar 

  • He, Y., Wang, J., Zhang, X., et al. (2019). Encoding transformation-based differential evolution algorithm for solving knapsack problem with single continuous variable. Swarm and Evolutionary Computation, 50, 100507.

    Article  Google Scholar 

  • Ishibuchi, H., Sakane, Y., Tsukamoto, N., & Nojima, Y. (2009). Evolutionary many-objective optimization by NSGAII and MOEA/D with large populations. In Proceedings of IEEE international conference on systems, man and cybernetics (pp. 1758–1763).

  • Iwata, S., & Fukuyama, Y. (2018). Differential evolutionary particle swarm optimization for load adjustment distribution state estimation using correntropy. IEEE Transactions on Power and Energy, 138(6), 423–431.

    Article  Google Scholar 

  • Jalota, H., & Thakur, M. (2017). Genetic algorithm designed for solving linear or nonlinear mixed-integer constrained optimization problems. In International proceedings on advances in soft computing, intelligent systems and applications (pp. 277–290).

  • Ji, B., Yuan, X. H., Yuan, Y. B., et al. (2019). Exact and heuristic methods for optimizing lock-quay system in inland waterway. European Journal of Operational Research, 277(2), 740–755.

    Article  Google Scholar 

  • Jozefowiez, N., Semet, F., & Talbi, E. G. (2005). Enhancements of NSGA II and its application to the vehicle routing problem with route balancing. In Proceedings of international conference on artificial evolution (evolution artificielle) (pp. 131–142).

  • Kamari, A., Nikookar, M., Sahranavard, L., & Mohammadi, A. H. (2014). Efficient screening of enhanced oil recovery methods and predictive economic analysis. Neural Computing and Applications, 25(3–4), 815–824.

    Article  Google Scholar 

  • Kesavan, P., Allgor, R. J., Gatzke, E. P., & Barton, P. I. (2004). Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Mathematical Programming, 100(3), 517–535.

    Article  Google Scholar 

  • Khalili-Damghani, K., Tavana, M., & Amirkhan, M. (2014). A fuzzy bi-objective mixed-integer programming method for solving supply chain network design problems under ambiguous and vague conditions. International Journal of Advanced Manufacturing Technology, 73, 1567–1595.

    Article  Google Scholar 

  • Kirches, C. (2010). Fast numerical methods for mixed-integer nonlinear model-predictive control. Berlin: Vieweg.

    Google Scholar 

  • Koutras, V. P., Platis, A. N., & Gravvanis, G. A. (2007). On the optimization of free resources using non-homogeneous Markov chain software rejuvenation model. Reliability Engineering and System Safety, 92(12), 1724–1732.

    Article  Google Scholar 

  • Kronqvist, J., Bernal, D. E., Lundell, A., & Grossmann, I. E. (2019). A review and comparison of solvers for convex MINLP. Optimization and Engineering, 20(2), 397–455.

    Article  Google Scholar 

  • Kucukoglu, I., Dewil, R., & Cattrysse, D. (2019). Hybrid simulated annealing and tabu search method for the electric travelling salesman problem with time windows and mixed charging rates. Expert Systems with Applications, 134, 279–303.

    Article  Google Scholar 

  • Lei, Y., Li, S., Zhang, X., et al. (2011). Optimal control of polymer flooding based on mixed-integer iterative dynamic programming. International Journal of Control, 84(11), 1903–1914.

    Article  Google Scholar 

  • Li, X., & Aneja, Y. P. (2020). A new branch-and-cut approach for the generalized regenerator location problem. Annals of Operations Research, 295, 229–255.

    Article  Google Scholar 

  • Limbourg, P., & Aponte, D. E. S. (2005). An optimization algorithm for imprecise multi-objective problem functions. In Proceedings of 2005 IEEE congress on evolutionary computation (pp. 459–466).

  • Liu, Q., Wang, X. Y., Fu, Q. M., et al. (2012). Double elite coevolutionary genetic algorithm. Journal of Software, 23(4), 765–775.

    Article  Google Scholar 

  • Liu, Z., Li, S., & Ge, Y. (2019a). A switch control based dynamic optimization of polymer flooding. In Proceedings of Chinese control conference (pp. 2094–2099).

