An Improved Atomic Orbital Search Algorithm Utilizing Firefly Algorithm for Optimization Problems

Subject Areas : Meta-heurestics

saeedeh Ghaemifard ¹

1 -

Received: 2025-03-09 Accepted : 2025-06-22 Published : 2025-01-13

Keywords: Mathematical modelling, Numerical methods, Hybrid algorithm, Atomic Orbital Search algorithm, Firefly Algorithm, Engineering optimization,

Abstract :

The Atomic Orbital Search inspiration from the rules of quantum mechanics and Firefly Algorithm is a metaheuristic technique which is widely used for solving the optimization problems. Both algorithms collectively improved the performance of search. This article goals to optimize engineering design problems utilizing a new hybrid optimizer; AOS-FA (Atomic Orbital Search-Firefly Algorithm). Incorporating the FA methodology into the basic AOS framework has successfully addressedthe issue of local optima trap and significantly enhanced the quality of solutions generated by the algorithm. The FA algorithm is work on the combinatorial optimization and utilized as application of AOS algorithm. Hence, we merge these two algorithms and make a hybrid algorithm. The purpose of the suggested hybridization method was to promote the improvement of the exploration-exploitation manners of the AOS search. To analyse the viability of the suggested hybridized algorithm in real-world usages, it is studied for five constrained engineering design issues, and the performance was determined with other outstanding metaheuristics extracted from the publications.

References:

1. Ghaemifard, S. and Ghannadiasl, A., "Usages of metaheuristic algorithms in investigating civil infrastructure optimization models; a review", AI in Civil Engineering, Vol. 3, No. 1, (2024), 17 DOI: 10.1007/s43503-024-00036-4.

2. Houssein, E.H., Mahdy, M.A., Shebl, D. and Mohamed, W.M., A survey of metaheuristic algorithms for solving optimization problems, in Metaheuristics in machine learning: Theory and applications. 2021, Springer.515-543.

3. Ghaemifard, S. and Ghannadiasl, A., "Design and analysis of offshore wind turbines: Problem formulation and optimization techniques", Journal of Marine Science and Application, Vol. 23, No. 4, (2024), 707-722 DOI: 10.1007/s11804-024-00473-8.

4. Ghannadiasl, A. and Ghaemifard, S., "Meta-heuristic algorithms: An appropriate approach in crack detection", Innovative Infrastructure Solutions, Vol. 9, No. 7, (2024), 263 DOI: 10.1007/s41062-024-01583-6.

5. Shahebrahimi, S.S., Lork, A., Shayegan, D.S. and AmirKardoust, A.A., "Solving the problem of multi-stakeholder construction site layout using metaheuristic algorithms", Iranian Journal of Optimization, Vol. 3, No. 15, (2024), 235-249.

6. Fazli, M., khiabani, F.M. and Daneshian, B., "Meta-heuristic algorithms to solve the problem of terminal facilities on a real scale", Iranian Journal of Optimization, Vol. 1, No. 14, (2022), 51-65.

7. Dorigo, M., Birattari, M. and Stutzle, T., "Ant colony optimization", IEEE computational intelligence magazine, Vol. 1, No. 4, (2006), 28-39.

8. Karaboga, D., "Artificial bee colony algorithm", scholarpedia, Vol. 5, No. 3, (2010), 6915.

9. Eberhart, R. and Kennedy, J., "Particle swarm optimization", in Proceedings of the IEEE international conference on neural networks, Citeseer. Vol. 4, No., (1995), 1942-1948.

10. Jia, H., Peng, X. and Lang, C., "Remora optimization algorithm", Expert systems with applications, Vol. 185, No., (2021), 115665.

11. Hussien, A.G., "An enhanced opposition-based salp swarm algorithm for global optimization and engineering problems", Journal of Ambient Intelligence and Humanized Computing, Vol. 13, No. 1, (2022), 129-150.

12. Assiri, A.S., Hussien, A.G. and Amin, M., "Ant lion optimization: Variants, hybrids, and applications", IEEE Access, Vol. 8, No., (2020), 77746-77764.

13. Mirjalili, S., Mirjalili, S.M. and Lewis, A., "Grey wolf optimizer", Advances in engineering software, Vol. 69, No., (2014), 46-61.

14. Yang, X.S. and Hossein Gandomi, A., "Bat algorithm: A novel approach for global engineering optimization", Engineering computations, Vol. 29, No. 5, (2012), 464-483.

15. Gandomi, A.H. and Alavi, A.H., "Krill herd: A new bio-inspired optimization algorithm", Communications in nonlinear science and numerical simulation, Vol. 17, No. 12, (2012), 4831-4845.

16. Mirjalili, S. and Lewis, A., "The whale optimization algorithm", Advances in engineering software, Vol. 95, No., (2016), 51-67.

17. Holland, J.H., "Genetic algorithms", Scientific american, Vol. 267, No. 1, (1992), 66-73.

18. Beyer, H.-G. and Schwefel, H.-P., "Evolution strategies–a comprehensive introduction", Natural computing, Vol. 1, No., (2002), 3-52.

19. Banzhaf, W., Koza, J., Ryan, C., Spector, L. and Jacob, C., "Genetic programming", IEEE Intelligent Systems and their Applications, Vol. 15, No. 3, (2000), 74-84.

20. Price, K.V., Differential evolution, in Handbook of optimization: From classical to modern approach. 2013, Springer.187-214.

21. Jaderyan, M. and Khotanlou, H., "Virulence optimization algorithm", Applied Soft Computing, Vol. 43, No., (2016), 596-618.

22. Hatamlou, A., "Black hole: A new heuristic optimization approach for data clustering", Information sciences, Vol. 222, No., (2013), 175-184.

23. Sinha, N., Chakrabarti, R. and Chattopadhyay, P., "Evolutionary programming techniques for economic load dispatch", IEEE transactions on evolutionary computation, Vol. 7, No. 1, (2003), 83-94.

24. Rashedi, E., Nezamabadi-Pour, H. and Saryazdi, S., "Gsa: A gravitational search algorithm", Information sciences, Vol. 179, No. 13, (2009), 2232-2248.

25. Ghasemi, M., Golalipour, K., Zare, M., Mirjalili, S., Trojovský, P., Abualigah, L. and Hemmati, R., "Flood algorithm (fla): An efficient inspired meta-heuristic for engineering optimization", The Journal of Supercomputing, Vol. 80, No. 15, (2024), 22913-23017 DOI: 10.1007/s11227-024-06291-7.

26. Kaveh, A. and Dadras, A., "A novel meta-heuristic optimization algorithm: Thermal exchange optimization", Advances in engineering software, Vol. 110, No., (2017), 69-84.

27. Kaveh, A. and Khayatazad, M., "A new meta-heuristic method: Ray optimization", Computers & Structures, Vol. 112, No., (2012), 283-294.

28. Geem, Z.W., Kim, J.H. and Loganathan, G.V., "A new heuristic optimization algorithm: Harmony search", simulation, Vol. 76, No. 2, (2001), 60-68.

