Abstract
Simulated annealing (SA) is one of the oldest and the most well-known metaheuristics for optimization problems. One exclusive merit of this algorithm is that it does not get stuck in any local optima. However, due to strictly random process and some unnecessary moves, the convergence speed of SA is relatively slow. To alleviate this weakness, in this paper, a hybrid metaheuristic algorithm, comprising SA and D-number to better explore the search space is introduced. Within this proposed framework, D-number is embedded in SA and works as a stochastic engine (random generator) to reduce redundant moves, particularly during high temperatures. Mathematically, in the new approach, the probability of accepting inferior solution can be checked by D-number instead of uniformly distributed random variable. The results derived from hybrid SA show this search mechanism allows some non-improving moves to be avoided. Consequently, D-number as a high quality random generator in SA results in a good performance with low implementation effort in some cases. Traveling salesman problem (TSP) as an illustrative application is selected to verify the performance of this hybrid SA. On the whole, the result derived from the combination of SA and D-numbers is relatively encouraging.
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MATLAB codes can be obtained from: https://matlabhome.ir/, https://matlabsite.com/.
References
Abbasi, B., Mahlooji, H.: Improving response surface methodology by using artificial neural network and simulated annealing. Expert Syst. Appl. 39(3), 3461–3468 (2012)
Abdel-Basset, M., Ding, W., El-Shahat, D.: A hybrid Harris Hawks optimization algorithm with simulated annealing for feature selection. Artif. Intell. Rev. 54(1), 593–637 (2021)
Alberdi, R., Khandelwal, K.: Comparison of robustness of metaheuristic algorithms for steel frame optimization. Eng. Struct. 102, 40–60 (2015)
Bandyopadhyay, R., Basu, A., Cuevas, E., Sarkar, R.: Harris Hawks optimisation with simulated annealing as a deep feature selection method for screening of COVID-19 CT-scans. Appl. Soft Comput. 111, 107698 (2021)
Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Comput. Surv. (CSUR) 35(3), 268–308 (2003)
Boussaïd, I., Lepagnot, J., Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)
Černý, V.: Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45, 41–51 (1985)
Črepinšek, M., Liu, S.H., Mernik, M.: Exploration and exploitation in evolutionary algorithms: a survey. ACM Comput. Surv. (CSUR) 45(3), 1–33 (2013)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. In: Classic Works of the Dempster–Shafer Theory of Belief Functions, pp. 57–72. Springer, Berlin (2008)
Deng, Y.: D numbers: theory and applications. J. Inf. Comput. Sci. 9(9), 2421–2428 (2012)
Deng, X., Jiang, W.: D number theory based game-theoretic framework in adversarial decision making under a fuzzy environment. Int. J. Approx. Reason. 106, 194–213 (2019)
Deng, X., Jiang, W.: A framework for the fusion of non-exclusive and incomplete information on the basis of D number theory. Appl. Intell. 53(10), 11861–11884 (2023)
Deng, X., Hu, Y., Deng, Y., Mahadevan, S.: Environmental impact assessment based on D numbers. Expert Syst. Appl. 41(2), 635–643 (2014)
Emami, H.: Seasons optimization algorithm. Eng. Comput. 38(2), 1845–1865 (2022)
Ezugwu, A.E., Shukla, A.K., Nath, R., Akinyelu, A.A., Agushaka, J.O., Chiroma, H., Muhuri, P.K.: Metaheuristics: a comprehensive overview and classification along with bibliometric analysis. Artif. Intell. Rev. 54(6), 4237–4316 (2021)
Formato, R.A.: Central force optimization: a new deterministic gradient-like optimization metaheuristic. Opsearch 46(1), 25–51 (2009)
Franzin, A., Stützle, T.: Revisiting simulated annealing: a component-based analysis. Comput. Oper. Res. 104, 191–206 (2019)
Ganesan, T., Vasant, P., Elamvazuthi, I.: Multiobjective optimization using particle swarm optimization with non-Gaussian random generators. Intell. Dec. Technol. 10(2), 93–103 (2016)
