Abstract
Nature inspired meta-heuristic algorithms are an integral part of modern optimization techniques. One such algorithm is bat algorithm which is inspired from echolocation behavior of bats and has been successfully applied to non-linear single-objective optimization problems. In this paper, a multi-objective extension of bat algorithm is proposed using the concepts of Pareto non-dominance and elitism. The novel algorithm is tested using thirty multi-objective test problems. The performance is measured using metrics namely, hyper-volume ratio, generational distance and spacing. The newly developed algorithm is then applied to a real-world multi-objective optimization problem of a phthalic anhydride reactor. It shows faster convergence for test problems as well as the industrial optimization problem than two popular nature inspired meta-heuristic algorithms, i.e. multi-objective non-dominated sorting particle swarm optimization and real-coded elitist non-dominated sorting genetic algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- A min :
-
Minimum loudness
- A max :
-
Maximum loudness
- A i :
-
Loudness of ith bat
- At :
-
Average loudness of all the bats at tth iteration
- BA:
-
Bat algorithm
- e :
-
Exponential
- f i :
-
Frequency of ith bat
- f min :
-
Minimum frequency
- f max :
-
Maximum frequency
- Ng :
-
Bats having rank 1
- Np :
-
Total number of solutions (bats) in the population
- NV :
-
Total number of decision variables
- NSPSO:
-
Non-dominated sorting PSO
- NSBAT-II:
-
Elitist non-dominated sorting BA
- r :
-
Pulse emission rate
- r i :
-
Pulse emission rate at ith bat
- r i,t :
-
Pulse emission rate at tth iteration
- RNSGA-II:
-
Real-coded NSGA-II
- t :
-
Number of iterations
- t max :
-
User-specified maximum number of iterations
- v i :
-
Current velocity of ith bat
- \(v_{i,j}^{0}\) :
-
Initial ith bat velocity
- \(v_{i,j}^{t}\) :
-
jth component of ith bat velocity at tth iteration
- w i :
-
Inertia weight for ith bat
- \(\varvec{x}_{i}\) :
-
Current position of ith bat
- \(x_{j}^{low}\) :
-
Lower value of decision variable
- \(x_{j}^{high}\) :
-
Upper of decision variable
- \(x_{g}^{best}\) :
-
Total number of solution having rank 1
- \(x_{i,j}^{0}\) :
-
jth component of ith position at 0th iteration
- \(x_{i,j}^{t}\) :
-
jth component of ith position at tth iteration
- \(\varepsilon\) :
-
Random number between −1 and 1
References
Aggarwal M, Hanmandlu M (2015) Representing uncertainty with information sets. IEEE Trans Fuzzy Syst 6706:1–15. doi:10.1109/TFUZZ.2015.2417593
Ali M, Siarry P, Pant M (2011) An efficient differential evolution based algorithm for solving multi-objective optimization problems. Eur J Oper Res 217:404–416. doi:10.1016/j.ejor.2011.09.025
Bahmani-Firouzi B, Azizipanah-Abarghooee R (2014) Optimal sizing of battery energy storage for micro-grid operation management using a new improved bat algorithm. Int J Electr Power Energy Syst 56:42–54. doi:10.1016/j.ijepes.2013.10.019
Bansal JC, Sharma H, Jadon SS, Clerc M (2014) Spider monkey optimization algorithm for numerical optimization. Memet Comput 6(1):31–47. doi:10.1007/s12293-013-0128-0
Banzhaf W, Nordin P, Keller RE, Francone FD (2011) Genetic programming: an introduction, vol 31. Morgan Kaufmann, San Francisco. doi:10.1109/5254.846288
Bhat GR, Gupta SK (2008) MO optimization of phthalic anhydride industrial catalytic reactors using guided GA with the adapted jumping gene operator. Chem Eng Res Des 86:959–976. doi:10.1016/j.cherd.2008.03.012
Carmelo Filho JA, De Lima Neto FB, Lins AJ, Nascimento AI, Lima MP (2008) A novel search algorithm based on fish school behavior. In: Conference proceedings—IEEE international conference on systems, man and cybernetics, pp 2646–2651. doi:10.1109/ICSMC.2008.4811695
Černý V (1985) Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Optim Theory Appl 45(1):41–51. doi:10.1007/BF00940812
Chawla M, Duhan M (2015) Bat algorithm: a survey of the state-of-the-art. Appl Artif Intell 29(6):617–634. doi:10.1080/08839514.2015.1038434
