Abstract
In this paper, a novel hybrid optimization algorithm is introduced by hybridizing a Harris hawks optimization algorithm(HHO) and simulated annealing for the purpose of accelerating its global convergence performance and optimizing structural design problems. This paper is the first research study in which the hybrid Harris hawks simulated annealing algorithm (HHOSA) is used for the optimization of design parameters for highway guardrail systems. The HHOSA is evaluated using the well-known benchmark problems such as the three-bar truss problem, cantilever beam problem, and welded beam problem. Finally, a guardrail system that has an H1 containment level as a case study is optimized to investigate the performance of the HHOSA. The guardrail systems are designed with different cross-sections and distances between the posts. TB11 and TB42 crash analyses are performed according to EN 1317 standards. Twenty-five different designs are evaluated considering weight, the guardrail working width, and the acceleration severity index (ASI). As a result of this research, the optimum design of a guardrail is obtained, which has a minimum weight and acceleration severity index value (ASI). The results show that the HHOSA is a highly effective approach for optimizing real-world design problems.
References
1 B. S. Yildiz , A. R.Yildiz: Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes, Materials Testing59 (2017), No. 5, pp. 425–429 DOI: 10.3139/120.111024Search in Google Scholar
2 A. R. Yildiz : Cuckoo search algorithm for the selection of optimal machining parameters in milling operations, International Journal of Advanced Manufacturing Technology64 (2013), No. 1–4, pp. 55–61 DOI: 10.1007/s00170-012-4013-7Search in Google Scholar
3 A. R. Yildiz : A comparative study of population-based optimization algorithms for turning operations, Information Sciences210 (2012), pp. 81–88 DOI: 10.1016/j.ins.2012.03.005Search in Google Scholar
4 A. R. Yildiz : A novel hybrid immune algorithm for global optimization in design and manufacturing, Robotics and Computer-Integrated Manufacturing25 (2009), No. 2, pp. 261–270 DOI: 10.1016/j.rcim.2007.08.002Search in Google Scholar
5 A. R. Yildiz : An effective hybrid immune-hill climbing optimization approach for solving design and manufacturing optimization problems in industry, Journal of Materials Processing Technology209 (2009), No. 6, pp. 2773–2780 DOI: 10.1016/j.jmatprotec.2008.06.028Search in Google Scholar
6 A. R. Yildiz : A new hybrid bee colony optimization approach for robust optimal design and manufacturing, Applied Soft Computing13 (2013), No. 5, pp. 2906–2912 DOI: 10.1016/j.asoc.2012.04.013Search in Google Scholar
7 A. R. Yildiz : A new hybrid differential evolution algorithm for the selection of optimal machining parameters in milling operations, Applied Soft Computing13 (2013), No. 3, pp. 1561–1566 DOI: 10.1016/j.asoc.2011.12.016Search in Google Scholar
8 A. R. Yildiz : Hybrid Taguchi-differential evolution algorithm for optimization of multi-pass turning operations, Applied Soft Computing13 (2013), No. 3, pp. 1433–1439 DOI: 10.1016/j.asoc.2012.01.012Search in Google Scholar
9 A. R. Yildiz , F.Ozturk: Hybrid enhanced genetic algorithm to select optimal machining parameters in turning operation, Journal of Engineering Manufacture220 (2006), No. 12, pp. 2041–2053 DOI: 10.1243/09544054JEM570Search in Google Scholar
10 A. R. Yildiz , K.Solanki: Multi-objective optimization of vehicle crashworthiness using new particle swarm based approach, International Journal of Advanced Manufacturing Technology59 (2012), No. 1–4, pp. 367–376 DOI: 10.1007/s00170-011-3496-ySearch in Google Scholar
11 A. R. Yildiz : Hybrid Taguchi-Harmony Search Algorithm for Solving Engineering Optimization Problems, International Journal of Industrial Engineering Theory, Applications and Practice15 (2008), No. 3, pp. 286–293Search in Google Scholar