Geng, X., Chen, Z., Yang, W., Shi, D., Zhao, K.: Solving the traveling salesman problem based on an adaptive simulated annealing algorithm with greedy search. Appl. Soft Comput. 11(4), 3680–3689 (2011)
Goh, S.L., Kendall, G., Sabar, N.R., Abdullah, S.: An effective hybrid local search approach for the post enrolment course timetabling problem. Opsearch 57, 1131–1163 (2020)
Golden, B.L., Skiscim, C.C.: Using simulated annealing to solve routing and location problems. Naval Res. Logist. Q. 33(2), 261–279 (1986)
Grobelny, J., Michalski, R.: A novel version of simulated annealing based on linguistic patterns for solving facility layout problems. Knowl. Based Syst. 124, 55–69 (2017)
Gunawan, A., Ng, K.M., Poh, K.L.: A hybridized Lagrangian relaxation and simulated annealing method for the course timetabling problem. Comput. Oper. Res. 39(12), 3074–3088 (2012)
Hasan, M., Islam, M.R., Mugdha, A.G.: Solving maximum clique problem using chemical reaction optimization. OPSEARCH 1–37 (2023)
Hussain, K., Mohd Salleh, M.N., Cheng, S., Shi, Y.: Metaheuristic research: a comprehensive survey. Artif. Intell. Rev. 52(4), 2191–2233 (2019)
Jeong, S.J., Kim, K.S., Lee, Y.H.: The efficient search method of simulated annealing using fuzzy logic controller. Expert Syst. Appl. 36(3), 7099–7103 (2009)
Kirkpatrick, S., Gelatt, C.D., Jr., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Lai, H., Liao, H.: A multi-criteria decision making method based on DNMA and CRITIC with linguistic D numbers for blockchain platform evaluation. Eng. Appl. Artif. Intell. 101, 104200 (2021)
Lamperti, R.D., de Arruda, L.V.R.: A strategy based on wave swarm for the formation task inspired by the traveling salesman problem. Eng. Appl. Artif. Intell. 126, 106884 (2023)
Liang, J., Guo, S., Du, B., Li, Y., Guo, J., Yang, Z., Pang, S.: Minimizing energy consumption in multi-objective two-sided disassembly line balancing problem with complex execution constraints using dual-individual simulated annealing algorithm. J. Clean. Prod. 284, 125418 (2021)
Mirjalili, S., Mirjalili, S.M., Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)
Mousavi-Nasab, S.H., Sotoudeh-Anvari, A.: An extension of best-worst method with D numbers: Application in evaluation of renewable energy resources. Sustain. Energy Technol. Assess. 40, 100771 (2020)
Nino-Ruiz, E.D., Yang, X.S.: Improved Tabu Search and Simulated Annealing methods for nonlinear data assimilation. Appl. Soft Comput. 83, 105624 (2019)
Oztas, G.Z., Erdem, S.: A penalty-based algorithm proposal for engineering optimization problems. Neural Comput. Appl. 35(10), 7635–7658 (2023)
Pradeepmon, T.G., Panicker, V.V., Sridharan, R.: A variable neighbourhood search enhanced estimation of distribution algorithm for quadratic assignment problems. Opsearch 58(1), 203–233 (2021)
Rabbouch, B., Saâdaoui, F., Mraihi, R.: Empirical-type simulated annealing for solving the capacitated vehicle routing problem. J. Exp. Theor. Artif. Intell. 32(3), 437–452 (2020)
Rajwar, K., Deep, K., Das, S.: An exhaustive review of the metaheuristic algorithms for search and optimization: taxonomy, applications, and open challenges. Artif. Intell. Rev. 1–71 (2023)
Rathod, V.: Multi-drill path sequencing models: a comparative study. Opsearch 60(1), 554–570 (2023)
Rosen, S.L., Harmonosky, C.M.: An improved simulated annealing simulation optimization method for discrete parameter stochastic systems. Comput. Oper. Res. 32(2), 343–358 (2005)
Salama, M., Srinivas, S.: Adaptive neighborhood simulated annealing for sustainability-oriented single machine scheduling with deterioration effect. Appl. Soft Comput. 110, 107632 (2021)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Silva, M.A.L., de Souza, S.R., Souza, M.J.F., de Franca Filho, M.F.: Hybrid metaheuristics and multi-agent systems for solving optimization problems: a review of frameworks and a comparative analysis. Appl. Soft Comput. 71, 433–459 (2018)
Singh, H.K., Ray, T., Smith, W.: C-PSA: Constrained Pareto simulated annealing for constrained multi-objective optimization. Inf. Sci. 180(13), 2499–2513 (2010)
Singh, P., Kamthane, A.R., Tanksale, A.N.: Metaheuristics for the distance constrained generalized covering traveling salesman problem. Opsearch 58(3), 575–609 (2021)