  • Liu, Z., Li, S., & Han, L. (2019b). A fuzzy multi-objective strategy of polymer flooding based on possibilistic programming. Lecture Notes in Electrical Engineering, 529, 247–256.

    Article  Google Scholar 

  • Liu, Z., Li, S., & Zhao, K. (2020). Extended multi-interval Legendre–Gauss–Radau pseudospectral method for mixed-integer optimal control problem in engineering. International Journal of Systems Science. https://doi.org/10.1080/00207721.2020.1849862.

  • Masri, H., & Abdulla, Y. (2018). A multiple objective stochastic programming model for working capital management. Technological Forecasting and Social Change, 131, 141–146.

    Article  Google Scholar 

  • Melo, W., Fampa, M., & Raupp, F. (2018). Integrality gap minimization heuristics for binary mixed integer nonlinear programming. Journal of Global Optimization, 71(3), 593–612.

    Article  Google Scholar 

  • Melo, W., Fampa, M., & Raupp, F. (2020a). An overview of MINLP algorithms and their implementation in Muriqui Optimizer. Annals of Operations Research, 286, 217–241.

    Article  Google Scholar 

  • Melo, W., Fampa, M., & Raupp, F. (2020b). Two linear approximation algorithms for convex mixed integer nonlinear programming. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03722-5.

  • Miettinen, K., Ruiz, F., & Wierzbicki, A. P. (2008). Introduction to multiobjective optimization: interactive approaches. In Multiobjective optimization (pp. 27–57). Berlin: Springer.

  • Mykkeltvedt, T. S., Raynaud, X., & Lie, K. A. (2017). Fully implicit higher-order schemes applied to polymer flooding. Computational Geosciences, 2, 1–22.

    Google Scholar 

  • Naderi, B., Govindan, K., & Soleimani, H. (2020). A Benders decomposition approach for a real case supply chain network design with capacity acquisition and transporter planning: wheat distribution network. Annals of Operations Research, 291, 685–705.

    Article  Google Scholar 

  • Naujoks, B., Beume, N., & Emmerich, M. (2005). Multi-objective optimisation using S-metric selection: Application to three-dimensional solution spaces. IEEE Congress on Evolutionary Computation, 2015, 1282–1289.

    Google Scholar 

  • Neto, T., Constantino, M., Martins, I., & Pedroso, J. P. (2020). A multi-objective Monte Carlo tree search for forest harvest scheduling. European Journal of Operational Research, 282(3), 1115–1126.

    Article  Google Scholar 

  • Neumann, F., & Witt, C. (2010). Bioinspired computation in combinatorial optimization. Berlin Heidelberg: Springer.

    Book  Google Scholar 

  • Neumann, F., & Witt, C. (2012). Bioinspired computation in combinatorial optimization: Algorithms and their computational complexity. In Proceedings of the 14th annual conference companion on genetic and evolutionary computation (pp. 1035–1058).

  • Niakan, F., Vahdani, B., & Mohammadi, M. (2015). A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach. Engineering Optimization, 47(12), 1670–1688.

    Article  Google Scholar 

  • Pal, B. B., Moitra, B. N., & Sen, S. (2017). A linear goal programming approach to multiobjective fractional programming with interval parameter sets. International Journal of Mathematics in Operational Research, 3(6), 697–714.

    Article  Google Scholar 

  • Pei, J. Y., & Shan, P. (2019). A multi-objective hybrid differential optimization algorithm for flow-shop scheduling problem. International Journal of Simulation Modelling, 18(3), 500–509.

    Article  Google Scholar 

  • Pierro, F. D., Khu, S. T., & Savic, D. (2007). An investigation on preference order ranking scheme for multiobjective evolutionary optimization. IEEE Transactions on Evolutionary Computation, 11(1), 17–45.

    Article  Google Scholar 

  • Pistikopoulos, E. N., & Floudas, C. A. (1998). Nonlinear and mixed-integer optimization: Fundamentals and applications. Journal of Global Optimization, 2, 108–110.