29. Ghorbani, N. and Babaei, E., "Exchange market algorithm", Applied Soft Computing, Vol. 19, No., (2014), 177-187.

30. Kannan, B. and Kramer, S.N., "An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design", Vol., No., (1994).

31. Hakli, H. and Ortacay, Z., "An improved scatter search algorithm for the uncapacitated facility location problem", Computers & Industrial Engineering, Vol. 135, No., (2019), 855-867.

32. Ghannadiasl, A. and Ghaemifard, S., "Parameter selection for pso-based hybrid algorithms and its effect on crack detection in cantilever beams", Numerical Methods in Civil Engineering, Vol. 9, No. 2, (2024), 17-28 DOI: 10.61186/nmce.2311.1036.

33. Ghajarnia, N., Haddad, O.B. and Mariño, M.A., "Performance of a novel hybrid algorithm in the design of water networks", Proceedings of the Institution of Civil Engineers - Water Management, Vol. 164, No. 4, (2011), 173-191 DOI: 10.1680/wama.1000028.

34. Ghannadiasl, A. and Ghaemifard, S., "Crack detection of the cantilever beam using new triple hybrid algorithms based on particle swarm optimization", Frontiers of Structural and Civil Engineering, Vol. 16, No. 9, (2022), 1127-1140.

35. Khorram, S. and Bahrami, A., "Application of a new fuzzy dematel–todim hybrid algorithm in port dredging project management", Proceedings of the Institution of Civil Engineers - Maritime Engineering, Vol. 173, No. 3, (2020), 79-95 DOI: 10.1680/jmaen.2017.22.

36. Euchi, J. and Sadok, A., "Optimising the travel of home health carers using a hybrid ant colony algorithm", Proceedings of the Institution of Civil Engineers - Transport, Vol. 176, No. 6, (2023), 325-336 DOI: 10.1680/jtran.19.00114.

37. Hemagowri, J. and Tamil Selvan, P., "A hybrid evolutionary algorithm of optimized controller placement in sdn environment", Computer Assisted Methods in Engineering and Science, Vol., No., (2023), DOI: 10.24423/cames.498.

38. Abouhabaga, O.O.F., Gadallah, M.H., Kouta, H.K. and Zaghloul, M.A., "Hybrid inner-outer algorithm for solving real-world mechanical optimization problems", Journal of Engineering and Applied Science, Vol. 68, No. 1, (2021), 2 DOI: 10.1186/s44147-021-00004-0.

39. Fasina, E., Sawyerr, B.A., Abdullahi, Y.U. and Oke, S.A., "A comparison of two hybrid optimization techniques: The taguchi-bbd-firefly and the taguchi-regression-firefly methods on the is 2062-e250 steel plates boring problem", Journal of Engineering and Applied Science, Vol. 70, No. 1, (2023), 47 DOI: 10.1186/s44147-023-00215-7.

40. Chen, S. and Zheng, J., "A hybrid grey wolf optimizer for engineering design problems", Journal of Combinatorial Optimization, Vol. 47, No. 5, (2024), 86 DOI: 10.1007/s10878-024-01189-9.

41. Eiben, A.E. and Schippers, C.A., "On evolutionary exploration and exploitation", Fundamenta Informaticae, Vol. 35, No. 1-4, (1998), 35-50.

42. Azizi, M., "Atomic orbital search: A novel metaheuristic algorithm", Applied Mathematical Modelling, Vol. 93, No., (2021), 657-683.

43. Yang, X.-S., "Firefly algorithms for multimodal optimization", in International symposium on stochastic algorithms, Springer. Vol., No., (2009), 169-178.
44. Schmit Jr, L. and Farshi, B., "Some approximation concepts for structural synthesis", AIAA journal, Vol. 12, No. 5, (1974), 692-699.

45. Farshi, B. and Alinia-Ziazi, A., "Sizing optimization of truss structures by method of centers and force formulation", international Journal of Solids and Structures, Vol. 47, No. 18-19, (2010), 2508-2524.

46. Borges, A.d.Á., "Otimização de forma e paramétrica de estruturas treliçadas através dos métodos meta-heurísticos harmony search e firefly algorithm", Vol., No., (2013).
47. Ghaemifard, S. and Ghannadiasl, A., "A comparison of metaheuristic algorithms for structural optimization: Performance and efficiency analysis", Advances in Civil Engineering, Vol. 2024, No. 1, (2024), 2054173 DOI: https://doi.org/10.1155/2024/2054173.

48. Cheng, M.-Y. and Prayogo, D., "Symbiotic organisms search: A new metaheuristic optimization algorithm", Computers & Structures, Vol. 139, No., (2014), 98-112.

49. Bayzidi, H., Talatahari, S., Saraee, M. and Lamarche, C.-P., "Social network search for solving engineering optimization problems", Computational Intelligence and Neuroscience, Vol. 2021, No., (2021), 1-32.

50. Wang, G.G., "Adaptive response surface method using inherited latin hypercube design points", J. Mech. Des., Vol. 125, No. 2, (2003), 210-220.

51. Gandomi, A.H., Yang, X.-S. and Alavi, A.H., "Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems", Engineering with Computers, Vol. 29, No., (2013), 17-35.

52. Akay, B. and Karaboga, D., "Artificial bee colony algorithm for large-scale problems and engineering design optimization", Journal of intelligent manufacturing, Vol. 23, No., (2012), 1001-1014.

53. Sandgren, E., "Nonlinear integer and discrete programming in mechanical design optimization", Vol., No., (1990).

54. Rao, S.S., "Engineering optimization: Theory and practice, John Wiley & Sons, (2019).

55. Bayzidi, H., Talatahari, S., Saraee, M. and Lamarche, C.-P., "Social network search for solving engineering optimization problems", Computational Intelligence and Neuroscience, Vol. 2021, No., (2021), 8548639 DOI: 10.1155/2021/8548639.

56. Gandomi, A.H. and Roke, D.A., "Engineering optimization using interior search algorithm", in 2014 IEEE symposium on swarm intelligence, IEEE. Vol., No., (2014), 1-7.

57. Sadollah, A., Bahreininejad, A., Eskandar, H. and Hamdi, M., "Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems", Applied Soft Computing, Vol. 13, No. 5, (2013), 2592-2612.

58. Kamalinejad, M., Arzani, H. and Kaveh, A., "Quantum evolutionary algorithm with rotational gate and h ϵ-gate updating in real and integer domains for optimization", Acta Mechanica, Vol. 230, No. 8, (2019), 2937-2961.

59. Mezura-Montes, E. and Coello, C.A.C., "An empirical study about the usefulness of evolution strategies to solve constrained optimization problems", International Journal of General Systems, Vol. 37, No. 4, (2008), 443-473.

60. Mirjalili, S., Mirjalili, S.M. and Hatamlou, A., "Multi-verse optimizer: A nature-inspired algorithm for global optimization", Neural Computing and Applications, Vol. 27, No., (2016), 495-513.

61. Kaveh, A. and Talatahari, S., "An improved ant colony optimization for constrained engineering design problems", Engineering Computations, Vol. 27, No. 1, (2010), 155-182.

62. Kaveh, A. and Talatahari, S., "A novel heuristic optimization method: Charged system search", Acta Mechanica, Vol. 213, No. 3-4, (2010), 267-289.

63. Askarzadeh, A., "A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm", Computers & Structures, Vol. 169, No., (2016), 1-12.

