A novel chaotic Henry gas solubility optimization algorithm for solving real-world engineering problems

Abstract

The paper proposes a novel metaheuristic based on integrating chaotic maps into a Henry gas solubility optimization algorithm (HGSO). The new algorithm is named chaotic Henry gas solubility optimization (CHGSO). The hybridization is aimed at enhancement of the convergence rate of the original Henry gas solubility optimizer for solving real-life engineering optimization problems. This hybridization provides a problem-independent optimization algorithm. The CHGSO performance is evaluated using various conventional constrained optimization problems, e.g., a welded beam problem and a cantilever beam problem. The performance of the CHGSO is investigated using both the manufacturing and diaphragm spring design problems taken from the automotive industry. The results obtained from using CHGSO for solving the various constrained test problems are compared with a number of established and newly invented metaheuristics, including an artificial bee colony algorithm, an ant colony algorithm, a cuckoo search algorithm, a salp swarm optimization algorithm, a grasshopper optimization algorithm, a mine blast algorithm, an ant lion optimizer, a gravitational search algorithm, a multi-verse optimizer, a Harris hawks optimization algorithm, and the original Henry gas solubility optimization algorithm. The results indicate that with selecting an appropriate chaotic map, the CHGSO is a robust optimization approach for obtaining the optimal variables in mechanical design and manufacturing optimization problems.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Wang ZG, Wong YS, Rahman M (2004) Optimisation of multi-pass milling using genetic algorithm and genetic simulated annealing. Int J Adv Manuf Technol 24(9–10):727–732

    Article  Google Scholar 

  2. 2.

    Armarego EJA, Smith AJR, Wang J (1994) Computer-aided constrained optimization analyses and strategies for multipass helical tooth milling operations. CIRP Ann Manuf Technol 43(1):437–442

    Article  Google Scholar 

  3. 3.

    Gupta R, Batra JL, Lal GK (1995) Determination of optimal subdivision of depth of cut in multipass turning with constraints. Int J Prod Res 33(9):2555–2565

    MATH  Article  Google Scholar 

  4. 4.

    Tolouei-Rad M, Bidhendi IM (1997) On the optimization of machining parameters for milling operations. Int J Mach Tools Manuf 37(1):1–16

    Article  Google Scholar 

  5. 5.

    Yildiz AR, Yildiz BS, Sait SM, Li XY (2019) The Harris hawks, grasshopper and multi-verse optimization algorithms for the selection of optimal machining parameters in manufacturing operations. Mater Test 61(8):725–733

    Article  Google Scholar 

  6. 6.

    Yildiz AR, Yildiz BS, Sait SM, Bureerat S, Pholdee N (2019) A new hybrid Harris hawks Nelder-Mead optimization algorithm for solving design and manufacturing problems. Mater Tes 61(8):735–743

    Article  Google Scholar 

  7. 7.

    Yildiz BS, Yildiz AR, Bureerat S, Pholdee N, Sait SM, Patel V (2020) The Henry gas solubility optimization algorithm for optimum structural design of automobile brake components. Mater Test 62(3):261–264

    Article  Google Scholar 

  8. 8.

    Yildiz BS (2020) The spotted hyena optimization algorithm for weight-reduction ofautomobile brake components. Mater Test 62(4):383–388

    Article  Google Scholar 

  9. 9.

    Yildiz BS, Pholdee N, Bureerat S, Sait SM, Yildiz AR (2020) Sine-cosine optimization algorithm for the conceptual design of automobile components. Mater Test 62(7):744–748

    Article  Google Scholar 

  10. 10.

    Yildiz BS (2020) The mine blast algorithm for the structural optimization of electrical vehicle components. Mater Test 62(5):497–501

    Article  Google Scholar 

  11. 11.

    Panagant N, Pholdee N, Bureerat S, Yildiz AR, Sait SM (2020) Seagull optimization algorithm for solving real-world design optimization problems. Mater Test 62(6):640–644

    Article  Google Scholar 

  12. 12.

    Yildiz BS, Yildiz AR (2019) The Harris hawks optimization algorithm, salp swarm algorithm, grasshopper optimization algorithm and dragonfly algorithm for structural design optimization of vehicle components. Mater Test 61(8):744–748

    Article  Google Scholar 

  13. 13.