12 A. R. Yildiz : A novel particle swarm optimization approach for product design and manufacturing, International Journal of Advanced Manufacturing Technology40 (2009), No. 5–6, pp. 617–628 DOI: 10.1007/s00170-008-1453-1Search in Google Scholar
13 O. F. Sonmez : Shape optimization of 2D structures using simulated annealing, Computer Methods in Applied Mechanics and Engineering, 196(2007), pp. 3279–3299 DOI: 10.1016/j.cma.2007.01.019Search in Google Scholar
14 S. Bureerat , N.Pholdee: Inverse problem based differential evolution for efficient structural health monitoring of trusses, Applied Soft Computing66 (2018), pp. 462–472 DOI: 10.1016/j.asoc.2018.02.046Search in Google Scholar
16 N. Panagant , S.Bureerat: Truss topology, shape and sizing optimization by fully stressed design based on hybrid grey wolf optimization and adaptive differential evolution, Engineering Optimization50 (2018), No. 10, pp. 1645–1661 DOI: 10.1080/0305215X.2017.1417400Search in Google Scholar
17 A. R. Yildiz : A novel hybrid whale nelder mead algorithm for optimization of design and manufacturing problems, International Journal of Advanced Manufacturing Technology (2019) DOI: 10.1007/s00170-019-04532-1Search in Google Scholar
18 A. R. Yıldız : A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing, Applied Soft Computing13 (2013), pp. 2906–2912 DOI: 10.1016/j.asoc.2012.04.013Search in Google Scholar
19 B. S. Yildiz , A. R.Yildiz: Comparison of grey wolf, whale, water cycle, ant lion and sine-cosine algorithms for the optimization of a vehicle engine connecting rod, Materials Testing60 (2018), No. 3, pp. 311–315 DOI: 10.3139/120.111153Search in Google Scholar
20 T. Güler , A.Demirci, A. R.Yıldız, U.Yavuz: Lightweight design of an automobile hinge component using glass fiber polyamide composites, Materials Testing60 (2018), No. 3, pp. 306–310 DOI: 10.3139/120.111152Search in Google Scholar
21 A. R. Yildiz : A New Design Optimization Framework based on immune Algorithm and Taguchi Method, Computers in Industry60 (2009), pp. 613–620 DOI: 10.1016/j.compind.2009.05.016Search in Google Scholar
22 B. S. Yildiz , H.Lekesiz: Fatigue-based structural optimisation of vehicle components, International Journal of Vehicle Design73 (2017), pp. 54–62 DOI: 10.1504/IJVD.2017.10003398Search in Google Scholar
23 H. Abderazek , A. R.Yildiz, S.Mirjalili: Comparison of recent optimization algorithms for design optimization of a cam-follower mechanism, Knowledge-Based Systems (2019) DOI: 10.1016/j.knosys.2019.105237Search in Google Scholar
24 A. R. Yildiz : Designing of optimum vehicle components using new generation optimization methods, Journal of Polytechnic20 (2017), No. 2, pp. 319–323 DOI: 10.2339/2017.20.2.325-332Search in Google Scholar
25 H. Gokdağ , A. R.Yildiz: Structural damage detection using modal parameters and particle swarm optimizationMaterials Testing54 (2012), No. 6, pp. 416–420 DOI: 10.3139/120.110346Search in Google Scholar
26 F. Hamza , H.Abderazek, S.Lakhdar, D.Ferhat, A. R.Yildiz: Optimum design of cam-roller follower mechanism using a new evolutionary algorithm, The International Journal of Advanced Manufacturing Technology99 (2018), No. 5–8, pp. 1261–1282 DOI: 10.1007/s00170-018-2543-3Search in Google Scholar
27 N. Pholdee , S.Bureerat, A. R.Yildiz: Hybrid real-code population-based incremental learning and differential evolution for many-objective optimisation of an automotive floor-frame, International Journal of Vehicle Design73 (2017), No. 1–3, pp. 20–53 DOI: 10.1504/IJVD.2017.082578Search in Google Scholar