64. Kaveh, A., Motie Share, M.A. and Moslehi, M., "Magnetic charged system search: A new meta-heuristic algorithm for optimization", Acta Mechanica, Vol. 224, No. 1, (2013), 85-107.

65. Azizi, M., Talatahari, S., Khodadadi, N. and Sareh, P., "Multiobjective atomic orbital search (moaos) for global and engineering design optimization", IEEE Access, Vol. 10, No., (2022), 67727-67746 DOI: 10.1109/ACCESS.2022.3186696.

66. Elaziz, M.A., Abualigah, L., Yousri, D., Oliva, D., Al-Qaness, M.A., Nadimi-Shahraki, M.H., Ewees, A.A., Lu, S. and Ali Ibrahim, R., "Boosting atomic orbit search using dynamic-based learning for feature selection", Mathematics, Vol. 9, No. 21, (2021), 2786.

Full-Text:



Islamic Azad University Rasht Branch ISSN: 2588-5723        E-ISSN:2008-5427

Optimization Iranian Journal of Optimization Volume 16, Issue 2, 2024,103-  Research Paper

 

 

                                        Online version is available on: https://sanad.iau.ir/journal/ijo

 

An Improved Atomic Orbital Search Algorithm Utilizing Firefly Algorithm for Optimization Problems 

Saeedeh Ghaemifard * 

Ph.D. student, Department of Civil Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

 

Revise Date: ...............                     Abstract

Accept Date: ....................                 

Keywords:  mathematical modelling numerical methods hybrid algorithm Atomic Orbital Search algorithm engineering optimization firefly algorithm

The Atomic Orbital Search inspiration from the rules of quantum mechanics and Firefly Algorithm is a metaheuristic technique which is widely used for solving the optimization problems. Both algorithms collectively improved the performance of search. This article goals to optimize engineering design problems utilizing a new hybrid optimizer; AOS-FA (Atomic Orbital Search-Firefly Algorithm). Incorporating the FA methodology into the basic AOS framework has successfully addressed the issue of local optima trap and significantly enhanced the quality of solutions generated by the algorithm. The FA algorithm is work on the combinatorial optimization and utilized as application of AOS algorithm. Hence, we merge these two algorithms and make a hybrid algorithm. The purpose of the suggested hybridization method was to promote the improvement of the exploration-exploitation manners of the AOS search. To analyse the viability of the suggested hybridized algorithm in real-world usages, it is studied for five constrained engineering design issues, and the performance was determined with other outstanding metaheuristics extracted from the publications.

 

 

 

 

 

 

*Correspondence E‐mail: [e-mail]

 

INTRODUCTION

   Optimization is an essential part of engineering design, and hence in numerous real-world challenging problems with various frameworks, Meta-Heuristics (MHs) have become increasingly fascinating as a robust instrument for optimization. Engineering regulation can achieve stable and efficient mechanisms by using well-designed optimum models. These models are developed based on mathematical theorems and approaches. Although optimization methods were used by historical figures such as Newton, Lagrange, and Cauchyeski for smaller-scale issues, modern engineers rely on improved and hybrid versions of these algorithms to effectively solve more extensive and more complex engineering design problems(Ghaemifard & Ghannadiasl, 2024c). Over the past two decades, the rise of environmental and global phenomena due to techno-logical advancements and population growth has made complicated engineering designs more challenging. As a solution, metaheuristic optimization algorithms have become a popular choice for achieving reasonable solutions in less time (Ghaemifard & Ghannadiasl, 2024b; Houssein, Mahdy, Shebl, & Mohamed, 2021). Numerous metaheuristic optimization algorithms have been developed and proven effective in improving optimization processes beyond their predecessors, despite their unique processes and textures. To tackle global optimization problems, meta-heuristic algorithms are a frequently employed solution. The optimal solution is primarily achieved by simulating of both nature and human intelligence. By conducting a global search, they can identify an approximate solution that closely approximates the optimal solution to some degree. Exploration and exploitation are the fundamental principles of MHs. Exploration is crucial in order to thoroughly search the entire space and locate the optimal solution, which could potentially be located anywhere within it. To maximize the use of valuable information, it is essential to engage in effective exploitation. Optimal solutions are generally correlated in specific ways. Utilize these correlations to regulate gradually and search slowly from the initial answer to get the optimal solution. MHs strive to achieve a harmonious balance between exploration and exploitation. MHs have gained significant attention from scholars in recent years due to their numerous advantages, including their simple and intuitive operation, as well as their fast-running speed (Fazli, Khiabani, & Daneshian, 2022; Ghaemifard & Ghannadiasl, 2024c; Ghannadiasl & Ghaemifard, 2024a; Shahebrahimi, Lork, Shayegan, & Amir). There have been numerous proposals for meta-heuristic algorithms, totalling in the hundreds. MHs can be categorized into four groups based on various design inspirations: evolutionary, physical, swarm-based, and human-based algorithms. Swarm-based algorithms are a powerful tool in optimization, and computational intelligence has made great strides in recent years. These algorithms include Ant Colony Optimization (Dorigo, Birattari, & Stutzle, 2007), Artificial Bee Colony (Karaboga, 2010), Particle Swarm Optimization (Eberhart & Kennedy, 1995), Remora Optimization Algorithm (Jia, Peng, & Lang, 2021), Slap Swarm Algorithm (Hussien, 2022), Ant Lion Optimizer (Assiri, Hussien, & Amin, 2020), Grey Wolf Optimization (Mirjalili, Mirjalili, & Lewis, 2014), Bat Algorithm (X. S. Yang & Hossein Gandomi, 2012), Krill Herd (X. S. Yang & Hossein Gandomi, 2012), and Whale Optimization Algorithm (Mirjalili & Lewis, 2016). Each of these algorithms has its strengths and weaknesses, and researchers continue to explore new variations and combinations to push the boundaries of what is possible in optimization. Whether you are working in engineering, finance, or any other field where optimization is critical, these swarm-based algorithms offer robust solutions that can help you achieve your goals. There are several types of evolutionary algorithms available for use, including Genetic Algorithm (GA) (Holland, 1992), Evolution Strategy (ES) (Beyer & Schwefel, 2002), Genetic Programming (GP) (Banzhaf, Koza, Ryan, Spector, & Jacob, 2000), Differential Evolution (DE) (Price, 2013), Virulence Optimization Algorithm (VOA) (Jaderyan & Khotanlou, 2016), Black Hole Algorithm (BH) (Hatamlou, 2013), Evolutionary Programming (EP) (Sinha, Chakrabarti, & Chattopadhyay, 2003), Gravitational Search Algorithm (GSA) (Rashedi, Nezamabadi-Pour, & Saryazdi, 2009). Several physical-based algorithms have been developed to address optimization problems. These include Simulated Annealing, Flood algorithm (FLA) (Ghasemi et al., 2024), Thermal Exchange Optimization (TEO) (Ali Kaveh & Dadras, 2017) and Ray Optimization (RO) (A Kaveh & Khayatazad, 2012). Harmony Search (HS) (Geem, Kim, & Loganathan, 2001), and Exchanged Market Algorithm (EMA) (Ghorbani & Babaei, 2014) are categorized as human‐based algorithms. MHs can be significantly optimized through the use of these algorithms. Researchers have proposed various methods to enhance the convergence performance and efficiency of metaheuristic algorithms. To achieve this goal, improved versions such as those developed by (Ghannadiasl & Ghaemifard, 2024b; Hakli & Ortacay, 2019; Kannan & Kramer, 1994) have been expressed, as well as hybrid versions that combine multiple algorithms such as those developed (Abouhabaga, Gadallah, Kouta, & Zaghloul, 2021; Chen & Zheng, 2024; Euchi & Sadok, 2021; Fasina, Sawyerr, Abdullahi, & Oke, 2023; Ghajarnia, Bozorg Haddad, & Mariño, 2011; Ghannadiasl & Ghaemifard, 2022; Hemagowri & Selvan, 2023; Khorram & Bahrami, 2020). These approaches have shown promise in producing solutions with fewer iterations. Fig.1a, displays a comparison graph of the number and percentage of studies conducted on hybrid optimization algorithms over the years. Over the past two decades, there has been a significant rise in the utilization of hybrid metaheuristic optimization algorithms. Fig.1b presents the findings of a study that analysed the distribution of hybrid optimization studies across various fields using data obtained from the Web of Science database. While approximately 50% of studies fall outside of the fields represented in the Fig.1, it is clear that hybrid optimization algorithms are widely utilized in multidisciplinary engineering research. The purpose of this paper is to introduce an innovative algorithm called AOS-FA (Atomic Orbital Search­-­Firefly optimization). The purpose of developing this algorithm was to test its effectiveness in achieving global optimum solutions and enhancing overall performance. In Fig.2, the main objective and general process of the paper are presented. There are five sections to the remainder of the study. We describe the procedures in Sect. 2. In Section 3, numerical examples are shown, and the effectiveness of recommendation algorithms is assessed. Conclusions and upcoming projects are discussed in Sect. 4.