    Khalilpourazari S, Khalilpourazary S (2017) A lexicographic weighted Tchebycheff approach for multi-constrained multi-objective optimization of the surface grinding process. Eng Optim 49(5):878–895

    MathSciNet  Article  Google Scholar 

  14. 14.

    Taylor FW (1906) On the art of cutting metals. American society of mechanical engineers

  15. 15.

    Wang ZG, Rahman M, Wong YS, Sun J (2005) Optimization of multi-pass milling using parallel genetic algorithm and parallel genetic simulated annealing. Int J Mach Tools Manuf 45(15):1726–1734

    Article  Google Scholar 

  16. 16.

    Gilbert, W. W. (1950). Economics of machining. Machining-Theory and Practice, 465–485

  17. 17.

    Okushima K, Hitomi K (1964) A study of economical machining: an analysis of the maximum-profit cutting speed. Int J Prod Res 3(1):73–78

    Article  Google Scholar 

  18. 18.

    Ermer DS (1971) Optimization of the constrained machining economics problem by geometric programming. J Eng Ind 93(4):1067–1072

    Article  Google Scholar 

  19. 19.

    Petropoulos PG (1973) Optimal selection of machining rate variables by geometric programming. Int J Prod Res 11(4):305–314

    Article  Google Scholar 

  20. 20.

    Boothroyd G (1976) Maximum rate of profit criteria in machining. Trans ASME J Eng Ind 1:217–220

    Article  Google Scholar 

  21. 21.

    Hati SK, Rao SS (1976) Determination of optimum machining conditions—deterministic and probabilistic approaches. J Eng Ind 98(1):354–359

    Article  Google Scholar 

  22. 22.

    Iwata K, Murotsu Y, Oba F (1977) Optimization of cutting conditions for multi-pass operations considering probabilistic nature in machining processes. J Eng Ind 99(1):210–217

    Article  Google Scholar 

  23. 23.

    Lambert BK, Walvekar AG (1978) Optimization of multi-pass machining operations. Int J Prod Res 16(4):259–265

    Article  Google Scholar 

  24. 24.

    Chen MC, Tsai DM (1996) A simulated annealing approach for optimization of multi-pass turning operations. Int J Prod Res 34(10):2803–2825

    MATH  Article  Google Scholar 

  25. 25.

    Ermer DS, Kromodihardjo S (1981) Optimization of multipass turning with constraints. J Eng Ind 103(4):462–468

    Article  Google Scholar 

  26. 26.

    Gopalakrishnan B, Al-Khayyal F (1991) Machine parameter selection for turning with constraints: an analytical approach based on geometric programming. Int J Prod Res 29(9):1897–1908

    MATH  Article  Google Scholar 

  27. 27.

    Shin YC, Joo YS (1992) Optimization of machining conditions with practical constraints. Int J Prod Res 30(12):2907–2919

    Article  Google Scholar 

  28. 28.

    Tan FP, Creese RC (1995) A generalized multi-pass machining model for machining parameter selection in turning. Int J Prod Res 33(5):1467–1487

    MATH  Article  Google Scholar 

  29. 29.

    Agapiou JS (1992) The optimization of machining operations based on a combined criterion, part 2: multipass operations. J Eng Ind 114(4):508–513

    Article  Google Scholar 

  30. 30.

    Armarego EJA, Smith AJR, Wang J (1993) Constrained optimization strategies and CAM software for single-pass peripheral milling. Int J Prod Res 31(9):2139–2160

    Article  Google Scholar 

  31. 31.

    Yildiz AR, Abderazek H, Mirjalili S (2019) A comparative study of recent non-traditional methods for mechanical design optimization. Arch Comput Methods Eng 27:1031–1048

    MathSciNet  Article  Google Scholar 

  32. 32.

    Karaduman A, Yildiz BS, Yildiz AR (2020) Experimental and numerical fatigue based design optimisation of clutch diaphragm spring in the automotive industry. Int J Veh Des 80(2/3/4):330–345

    Article  Google Scholar 

  33. 33.

    Abderazek H, Yildiz AR, Sait SM (2019) Optimal design of planetary gear train for automotive transmissions using advanced meta-heuristics. Int J Veh Des 80(2/3/4):121–136

    Article  Google Scholar 

  34. 34.

