Skip to main content
SpringerLink
Log in

African vultures optimization algorithm based Choquet fuzzy integral for global optimization and engineering design problems

  • Published:
Artificial Intelligence Review Aims and scope Submit manuscript

Abstract

Addressing complex optimization problems demands innovative solutions capable of navigating the interdependencies among variables, a reality often oversimplified by traditional metaheuristics. To address this challenge, this paper presents an enhanced African Vultures Optimization Algorithm, termed ci-AVOA, that incorporates the Choquet Integral, a powerful operator adept at considering criteria significance and interconnectedness in optimization scenarios. Unlike its predecessor, the ci-AVOA treats optimization problems in their true complexity by recognizing and accounting for the relationships between variables. The performance of ci-AVOA is evaluated on ten CEC2020 benchmark functions and four engineering design problems, pitted against other renowned optimization algorithms and the original AVOA. Across low and high dimensional benchmark functions, ci-AVOA consistently outperforms its counterparts, underpinning its superiority. This superior performance is further validated using non-parametric statistical tests, solidifying ci-AVOA as an effective and robust tool for tackling complex optimization problems. In essence, this study provides a significant contribution by augmenting a well-known metaheuristic with the Choquet Integral to devise a superior algorithm, ci-AVOA. This innovation extends the problem-solving capabilities of metaheuristics, promising more accurate and robust solutions for complex, real-world optimization problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32

Similar content being viewed by others

Abbreviations

\(\mu\) :

Fuzzy mesure

\(\textit{ci}-AVOA\) :

Choquet Integral African Vultures Optimization Algorithm

AOAVOA :

Improved Hybrid Acquila Optimizer and African Vultures Optimization Algorithm

AOS :

Atomic Orbital Search

AVG :

Average results

AVOA :

African Vultures Optimization Algorithm

Best :

Best Results

\(BestVulture_1\) :

First best vulture

\(BestVulture_2\) :

Second best vulture

\(CEC'2020\) :

CEC’2020 Competition Benchmark

Cos :

Function of cosine

F :

Starvation rate

GA :

Genetic algorithm

GWO :

Grey Wolf Optimizer

h :

A random number between [− 2,2]

IAVOA :

Improved African Vultures Optimization Algorithm

iter :

Number of iterations

L1:

Probability parameter to select the first best vulture

L2:

Probability parameter to select the second best vulture

lb :

The lower bound of search spaces

\(max_{iter}\) :

Maximum number of iterations

Mean :

Average results

N :

Number of vultures

\(P_1\) :

A random number between [0,1]

\(P_2\) :

A random number between [0,1]

\(P_3\) :

A random number between [0,1]

\(P_i\) :

Vulture position vector

PSO :

Particle Swarm Optimization

\(R_i\) :

One best vulture selected

\(rand_1\) :

A random number between [0,1]

\(rand_2\) :

A random number between [0,1]

\(rand_3\) :

A random number between [0,1]

\(rand_4\) :

A random number between [0,1]

\(rand_5\) :

A random number between [0,1]

\(rand_6\) :

A random number between [0,1]

\(rand_{p1}\) :

A random number between [0,1]

\(rand_{p2}\) :

A random number between [0,1]

\(rand_{p3}\) :

A random number between [0,1]

Sin :

Function of sine

SSA :

Salp Swarm Optimization

STD :

Standard Deviation

TAVOA :

Enhanced African Vultures Optimization Algorithm with tent map and time varying mechanism

ub :

The Upper bound of search spaces

\(V_i\) :

Fitness value of best vultures

w :

A parameter that determines the probability of entering the exploration and exploitation phases

z :

A random number between [− 1,1]

References

  • Abdollahzadeh B, Gharehchopogh FS, Mirjalili S (2021) African vultures optimization algorithm: a new nature-inspired metaheuristic algorithm for global optimization problems. Comput Ind Eng 158:107408

    Google Scholar 

  • Abualigah L, Elaziz MA, Sumari P, Khasawneh AM, Alshinwan M, Mirjalili S, Shehab M, Abuaddous HY, Gandomi AH (2022) Black hole algorithm: a comprehensive survey. Appl Intell. https://doi.org/10.1007/s10489-021-02980-5

    Article  Google Scholar 

  • Alkan N, Kahraman C (2020) Fuzzy metaheuristics: a state-of-the-art review. In: International Conference on Intelligent and Fuzzy Systems, pp. 1447–1455. Springer

  • Arriola ER, Ubando AT, Chen W-H (2022) A bibliometric review on the application of fuzzy optimization to sustainable energy technologies. Int J Energy Res 46(1):6–27

    Google Scholar 

  • Askari Q, Younas I, Saeed M (2020) Political optimizer: a novel socio-inspired meta-heuristic for global optimization. Knowl-Based Syst 195:105709

