An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems
Abstract
:1. Introduction
2. Gray Wolf Optimization
2.1. Social Hierarchy
2.2. Encircling the Prey
2.3. Attacking the Prey
3. Improved Gray Wolf Optimization Algorithm (IGWO)
3.1. Tent Chaos Initialization
3.2. Gaussian Perturbation
3.3. Cosine Control Factor
4. The Simulation Results
4.1. Optimization Function and Experimental Environment
4.2. Analysis of Different Strategies
4.3. Analysis of Experimental Results
4.3.1. Compared with Other Algorithms
4.3.2. Convergence Analysis
4.3.3. Numerical Result Test
5. Application to Solve Engineering Optimization Problem
5.1. Pressure Vessel Design Problem
5.2. Spring Design Problem
5.3. Welded Beam Design Problem
5.4. Three Truss Design Problem
6. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Begum, R.A.; Siwar, C.; Pereira, J.J.; Jaafar, A.H. A benefit–cost analysis on the economic feasibility of construction waste minimisation: The case of Malaysia. Resour. Conserv. Recycl. 2006, 48, 86–98. [Google Scholar] [CrossRef]
- Huang, J.; Zhu, Y.; Kelly, J.T.; Jang, C.; Wang, S.; Xing, J.; Yu, L. Large-scale optimization of multi-pollutant control strategies in the Pearl River Delta region of China using a genetic algorithm in machine learning. Sci. Total Environ. 2020, 722, 137701. [Google Scholar] [CrossRef] [PubMed]
- Guo-Chu, C.; Jin-Shou, Y.U. Particle swarm optimization algorithm. Inf. Control 2005, 186, 454–458. [Google Scholar]
- Mirjalili, S. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowl. Based Syst. 2015, 89, 228–249. [Google Scholar] [CrossRef]
- Mirjalili, S. The Ant Lion Optimizer. Adv. Eng. Softw. 2015, 83, 80–98. [Google Scholar] [CrossRef]
- Mirjalili, S. SCA: A Sine Cosine Algorithm for solving optimization problems. Knowl. Based Syst. 2016, 96, 120–133. [Google Scholar] [CrossRef]
- Gandomi, A.H.; Yang, X.-S. Chaotic bat algorithm. J. Comput. Sci. 2014, 5, 224–232. [Google Scholar] [CrossRef]
- Yang, X.-S. Flower Pollination Algorithm for Global Optimization. In International Conference on Unconventional Computing and Natural Computation; Springer: Berlin, Heidelberg, 2012; Volume 7445, pp. 240–249. [Google Scholar]
- Mirjalili, S.; Gandomi, A.H.; Mirjalili, S.Z.; Saremi, S.; Faris, H.; Mirjalili, S.M. Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Adv. Eng. Softw. 2017, 114, 163–191. [Google Scholar] [CrossRef]
- Afshar, M.H. Extension of the constrained particle swarm optimization algorithm to optimal operation of mul-ti-reservoirs system. Int. J. Electr. Power Energy Syst. 2013, 51, 71–81. [Google Scholar] [CrossRef]
- Kim, N.S.; Janic, M.; Van Wee, B. Trade-off between carbon dioxide emissions and logistics costs based on multi-objective optimization. Transp. Res. Rec. 2009, 2139, 107–116. [Google Scholar] [CrossRef] [Green Version]
- Lin, W.; Yu, D.Y.; Zhang, C.; Liu, X.; Zhang, S.; Tian, Y.; Xie, Z. A multi-objective teaching−learning-based opti-mization algorithm to scheduling in turning processes for minimizing makespan and carbon footprint. J. Clean. Prod. 2015, 101, 337–347. [Google Scholar] [CrossRef]
- He, Y.; Li, Y.; Wu, T.; Sutherland, J.W. An energy-responsive optimization method for machine tool selection and operation sequence in flexible machining job shops. J. Clean. Prod. 2015, 87, 245–254. [Google Scholar] [CrossRef]
- Du, B.; Zhang, J.F.; Gao, Z.H.; Li, T.; Huang, Z.Q.; Zhang, N. Based on simulated annealing particle swarm algorithm of optimal allocation of water resources research. J. Drain. Irrig. Mach. Eng. 2020, 1–10. Available online: http://kns.cnki.net/kcms/detail/32.1814.th.20200927.0952.002.html (accessed on 1 February 2021).
- Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 2014, 69, 46–61. [Google Scholar] [CrossRef] [Green Version]
- Maharana, D.; Kotecha, P. Optimization of Job Shop Scheduling Problem with Grey Wolf Optimizer and JAYA Al-gorithm. In Smart Innovations in Communication and Computational Sciences; Springer: Singapore, Singapore, 2019; Volume 669, pp. 47–58. [Google Scholar]
- Precup, R.-E.; David, R.-C.; Petriu, E.M. Grey Wolf Optimizer Algorithm-Based Tuning of Fuzzy Control Systems With Reduced Parametric Sensitivity. IEEE Trans. Ind. Electron. 2016, 64, 527–534. [Google Scholar] [CrossRef]
- Sharma, Y.; Saikia, L.C. Automatic generation control of a multi-area ST—Thermal power system using Grey Wolf Optimizer algorithm based classical controllers. Int. J. Electr. Power Energy Syst. 2015, 73, 853–862. [Google Scholar] [CrossRef]
- Shakarami, M.; Davoudkhani, I.F. Wide-area power system stabilizer design based on Grey Wolf Optimization algorithm considering the time delay. Electr. Power Syst. Res. 2016, 133, 149–159. [Google Scholar] [CrossRef]
- Yao, X.; Li, Z.; Liu, L.; Cheng, X. Multi-Threshold Image Segmentation Based on Improved Grey Wolf Optimization Algorithm. IOP Conf. Series: Earth Environ. Sci. 2019, 252, 042105. [Google Scholar] [CrossRef] [Green Version]
- Yang, J.C.; Long, W. Improved Grey Wolf Optimization Algorithm for Constrained Mechanical Design Problems. Appl. Mech. Mater. 2016, 851, 553–558. [Google Scholar] [CrossRef]
- Chandar, S.K. Grey Wolf optimization-Elman neural network model for stock price prediction. Soft Comput. 2021, 25, 649–658. [Google Scholar] [CrossRef]
- Bansal, J.C.; Singh, S. A better exploration strategy in Grey Wolf Optimizer. J. Ambient. Intell. Humaniz. Comput. 2021, 12, 1099–1118. [Google Scholar] [CrossRef]
- Wang, J.-S.; Li, S.-X. An Improved Grey Wolf Optimizer Based on Differential Evolution and Elimination Mechanism. Sci. Rep. 2019, 9, 1–21. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Teng, Z.-J.; Lv, J.-L.; Guo, L.-W. An improved hybrid grey wolf optimization algorithm. Soft Comput. 2019, 23, 6617–6631. [Google Scholar] [CrossRef]
- Zhang, S.; Luo, Q.; Zhou, Y. Hybrid Grey Wolf Optimizer Using Elite Opposition-Based Learning Strategy and Simplex Method. Int. J. Comput. Intell. Appl. 2017, 16, 1750012. [Google Scholar] [CrossRef]
- Tian, D.P. Particle Swarm Optimization Based on Tent Chaotic Sequences. Comput. Eng. 2010, 4, 180–182. [Google Scholar]
- Shan, L.; Qiang, H.; Li, J.; Wang, Z. Chaos optimization algorithm based on Tent mapping. Control Decis. 2005, 2, 179–182. [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995. [Google Scholar]
- Li, C.; Luo, G.; Qin, K.; Li, C. An image encryption scheme based on chaotic tent map. Nonlinear Dyn. 2017, 87, 127–133. [Google Scholar] [CrossRef]
- Batra, I.; Ghosh, S. An Improved Tent Map-Adaptive Chaotic Particle Swarm Optimization (ITM-CPSO)-Based Novel Approach Toward Security Constraint Optimal Congestion Management. Iran. J. Sci. Technol. Trans. Electr. Eng. 2018, 42, 261–289. [Google Scholar] [CrossRef]
- Gokhale, S.; Kale, V. An application of a tent map initiated Chaotic Firefly algorithm for optimal overcurrent relay coordination. Int. J. Electr. Power Energy Syst. 2016, 78, 336–342. [Google Scholar] [CrossRef]
- Mitić, M.; Vuković, N.; Petrović, M.; Miljković, Z. Chaotic fruit fly optimization algorithm. Knowl. Based Syst. 2015, 89, 446–458. [Google Scholar] [CrossRef]
- Huang, Q.; Li, J.; Song, C.; Xu, C.; Lin, X. A whale optimization algorithm based on cosine control factor and polynomial variation. Control Decis. 2020, 35, 50–59. [Google Scholar]