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York. https://books.google.co.in/books/about/Multi_Objective_Optimization_Using_Evolu.html?id=OSTn4GSy2uQC&pgis=1\nhttps://books.google.com/books?id=OSTn4GSy2uQC&pgis=1
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197. doi:10.1109/4235.996017
Doğan B, Ölmez T (2015) A new metaheuristic for numerical function optimization: vortex search algorithm. Inf Sci 293:125–145. doi:10.1016/j.ins.2014.08.053
Dorigo M, Blum C (2005) Ant colony optimization theory: a survey. Theoret Comput Sci 344(2–3):243–278. doi:10.1016/j.tcs.2005.05.020
Duan H, Luo Q (2015) New progresses in swarm intelligence–based computation. Int J Bio-Inspired Comput 7(1):26–35. http://www.inderscienceonline.com/doi/abs/10.1504/IJBIC.2015.067981
Durillo JJ, Nebro AJ (2011) JMetal: a java framework for multi-objective optimization. Adv Eng Softw 42(10):760–771. doi:10.1016/j.advengsoft.2011.05.014
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 110–111:151–166. doi:10.1016/j.compstruc.2012.07.010
Fister I, Yang X-S, Fister I, Brest J, Fister D (2013) A brief review of nature-inspired algorithms for optimization. arXiv Preprint arXiv:1307.4186 80(3):1–7. http://arxiv.org/abs/1307.4186
Fong S, Wang X, Xu Q, Wong R, Fiaidhi J, Mohammed S (2015) Recent advances in metaheuristic algorithms: does the makara dragon exist? J Supercomput. doi:10.1007/s11227-015-1592-8
Gandomi AH, Alavi AH (2012) Krill Herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 17(12):4831–4845. doi:10.1016/j.cnsns.2012.05.010
Gandomi AH, Yang XS, Alavi AH (2013) Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 29(1):17–35. doi:10.1007/s00366-011-0241-y
Ghaemi M, Feizi-Derakhshi M-R (2014) Forest optimization algorithm. Expert Syst Appl 41(15):6676–6687. doi:10.1016/j.eswa.2014.05.009
Goel L, Gupta D, Panchal VK (2012) Hybrid bio-inspired techniques for land cover feature extraction: a remote sensing perspective. Appl Soft Comput J 12(2):832–849. doi:10.1016/j.asoc.2011.10.006
Goldberg David, Holland John (1988) Genetic algorithms and machine learning. Mach Learn 3:95–99. doi:10.1023/A:1022602019183
Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506. doi:10.1109/TEVC.2005.861417
Jordehi AR (2015) Chaotic bat swarm optimisation (CBSO). Appl Soft Comput 26:523–530. doi:10.1016/j.asoc.2014.10.010
Jun L, Liheng L, Xianyi W (2015) A double-subpopulation variant of the bat algorithm. Appl Math Comput 263:361–377. doi:10.1016/j.amc.2015.04.034
Karaboga D, Basturk B (2008) On the performance of artificial bee colony (ABC) algorithm. Appl Soft Comput 8(1):687–697. doi:10.1016/j.asoc.2007.05.007
Kaveh A, Farhoudi N (2013) A new optimization method: dolphin echolocation. Adv Eng Softw 59:53–70. doi:10.1016/j.advengsoft.2013.03.004
Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search. Acta Mech 213(3–4):267–289. doi:10.1007/s00707-009-0270-4
Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Neural Networks, 1995. Proceedings of IEEE international conference on, vol 4, pp 1942–1948. doi:10.1109/ICNN.1995.488968
Kirkpatrick S, Gelatt CDD, Vecchi MP, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. doi:10.1126/science.220.4598.671
Krishnanand KN, Ghose D (2009) Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions. Swarm Intell 3(2):87–124. doi:10.1007/s11721-008-0021-5
Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36–38):3902–3933. doi:10.1016/j.cma.2004.09.007
Lin K-C, Zhang K-Y, Huang Y-H, Hung JC, Yen N (2016) Feature selection based on an improved cat swarm optimization algorithm for big data classification. J Supercomput. doi:10.1007/s11227-016-1631-0
Mello RF, Andrade Filho JA, Senger LJ, Yang LT (2008) Grid job scheduling using route with genetic algorithm support. Telecommun Syst 38(3–4):147–160. doi:10.1007/s11235-008-9101-5
Meng XB, Gao XZ, Liu Y, Zhang H (2015) A novel bat algorithm with habitat selection and Doppler effect in echoes for optimization. Expert Syst Appl 42:6350–6364
Mirjalili S, Mirjalili SM, Yang XS (2013) Binary bat algorithm. Neural Comput Appl. doi:10.1007/s00521-013-1525-5
Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. doi:10.1016/j.advengsoft.2013.12.007
Niu J, Zhong W, Liang Y, Luo N, Qian F (2015) Fruit fly optimization algorithm based on differential evolution and its application on gasification process operation optimization. Knowl Based Syst 88:253–263. doi:10.1016/j.knosys.2015.07.027
Ramteke M, Ghune N, Trivedi V (2015) Simulated binary jumping gene: a step towards enhancing the performance of real-coded genetic algorithm. Inf Sci 325:429–454. doi:10.1016/j.ins.2015.07.033
Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248. doi:10.1016/j.ins.2009.03.004
Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl Soft Comput J 13(5):2592–2612. doi:10.1016/j.asoc.2012.11.026
Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Massachusetts Institute of Technology, Boston
Sedighizadeh M, Faramarzi H, Mahmoodi MM, Sarvi M (2014) Hybrid approach to FACTS devices allocation using multi-objective function with NSPSO and NSGA-II algorithms in fuzzy framework. Int J Electr Power Energy Syst 62:586–598. doi:10.1016/j.ijepes.2014.04.058
Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim. doi:10.1023/A:1008202821328
Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis. Technical Report TR-98-03, Department of Electrical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, Ohio
Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82. doi:10.1109/4235.585893
Wu TH, Chung SH, Chang CC (2010) A water flow-like algorithm for manufacturing cell formation problems. Eur J Oper Res 205(2):346–360. doi:10.1016/j.ejor.2010.01.020
Xie J, Zhou Y, Chen H (2013) A novel bat algorithm based on differential operator and lévy flights trajectory. Comput Intell Neurosci 1–23
Yang X (2011) Bat algorithm for multiobjective optimization. Int J Bio-Inspir Comput 3(5):267–274. doi:10.1504/IJBIC.2011.042259
Yang XS (2013) Multiobjective firefly algorithm for continuous optimization. Eng Comput 29(2):175–184. doi:10.1007/s00366-012-0254-1
Yang XS, He X (2013) Bat algorithm: literature review and applications. Int J Bio-Inspir Comput 5(3):141. doi:10.1504/IJBIC.2013.055093
Yang XS, Karamanoglu M, He X (2013) Multi-objective flower algorithm for optimization. Procedia Comput Sci 18:861–868. doi:10.1016/j.procs.2013.05.251
Yazdani M, Jolai F (2015) Lion optimization algorithm (LOA): a nature-inspired metaheuristic algorithm. J Comput Design Eng. doi:10.1016/j.jcde.2015.06.003
Yilmaz S, Kucuksille EU (2013) Improved bat algorithm (IBA) on continuous optimization problems. Lect Notes Softw Eng 1(3):279–283. doi:10.7763/LNSE.2013.V1.61
Yılmaz S, Küçüksille EU (2015) A new modification approach on bat algorithm for solving optimization problems. Appl Soft Comput 28:259–275. doi:10.1016/j.asoc.2014.11.029
Yu JQ, Li VOK (2015) A social spider algorithm for global optimization. Appl Soft Comput 30:614–627. doi:10.1016/j.asoc.2015.02.014
Zhao W, Wang L (2016) An effective bacterial foraging optimizer for global optimization. Inf Sci 329:719–735. doi:10.1016/j.ins.2015.10.001
Zhou Y, Li L, Ma M (2015a) A complex-valued encoding bat algorithm for solving 0–1 knapsack problem. Neural Process Lett 1–24. doi:10.1007/s11063-015-9465-y
Zhou Y, Luo Q, Chen H, He A, Wu J (2015b) A discrete invasive weed optimization algorithm for solving traveling salesman problem. Neurocomputing 151:1227–1236. doi:10.1016/j.neucom.2014.01.078
Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms—a comparative case study. Parallel Probl Solv Nat 1498:292–301. doi:10.1007/BFb0056872
Acknowledgments
The partial financial support from Science and Engineering Research Board, Department of Science and Technology, Government of India, New Delhi [through Grant SERB/F/1510/2014-2015, dated June 5, 2014] is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Prakash, S., Trivedi, V. & Ramteke, M. An elitist non-dominated sorting bat algorithm NSBAT-II for multi-objective optimization of phthalic anhydride reactor. Int J Syst Assur Eng Manag 7, 299–315 (2016). https://doi.org/10.1007/s13198-016-0467-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13198-016-0467-6