 

METHODS

The proposed AOS-FA method

A good meta-heuristic algorithm balances its exploration and exploitation functions to achieve optimal performance (Eiben & Schippers, 1998). The Atomic Orbital Search method boasts robust global optimization, adaptability, and robustness (Mahdi Azizi, 2021). The Firefly algorithm has strong local search abilities and fast convergence, but it often converges to a local optimum instead of a global optimal solution (X.-S. Yang, 2009).

 

 

 

(a)

(b)

 

Fig. 1. Web of Science citation report studies: (a) Number of published hybrid optimization studies, (b)  Distribution of hybrid optimization studies according to fields

 

This section presents the proposed algorithm hybrid AOS-FA, which combines the benefits of two metaheuristic algorithms: AOS and FA. FA has strong exploration capabilities, allowing it to visit all local and global modes and find suitable solutions, while AOS has high exploitation capabilities. The AOS-FA algorithm is based on three principles. Hybrid algorithms can supplement strengths and weaknesses. By establishing new populations that share the best individuals from both groups, this mixture can protect against early convergence while retaining helpful qualities from AOS and FA. Ultimately, the AOS-FA algorithm uses only the parameters from the original AOS and FA algorithms. Fig. 3 illustrated the pseudo-code of AOS-FA. In this hybrid algorithm first the AOS algorithm is run.

 

Fig.2. General process of the paper

 

To improve optimization algorithms, in the quantum-based atomic model, the AOS algorithm suggests using solution candidates (X) as electrons. The solution candidates () represent each electron, and decision variables ) are used to deter in the search space. In this hybrid algorithm first the AOS algorithm is run. To improve optimization algorithms, in the quantum-based atomic model, the AOS algorithm suggests using solution candidates (X) as electrons. In the search space, solution candidates () represent each electron, and decision variables () are used to determine their positions.

 

(1)

To describe the location of solution volunteers within the probe zone, the problem dimension is defined via d. Parameter m is the presenter of the number of solution candidates. Each electron has an energy state, according to the quantum atomic model. The objective subordinate is this energy state.

 

(2)

To determine the objective function values, refer to vector E. Energy level  pertains to the ith solution volunteers. At the same time, the number of electrons in the probe area is represented by m. The Probability Density Function is a mathematical model utilized to define the station of electrons around the nucleus in the quantum atomic model. According to probability theory, Probability Density Function expresses the probability of a variable happening within a special scope. The Probability Density Function analysis reveals that the solution candidates are distributed among the imaginary layers created to determine the position of electrons. In the following, the vectors of location and objective subordinate values of answer volunteers in imaginary layers are expressed as Eq3 and Eq4  respectively.

 

(3)

 

(4)

The electrons located near to the nucleus are supposed to be in the base mode of energy. The mathematical model utilizes the place and purpose subordinate amounts of solution candidates in each layer to define the binding energy essential for drawing an electron from its cover. To define the binding condition and binding energy of solution candidates in each imaginary layer, the situations and purpose subordinate amounts of all candidates in the layer are averaged. To achieve the intended purpose, the following mathematical equations are provided:

 

 

In the kth layer,  and  represent the condition and energy of binding, respectively. The location and objective subordinate amount of ith solution candidates are denoted by  and , while m represents the whole number of solution candidates in the probe area. To accurately assess the state and strength of binding of an atom, one must take into account the average positions and objective function values of all potential solutions in the search space. The location (Xi) and purpose subordinate amount () of each solution candidate within the atom can be analyzed to determine the atom's binding state BS and binding energy BE. Use the following mathematical equations to update the candidate positions:

 

=	(5)
=	(6)

To determine the amount of emitted energy, vectors , , and   are randomly generated with a uniform distribution between (0,1). and  represent the current and upcoming positions, while LE refers to the candidate with the lowest energy level in the atom. Additionally, BS denotes the binding state of the atom. photon absorption happened when the energy level of a solution candidate in a special layer is lower than the layer's binding energy. This procedure involves solution candidates absorbing a photon with energy levels β and γ to efficiently get both the binding situation of the layer and the lowest energy level condition of the electron within the layer. The mathematical equation to update the location of solution candidates is (Mahdi Azizi, 2021):

 

=

(7)

After running the AOS algorithm, it is time to run FA algorithm so that firstly, rank the population in FA algorithm. The attractiveness of a Firefly is shown in as Eq 9.

 

=

(8)

(9)

Let's consider a scenario with n fireflies, where xi represents the solution for each individual Firefly. The brightness of a firefly, denoted as i, is closely linked to the objective function f(xi). The objective function expressed in Eq 10, showed the brightness I of a firefly.

 

(10)

 

The dimmer Firefly is absorbed and moves towards the shining one, and parameter β expressed the specific level of attractiveness of each Firefly. β, is related to the spacing between Fireflies. The attractiveness function of the Firefly is showed as Eq11.

 

(11)

 

 represents the attractiveness of the Firefly when it is at r = 0, while γ represents the light absorption coefficient of the media. Firefly at location xi moves towards a brighter Firefly at location xj using Eq12 (X.-S. Yang, 2009).

 

(12)

 

When the Firefly xj is attracted,  affects the movement, while  is a randomization parameter. If  = 0, the movement is random The algorithm compares the Firefly's new location to the past one to define its fascination. If the new position seems more attractive, the Firefly will move; if not, it will stay in its current position. The stopping criteria for the FA are set by a pre-defined number of iterations or a fitness value deemed appropriate. According to Eq13, the Firefly that shines the brightest moves in a random pattern.