- Chatterjee, A.; Siarry, P. Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Comput. Oper. Res. 2006, 33, 859–871. [Google Scholar] [CrossRef]
- Derrac, J.; García, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput. 2011, 1, 3–18. [Google Scholar] [CrossRef]
- Mirjalili, S.; Mirjalili, S.M.; Hatamlou, A. Multi-verse optimizer: A nature-inspired algorithm for global optimization. Neural Comput. Appl. 2016, 27, 495–513. [Google Scholar] [CrossRef]
- Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A Gravitational Search Algorithm. Inf. Sci. 2009, 179, 2232–2248. [Google Scholar] [CrossRef]
- Chen, H.; Wang, M.; Zhao, X. A multi-strategy enhanced sine cosine algorithm for global optimization and con-strained practical engineering problems. Appl. Math. Comput. 2020, 369, 124872. [Google Scholar]
- Coello, C.A.C.; Montes, E.M. Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv. Eng. Inform. 2002, 16, 193–203. [Google Scholar] [CrossRef]
- Coello, C.A.C. Use of a self-adaptive penalty approach for engineering optimization problems. Comput. Ind. 2000, 41, 113–127. [Google Scholar] [CrossRef]
- Deb, K. GeneAS: A Robust Optimal Design Technique for Mechanical Component Design. In Evolutionary Algorithms in Engineering Applications; Springer: Berlin, Heidelberg, 1997; pp. 497–514. [Google Scholar]
- Mezura-Montes, E.; Coello, C.A.C. An empirical study about the usefulness of evolution strategies to solve con-strained optimization problems. Int. J. Gen. Syst. 2008, 37, 443–473. [Google Scholar] [CrossRef]
- Huang, F.-Z.; Wang, L.; He, Q. An effective co-evolutionary differential evolution for constrained optimization. Appl. Math. Comput. 2007, 186, 340–356. [Google Scholar] [CrossRef]
- Kaveh, A.; Talatahari, S. An improved ant colony optimization for constrained engineering design problems. Eng. Comput. 2010, 27, 155–182. [Google Scholar] [CrossRef]
- Mahdavi, M.; Fesanghary, M.; Damangir, E. An improved harmony search algorithm for solving optimization problems. Appl. Math. Comput. 2007, 188, 1567–1579. [Google Scholar] [CrossRef]
- Mirjalili, S.; Lewis, A. The Whale Optimization Algorithm. Adv. Eng. Softw. 2016, 95, 51–67. [Google Scholar] [CrossRef]
- Belegundu, A.D.; Arora, J.S. A study of mathematical programming methods for structural optimization. Part I: Theory. Int. J. Numer. Methods Eng. 1985, 21, 1583–1599. [Google Scholar] [CrossRef]
- Zhang, J.; Wang, J.S. Improved Whale Optimization Algorithm Based on Nonlinear Adaptive Weight and Golden Sine Operator. IEEE Access 2020, 8, 77013–77048. [Google Scholar] [CrossRef]
- Kaveh, A.; Khayatazad, M. A new meta-heuristic method: Ray Optimization. Comput. Struct. 2012, 112–113, 283–294. [Google Scholar] [CrossRef]
- Krohling, R.A.; Coelho, L.D.S. Coevolutionary Particle Swarm Optimization Using Gaussian Distribution for Solving Constrained Optimization Problems. IEEE Trans. Syst. Man, Cybern. Part B 2006, 36, 1407–1416. [Google Scholar] [CrossRef]
- He, Q.; Wang, L. An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng. Appl. Artif. Intell. 2007, 20, 89–99. [Google Scholar] [CrossRef]
- Sun, W.-Z.; Wang, J.-S.; Wei, X. An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance. Symmetry 2018, 10, 210. [Google Scholar] [CrossRef] [Green Version]
- Coello, C.A.C. Constraint-handling using an evolutionary multiobjective optimization technique. Civ. Eng. Environ. Syst. 2000, 17, 319–346. [Google Scholar] [CrossRef]
- Deb, K. An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 2000, 186, 311–338. [Google Scholar] [CrossRef]