    Article  Google Scholar 

  • Regina, S. B., Yalcin, C. K., & Mustafa, M. R. (2019). Algorithms for generating Pareto fronts of multi-objective integer and mixed-integer programming problems. Optimization and Control. arXiv:1903.07041 [math.OC], Cornell University.

  • Sadowski, K. L., Thierens, D., & Bosman, P. A. N. (2018). GAMBIT: A parameterless model-based evolutionary algorithm for mixed-integer problems. Evolutionary Computation, 26(1), 117–143.

    Article  Google Scholar 

  • Sahinidis, N., & Grossmann, I. E. (1991). MINLP model for cyclic multiproduct scheduling on continuous parallel lines. Computers and Chemical Engineering, 15(2), 85–103.

    Article  Google Scholar 

  • Schlegel, M., Stockmann, K., Binder, T., & Marquardt, W. (2005). Dynamic optimization using adaptive control vector parameterization. Computers and Chemical Engineering, 29(8), 1731–1751.

    Article  Google Scholar 

  • Sengupta, A., Pal, T. K., & Chakraborty, D. (2001). Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets and Systems, 119(1), 129–138.

    Article  Google Scholar 

  • Smotherman, M., & Zemoudeh, K. (1989). A non-homogeneous Markov model for phased-mission reliability analysis. IEEE Transactions on Reliability, 38(5), 585–590.

    Article  Google Scholar 

  • Sun, J., & Gong, D. (2013). Solving interval multi-objective optimization problems using evolutionary algorithms with lower limit of possibility degree. Chinese Journal of Electronics, 22(2), 269–272.

    Google Scholar 

  • Syrichas, A., & Crispin, A. (2017). Large-scale vehicle routing problems: Quantum annealing, tunings and results. Computers and Operations Research, 87, 52–62.

    Article  Google Scholar 

  • Talbi, E. G., Rahoual, M., Mabed, M. H., & Dhaenens, C. (2001). A hybrid evolutionary approach for multicriteria optimization problems: Application to the flow shop. In Evolutionary multi-criterion optimization (pp. 416–428). Berlin: Springer.

  • Tamura, K., & Yasuda, K. (2020). The spiral optimization algorithm: Convergence conditions and settings. IEEE Transactions on Systems Man Cybernetics-Systems, 50(1), 360–375.

    Article  Google Scholar 

  • Trespalacios, F., & Grossmann, I. E. (2014). Review of mixed-integer nonlinear and generalized disjunctive programming methods. Chemie Ingenieur Technik, 86(7), 991–1012.

    Article  Google Scholar 

  • Walther, A. (2007). Automatic differentiation of explicit Runge–Kutta methods for optimal control. Computational Optimization and Applications, 36(1), 83–108.

    Article  Google Scholar 

  • Wang, H., Laredo, D., Cuate, O., et al. (2019a). Enhanced directed search: A continuation method for mixed-integer multi-objective optimization problems. Annals of Operations Research, 279, 343–365.

    Article  Google Scholar 

  • Wang, Z., Zhang, J. H., & Yang, S. X. (2019b). An improved particle swarm optimization algorithm for dynamic job shop scheduling problems with random job arrivals. Swarm and Evolutionary Computation, 51, 100594.

    Article  Google Scholar 

  • Wu, H. C. (2009). The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions. Fuzzy Optimization and Decision Making, 196(1), 49–60.

    Google Scholar 

  • Wu, L. H., Wang, Y. N., & Chen, Z. L. (2007). Modified differential evolution algorithm for mixed-integer nonlinear programming problems. Journal of Chinese Computer Systems, 28(4), 666–669.

    Google Scholar 

  • Xu, G., Luo, K., Jing, G. X., et al. (2020). On convergence analysis of multi-objective particle swarm optimization algorithm. European Journal of Operational Research, 286(1), 32–38.

    Article  Google Scholar 

  • Xu, G., & Lv, Y. (2008). A new ranking approach for interval numbers in uncertain multiple-attribute decision making problems. Statistcs Decision, 19, 154–157.