 

(13)

 

NUMERICAL EXAMPLES

The results illustrated in this part can be used to compare the efficiency of the proposed algorithm investigated in this article. An Intel i5 (2.4 GHz) system with 8 GB of RAM was utilized for all of the simutlations. To validate and compare the algorithms detailed in this article with other algorithms, these results are compared with the results of some studies. In Table 1, the control parameters are defined for estimating different plans of the suggested algorithm. The control parameters have been set to assess the various processes of the suggested algorithm based on the standard range of each algorithm which stated in other articles.

 

Table 1: Control parameters of algorithms

Optimization parameters	Value
Gamma parameter of FA	1
Beta0 parameter of FA	2
Alpha parameter of FA	0.2
FotonRate parameter of AOS	0.1
LayerNumber parameter of AOS	5

 

In this section, we will delve into a comprehensive analysis of the prevailing engineering design issues. It is noteworthy to emphasize that the ensuing discussion focuses exclusively on the most renowned engineering design problems encountered in practical applications. Efficacious resolution of these problems typically necessitates a proactive methodology aimed at determining the optimal parameters for the most ideal design.

 

 

 

Procedure Atomic Orbital Search- Firefly Algorithm

 Objective function 𝑓(𝑥), 𝑥= (𝑥1, 𝑥2, …,𝑥𝑑)𝑇

Determine initial positions of solution candidates (Xi) in the search space with m candidates

                Evaluate fitness values (Ei) for initial solution candidates

               Determine the binding state(BS) and binding energy (BE) of the atom

              Determine the candidate with the lowest energy level in the atom (LE)

              While Iteration < Maximum number of iterations

Generate n as the number of imaginary layers

                                                Create imaginary layers

                         Sort solution candidates in an ascending or descending order

                         Distribute solution candidates in the imaginary layers by PDF

                                    For k=1: n

                   Determine the binding state (BSk) and binding energy (BEk) of the kth layer

                 Determine the candidate with the lowest energy level in the kth layer (LEk)

                                   For i=1: p

                                                  Generate

                                                  Determine PR

                                                  If

                                                 If  

 

                                               Else if

 

                                              end

                                              Else if

 

                                            End

                                           End

                                          End

Update binding state(BS) and binding energy (BE) OF ATOM

                                   Update candidate with the lowest energy level in the atom (LE)

End while

Rank the population 𝐗i, and update the current best.

Initialize a population of fireflies 𝑋𝑖 (𝑖=1, 2,…,𝑛)

Calculate the fitness value 𝑓(𝑋𝑖) to determine the light intensity 𝐼𝑖 at 𝑋𝑖

 Define light absorption coefficient 𝛾

while (𝑡<𝑀𝑎𝑥𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛_𝐹𝐴)
        for 𝑖=1:𝑛=1: all n fireflies
             for 𝑗=1:𝑛=1: all n fireflies
                  if (𝐼𝑗>𝐼𝑖)
                       Move firefly i towards j in all d-dimensions via Lévy flight.
                  end if.

Attractiveness varies with distance r via −𝑒−𝛾𝑟2.
                  Evaluate new solutions and update light intensity.
             end for j
        end for i
        Rank the fireflies and find the current best.   
   end while
   Output the best solution.
End procedure

Fig.3. Pseudo-code of AOS-FA Algorithm

Optimization of truss size

In this section, the size optimization of a 10-bar truss (Fig. 4) is studied. This truss has been numerically investigated by several researchers like Schmit Jr and Farshi (1974), Farshi and Alinia-Ziazi (2010). In analysing the 10-bar plane truss, displacement, and stress constraints are utilized with each other. The translations of nodes 5 and 6, located on the left, are constrained in the x and y directions. The two free nodes of the lower bars (2 and 4) obtain vertical loads (y-direction). 

 

 

Fig.4. Ten-bar truss

 

All bars, except number 9, have the same tension limit for traction and compression. Nodes 1 through 4 have the same displacement limit in the y-direction. The cross-sectional areas of the ten elements are considered as continuous design variables. In Table 2, the mechanical properties, loading, stresses and displacements, and design variables of the truss are presented in Table 3, while Table 4 details the decision-making criteria and constraints to arrive at the best option. Table 4 shows that the FA algorithm with an objective function value of 2298.77 provided the best solution for truss size optimization, outperforming AOS and AOS-FA. All optimization algorithms had 100 research agents and 300 iterations. Fig.6 presents the computational time and standard deviation values for the minimum mass obtained after four independent executions of each algorithm. Based on Table 4 and Fig.5, it is clear that the AOS-FA algorithm, despite its shorter runtime, was not effective in optimizing the truss size. It is noticed that in this problem, numerical method has better results against to AOS-FA although it has not good results against to other algorithms that mentioned.

 

Table 2: Mechanical properties of the considered truss

 

 

 

 

Table 3: Nodal loading components and constraints for 10 bar plane truss

Material	Aluminium
Density, ρ	2767.99 kg/m3
Young’s modulus, E	68.95109 N/m 2

 

 

 

 

 

 

 

 

 

 

 

Fig.5. Standard deviation and computational time (s) for ten-bar plane truss

 

 

Design of I-shaped beams

In the I-beam design problem (Fig. 7), the goal is to minimize the vertical deflection, while satisfying the cross-sectional area and stress constraints under given loads. The variables of this problem are the width of the flange b (= x1), the height of section h (= x2), the thickness of the web tw (= x3), and the thickness of the flange tf (= x4). The maximum vertical deflection of the beam is . The objective function when the modulus of elasticity is 523.104 kN/cm2 and L=5200 cm, is formulated as follows:

Minimize:

 

Loading
No.	X-direction		Y-direction
2 and 4	0		-444.82 KN
Tension restrictions
Bar		Value
9		±517.11 MPa
Displacement
NO.			Value
1,2,3,4			±50.8 mm (Y direction)
The range of design variables
64.5 mm²  20000 mm²

Subject to:

(14)

In this problem, the best statistical results were achieved after four independent implementations using 100 search agents and 300 iterations by algorithms mentioned in Tables 5 and 6, respectively. From Table 6, it can be understood that AOS obtained the shortest computational execution time. According to Table 5 can say that metaheuristic algorithms that used for this problem, has good performances and the obtained results against to ARSM method is good. This subject shows that metaheuristic algorithms have good results than numerical methods. Also, it is determined that AOS-FA is top from AOS.