- Coello, C.A.C. Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art. Comput. Methods Appl. Mech. Eng. 2002, 191, 1245–1287. [Google Scholar] [CrossRef]
- Sadollah, A.; Bahreininejad, A.; Eskandar, H.; Hamdi, M. Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems. Appl. Soft Comput. 2013, 13, 2592–2612. [Google Scholar] [CrossRef]
- Zhang, M.; Luo, W.; Wang, X. Differential evolution with dynamic stochastic selection for constrained optimization. Inf. Sci. 2008, 178, 3043–3074. [Google Scholar] [CrossRef]
- Gandomi, A.H.; Yang, X.-S.; Alavi, A.H. Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems. Eng. Comput. 2013, 29, 17–35. [Google Scholar] [CrossRef]
- Ray, T.; Saini, P. Engineering design optimization using a swarm with an intelligent information sharing among individuals. Eng. Optim. 2001, 33, 735–748. [Google Scholar] [CrossRef]
- Tsai, J.-F. Global optimization of nonlinear fractional programming problems in engineering design. Eng. Optim. 2005, 37, 399–409. [Google Scholar] [CrossRef]
Number | Name | Benchmark | Dim | Range | |
---|---|---|---|---|---|
F1 | Sphere | 30 | [−100, 100] | 0 | |
F2 | Schwefel’problem2.22 | 30 | [−10, 10] | 0 | |
F3 | Schwefel’problem1.2 | 30 | [−100, 100] | 0 | |
F4 | Schwefel’problem2.21 | 30 | [−100, 100] | 0 | |
F5 | Rosenbrock | 30 | [−30, 30] | 0 | |
F6 | Step | 30 | [−100, 100] | 0 | |
F7 | Noise | 30 | [−1.28, 1.28] | 0 |
Number | Name | Benchmark | Dim | Range | |
---|---|---|---|---|---|
F8 | Generalized Schwfel’s problem | 30 | [−500, 500] | −12, 569.5 | |
F9 | Rastrigin | 30 | [−5.12, 5.12] | 0 | |
F10 | Ackley | 30 | [−32, 32] | 0 | |
F11 | Griewank | 30 | [−600, 600] | 0 | |
F12 | Generalized penalized function 1 | 30 | [−50, 50] | 0 | |
F13 | Generalized Penalized Function 2 | 30 | [−50, 50] | 0 |
Number | Name | Benchmark | Dim | Range | |
---|---|---|---|---|---|
F14 | Shekel’s foxholes function | 2 | [−65.536, 65.536] | 1 | |
F15 | Kowalik’s Function | 4 | [−5, 5] | 0.0003 | |
F16 | Six-hump camelback | 2 | [−5, 5] | −1.0316 | |
F17 | Branin | 2 | [−5, 10] [10, 15] | 0.39788 | |
F18 | Goldstein–Price function | 2 | [−2, 2] | 3 | |
F19 | Hartmann 1 | 3 | [0, 1] | −3.86 | |
F20 | Hartmann 2 | 6 | [1, 6] | −3.32 | |
F21 | Shekel 1 | 4 | [0, 10] | −10.1532 | |
F22 | Shekel 2 | 4 | [0, 10] | −10.4028 | |
F23 | Shekel 3 | 4 | [0, 10] | −10.5363 |
Function | GWO | GWO1 | GWO2 | GWO3 | IGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|
ave | std | ave | std | ave | std | ave | std | ave | std | |
F1 | 1.55 × 10−27 | 2.95 × 10−27 | 1.18 × 10−30 | 2.15 × 10−30 | 1.72 × 10−38 | 6.97 × 10−38 | 3.25 × 10−28 | 4.78 × 10−28 | 8.34 × 10−40 | 2.35 × 10−39 |
F2 | 9.89 × 10−17 | 9.40 × 10−17 | 8.14 × 10−17 | 4.84 × 10−18 | 9.97 × 10−19 | 1.92 × 10−18 | 1.03 × 10−17 | 6.89 × 10−18 | 9.96 × 10−24 | 1.23 × 10−23 |
F3 | 2.47 × 10−5 | 6.57 × 10−5 | 3.03 × 10−5 | 7.30 × 10−5 | 3.19 × 10−10 | 1.70 × 10−9 | 3.74 × 10−5 | 7.15 × 10−5 | 5.95 × 10−8 | 9.72 × 10−8 |
F4 | 7.54 × 10−7 | 1.16 × 10−6 | 8.47 × 10−9 | 9.94 × 10−9 | 5.21 × 10−15 | 1.04 × 10−14 | 7.45 × 10−8 | 5.55 × 10−7 | 1.78 × 10−11 | 3.00 × 10−11 |
F5 | 2.73 × 101 | 7.53 × 10−1 | 2.98 × 102 | 2.46 × 10−1 | 2.98 × 102 | 2.39 × 10−1 | 2.98 × 102 | 2.09 × 10−1 | 2.72 × 101 | 7.30 × 10−1 |
F6 | 7.37 × 10−1 | 3.06 × 10−1 | 7.26 × 10−1 | 3.53 × 10−1 | 1.72 | 5.35 × 10−1 | 6.56 × 10−1 | 3.95 × 10−1 | 1.68 | 4.28 × 10−1 |
F7 | 2.40 × 10−3 | 1.40 × 10−3 | 2.20 × 10−3 | 9.42 × 10−4 | 7.55 × 10−4 | 4.52 × 10−4 | 2.00 × 10−3 | 1.60 × 10−3 | 8.87 × 10−4 | 5.52 × 10−4 |