    Panagan N, Pholdee N, Wansasueb K, Bureerat S, Yildiz AR, Sait SM (2019) Comparison of recent algorithms for many-objective optimisation of an automotive floor-frame. Int J Veh Des 80(2/3/4):176–208

    Article  Google Scholar 

  35. 35.

    Abderazek H, Yildiz AR, Sait SM (2019) Mechanical engineering design optimisation using novel adaptive differential evolution algorithm. Int J Veh Des 80(2/3/4):285–329

    Article  Google Scholar 

  36. 36.

    Aye CM, Pholdee N, Yildiz AR, Bureerat S, Sait SM (2019) Multi-surrogate assisted metaheuristics for crashworthiness optimisation. Int J Veh Des 80(2/3/4):223–240

    Article  Google Scholar 

  37. 37.

    Yildiz AR (2019) A novel hybrid whale nelder mead algorithm for optimization of design and manufacturing problems. Int J Adv Manuf Technol 105:5091–5104

    Article  Google Scholar 

  38. 38.

    Sarangkum R, Wansasueb K, Panagant N, Pholdee N, Bureerat S, Yildiz AR, Sait SM (2019) Automated design of aircraft fuselage stiffeners using multiobjective evolutionary optimisation. Int J Veh Des 80(2/3/4):162–175

    Article  Google Scholar 

  39. 39.

    Yildiz BS (2017) A comparative investigation of eight recent population-based optimisation algorithms for mechanical and structural design problems. Int J Veh Des 73(1–3):208–218

    Article  Google Scholar 

  40. 40.

    Holland JH (1975) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. The University of Michigan Press, Ann Arbor

    Google Scholar 

  41. 41.

    Chen MC, Chen KY (2003) Optimization of multipass turning operations with genetic algorithms: a note. Int J Prod Res 41(2003):3385–3388

    Article  Google Scholar 

  42. 42.

    Dhiman G, Kumar V (2019) Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems. Knowl-Based Syst 165:169–196

    Article  Google Scholar 

  43. 43.

    Hashim AF, Houssein EH, Mabrouk MS, Al-Atanaby W, Mirjalili S (2019) Henry gas solubility optimization: a novel physics-based algorithm. Future Gener Comput Syst 101:646–667

    Article  Google Scholar 

  44. 44.

    Saremi S, Mirjalili S, Lewis A (2017) Grasshopper optimization algorithm: theory and application. Adv Eng Softw 105:30–47

    Article  Google Scholar 

  45. 45.

    Mirjalili S, Gandomi AH, Mirjalili SZ, Saremi S, Faris H, Mirjalili SM (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163–191

    Article  Google Scholar 

  46. 46.

    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  47. 47.

    Mirjalili S (2016) Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput Appl 27(4):1053–1073

    MathSciNet  Article  Google Scholar 

  48. 48.

    Mirjalili S (2015) Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowl-Based Syst 89:228–249

    Article  Google Scholar 

  49. 49.

    Mirjalili S (2015) The ant lion optimizer. Adv Eng Softw 83:80–98

    Article  Google Scholar 

  50. 50.

    Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Micro machine and human science, 1995. MHS’95. Proceedings of the sixth international symposium on (pp. 39–43). IEEE

  51. 51.

    Yang XS, Deb S (2009) Cuckoo search via Lévy flights. In: Nature and biologically inspired computing, 2009. NaBIC 2009. World Congress on (pp. 210–214). IEEE

  52. 52.

    Mirjalili S, Mirjalili SM, Hatamlou A (2016) Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 27(2):495–513

    Article  Google Scholar 

  53. 53.

    Heidari A, Mirjalili S, Farris H, Aljarah I, Mafarja M, Chen H (2019) Harris hawks optimization: Algorithm and applications. Future Gener Comput Syst 97:849–872. https://doi.org/10.1016/j.future.2019.02.028

    Article  Google Scholar 

  54. 54.

    Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133

    Article  Google Scholar 

  55. 55.

    Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water cycle algorithm—a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110:151–166

    Article  Google Scholar 

  56. 56.