28 S. Karagöz , A. R.Yildiz: A comparison of recent metaheuristic algorithms for crashworthiness optimisation of vehicle thin-walled tubes considering sheet metal forming effects, International Journal of Vehicle Design73 (2017), No. 1–3, pp. 179–188 DOI: 10.1504/IJVD.2017.082593Search in Google Scholar
29 A. R. Yildiz , E.Kurtuluş, E.Demirci, B. S.Yildiz, S.Karagöz: Optimization of thin-wall structures using hybrid gravitational search and Nelder-Mead algorithm, Materials Testing58 (2016), No. 1, pp. 75–78 DOI: 10.3139/120.110823Search in Google Scholar
30 B. S. Yildiz : A comparative investigation of eight recent population-based optimisation algorithms for mechanical and structural design problems, International Journal of Vehicle Design73 (2017), No. 1–3, pp. 208–218 DOI: 10.1504/IJVD.2017.082603Search in Google Scholar
31 B. S. Yildiz , H.Lekesiz, A. R.Yildiz: Structural design of vehicle components using gravitational search and charged system search algorithms, Materials Testing58 (2016), No. 1, pp. 79–81 DOI: 10.3139/120.110819Search in Google Scholar
32 A. R. Yildiz : Comparison of evolutionary based optimization algorithms for structural design optimization, Engineering Applications of Artificial Intelligence26 (2013), No. 1, pp. 327–333 DOI: 10.1016/j.engappai.2012.05.014Search in Google Scholar
33 A. R. Yildiz : Optimization of cutting parameters in multi-pass turning using artificial bee colony-based approach, Information Sciences220 (2013), pp. 399–407 DOI: 10.1016/j.ins.2012.07.012Search in Google Scholar
34 A. R. Yildiz , K.Saitou: Topology synthesis of multi-component structural assemblies in continuum domains, Transactions of ASME, Journal of Mechanical Design133 (2011), No. 1, 011008-9 DOI: 10.1115/1.4003038Search in Google Scholar
35 A. R. Yıldız , U. A.Kılıçarpa, E.Demirci: Topography and topology optimization of diesel engine components for light-weight design in the automotive industry, Materials Testing61 (2019), No. 1, pp. 27–34 DOI: 10.3139/120.111277Search in Google Scholar
36 A. Demirci , A. R.Yıldız: An experimental and numerical investigation of the effects of geometry and spot welds on the crashworthiness of vehicle thin-walled structures, Materials Testing60 (2018), No. 6, pp. 553–561 DOI: 10.3139/120.111187Search in Google Scholar
37 A. Demirci , A. R.Yıldız: An investigation of the crash performance of magnesium, aluminum and advanced high strength steels and different cross-sections for vehicle thin-walled energy absorbers, Materials Testing60 (2018), No. 7–8, pp. 661–668 DOI: 10.3139/120.111201Search in Google Scholar
38 A. R. Yildiz : A new hybrid particle swarm optimization approach for structural design optimization in automotive industry, Journal of Automobile Engineering226 (2012), No. 10, pp. 1340–1351 DOI: 10.1177/0954407012443636Search in Google Scholar
39 B. S. Yildiz : Natural frequency optimization of vehicle components using the interior search algorithm, Materials Testing59 (2017), No. 5, pp. 456–458 DOI: 10.3139/120.111018Search in Google Scholar
40 A. R. Yildiz : Optimal structural design of vehicle components using topology design and optimization, Materials Testing50 (2008), No. 4, pp. 224–228 DOI: 10.3139/120.100880Search in Google Scholar
41 A. R. Yildiz , N.Öztürk, N.Kaya, F.Öztürk: Hybrid multi-objective shape design optimization using Taguchi's method and genetic algorithm, Structural and Multidisciplinary Optimization34 (2007), No. 4, pp. 317–332 DOI: 10.1007/s00158-006-0079-xSearch in Google Scholar
42 A. Demirci , A. R.Yıldız: A new hybrid approach for reliability-based design optimization of structural components, Materials Testing61 (2019), No. 2, pp. 111–119 DOI: 10.3139/120.111291Search in Google Scholar
43 A. R. Yildiz , E.Kurtuluş, E.Demirci, B. S.Yildiz, S.Karagöz: Optimization of thin-wall structures using hybrid gravitational search and Nelder-Mead algorithm, Materials Testing58 (2016), No. 1, pp. 75–78 DOI: 10.3139/120.110823Search in Google Scholar