F8 | 1.84 × 10−14 | 8.25 × 102 | −6.09 × 103 | 6.96 × 102 | −4.44 × 103 | 1.37 × 103 | −5.98 × 103 | 9.15 × 102 | −4.64 × 103 | 1.26 × 103 |
F9 | 2.41 | 3.30 | 1.99 | 3.53 | 0 | 0 | 3.51 × 10−2 | 1.40 × 10−1 | 0 | 0 |
F10 | 1.03 × 10−13 | 1.84 × 10−14 | 1.06 × 10−15 | 2.26 × 10−14 | 9.30 × 10−15 | 2.38 × 10−15 | 1.66 × 10−13 | 5.28 × 10−14 | 1.88 × 10−14 | 4.01 × 10−15 |
F11 | 3.20 × 10−3 | 6.70 × 10−3 | 1.20 × 10−4 | 5.00 × 10−3 | 0 | 0 | 3.50 × 10−6 | 9.40 × 10−6 | 0 | 0 |
F12 | 4.59 × 10−2 | 2.13 × 10−2 | 5.43 × 10−2 | 3.12 × 10−2 | 1.11 × 10−1 | 4.30 × 10−2 | 3.84 × 10−2 | 2.28 × 10−2 | 1.02 × 10−1 | 3.95 × 10−2 |
F13 | 6.41 × 10−1 | 2.37 × 10−1 | 6.48 × 10−1 | 2.23 × 10−1 | 1.10 | 2.32 × 10−1 | 5.21 × 10−1 | 1.92 × 10−1 | 1.05 | 2.24 × 10−1 |
F14 | 4.55 | 4.36 | 5.02 | 4.11 | 4.98 | 4.23 | 5.04 | 4.49 | 5.08 | 4.31 |
F15 | 6.50 × 10−3 | 9.30 × 10−3 | 4.40 × 10−3 | 8.10 × 10−3 | 5.19 × 10−4 | 1.86 × 10−4 | 5.10 × 10−3 | 8.60 × 10−3 | 4.94 × 10−4 | 1.22 × 10−4 |
F16 | −1.03 | 4.65 × 10−8 | −1.03 | 1.69 × 10−8 | −1.03 | 1.04 × 10−5 | −1.03 | 1.86 × 10−11 | −1.03 | 4.84 × 10−8 |
F17 | 3.98 × 10−1 | 3.88 × 10−6 | 3.98 × 10−1 | 1.63 × 10−6 | 3.98 × 10−1 | 1.63 × 10−5 | 3.98 × 10−1 | 1.16 × 10−4 | 3.98 × 10−1 | 3.83 × 10−4 |
F18 | 3.00 | 2.99 × 10−5 | 5.70 | 1.48 × 101 | 3.00 | 3.17 × 10−5 | 3.00 | 4.14 × 10−5 | 3.00× | 3.15 × 10−5 |
F19 | 4.56 | 2.90 × 10−3 | −3.86 | 2.40 × 10−3 | −3.86 | 2.90 × 10−3 | −3.86 | 2.70 × 10−3 | −3.86 | 2.90 × 10−3 |
F20 | −3.27 | 7.48 × 10−2 | −3.27 | 6.75 × 10−2 | −3.19 | 8.67 × 10−2 | −3.23 | 8.52 × 10−2 | −3.23 | 9.19 × 10−2 |
F21 | −9.39 | 2.00 | −8.47 | 2.68 | −8.28 | 2.49 | −9.81 | 1.29 | −7.87 | 2.69 |
F22 | −1.04 × 101 | 9.22 × 10−4 | −1.02 × 101 | 9.70 × 10−1 | −9.78 | 1.90 | −1.00 × 101 | 1.35 | −9.69 | 1.83 |
F23 | −1.04 × 101 | 9.87 × 10−1 | −1.05 × 101 | 7.63 × 10−4 | −1.03 × 101 | 1.07 | −1.05 × 101 | 2.36 × 10−7 | −1.02 × 101 | 1.37 |
Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|
F1 | ave | 8.34 × 10−40 | 1.55 × 10−27 | 2.70 × 10−12 | 2.34 × 103 | 8.76 × 10−3 | 4.64 | 4.22 × 101 | 2.67 × 10−7 |
std | 2.35 × 10−39 | 2.95 × 10−27 | 7.91 × 10−12 | 5.04 × 103 | 1.32 × 10−2 | 1.58 | 1.61 × 101 | 3.31 × 10−7 | |
F2 | ave | 9.96 × 10−24 | 9.89 × 10−17 | 9.26 × 10−10 | 2.92 × 101 | 1.12 × 10−1 | 1.05 × 101 | 8.40 | 2.72 |
std | 1.23 × 10−23 | 9.40 × 10−17 | 2.51 × 10−9 | 1.78 × 101 | 1.01 × 10−1 | 2.10 | 1.65 × 101 | 1.94 | |
F3 | ave | 5.95 × 10−8 | 2.47 × 10−5 | 5.49 × 10−1 | 2.03 × 104 | 5.61 × 10−3 | 7.05 | 6.02 × 101 | 1.38 × 103 |
std | 9.72 × 10−8 | 6.57 × 10−5 | 3.00 | 1.02 × 104 | 6.75 × 10−3 | 2.42 | 3.33 × 101 | 6.63 × 102 | |
F4 | ave | 1.78 × 10−11 | 7.54 × 10−7 | 1.00 × 10−3 | 6.78 × 101 | 7.66 × 10−2 | 1.23 | 2.61 | 1.14 × 101 |
std | 3.00 × 10−11 | 1.16 × 10−6 | 2.60 × 10−3 | 1.02 × 101 | 8.77 × 10−2 | 11.36 × 10−1 | 5.22 × 10−1 | 3.18 | |
F5 | ave | 2.72 × 101 | 2.73 × 101 | 7.36 | 2.68 × 106 | 5.38 × 10−3 | 4.98 × 102 | 2.48 × 104 | 3.11 × 102 |
std | 7.30 × 10−1 | 7.53 × 10−1 | 3.81 × 10−1 | 1.46 × 107 | 7.82 × 10−3 | 1.65 × 102 | 2.02 × 104 | 4.69 × 102 | |
F6 | ave | 1.68 | 7.37 × 10−1 | 4.53 × 10−1 | 1.35 × 103 | 8.57 × 10−3 | 6.07 | 5.04 × 101 | 2.11 × 10−7 |
std | 4.28 × 10−1 | 3.06 × 10−1 | 1.55 × 10−1 | 4.37 × 103 | 1.44 × 10−2 | 1.72 | 1.54 × 101 | 3.85 × 10−7 | |
F7 | ave | 8.87 × 10−4 | 2.40 × 10−3 | 2.50 × 10−3 | 2.83 | 1.12 × 10−1 | 1.52 × 102 | 2.99 × 103 | 1.74 × 10−1 |
std | 5.52 × 10−4 | 1.40 × 10−3 | 2.00 × 10−3 | 4.45 | 1.03 × 10−1 | 3.51 × 101 | 2.15 × 103 | 7.68 × 10−2 |
Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|