    Google Scholar 

  • Yan, W., Miao, R., & Li, S. (2007). Multi-period semi-variance portfolio selection: Model and numerical solution. Applied Mathematics and Computation, 194(1), 128–134.

    Article  Google Scholar 

  • Yang, S., Li, M., Liu, X., & Zheng, J. (2013). A grid-based evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 17(5), 721–736.

    Article  Google Scholar 

  • Yibing, L. V., Tiesong, H. U., & Wang, G. (2007). A penalty function method based on Kuhn-Tucker condition for solving linear bilevel programming. Applied Mathematics and Computation, 188, 803–813.

    Google Scholar 

  • Zarbakhshnia, N., Kannan, D., Mavi, R. K., & Soleimani, H. (2020). A novel sustainable multi-objective optimization model for forward and reverse logistics system under demand uncertainty. Annals of Operations Research, 295, 843–880.

  • Zhan, Z. H., Li, J., Cao, J., et al. (2013). Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems. IEEE Transactions on Cybernetics, 43(2), 445–463.

    Article  Google Scholar 

  • Zhang, L., Wang, S., Zhang, K., et al. (2018). Cooperative artificial bee colony algorithm with multiple populations for interval multi-objective optimization problems. IEEE Transactions on Fuzzy Systems, 27(5), 1052–1065.

    Article  Google Scholar 

  • Zhang, M., & Li, H. Q. (2018). A reference direction and entropy based evolutionary algorithm for many-objective optimization. Applied Soft Computing, 70, 108–130.

    Article  Google Scholar 

  • Zhang, Q., & Li, H. (2008). MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 11(6), 712–731.

    Article  Google Scholar 

  • Zhang, Q., Li, S. R., Zhang, X. D., & Lei, Y. (2010). Constraint aggregation based numerical optimal control. In Proceedings of the 29th Chinese control conference (pp. 1560–1565).

  • Zhu, Y., & Kuno, T. (2006). A disjunctive cutting-plane-based branch-and-cut algorithm for 0–1 mixed-integer convex nonlinear programs. Industrial and Engineering Chemistry Research, 45(1), 187–196.

    Article  Google Scholar 

  • Zou, X., Chen, Y., Liu, M., & Kang, L. (2008). A new evolutionary algorithm for solving many-objective optimization problems. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38(5), 1402–1412.

    Article  Google Scholar 

  • Zouache, D., Abdelaziz, F. B., Lefkir, M., & Chalabi, N. E. (2021). Guided Moth-Flame optimiser for multi-objective optimization problems. Annals of Operations Research, 296, 877–899.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China [Grant Number 61573378], and the BUPT Excellent Ph.D. Students Foundation [Grant Number CX2019113].

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Appendix

Appendix

A1. Parameters tuning

T, \(\varGamma \), \(\varDelta \varGamma \) for QA and \({{N}^{\max }}\), \({{V}_{f}}\), \({{D}^{\max }}\) for QKH are the parameters to be tuned. For each parameter, the value range is obtained by the analysis of algorithm mechanism and past optimization examples. Then several candidate values \(\Upsilon \) are selected within the parameter range. Five benchmark examples with different computational complexities (F03, F05, F12, F14 and F18) are carried out for testing each candidate value while other parameters are set as the mean value of the corresponding range. The standard deviation of the obtained solution \(\delta \) is adopted as the evaluation criterion. The maximum fitness evaluations for testing are set as \({4}\times {{10}^{3}}\) for F03, \({6}\times {{10}^{3}}\) for F05, \(2.5\times {{10}^{4}}\) for F12, \(3.5\times {{10}^{5}}\) for F14 and \(2.5\times {{10}^{4}}\) for F19. When an optimal candidate value of any parameter is obtained, then it is fixed in the following tuning. The research results of parameters tuning are shown in Table 15, where the bold characters are the optimal candidate values.

Table 15 Results of parameters tuning
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Table 16 The basic MOMINLP benchmark examples for Experiment study 3
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Table 17 Parameter nomenclatures and value settings for Experimental study 4
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Liu, Z., Li, S. A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm. Ann Oper Res 311, 853–898 (2022). https://doi.org/10.1007/s10479-021-03998-1

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