44 A. R. Yildiz , N.Öztürk, N.Kaya, F.Öztürk: Hybrid multi-objective shape design optimization using Taguchi's method and genetic algorithm, Structural and Multidisciplinary Optimization34 (2007), No. 4, pp. 317–332 DOI: 10.1007/s00158-006-0079-xSearch in Google Scholar
45 S. Saremi , S.Mirjalili, A.Lewis: Grasshopper optimisation algorithm: theory and application, Advances in Engineering Software105 (2017), pp. 30–47 DOI: 10.1016/j.advengsoft.2017.01.004Search in Google Scholar
46 A. Heidari , S.Mirjalili, H.Farris, I.Aljarah, M.Mafarja, H.Chen: Harris hawks optimization: Algorithm and applications, Future Generation Computer Systems97 (2019), pp. 849–872, DOI: 10.1016/j.future.2019.02.028Search in Google Scholar
47 S. Mirjalili , S. M.Mirjalili, A.Lewis: Grey wolf optimizer, Advances in Engineering Software69 (2014), pp. 46–61 DOI: 10.1016/j.advengsoft.2013.12.007Search in Google Scholar
48 S. Mirjalili : Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications27 (2016), No. 4, pp. 1053–1073 DOI: 10.1007/s00521-015-1920-1Search in Google Scholar
49 S. Mirjalili : Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowledge-Based Systems89 (2015), pp. 228–249 DOI: 10.1016/j.knosys.2015.07.006Search in Google Scholar
50 S. Mirjalili , A. H.Gandomi, S. Z.Mirjalili, S.Saremi, H.Faris, S. M.Mirjalili: salp swarm optimization Algorithm: A bio-inspired optimizer for engineering design problems, Advances in Engineering Software114 (2017), pp. 163–191 DOI: 10.1016/j.advengsoft.2017.07.002Search in Google Scholar
51 S. Mirjalili : The ant lion optimizer, Advances in Engineering Software83 (2015), pp. 80–98 DOI: 10.1016/j.advengsoft.2015.01.010Search in Google Scholar
52 X.-S. Yang , S.Deb: Cuckoo search via Lévy flights, World Congress on Nature and Biologically Inspired Computing, NaBIC (2009), pp. 210–214 DOI: 10.1109/NABIC.2009.5393690Search in Google Scholar
53 S. Mirjalili , S. M.Mirjalili, A.Hatamlou: Multi-verse optimizer: a nature-inspired algorithm for global optimization, Neural Computing and Applications27 (2016), No. 2, pp. 495–513 DOI: 10.1007/s00521-015-1870-7Search in Google Scholar
54 M. Kiani , A. R.Yildiz: A comparative study of non-traditional methods for vehicle crashworthiness and NVH optimization, Archives of Computational Methods in Engineering23 (2016), No. 4, pp. 723–734 DOI: 10.1007/s11831-015-9155-ySearch in Google Scholar
55 S. Mirjalili : SCA: a sine cosine algorithm for solving optimization problems, Knowledge-Based Systems96 (2016), pp. 120–133 DOI: 10.1016/j.knosys.2015.12.022Search in Google Scholar
56 A. R. Yildiz , H.Abderazek, S.Mirjalili: A comparative study of recent non-traditional methods for mechanical design optimization, Archives of Computational Methods in Engineering (2019), pp. 1–18 DOI: 10.1007/s11831-019-09343-xSearch in Google Scholar
57 A. Sadollah , H.Eskandar, A.Bahreininejad, J. H.Kim: Water cycle algorithm for solving multi-objective optimization problems, Soft Computing19 (2015), No. 9, pp. 2587–2603 DOI: 10.1007/s00500-014-1424-4Search in Google Scholar
58 A. Tabiei , J.Wu: Roadmap for Crashworthiness Finite Element Simulation of Roadside Safety Structures, Finite Elements in Analysis and Design34 (2000), No. 2, pp. 145–157 DOI: 10.1016/S0168-874X(99)00035-9Search in Google Scholar
59 C. A. Plaxico , M. H.Ray, K.Hiranmayee: Comparison of the impact performance of the G4(1W) and G4(2W) guardrail systems under NCHRP report 350 Test 3–11 conditions, Transportation Research Record, Paper No.00-0525, Transportation Research Board, Washington D. C., USA (2000)Search in Google Scholar
60 E. Kurutulus : Development of Next Generation Guardrail Systems Using Crash Analysis and Heuristic Optimization Methods, MSc Thesis, Uludag University, Bursa, TurkeySearch in Google Scholar