    Sadollah A, Eskandar H, Bahreininejad A, Kim JH (2015) Water cycle algorithm for solving multi-objective optimization problems. Soft Comput 19(9):2587–2603

    Article  Google Scholar 

  57. 57.

    Dhiman G, Kaur A (2019) STOA: A bio-inspired based optimization algorithm for industrial engineering problems. Eng Appl Artif Intell 82:148–174

    Article  Google Scholar 

  58. 58.

    Yıldız BS, Yıldız AR (2017) Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes. Mater Test 59(5):425–429

    Article  Google Scholar 

  59. 59.

    Wen XM, Tay AAO, Nee AYC (1992) Micro-computer-based optimization of the surface grinding process. J Mater Process Technol 29(1–3):75–90

    Article  Google Scholar 

  60. 60.

    Saravanan R, Asokan P, Sachidanandam M (2002) A multi-objective genetic algorithm (GA) approach for optimization of surface grinding operations. Int J Mach Tools Manuf 42(12):1327–1334

    Article  Google Scholar 

  61. 61.

    Baskar N, Saravanan R, Asokan P, Prabhaharan G (2004) Ants colony algorithm approach for multi-objective optimization of surface grinding operations. Int J Adv Manuf Technol 23(5–6):311–317

    Article  Google Scholar 

  62. 62.

    Krishna AG, Rao KM (2006) Multi-objective optimization of surface grinding operations using scatter search approach. Int J Adv Manuf Technol 29(5–6):475–480

    Article  Google Scholar 

  63. 63.

    Lee KM, Hsu MR, Chou JH, Guo CY (2011) Improved differential evolution approach for optimization of surface grinding process. Expert Syst Appl 38(5):5680–5686

    Article  Google Scholar 

  64. 64.

    Zhang G, Liu M, Li J, Ming W, Shao X, Huang Y (2014) Multi-objective optimization for surface grinding process using a hybrid particle swarm optimization algorithm. Int J Adv Manuf Technol 71(9–12):1861–1872

    Article  Google Scholar 

  65. 65.

    Krishna, A. G. (2007). Retracted: optimization of surface grinding operations using a differential evolution approach. Doi: https://doi.org/10.1016/j.jmatprotec.2006.10.010

  66. 66.

    Lin X, Li H (2008) Enhanced Pareto particle swarm approach for multi-objective optimization of surface grinding process. In: Intelligent information technology application, 2008. IITA’08. Second international symposium on (Vol. 2, pp. 618–623). IEEE

  67. 67.

    Gupta R, Shishodia KS, Sekhon GS (2001) Optimization of grinding process parameters using enumeration method. J Mater Process Technol 112(1):63–67

    Article  Google Scholar 

  68. 68.

    Slowik A, Slowik J (2008) Multi-objective optimization of surface grinding process with the use of evolutionary algorithm with remembered Pareto set. Int J Adv Manuf Technol 37(7–8):657–669

    Article  Google Scholar 

  69. 69.

    Pawar PJ, Rao RV, Davim JP (2010) Multiobjective optimization of grinding process parameters using particle swarm optimization algorithm. Mater Manuf Processes 25(6):424–431

    Article  Google Scholar 

  70. 70.

    Rao RV, Pawar PJ (2010) Grinding process parameter optimization using non-traditional optimization algorithms. Proc Instit Mech Eng Part B J Eng Manuf 224(6):887–898

    Article  Google Scholar 

  71. 71.

    Pawar PJ, Rao RV (2013) Parameter optimization of machining processes using teaching–learning-based optimization algorithm. Int J Adv Manuf Technol 67(5–8):995–1006

    Article  Google Scholar 

  72. 72.

    Huang J, Gao L, Li X (2015) An effective teaching-learning-based cuckoo search algorithm for parameter optimization problems in structure designing and machining processes. Appl Soft Comput 36:349–356

    Article  Google Scholar 

  73. 73.

    Khalilpourazari S, Khalilpourazary S (2018) A Robust Stochastic Fractal Search approach for optimization of the surface grinding process. Swarm Evolut Comput 38:173–186

    Article  Google Scholar 

  74. 74.

    Krasnogor N, Smith J (2005) A tutorial for competent memetic algorithms: model, taxonomy and design issues. IEEE Trans Evol Comput 9(5):474–488

    Article  Google Scholar 

  75. 75.