 

 

 

 

Fig.7. Beam design problem

 

Table 4: Optimum design of cross-sections (cm²) for ten-bar plane truss

	(15)
	(16)

Present paper					Schmit Jr and Farshi (1974)	Borges (2013)		Ghaemifard and Ghannadiasl (2024a)
Member	FA	AOS	AOS-FA	Numerical method	PSO	HS	GWO
1	193.699	164.245	197.142	215.676	180.150	196.180	198.94
2	0.645	5.673	0.645	0.645	0.645	1.128	0.71
3	159.320	192.922	164.340	156.5158	151.150	144.560	156.27
4	93.244	187.862	106.074	91.9998	99.269	103.170	95.80
5	0.645	0.645	0.645	0.645	0.645	0.645	0.65
6	3.517	0.645	0.645	0.645	3.726	3.560	0.64
`7	47.892	61.004	52.977	54.1289	47.710	48.884	54.72
8	140.162	167.049	139.323	133.8061	139.810	138.040	133.59
9	134.560	99.228	121.686	127.032	146.780	138.020	134.10
10	0.645	0.645	0.645	0.645	0.645	0.673	0.74
Mass (kg)	2298.77	2700.16	2314.5	2308.3315	2301.41	2302.60	2303.41

 

Pressure vessel design

The goal of designing pressure vessels is to meet production needs while reducing container costs. The key design variables are head thickness (Th), shell thickness (Ts), container length (L), and inner radius (R). in this problem, Ts and Th are integers of 0.625, while R and L are continuous variables. Fig.8 shows the optimal structure design schematic.

 

 

            Fig.8. Schematic of the pressure vessel

 

Mathematical formulation for this problem is:

 Consider:

 

(17)

 

Minimize:

 

 

Subject to:

0.6224[( ((2.856

(18)

 

Variable Range:

 

 

	(19)
	(20)
	(21)
	(22)

 

Table 5: Best results for the optimal design of I-Shaped Beam

(23)

Present work						Another research
		FA	AOS		AOS-FA		SOS (Cheng & Prayogo, 2014)	SNS  (Bayzidi, Talatahari, Saraee, & Lamarche, 2021b)	ARSM (Wang, 2003)
Variables	x1	80	80	80		80	80	80
	x2	50	50	50		50	50	37.05
	x3	0.9	0.9	0.9		0.9	0.9	1.71
	x4	2.32	2.32	2.32		2.3217	2.3217	2.31
Constraints	g1	0.0766	-21.89	0.07668		-0.000222	0	---
	g2	4.4285	5.9667	4.4285		-1.57	-1.5702	---
FCost		0.0131	0.0130		0.0130		0.0130	0.0130	0.0157

 

Method	Standard deviation	Mean	Max	Min	Time(s)
FA	11493.9	1.96171	1.96171	1.96171	35.04
AOS	33.6697	31.5419	80	0.9	0.0009688
AOS-FA	0	1.9622	1.9622	1.9622	3.0606
Ghaemifard and Ghannadiasl (2024a)	0	1.99466e+16	1.99466e+16	1.99466e+16	11.862

Table 6: The statistical results of each algorithm

 

 

 

 

 

 

Table 7, presents the results of pressure vessel design issues. AOS-FA is a cost-effective algorithm that produces excellent results. The statistical results of the Pressure Vessel Design problem are presented in Table 8. It shows that the AOS-FA has the shortest computational execution time. According to the Table, it is determined that AOS obtained optimal result contrast to AOS-FA and FA and FA has better result from AOS-FA. Although AOS-FA algorithm has optimal result contrast to NLP method which shows hybrid algorithm has good performance.

 

 

Tubular column design

In this section, the design of a tubular column that is uniform in shape and can withstand the pressure at minimum cost was presented. The optimization variables for this problem are the average diameter of the column d (x1) and the thickness of the tube t (x2). The object has a yield stress of 500 kgf/cm2, modulus of elasticity of 0.85 × 106 kgf/cm2, and density of 0.0025 kgf/cm3. The formula for this problem is:

 

(24)

 

Table 7: Variables design of problem

	Algorithm	X1	X2	X3	X4	Optimal cost
Present study	FA	0.7967	0.3938	41.281408	187.031440	5979.983
	AOS	0.7782	0.3933	40.714312	195.311948	5888.6
	AOS-FA	0.9950	0.4933	51.2635	178.6489	6404.9
Other research	CS (Amir Hossein Gandomi, Yang, & Alavi, 2013)	0.8125	0.4375	42.0984456	176.6365	6059.714
	ABC (Akay & Karaboga, 2012)	0.8125	0.4375	42.098446	176.6365	6059.714
	NLP (Sandgren, 1990)	1.125	0.625	48.97	106.72	7982.5

 

 

Table 8: Statistical results of the Pressure vessel design problem

	FA	AOS	AOS-FA
Std	98.8673	17.5656	0
Min	5917.84	0.493384	6443.12
Max	6093.99	185.38	6446.12
Mean	5979.98	62.3957	6443.12
Time	3.34	1.97	1.32

 

The constraints on the stresses in the columns are:

 

 

Where the range of variables is:

	(25)
	(26)
	(27)
	(28)
	(29)
	(30)

 

Fig.9. Model of the tubular column

We compared the optimal solution obtained by AOS-FA with other papers and found that AOS-FA had good results in this problem. The particular information is shown in Table 9 and 10 respectively.

Welded beam design

This is an engineering optimization problem involving the design of a welded beam to minimize cost, with four variables to optimize and seven constraints, as illustrated in Fig.10. The formula for this problem is:

Minimize:

 

(31)

It was denoting the thickness of the weld, height, length, and reinforcement thickness as h, l, t, and b showed as x1, x2, x3, and x4, respectively. 

 

 

 

Fig. 10. Welded beam design problem

 

 

 

 

 

Table 9: Statistical results of the Pressure vessel design problem

	(32)
	(33)
	(34)
	(35)
	(36)
	(37)
	(38)
	(39)
	(40)

			Present work		Another research
	Exact value  (Rao, 2019)		FA	AOS	AOS-FA	SNS (Bayzidi, Talatahari, Saraee, & Lamarche, 2021a)	ISA (Amir H Gandomi & Roke, 2014)
Variables	x1	5.44	5.4521	5.4520	5.4526	5.4513	5.4511
	x2	0.293	0.2916	0.2916	0.2916	0.2919	0.2919
Constraints	g1	---	-9.9747e-09	-2.0701e-04	-1.2823e-04	-0.024	-2.5e-10
	g2	---	-3.2983e-07	-0.0041	-0.0121	-0.109	-1.8e-10
	g3	---	-0.6332	-0.6332	-0.6332	-0.633	-0.633
	g4	---	-0.6106	-0.6106	-0.6105	-0.610	-0.6106
	g5	---	-0.6332	-0.6332	-0.6332	-0.315	-0.3149
	g6	---	-0.3185	-0.3185	-0.3184	-0.635	-0.635
FCost		26.53	26.4864	26.4882	26.4886	26.532	26.4994

 

Table 10: The statistical results of each algorithm

Method	Standard deviation	Mean	Max	Min	Time(s)
FA	2.66919e-05	26.4864	26.4864	26.4864	35.154
AOS	3.64895	2.87187	5.45206	0.291671	7.2689
AOS-FA	0	26.4886	26.4886	26.4886	27.618

 

 

Table 11 shows the best results achieved after four independent implementations, using 100 search agents and 300 iterations with the mentioned algorithms. According to the Table 11 can say that performance of the AOS-FA algorithm is better than the FA algorithm and that is better than the AOS algorithm. Also, it is funded that metaheuristic algorithms have good result against to exact value which is showed the robust of these algorithms. Figure 11 displays the statistical results from four independent algorithm executions and their corresponding computational times. The problem is evaluated using the studied algorithms in Table 11 and compared to other literature. For instance, Sadollah, Bahreininejad, Eskandar, and Hamdi (2013) achieved the best cost value of 1.724853 when investigating MBA for this problem. In previous studies, different algorithms were evaluated for welded beam optimization. Kamalinejad, Arzani, and Kaveh (2019) and Mezura-Montes and Coello (2008) obtained 1.742706 and 1.737300 values for QEA and ES algorithms, respectively. Mirjalili, Mirjalili, and Hatamlou (2016) used MVO and achieved a result of 1.72645. The best cost values were obtained by ICO (A Kaveh & S Talatahari, 2010), CSS (A Kaveh & Siamak Talatahari, 2010), CSA (Askarzadeh, 2016), and MCSS (A Kaveh, Motie Share, & Moslehi, 2013)  algorithms, which resulted in 1.724918, 1.724866, 1.7248523, and 1.724855, respectively. Additionally, FA algorithm achieved a low standard deviation value, and AOS was the fastest algorithm in computational execution time, as shown in Fig.11.