F8 | ave | −4.64 × 103 | 1.84 × 10−14 | −2.16 × 103 | −8.61 × 103 | −5.57 × 102 | -Inf | −4.73 × 101 | −7.40 × 103 |
std | 1.26 × 103 | 8.25 × 102 | 1.77 × 102 | 8.71 × 102 | 4.79 × 10−3 | --- | 8.83 | 6.39 × 102 | |
F9 | ave | 0 | 2.41 | 5.52 × 10−1 | 1.64 × 102 | 5.70 × 10−1 | 4.22 × 101 | 1.90 × 102 | 5.43 × 101 |
std | 0 | 3.30 | 2.92 | 2.98 × 101 | 6.01 × 10−1 | 7.28 | 3.08 × 101 | 2.24 × 101 | |
F10 | ave | 1.88 × 10−14 | 1.03 × 10−13 | 5.50 × 10−6 | 1.59 × 101 | 3.96 | 3.44 | 5.60 | 2.46 |
std | 4.01 × 10−15 | 1.84 × 10−14 | 2.76 × 10−5 | 6.56 | 7.34 | 2.51 × 10−1 | 6.66 × 10−1 | 8.54 × 10−1 | |
F11 | ave | 0 | 3.20 × 10−3 | 8.05 × 10−2 | 3.70 × 101 | 3.36 × 10−3 | 2.32 × 10−1 | 8.64 × 10−1 | 1.95 × 10−2 |
std | 0 | 6.70 × 10−3 | 1.45 × 10−1 | 5.06 × 101 | 3.83 × 10−3 | 8.29 × 10−2 | 1.12 × 10−1 | 1.67 × 10−2 | |
F12 | ave | 1.02 × 10−1 | 4.59 × 10−2 | 9.27 × 10−2 | 1.55 × 102 | 2.27 × 10−1 | 2.33 × 10−1 | 2.06 | 6.16 |
std | 3.95 × 10−2 | 2.13 × 10−2 | 4.55 × 10−2 | 4.58 × 102 | 5.16 × 10−1 | 9.85 × 10−2 | 7.12 × 10−1 | 2.26 | |
F13 | ave | 1.05 | 6.41 × 10−1 | 3.00 × 10−1 | 1.37 × 107 | 1.38 × 10−2 | 2.93 | 8.20 | 1.20 × 101 |
std | 2.24 × 10−1 | 2.37 × 10−1 | 1.08 × 10−1 | 7.49 × 107 | 1.19 × 10−2 | 7.01 × 10−1 | 2.79 | 1.54 × 101 |
Function | Index | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|---|
F14 | ave | 5.08 | 4.55 | 1.94 | 3.13 | 9.98 × 10−1 | 1.26 × 101 | 1.27 × 101 | 1.26 |
std | 4.310 | 4.36 | 9.97 × 10−1 | 2.51 | 8.07 × 10−6 | 3.48 × 10−1 | 1.88 × 10−14 | 6.35 × 10−1 | |
F15 | ave | 4.94 × 10−4 | 6.50 × 10−3 | 1.00 × 10−3 | 1.06 × 10−3 | 5.30 × 10−3 | 3.70 × 10−3 | 6.31 × 10−3 | 2.20 × 10−3 |
std | 1.22 × 10−4 | 9.30 × 10−3 | 3.66 × 10−4 | 4.24 × 10−4 | 5.47 × 10−3 | 6.10 × 10−3 | 1.01 × 10−2 | 4.90 × 10−3 | |
F16 | ave | −1.03 | −1.03 | −1.03 | −1.03 | 1.22 × 101 | −1.03 | −1.03 | −1.03 |
std | 4.84 × 10−8 | 4.65 × 10−8 | 4.28 × 10−5 | 6.78 × 10−16 | 3.29 × 101 | 2.11 × 10−9 | 4.99 × 10−7 | 1.68 × 10−14 | |
F17 | ave | 3.98 × 10−1 | 3.98 × 10−1 | 4.01 × 10−1 | 3.98 × 10−1 | 3.20 | 3.99 × 10−1 | 3.98 × 10−1 | 3.81 |
std | 3.83 × 10−4 | 3.88 × 10−6 | 3.00 × 10−3 | 0 | 3.30 | 2.90 × 10−3 | 3.54 × 10−10 | 3.57 × 10−2 | |
F18 | ave | 3.00 | 3.00 | 3.00 | 3.00 | 8.14 × 103 | 1.83 × 101 | 6.60 | 3.00 |
std | 3.15 × 10−5 | 2.99 × 10−5 | 1.54 × 10−4 | 2.23 × 10−15 | 1.88 × 104 | 2.53 × 101 | 1.54 × 101 | 2.45 × 10−13 | |
F19 | ave | −3.86 | 4.56 | −3.85 | −3.86 | −1.08 | −3.42 | −3.86 | −3.86 |
std | 2.90 × 10−3 | 2.90 × 10−3 | 3.10 × 10−3 | 2.71 × 10−15 | 8.23 × 10−1 | 9.85 × 10−1 | 2.86 × 10−7 | 5.68 × 10−9 | |
F20 | ave | −3.23 | −3.27 | −2.93 | −3.24 | −5.03 × 10−1 | −3.16 | −2.93 | −3.22 |
std | 9.19 × 10−2 | 7.48 × 10−2 | 2.17 × 10−1 | 7.08 × 10−2 | 5.34 × 10−1 | 3.74 × 10−1 | 2.17 × 10−1 | 6.12 × 10−2 | |
F21 | ave | −7.87 | −9.39 | −1.93 | −7.72 | −2.49 × 10−1 | −5.23 | −1.02 × 101 | −7.15 |
std | 2.69 | 2.00 | 1.57 | 3.12 | 3.82 × 10−1 | 9.31 × 10−1 | 3.92 × 10−4 | 3.56 | |
F22 | ave | −9.69 | −1.04 × 101 | −4.11 | −7.46 | −1.52 × 10−1 | −5.09 | −1.04 × 101 | −8.92 |
std | 1.83 | 9.22 × 10−4 | 1.66 | 3.49 | 1.20 × 10−1 | 5.23 × 10−7 | 5.50 × 10−3 | 2.78 | |
F23 | ave | −1.02 × 101 | −1.04 × 101 | −3.82 | −7.58 | −2.45 × 10−1 | −5.30 | −1.05 × 101 | −8.34 |
std | 1.37 | 9.87 × 10−1 | 1.59 | 3.74 | 2.23 × 10−1 | 9.38 × 10−1 | 3.26 × 10−3 | 3.46 |
Function | IGWO vs. GWO p-Value Win | IGWO vs. SCA p-Value Win | IGWO vs. MFO p-Value Win | IGWO vs. PSO p-Value Win | IGWO vs. BA p-Value Win | IGWO vs. FPA p-Value Win | 1GWO vs. SSA p-Value Win |
---|---|---|---|---|---|---|---|