    Google Scholar 

  • Azizi M (2021) Atomic orbital search: a novel metaheuristic algorithm. Appl Math Model 93:657–683

    MathSciNet  MATH  Google Scholar 

  • Barbosa HJ, Lemonge AC (2002) An adaptive penalty scheme in genetic algorithms for constrained optimization problems. In: Proceedings of the 4th Annual Conference on Genetic and Evolutionary Computation, pp. 287–294

  • Bianchi L, Gambardella LM, Dorigo M (2002) Solving the homogeneous probabilistic traveling salesman problem by the aco metaheuristic. International workshop on ant algorithms. Springer, pp 176–187

    Google Scholar 

  • Boissiere J, Martin F, Teypaz N, Mauris G, Cung V (2007) Using Choquet-integral for guiding Tabu search in multi-criteria public transport network design. IFAC Proc Vol 40(18):617–622

    Google Scholar 

  • Branke J, Corrente S, Greco S, Słowiński R, Zielniewicz P (2016) Using Choquet integral as preference model in interactive evolutionary multiobjective optimization. Eur J Oper Res 250(3):884–901

    MathSciNet  MATH  Google Scholar 

  • Carbas S, Toktas A, Ustun D (eds) (2021) Nature-inspired metaheuristic algorithms for engineering optimization applications. Springer tracts in nature-inspired computing. Springer, Singapore

    Google Scholar 

  • Chakraborty S, Sharma S, Saha AK, Saha A (2022) A novel improved whale optimization algorithm to solve numerical optimization and real-world applications. Artif Intelli Rev. https://doi.org/10.1007/s10462-021-10114-z

    Article  Google Scholar 

  • Cheikh-Graiet SB, Dotoli M, Hammadi S (2020) A Tabu search based metaheuristic for dynamic carpooling optimization. Comput Ind Eng 140:106217

    Google Scholar 

  • Chopard B, Tomassini M, Chopard B, Tomassini M (2018) Performance and limitations of metaheuristics. An Introduction to Metaheuristics for Optimization, 191–203

  • Choquet G (1954) Theory of capacities. Annales de L’institut Fourier, pp 131–295

    MATH  Google Scholar 

  • Costilla-Enriquez N, Weng Y, Zhang B (2020) Combining Newton-Raphson and stochastic gradient descent for power flow analysis. IEEE Trans Power Syst 36(1):514–517

    Google Scholar 

  • Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2–4):311–338

    MATH  Google Scholar 

  • Dekkers A, Aarts E (1991) Global optimization and simulated annealing. Math Program 50(1):367–393

    MathSciNet  MATH  Google Scholar 

  • Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1(1):3–18

    Google Scholar 

  • Du D, Pardalos PM (1998) Handbook of combinatorial optimization, vol 4. Springer, New York

    MATH  Google Scholar 

  • Fan J, Li Y, Wang T (2021) An improved african vultures optimization algorithm based on tent chaotic mapping and time-varying mechanism. PLoS ONE 16(11):0260725

    Google Scholar 

  • Faris H, Aljarah I, Al-Betar MA, Mirjalili S (2018) Grey wolf optimizer: a review of recent variants and applications. Neural Comput Appl 30(2):413–435

    Google Scholar 

  • Ganesan T, Elamvazuthi I (2022) Bilevel optimization of taxing strategies for carbon emissions using fuzzy random matrix generators. Smart Cities and Machine Learning in Urban Health, IGI Global. 210–234

  • Gharehchopogh FS (2022) An improved Harris hawks optimization algorithm with multi-strategy for community detection in social network. J Bionic Eng. https://doi.org/10.1007/s42235-022-00303-z

    Article  Google Scholar 

  • Gharehchopogh FS (2023) Quantum-inspired metaheuristic algorithms: comprehensive survey and classification. Artif Intell Rev 56(6):5479–5543

    Google Scholar 

  • Gharehchopogh FS, Abdollahzadeh B, Arasteh B (2022) An improved farmland fertility algorithm with hyper-heuristic approach for solving travelling salesman problem. Tech Science Press

    Google Scholar 

  • Gharehchopogh FS, Ucan A, Ibrikci T, Arasteh B, Isik G (2023) Slime mould algorithm: a comprehensive survey of its variants and applications. Arch Comput Methods Eng. https://doi.org/10.1007/s11831-023-09883-3

    Article  Google Scholar 

  • Gharehchopogh FS, Namazi M, Ebrahimi L, Abdollahzadeh B (2023) Advances in sparrow search algorithm: a comprehensive survey. Arch Comput Methods Eng 30(1):427–455

    Google Scholar 

  • Grabisch M (1997) K-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167–189