    Vaz AIF, Vicente LN (2007) A particle swarm pattern search method for bound constrained global optimization. J Glob Optim 39(2):197–219

    MathSciNet  MATH  Article  Google Scholar 

  76. 76.

    Kang F, Junjie LJ, Li H (2013) Artificial bee colony algorithm and pattern search hybridized for global optimization. Appl Soft Comput 13:1781–1791

    Article  Google Scholar 

  77. 77.

    Yıldız AR, Kurtuluş E, Demirci E, Yıldız BS, Karagöz S (2016) Optimization of thin-wall structures using hybrid gravitational search and Nelder-Mead algorithm. Mater Test 58(1):75–78

    Article  Google Scholar 

  78. 78.

    Tavazoei MS, Haeri M (2007) Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Appl Math Comput 187:1076–1085

    MathSciNet  MATH  Article  Google Scholar 

  79. 79.

    Hilborn RC (2004) Chaos and nonlinear dynamics: an introduction for scientists and engineers, 2nd edn. Oxford Univ Press, New York

    Google Scholar 

  80. 80.

    He D, He C, Jiang L, Zhu H, Hu G (2001) Chaotic characteristic of a one-dimensional iterative map with infinite collapses. IEEE Trans Circuits Syst 48(7):900–906

    MathSciNet  MATH  Article  Google Scholar 

  81. 81.

    Erramilli A, Singh RP, Pruthi P (1994) Modeling packet traffic with chaotic maps. Royal Institute of Technology, Stockholm-Kista

    Google Scholar 

  82. 82.

    May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467

    MATH  Article  Google Scholar 

  83. 83.

    Li Y, Deng S, Xiao D (2011) A novel Hash algorithm construction based on chaotic neural network. Neural Comput Appl 20:133–141

    Article  Google Scholar 

  84. 84.

    Tomida AG (2008) Matlab toolbox and GUI for analyzing one-dimensional chaotic maps. In: International conference on computational sciences and its applications ICCSA, IEEE Press. p. 321–330

  85. 85.

    Devaney RL (1987) An introduction to chaotic dynamical systems. Addison-Wesley

  86. 86.

    Peitgen H, Jurgens H, Saupe D (1992) Chaos and fractals. Springer-Verlag, Berlin

    Google Scholar 

  87. 87.

    Ott E (2002) Chaos in dynamical systems. Cambridge University Press, Cambridge

    Google Scholar 

  88. 88.

    Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering op- timization problems. Appl Soft Comput 13:2592–2612

    Article  Google Scholar 

  89. 89.

    Gandomi AH, Yang X-S, Alavi AH (2013) Cuckoo search algorithm: a meta- heuristic approach to solve structural optimization problems. Eng Comput 29:17–35

    Article  Google Scholar 

  90. 90.

    Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic se- lection for constrained optimization. Inf Sci 178:3043–3074

    Article  Google Scholar 

  91. 91.

    Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differen- tial evolution for constrained numerical and engineering optimization. Appl Soft Comput 10:629–640

    Article  Google Scholar 

  92. 92.

    Ray T, Saini P (2001) Engineering design optimization using a swarm with an intel- ligent information sharing among individuals. Eng Optim 33:735–748

    Article  Google Scholar 

  93. 93.

    Tsai J-F (2005) Global optimization of nonlinear fractional programming problems in engineering design. Eng Optim 37:399–409

    MathSciNet  Article  Google Scholar 

  94. 94.

    Carlos A, Coello C (2000) Constraint-handling using an evolutionary multiobjective optimization technique. Civ Eng Syst 17:319–346

    Article  Google Scholar 

  95. 95.

    Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Method Appl Mech Eng 186:311–338

    MATH  Article  Google Scholar 

  96. 96.

    Ragsdell K, Phillips D (1976) Optimal design of a class of welded structures using geometric programming. ASME J Eng Ind 98:1021–1025

    Article  Google Scholar 

  97. 97.

    He Q, Wang L (2007) An effective co-evolutionary particle swarm optimiza- tion for constrained engineering design problems. Eng Appl Artif Intell 20:89–99

    Article  Google Scholar 

  98. 98.