Sotoudeh-Anvari, A.: A critical review on theoretical drawbacks and mathematical incorrect assumptions in fuzzy OR methods: review from 2010 to 2020. Appl. Soft Comput. 93, 106354 (2020)
Sotoudeh-Anvari, A.: A state-of-the-art review on D number (2012–2022): a scientometric analysis. Eng. Appl. Artif. Intell. 127, 107309 (2024)
Sotoudeh-Anvari, A., Hafezalkotob, A.: A bibliography of metaheuristics-review from 2009 to 2015. Int. J. Knowl. Based Intell. Eng. Syst. 22(1), 83–95 (2018)
Tabak, A., İlhan, İ: An effective method based on simulated annealing for automatic generation control of power systems. Appl. Soft Comput. 126, 109277 (2022)
Talbi, E.G.: Metaheuristics: From Design to Implementation, vol. 74. John Wiley & Sons, London (2009)
Toscano, R., Lyonnet, P.: A new heuristic approach for non-convex optimization problems. Inf. Sci. 180(10), 1955–1966 (2010)
Vallada, E., Villa, F., Fanjul-Peyro, L.: Enriched metaheuristics for the resource constrained unrelated parallel machine scheduling problem. Comput. Oper. Res. 111, 415–424 (2019)
Vasan, A., Raju, K.S.: Comparative analysis of simulated annealing, simulated quenching and genetic algorithms for optimal reservoir operation. Appl. Soft Comput. 9(1), 274–281 (2009)
Vincent, F.Y., Lin, S.W., Lee, W., Ting, C.J.: A simulated annealing heuristic for the capacitated location routing problem. Comput. Ind. Eng. 58(2), 288–299 (2010)
Waissi, G.R., Kaushal, P.: A polynomial matrix processing heuristic algorithm for finding high quality feasible solutions for the TSP. Opsearch 57(1), 73–87 (2020)
Wang, Y., Tian, D., Li, Y. (2013). An improved simulated annealing algorithm for traveling salesman problem. In: Proceedings of the 2012 International Conference on Information Technology and Software Engineering, pp. 525–532. Springer, Berlin
Wang, N., Liu, X., Wei, D.: A modified D numbers’ integration for multiple attributes decision making. Int. J. Fuzzy Syst. 20, 104–115 (2018)
Wang, M., Tian, Y., Zhang, K.: The fuzzy Weighted Influence Nonlinear Gauge System method extended with D numbers and MICMAC. Complex Intell. Syst. 9(1), 719–731 (2023)
Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1(1), 67–82 (1997)
Xia, J., Feng, Y., Liu, L., Liu, D., Fei, L.: On entropy function and reliability indicator for D numbers. Appl. Intell. 49, 3248–3266 (2019)
Yanar, T.A., Akyürek, Z.: Fuzzy model tuning using simulated annealing. Expert Syst. Appl. 38(7), 8159–8169 (2011)
Yi, H., Yang, X.: A metaheuristic algorithm based on simulated annealing for optimal sizing and techno-economic analysis of PV systems with multi-type of battery energy storage. Sustain. Energy Technol. Assess. 53, 102724 (2022)
Zeng, Y., Zhang, Z., Wu, T., Liang, W.: Integrated optimization and engineering application for disassembly line balancing problem with preventive maintenance. Eng. Appl. Artif. Intell. 127, 107416 (2024)
Zhong, Y., Wang, L., Lin, M., Zhang, H.: Discrete pigeon-inspired optimization algorithm with Metropolis acceptance criterion for large-scale traveling salesman problem. Swarm Evol. Comput. 48, 134–144 (2019)
Zhou, X., Deng, X., Deng, Y., Mahadevan, S.: Dependence assessment in human reliability analysis based on D numbers and AHP. Nucl. Eng. Des. 313, 243–252 (2017)
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We are thankful to Editor-in-Chief, Prof. Angappa Gunasekaran and two anonymous reviewers for their valuable comments.
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Writing—original draft preparation, Writing—review and editing, Resources, Methodology: Alireza Sotoudeh-Anvari. Formal analysis and investigation, Supervision, Conceptualization: Seyed Mojtaba Sajadi. All authors have read and agreed to the published version of the manuscript.
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Sotoudeh-Anvari, A., Sajadi, S.M. Hybridization of simulated annealing and D-numbers as a stochastic generator. OPSEARCH (2024). https://doi.org/10.1007/s12597-024-00772-2
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DOI: https://doi.org/10.1007/s12597-024-00772-2