F1 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 |
F2 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 |
F3 | 1.78 × 10−10 | 3.029 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 |
F4 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.01 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 |
F5 | 4.13 × 10−2 | 2.58 × 10−11 | 2.58 × 10−11 | 2.58 × 10−11 | 8.26 × 10−6 | 2.57 × 10−11 | 1.26 × 10−10 |
F6 | 2.97 × 10−11 | 2.50 × 10−2 | 2.70 × 10−2 | 2.97 × 10−11 | 2.97 × 10−11 | 4.89 × 10−11 | 2.97 × 10−11 |
F7 | 7.27 × 10−6 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 | 3.02 × 10−11 |
F8 | 3.01 × 10−11 | 1.81 × 10−1 | 3.01 × 10−11 | 2.99 × 10−11 | 3.01 × 10−11 | 3.02 × 10−11 | 1.21 × 10−10 |
F9 | 1.19 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.20 × 10−12 | 1.21 × 10−12 |
F10 | 1.58 × 10−11 | 1.68 × 10−11 | 1.68 × 10−11 | 5.65 × 10−13 | 1.63 × 10−11 | 1.68 × 10−11 | 1.68 × 10−11 |
F11 | 1.10 × 10−2 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 | 1.21 × 10−12 |
F12 | 1.06 × 10−7 | 3.01 × 10−11 | 3.01 × 10−11 | 3.01 × 10−11 | 9.12 × 10−1 | 1.61 × 10−10 | 3.01 × 10−11 |
F13 | 4.50 × 10−11 | 3.01 × 10−11 | 3.01 × 10−11 | 3.01 × 10−11 | 8.14 × 10−5 | 4.07 × 10−11 | 1.86 × 10−6 |
F14 | 3.26 × 10−1 | 8.26 × 10−1 | 6.58 × 10−1 | 4.76 × 10−4 | 5.12 × 10−11 | 2.30 × 10−12 | 1.79 × 10−4 |
F15 | 1.84 × 10−2 | 7.08 × 10−8 | 7.64 × 10−8 | 5.07 × 10−10 | 1.10 × 10−4 | 9.50 × 10−3 | 8.84 × 10−7 |
F16 | 1.06 × 10−11 | 2.36 × 10−12 | 1.61 × 10−1 | 2.36 × 10−12 | 6.50 × 10−14 | 2.49 × 10−12 | 1.61 × 10−1 |
F17 | 8.44 × 10−7 | 2.04 × 10−9 | 8.59 × 10−7 | 2.42 × 10−11 | 1.20 × 10−7 | 8.59 × 10−7 | 3.71 × 10−7 |
F18 | 3.79 × 10−1 | 1.76 × 10−1 | 1.21 × 10−12 | 3.01 × 10−11 | 1.21 × 10−12 | 5.96 × 10−11 | 1.21 × 10−12 |
F19 | 8.00 × 10−3 | 1.20 × 10−8 | 1.20 × 10−12 | 1.20 × 10−12 | 3.71 × 10−7 | 7.35 × 10−11 | 1.20 × 10−12 |
F20 | 2.23 × 10−1 | 1.01 × 10−8 | 1.50 × 10−3 | 1.21 × 10−12 | 6.25 × 10−4 | 5.53 × 10−8 | 1.40 × 10−2 |
F21 | 5.11 × 10−1 | 1.95 × 10−10 | 2.48 × 10−1 | 3.01 × 10−11 | 2.55 × 10−1 | 3.79 × 10−1 | 1.02 × 10−1 |
F22 | 5.07 × 10−10 | 3.68 × 10−11 | 5.73 × 10−2 | 3.01 × 10−11 | 7.73 × 10−11 | 5.57 × 10−10 | 6.35 × 10−2 |
F23 | 3.01 × 10−11 | 4.50 × 10−11 | 1.70 × 10−1 | 3.01 × 10−11 | 8.82 × 10−10 | 3.01 × 10−11 | 1.23 × 10−9 |
Function | IGWO | GWO | SCA | MFO | PSO | BA | FPA | SSA |
---|---|---|---|---|---|---|---|---|
F1 | 1.5 | 1.5 | 3 | 8 | 5 | 6 | 7 | 4 |
F2 | 1.5 | 1.5 | 3 | 8 | 4 | 7 | 6 | 5 |
F3 | 1 | 2 | 4 | 8 | 3 | 5 | 6 | 7 |
F4 | 1 | 2 | 3 | 8 | 4 | 5 | 6 | 7 |
F5 | 3 | 4 | 2 | 8 | 1 | 6 | 7 | 5 |
F6 | 5 | 4 | 3 | 8 | 2 | 6 | 7 | 1 |
F7 | 1 | 2 | 3 | 6 | 4 | 7 | 8 | 5 |
F8 | 3 | 7 | 4 | 1 | 5 | -- | 6 | 2 |
F9 | 1 | 4 | 2 | 7 | 3 | 5 | 8 | 6 |
F10 | 1 | 2 | 3 | 8 | 6 | 5 | 7 | 4 |
F11 | 1 | 2 | 5 | 8 | 3 | 6 | 7 | 4 |
F12 | 3 | 1 | 2 | 8 | 4 | 5 | 6 | 7 |
F13 | 4 | 3 | 2 | 8 | 1 | 5 | 6 | 7 |
F14 | 6 | 5 | 3 | 4 | 1 | 7 | 8 | 2 |
F15 | 1 | 8 | 2 | 3 | 6 | 5 | 7 | 4 |
F16 | 1.5 | 5 | 5 | 5 | 8 | 5 | 5 | 1.5 |
F17 | 2.5 | 2.5 | 6 | 2.5 | 7 | 5 | 2.5 | 8 |
F18 | 3 | 3 | 3 | 3 | 8 | 7 | 6 | 3 |
F19 | 2 | 8 | 5 | 3.5 | 7 | 6 | 3.5 | 1 |
F20 | 3 | 1 | 6.5 | 2 | 8 | 5 | 6.5 | 4 |
F21 | 3 | 2 | 7 | 4 | 8 | 6 | 1 | 5 |
F22 | 3 | 1.5 | 7 | 5 | 8 | 6 | 1.5 | 4 |
F23 | 3 | 2 | 7 | 5 | 8 | 6 | 1 | 4 |
Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|
Ts | Th | R | L | ||
MVO [37] | 0.8125 | 0.4375 | 42.0907382 | 176.738690 | 6060.8066 |
GSA [38] | 1.125000 | 0.6250 | 55.988659 | 84.4542025 | 8538.8359 |
PSO [39] | 0.812500 | 0.437500 | 42.091266 | 176.746500 | 6061.0777 |