    Coello Coello CA (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Method Appl Mech Eng 191:1245–1287

    MathSciNet  MATH  Article  Google Scholar 

  99. 99.

    Coello Coello CA, Mezura ME (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inf 16:193–203

    Article  Google Scholar 

  100. 100.

    Siddall JN (1972) Analytical decision-making in engineering design. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  101. 101.

    Wang GG (2003) Adaptive response surface method using inherited Latin hypercube design points. J Mech Des 125:210–220

    Article  Google Scholar 

  102. 102.

    Cheng M-Y, Prayogo D (2014) Symbiotic organisms search: a new metaheuristic optimization algorithm. Comput Struct 139:98–112

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ali Riza Yildiz.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

Cantilever problem

The problem can be formulated as follows:

Minimize:

$$f\left(X\right)=0.0624\left({x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}\right).$$

Subject to:

$$\mathrm{g}\left(x\right)=\frac{61}{{x}_{1}^{3}}+\frac{37}{{x}_{2}^{3}}+\frac{19}{{x}_{3}^{3}}+\frac{7}{{x}_{4}^{3}}+\frac{1}{{x}_{5}^{3}}-1\le 0.$$

The design variables are the heights (or widths) of the different beam elements, and the thickness is held fixed (here \(t\) = 2/3). The bound constraints are set as \(0.01\le {x}_{j}\le 100\).

Welded beam design problem

The problem can be formulated as follows:

Minimize: \(f\left(x\right)=1.10471{h}^{2}l+0.04811tb\left(14.0+l\right).\)

Subject to:

$${g}_{1}\left(x\right)=\tau \left(x\right)-{\tau }_{max}\le 0, {g}_{2}\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}\le 0,$$
$${g}_{3}\left(x\right)=h-b\le 0, {g}_{4}\left(x\right)=0.1047{h}^{2}+0.04811tb\left(14+l\right)-5\le 0,$$
$${g}_{5}\left(x\right)=0.125-h\le 0, {g}_{6}\left(x\right)=\delta \left(x\right)-{\delta }_{max}\le 0,$$
$${g}_{7}\left(x\right)=P-{P}_{c}(x)\le 0,$$

where \(\tau = \sqrt {(\tau ^{\prime})^{2} {\text{ + }}\left( {\tau ^{\prime\prime}} \right)^{{\text{2}}} {\text{ + 2}}\tau ^{\prime}\tau ^{\prime\prime}\frac{{\text{l}}}{{{\text{2}}R}}} ,\;\tau ^{\prime}{\text{ = }}\frac{P}{{\sqrt {\text{2}} hl}},\tau ^{\prime\prime}{\text{ = }}\frac{{MR}}{J},M{\text{ = }}P\left( {L{\text{ + }}\frac{{\text{1}}}{{\text{2}}}} \right),R{\text{ = }}\sqrt {\frac{{{\text{l}}^{{\text{2}}} }}{{\text{4}}}{\text{ + }}\frac{{(h{\text{ + }}t)^{{\text{2}}} }}{{\text{4}}}} ,\)

$$J = 2\left\{ {\sqrt 2 hl\left[ {\frac{{l^{2} }}{{12}} + \frac{{(h + t)^{2} }}{4}} \right]} \right\},\sigma = \frac{{6PL}}{{bt^{2} }},P_{c} = \frac{{4.013E\sqrt {t^{2} b^{6} /36} }}{{bt^{2} }}\left( {1 - \frac{t}{{2L}}\sqrt {\frac{E}{{4G}}} } \right),$$
$$P=6000lp, l=14in, E=30\times {10}^{6}psi, G=12\times {10}^{6}psi, {\tau }_{max}=13600 psi, {\delta }_{max}=0.25in, 0.1\le h,b\le 2,$$

and \(1\le l,t\le 10.\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yıldız, B.S., Pholdee, N., Panagant, N. et al. A novel chaotic Henry gas solubility optimization algorithm for solving real-world engineering problems. Engineering with Computers (2021). https://doi.org/10.1007/s00366-020-01268-5

Download citation

Keywords

  • Hybrid metaheuristics
  • Henry gas solubility optimization
  • Chaotic maps
  • Mechanical and manufacturing design
  • Diaphragm spring