61 M. Vesenjak , M.Borovinsek, Z.Ren: Computational simulations of road safety barriers using LS-DYNA, Proc. of the LS-DYNA Anwenderforum, B-III, pp. 11–17Search in Google Scholar
62 A. R. Yildiz , B. S.Yildiz, M. S.Sait, X. Y.Li: The Harris hawks, grasshopper and multi-verse optimization algorithms for the selection of optimal machining parameters in manufacturing operations, Materials Testing, 61 (2019) No. 8, pp. 725–733 DOI: 10.3139/120.111377Search in Google Scholar
63 B. S. Yildiz , A. R.Yildiz: The Harris hawks optimization algorithm, salp swarm algorithm, grasshopper optimization algorithm and dragonfly algorithm for structural design optimization of vehicle components, Materials Testing61 (2019), No. 8, pp. 744–748DOI 10.3139/120.111379Search in Google Scholar
64 A. R. Yildiz , B. S.Yildiz, M. S.Sait, S.Bureerat, N.Pholdee: A new hybrid Harris hawks Nelder-Mead optimization algorithm for solving design and manufacturing problems, Materials Testing61 (2019), No. 8, pp. 735–743 DOI: 10.3139/120.111378Search in Google Scholar
65 S. Saremi , S. M.Mirjalili, S.Mirjalili: Chaotic krill herd optimization algorithm, Procedia Technology12 (2014), pp. 180–185 DOI: 10.1016/j.protcy.2013.12.473Search in Google Scholar
66 S. Mirjalili , A. H.Gandomi: Chaotic gravitational constants for the gravitational search algorithm, Applied Soft Computing53 (2017), pp. 407–419 DOI: 10.1016/j.asoc.2017.01.008Search in Google Scholar
67 L. Coelho , V. CMariani: Use of chaotic sequences in a biologically inspired algorithm for engineering design optimization, Expert Syst Appl34 (2008), No. 3, pp. 1905–1913 DOI: 10.1016/j.eswa.2007.02.002Search in Google Scholar
68 M. S. Tavazoei , M.Haeri: Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms, Appl. Math. Comput.187 (2007), No. 2, pp. 1076–1085 DOI: 10.1016/j.amc.2006.09.087Search in Google Scholar
69 R. C. Hilborn : Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, 2nd Ed., Oxford Univ. Press, New York, USA (2004)Search in Google Scholar
70 D. He , C.He, L.Jiang, H.Zhu, G.Hu: Chaotic characteristic of a one-dimensional iterative map with infinite collapses, IEEE Trans Circuits Syst48 (2001), No. 7, pp. 900–906 DOI: 10.1109/81.933333Search in Google Scholar
71 S. Kirkpatrick , C. D.GelattJr., M. P.Vecchi: Optimisation by simulated annealing, Science, 220(1983), 671–680 DOI: 10.1126/science.220.4598.671Search in Google Scholar
72 A. R. Yildiz : Hybrid immune-simulated annealing algorithm for optimal design and manufacturing, International Journal of Materials and Product Technology, 34 (3) (2009), pp. 217–226 DOI: 10.1504/IJMPT.2009.024655Search in Google Scholar
73 Y. Lin , Z.Bian, X.Liu: Developing a dynamic neighborhood structure for an adaptive hybrid simulated annealing –tabu search algorithm to solve the sym- metrical traveling salesman problem, Appl. Soft Comput.49 (2016), pp. 937–952 DOI: 10.1016/j.asoc.2016.08.036Search in Google Scholar
74 P. Vasant : Hybrid simulated annealing and genetic algorithms for industrial production management problems, Int. J. Comput. Methods7 (2010), No. 2, pp. 279–297 DOI: 10.1142/S0219876210002209Search in Google Scholar
75 Z. Li , P.Schonfeld, Hybrid simulated annealing and genetic algorithm for op- timizing arterial signal timings under oversaturated traffic conditions, J. Adv. Transp.49 (2015), No. 1, pp. 153–170 DOI: 10.1002/atr.1274Search in Google Scholar
76 Y. Li , H.Guo, L.Wang, J.Fu, A hybrid genetic-simulated annealing algorithm for the location-in- ventory-routing problem considering returns under E-supply chain environment, Sci. World J.2013 (2013) DOI: 10.1155/2013/125893Search in Google Scholar
77 L. Junghans , N.Darde, Hybrid single objective genetic algorithm coupled with the simulated annealing optimization method for building optimization, Energy Build.86 (2015), pp. 651–662 DOI: 10.1016/j.enbuild.2014.10.039Search in Google Scholar