MSCA [40] | 0.776256 | 0.399600 | 40.325450 | 199.9213 | 5935.7161 |
GA (Coello) [41] | 0.812500 | 0.4345 | 40.323900 | 200.0000 | 6288.7445 |
GA (Coello and Montes) [42] | 0.812500 | 0.4375 | 42.097397 | 176.654050 | 6059.9463 |
GA (Deb et al.) [43] | 0.937500 | 0.50000 | 48.329000 | 112.679000 | 6410.3811 |
ES [44] | 0.812500 | 0.437500 | 42.098087 | 176.640518 | 6059.745605 |
DE (Huang et al.) [45] | 0.8125 | 0.4375 | 42.098411 | 176.637690 | 6059.7340 |
ACO (Kaveh et al.) [46] | 0.8125 | 0.4375 | 42.103624 | 176.572656 | 6059.0888 |
HIS [47] | 1.125000 | 0.625000 | 58.29015 | 43.69268 | 7197.7300 |
MFO | 0.8125 | 0.4375 | 42.098445 | 176.636596 | 6059.7143 |
WOA [48] | 0.812500 | 0.437500 | 42.098209 | 176.638998 | 6059.7410 |
IGWO [21] | 0.8125 | 0.4375 | 42.0984456 | 176.636596 | 6059.7143 |
GWO | 0.7852686 | 0.3891504 | 40.67564 | 195.6436 | 5913.8838 |
IGWO | 0.7784458 | 0.3854034 | 40.33393 | 199.8019 | 5888.6000 |
Algorithm | Optimum Variables | Optimum Cost | ||
---|---|---|---|---|
d | N | D | ||
Mathematical optimization method (Belegundu) [49] | 0.053396 | 0.3177 | 14.0260 | 0.0127303 |
GSA (Kaveh) [46] | 0.050000 | 0.317312 | 14.22867 | 0.0128739 |
GSA (Rashedi) [38] | 0.050276 | 0.323680 | 13.525410 | 0.0127022 |
SCA [6] | 0.050780 | 0.334779 | 12.72269 | 0.127097 |
MVO [37] | 0.05000 | 0.315956 | 14.22623 | 0.128169 |
GWO | 0.05000 | 0.31739 | 14.0351 | 0.012699 |
IGWO | 0.05159 | 0.354337 | 11.4301 | 0.012700 |
Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|
h | l | t | b | ||
NGS-WOA [50] | 0.202369 | 3.544214 | 9.04821 | 0.205723 | 1.72802 |
WOA [48] | 0.205396 | 33.484293 | 9.037426 | 0.206276 | 1.730499 |
RO [51] | 0.203687 | 3.528467 | 9.004233 | 0.207241 | 1.735344 |
MVO [37] | 0.20722744 | 3.393969312 | 9.018874001 | 0.207225774 | 1.7250 |
CPSO [52] | 0.205463 | 3.473193 | 9.044502 | 0.205695 | 1.72645 |
CPSO [53] | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.73148 |
HS [54] | 0.2442 | 6.2231 | 8.2915 | 0.2433 | 2.3807 |
GSA [38] | 0.182129 | 3.856979 | 10.0000 | 0.202376 | 1.87995 |
GA [55] | 0.1829 | 4.0483 | 9.3666 | 0.2059 | 1.82420 |
GA [56] | 0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.43312 |
Coello [40] | 0.208800 | 3.420500 | 8.997500 | 0.2100 | 1.74831 |
Coello and Monters [41] | 0.205986 | 3.471328 | 9.020224 | 0.206480 | 1.72822 |
GWO | 0.20527 | 3.4819 | 9.0389 | 0.20583 | 1.7269 |
IGWO | 0.20496 | 3.4872 | 9.0366 | 0.20573 | 1.7254 |
Algorithm | Optimum Variables | Optimum Cost | |
---|---|---|---|
X1 | X2 | ||
PSO-DE [39] | 0.7886751 | 0.4082482 | 263.8958433 |
MBA [57] | 0.7885650 | 0.4085597 | 263.8958522 |
DEDS [58] | 0.78867513 | 0.40824828 | 263.8958434 |
CS [59] | 0.78867 | 0.40902 | 263.9716 |
Ray and Sain [60] | 0.795 | 0.395 | 264.3 |
Tsa [61] | 0.788 | 0.408 | 263.68 |
WOA [48] | 0.789050544 | 0.407187512 | 263.8959474 |
MFO | 0.788244770931922 | 0.409466905784741 | 263.895979682 |
GWO | 0.78769 | 0.41108 | 263.9011 |
IGWO | 0.78846 | 0.40884 | 263.8959 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Li, Y.; Lin, X.; Liu, J. An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems. Sustainability 2021, 13, 3208. https://doi.org/10.3390/su13063208
Li Y, Lin X, Liu J. An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems. Sustainability. 2021; 13(6):3208. https://doi.org/10.3390/su13063208
Chicago/Turabian StyleLi, Yu, Xiaoxiao Lin, and Jingsen Liu. 2021. "An Improved Gray Wolf Optimization Algorithm to Solve Engineering Problems" Sustainability 13, no. 6: 3208. https://doi.org/10.3390/su13063208