 

 

 

 

 

 

 

Table 11: The best result of algorithms

		Exact value (Rao, 2019)	FA	AOS	AOS-FA
Variables	x1	0.2455	0.20572	0.18364	0.20157
	x2	6.1960	3.47049	4.00818	3.54910
	x3	8.2730	9.03662	9.06541	9.07004
	x4	0.2455	0.20572	0.205644	0.20557
Constraints	g1	---	-0.0081	-7.6980	0
	g2	---	-0.0038	-0	0
	g3	---	-9.2473e-08	-0.0220	0
	g4	---	-3.4330	-3.3813	-3.4121
	g5	---	-0.0807	-0.0586	-0.0717
	g6	---	-0.2355	-0.2357	-0.2356
	g7	---	-0.0072	-5.0511	0
FCost		2.386	1.7249	1.7645	1.7212

 

(a)	(b)
(c)	(d)

Fig.11. The statistical results of each algorithm (a) Std, (b) Mean, (c) Max-Min, (d) best value and time

 

 

CONCLUSION

The AOS algorithm outperforms other alternative meta-heuristics in converging to the global best for various mathematical test functions. It also excels in generating superior results with fewer function evaluations, showcasing its efficiency in addressing computational complexity issues. To boost the algorithm's performance, several researchers have introduced various enhancements (M. Azizi, Talatahari, Khodadadi, & Sareh, 2022; Elaziz et al., 2021). Additionally, the Firefly algorithm, inspired by the flashing behavior and bioluminescent communication of fireflies, is susceptible to premature convergence. Studies recommend adjusting constant parameters to alleviate this issue. This paper introduces a hybrid algorithm called AOS-FA for optimal engineering design, combining Atomic Orbital Search and the Firefly Algorithm based on quantum mechanics principles. The algorithm is evaluated on five well-known constrained design

problems across different engineering fields, demonstrating that AOS-FA surpasses most recent meta-heuristic algorithms in the literature in terms of performance. Furthermore, the suggested AOSFA might be very useful in resolving additional challenging optimization issues like feature selection, picture segmentation, path planning, traveling salesman issues, and flow shop scheduling issues.

 

 

 

 

 

REFERENCES

Abouhabaga, O. O. F., Gadallah, M. H., Kouta, H. K., & Zaghloul, M. A. (2021). Hybrid inner-outer algorithm for solving real-world mechanical optimization problems. Journal of Engineering and Applied Science, 68, 1-18.

Akay, B., & Karaboga, D. (2012). Artificial bee colony algorithm for large-scale problems and engineering design optimization. Journal of intelligent manufacturing, 23, 1001-1014.

Askarzadeh, A. (2016). A novel metaheuristic method for solving constrained engineering optimization problems: crow search algorithm. Computers & Structures, 169, 1-12.

Assiri, A. S., Hussien, A. G., & Amin, M. (2020). Ant lion optimization: variants, hybrids, and applications. IEEe Access, 8, 77746-77764.

Azizi, M. (2021). Atomic orbital search: A novel metaheuristic algorithm. Applied Mathematical Modelling, 93, 657-683.

Azizi, M., Talatahari, S., Khodadadi, N., & Sareh, P. (2022). Multiobjective Atomic Orbital Search (MOAOS) for Global and Engineering Design Optimization. IEEE Access, 10, 67727-67746. doi:10.1109/ACCESS.2022.3186696

Banzhaf, W., Koza, J., Ryan, C., Spector, L., & Jacob, C. (2000). Genetic programming. IEEE Intelligent Systems and their Applications, 15(3), 74-84.

Bayzidi, H., Talatahari, S., Saraee, M., & Lamarche, C.-P. (2021a). Social Network Search for Solving Engineering Optimization Problems. Computational Intelligence and Neuroscience, 2021, 8548639. doi:10.1155/2021/8548639

Bayzidi, H., Talatahari, S., Saraee, M., & Lamarche, C.-P. (2021b). Social network search for solving engineering optimization problems. Computational Intelligence and Neuroscience, 2021, 1-32.

Beyer, H.-G., & Schwefel, H.-P. (2002). Evolution strategies–a comprehensive introduction. Natural computing, 1, 3-52.

Borges, A. d. Á. (2013). Otimização de forma e paramétrica de estruturas treliçadas através dos métodos meta-heurísticos harmony search e firefly algorithm.

Chen, S., & Zheng, J. (2024). A hybrid grey wolf optimizer for engineering design problems. Journal of Combinatorial Optimization, 47(5), 86.

Cheng, M.-Y., & Prayogo, D. (2014). Symbiotic organisms search: a new metaheuristic optimization algorithm. Computers & Structures, 139, 98-112.

Dorigo, M., Birattari, M., & Stutzle, T. (2007). Ant colony optimization. IEEE computational intelligence magazine, 1(4), 28-39.

Eberhart, R., & Kennedy, J. (1995). Particle swarm optimization. Paper presented at the Proceedings of the IEEE international conference on neural networks.

Eiben, A. E., & Schippers, C. A. (1998). On evolutionary exploration and exploitation. Fundamenta Informaticae, 35(1-4), 35-50.

Elaziz, M. A., Abualigah, L., Yousri, D., Oliva, D., Al-Qaness, M. A., Nadimi-Shahraki, M. H., . . . Ali Ibrahim, R. (2021). Boosting atomic orbit search using dynamic-based learning for feature selection. Mathematics, 9(21), 2786.

Euchi, J., & Sadok, A. (2021). Optimising the travel of home health carers using a hybrid ant colony algorithm. Paper presented at the Proceedings of the institution of civil engineers-transport.

Farshi, B., & Alinia-Ziazi, A. (2010). Sizing optimization of truss structures by method of centers and force formulation. international Journal of Solids and Structures, 47(18-19), 2508-2524.

Fasina, E., Sawyerr, B. A., Abdullahi, Y. U., & Oke, S. A. (2023). A comparison of two hybrid optimization techniques: the Taguchi-BBD-firefly and the Taguchi-regression-firefly methods on the IS 2062-E250 steel plates boring problem. Journal of Engineering and Applied Science, 70(1), 47.

Fazli, M., Khiabani, F. M., & Daneshian, B. (2022). meta-heuristic algorithms to solve the problem of terminal facilities on a real scale. Iranian Journal of Optimization, 14(1), 51-65.