78 A. Sadollah , A.Bahreininejad, H.Eskandar, M.Hamdi: Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems, Applied Soft Computing13 (2013), No. 5, pp. 2592–2612 DOI: 10.1016/j.asoc.2012.11.026Search in Google Scholar
79 A. H. Gandomi , X.-S.Yang, A. H.Alavi: Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems, Engineering with Computers29 (2013) No. 1, pp. 17–35 DOI: 10.1007/s00366-011-0241-ySearch in Google Scholar
80 M. Zhang , W.Luo, X. WangX: Differential evolution with dynamic stochastic selection for constrained optimization, Information Sciences178 (2008), No. 15, pp. 3043–3074 DOI: 10.1016/j.ins.2008.02.014Search in Google Scholar
81 H. Liu H , Z.Cai, Y.Wang: Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization, Applied Soft Computing10 (2010), No. 2, pp. 629–640 DOI: 10.1016/j.asoc.2009.08.031Search in Google Scholar
82 T. Ray , P.Saini: Engineering design optimization using a swarm with an intelligent information sharing among individuals, Engineering Optimization33 (2001), No. 6, pp. 735–748 DOI: 10.1080/03052150108940941Search in Google Scholar
83 J.-F. Tsai : Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization37 (2005), No. 4, pp. 399–409 DOI: 10.1080/03052150500066737Search in Google Scholar
84 C. A. Coello Coello : Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering and Environmental Systems17 (2000), No. 4, pp. 319–346 DOI: 10.1080/02630250008970288Search in Google Scholar
85 K. Deb : An efficient constraint handling method for genetic algorithms, Computer Methods in Applied Mechanics and Engineering186 (2000), No. 2–4, pp. 311–338 DOI: 10.1016/S0045-7825(99)00389-8Search in Google Scholar
86 K. Ragsdell , D.Phillips: Optimal design of a class of welded structures using geometric programming, Journal of Engineering for Industry98 (1976), No. 3, pp. 1021–1025 DOI: 10.1115/1.3438995Search in Google Scholar
87 Q. He , L.Wang: An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering Applications of Artificial Intelligence20 (2007), No. 1, pp. 89–99 DOI: 10.1016/j.engappai.2006.03.003Search in Google Scholar
88 C. A. Coello Coello : Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art, Computer Methods in Applied Mechanics and Engineering191 (2002), No. 11–12, pp. 1245–1287 DOI: 10.1016/S0045-7825(01)00323-1Search in Google Scholar
89 C. A. Coello Coello , E. MezuraMontes: Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced Engineering Informatics16 (2002), No. 3, pp. 193–203 DOI: 10.1016/S1474-0346(02)00011-3Search in Google Scholar
90 J. N. Siddall : Analytical decision-making in engineering design, Englewood Cliffs, Prentice-Hall, Upper Saddle River, New Jersey, USA (1972)Search in Google Scholar
91 A. H. Gandomi , X.-S.Yang, A. H.Alavi: Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems, Engineering with Computers29 (2013) No. 1, pp. 17–35 DOI: 10.1007/s00366-011-0241-ySearch in Google Scholar
92 G. G. Wang : Adaptive response surface method using inherited latin hypercube design points, Journal of Mechanical Design125 (2003), No. 2, pp. 210–220 DOI: 10.1115/1.1561044Search in Google Scholar
93 M.-Y. Cheng , D.Prayogo: Symbiotic organisms search: a new metaheuristic optimization algorithm, Computers & Structures139 (2014), pp. 98–112 DOI: 10.1016/j.compstruc.2014.03.007Search in Google Scholar
94 A. Kaveh , M.Khayatazad: A new meta-heuristic method: ray optimization, Computers & Structures112 (2012), pp. 283–294 DOI: 10.1016/j.compstruc.2012.09.003Search in Google Scholar
95 H. Chickermane , H.Gea: Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering39 (1996), No. 5, pp. 829–846 DOI: 10.1002/(SICI)1097-0207(19960315)39:5Search in Google Scholar
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