Gandomi, A. H., & Roke, D. A. (2014). Engineering optimization using interior search algorithm. Paper presented at the 2014 IEEE symposium on swarm intelligence.

Gandomi, A. H., Yang, X.-S., & Alavi, A. H. (2013). Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Engineering with Computers, 29, 17-35.

Geem, Z. W., Kim, J. H., & Loganathan, G. V. (2001). A new heuristic optimization algorithm: harmony search. simulation, 76(2), 60-68.

Ghaemifard, S., & Ghannadiasl, A. (2024a). A Comparison of Metaheuristic Algorithms for Structural Optimization: Performance and Efficiency Analysis. Advances in Civil Engineering, 2024(1), 2054173. doi:https://doi.org/10.1155/2024/2054173

Ghaemifard, S., & Ghannadiasl, A. (2024b). Design and Analysis of Offshore Wind Turbines: Problem Formulation and Optimization Techniques. Journal of Marine Science and Application, 23(4), 707-722.

Ghaemifard, S., & Ghannadiasl, A. (2024c). Usages of metaheuristic algorithms in investigating civil infrastructure optimization models; a review. AI in Civil Engineering, 3(1), 17.

Ghajarnia, N., Bozorg Haddad, O., & Mariño, M. A. (2011). Performance of a novel hybrid algorithm in the design of water networks. Paper presented at the Proceedings of the Institution of Civil Engineers-Water Management.

Ghannadiasl, A., & Ghaemifard, S. (2022). Crack detection of the cantilever beam using new triple hybrid algorithms based on Particle Swarm Optimization. Frontiers of Structural and Civil Engineering, 16(9), 1127-1140.

Ghannadiasl, A., & Ghaemifard, S. (2024a). Meta-heuristic algorithms: an appropriate approach in crack detection. Innovative Infrastructure Solutions, 9(7), 263.

Ghannadiasl, A., & Ghaemifard, S. (2024b). Parameter Selection for PSO-Based Hybrid Algorithms and Its Effect on Crack Detection in Cantilever Beams. Numerical Methods in Civil Engineering, 9(2), 17-28.

Ghasemi, M., Golalipour, K., Zare, M., Mirjalili, S., Trojovský, P., Abualigah, L., & Hemmati, R. (2024). Flood algorithm (FLA): an efficient inspired meta-heuristic for engineering optimization. The Journal of Supercomputing, 80(15), 22913-23017.

Ghorbani, N., & Babaei, E. (2014). Exchange market algorithm. Applied Soft Computing, 19, 177-187.

Hakli, H., & Ortacay, Z. (2019). An improved scatter search algorithm for the uncapacitated facility location problem. Computers & Industrial Engineering, 135, 855-867.

Hatamlou, A. (2013). Black hole: A new heuristic optimization approach for data clustering. Information sciences, 222, 175-184.

Hemagowri, J., & Selvan, P. T. (2023). A hybrid evolutionary algorithm of optimized controller placement in sdn environment. Computer Assisted Methods in Engineering and Science, 30(4), 539-556.

Holland, J. H. (1992). Genetic algorithms. Scientific american, 267(1), 66-73.

Houssein, E. H., Mahdy, M. A., Shebl, D., & Mohamed, W. M. (2021). A survey of metaheuristic algorithms for solving optimization problems. In Metaheuristics in machine learning: theory and applications (pp. 515-543): Springer.

Hussien, A. G. (2022). An enhanced opposition-based salp swarm algorithm for global optimization and engineering problems. Journal of Ambient Intelligence and Humanized Computing, 13(1), 129-150.

Jaderyan, M., & Khotanlou, H. (2016). Virulence optimization algorithm. Applied Soft Computing, 43, 596-618.

Jia, H., Peng, X., & Lang, C. (2021). Remora optimization algorithm. Expert Systems with Applications, 185, 115665.

Kamalinejad, M., Arzani, H., & Kaveh, A. (2019). Quantum evolutionary algorithm with rotational gate and H ϵ-gate updating in real and integer domains for optimization. Acta Mechanica, 230(8), 2937-2961.

Kannan, B., & Kramer, S. N. (1994). An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design.

Karaboga, D. (2010). Artificial bee colony algorithm. scholarpedia, 5(3), 6915.

Kaveh, A., & Dadras, A. (2017). A novel meta-heuristic optimization algorithm: thermal exchange optimization. Advances in engineering software, 110, 69-84.

Kaveh, A., & Khayatazad, M. (2012). A new meta-heuristic method: ray optimization. Computers & structures, 112, 283-294.

Kaveh, A., Motie Share, M. A., & Moslehi, M. (2013). Magnetic charged system search: a new meta-heuristic algorithm for optimization. Acta Mechanica, 224(1), 85-107.

Kaveh, A., & Talatahari, S. (2010). An improved ant colony optimization for constrained engineering design problems. Engineering Computations, 27(1), 155-182.

Kaveh, A., & Talatahari, S. (2010). A novel heuristic optimization method: charged system search. Acta Mechanica, 213(3-4), 267-289.

Khorram, S., & Bahrami, A. (2020). Application of a new fuzzy Dematel–Todim hybrid algorithm in port dredging project management. Paper presented at the Proceedings of the Institution of Civil Engineers-Maritime Engineering.

Mezura-Montes, E., & Coello, C. A. C. (2008). An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. International Journal of General Systems, 37(4), 443-473.

Mirjalili, S., & Lewis, A. (2016). The whale optimization algorithm. Advances in engineering software, 95, 51-67.

Mirjalili, S., Mirjalili, S. M., & Hatamlou, A. (2016). Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Computing and Applications, 27, 495-513.

Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46-61.

Price, K. V. (2013). Differential evolution. In Handbook of optimization: From classical to modern approach (pp. 187-214): Springer.

Rao, S. S. (2019). Engineering optimization: theory and practice: John Wiley & Sons.

Rashedi, E., Nezamabadi-Pour, H., & Saryazdi, S. (2009). GSA: a gravitational search algorithm. Information sciences, 179(13), 2232-2248.

Sadollah, A., Bahreininejad, A., Eskandar, H., & Hamdi, M. (2013). Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems. Applied Soft Computing, 13(5), 2592-2612.

Sandgren, E. (1990). Nonlinear integer and discrete programming in mechanical design optimization.

Schmit Jr, L., & Farshi, B. (1974). Some approximation concepts for structural synthesis. AIAA journal, 12(5), 692-699.

Shahebrahimi, S., Lork, A., Shayegan, D. S., & Amir, A. Solving the Problem of Multi-Stakeholder Construction Site Layout Using Metaheuristic Algorithms.

Sinha, N., Chakrabarti, R., & Chattopadhyay, P. K. (2003). Evolutionary programming techniques for economic load dispatch. IEEE Transactions on evolutionary computation, 7(1), 83-94.

Wang, G. G. (2003). Adaptive response surface method using inherited latin hypercube design points. J. Mech. Des., 125(2), 210-220.

Yang, X.-S. (2009). Firefly algorithms for multimodal optimization. Paper presented at the International symposium on stochastic algorithms.

Yang, X. S., & Hossein Gandomi, A. (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering computations, 29(5), 464-483.

 

 

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