    MathSciNet  MATH  Google Scholar 

  • Greco S, Pavone MF, Talbi E-G, Vigo D (eds) (2021) Metaheuristics for combinatorial optimization. Advances in intelligent systems and computing, vol 1332. Springer

    Google Scholar 

  • Gurrola-Ramos J, Hernàndez-Aguirre A, Dalmau-Cedeño O (2020) Colshade for real-world single-objective constrained optimization problems. In: 2020 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE

  • Ho PY, Shimizu K (2007) Evolutionary constrained optimization using an addition of ranking method and a percentage-based tolerance value adjustment scheme. Inf Sci 177(14):2985–3004

    Google Scholar 

  • Hussain K, Salleh MNM, Cheng S, Shi Y (2019) On the exploration and exploitation in popular swarm-based metaheuristic algorithms. Neural Comput Appl 31:7665–7683

    Google Scholar 

  • Khajehzadeh M, Taha MR, El-Shafie A, Eslami M (2011) A survey on meta-heuristic global optimization algorithms. Res J Appl Sci Eng Technol 3(6):569–578

    MATH  Google Scholar 

  • Khalouli S, Ghedjati F, Hamzaoui A (2008) Hybrid approach using ant colony optimization and fuzzy logic to solve multi-criteria hybrid flow shop scheduling problem. In: Proceedings of the 5th International Conference on Soft Computing as Transdisciplinary Science and Technology, pp. 44–50

  • Kumar A, Wu G, Ali MZ, Mallipeddi R, Suganthan PN, Das S (2020) A test-suite of non-convex constrained optimization problems from the real-world and some baseline results. Swarm Evol Comput 56:100693

    Google Scholar 

  • Kumar A, Das S, Zelinka I (2020) A modified covariance matrix adaptation evolution strategy for real-world constrained optimization problems. In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion, pp. 11–12

  • Kumar A, Das S, Zelinka I (2020) A self-adaptive spherical search algorithm for real-world constrained optimization problems. In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion, pp. 13–14

  • Liu R, Wang T, Zhou J, Hao X, Xu Y, Qiu J (2022) Improved African vulture optimization algorithm based on quasi-oppositional differential evolution operator. IEEE Access 10:95197–95218

    Google Scholar 

  • Mafarja M, Heidari AA, Faris H, Mirjalili S, Aljarah I (2020) Dragonfly algorithm: theory, literature review, and application in feature selection. Nature-inspired optimizers, 47–67

  • Manita G, Korbaa O (2020) Binary political optimizer for feature selection using gene expression data. Comput Intell Neurosci. https://doi.org/10.1155/2020/8896570

    Article  Google Scholar 

  • Mirjalili S (2019) Genetic algorithm. Evolutionary algorithms and neural networks. Springer, pp 43–55

    MATH  Google Scholar 

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

    Google Scholar 

  • 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

    Google Scholar 

  • Mishra D, Naik B (2019) Detecting intrusive behaviors using swarm-based fuzzy clustering approach. In: Nayak J, Abraham A, Krishna BM, Chandra Sekhar GT, Das AK (eds) Soft computing in data analytics. Advances in intelligent systems and computing. Springer, pp 837–846

    Google Scholar 

  • Mohammadzadeh H, Gharehchopogh FS (2021) A multi-agent system based for solving high-dimensional optimization problems: A case study on email spam detection. Int J Commun Syst 34(3):4670

    Google Scholar 

  • Morales-Castañeda B, Zaldivar D, Cuevas E, Fausto F, Rodríguez A (2020) A better balance in metaheuristic algorithms: Does it exist? Swarm Evol Comput 54:100671

    Google Scholar 

  • Nama S, Sharma S, Saha AK, Gandomi AH (2022) A quantum mutation-based backtracking search algorithm. Artif Intell Rev. https://doi.org/10.1007/s10462-021-10078-0

    Article  Google Scholar 

  • Naseri TS, Gharehchopogh FS (2022) A feature selection based on the farmland fertility algorithm for improved intrusion detection systems. J Netw Syst Manag 30(3):40

    Google Scholar 

  • Nayak J, Naik B, Behera HS, Abraham A (2017) Hybrid chemical reaction based metaheuristic with fuzzy c-means algorithm for optimal cluster analysis. Expert Syst Appl 79:282–295

    Google Scholar 

  • Neuhäuser M (2011). In: Lovric M (ed) Wilcoxon-Mann-Whitney Test. Springer, Heidelberg, pp 1656–1658

  • Nssibi M, Manita G, Korbaa O (2021) Binary Giza pyramids construction for feature selection. Procedia Comput Sci 192:676–687

    Google Scholar 

  • Opara KR, Arabas J (2019) Differential evolution: a survey of theoretical analyses. Swarm Evol Comput 44:546–558

    Google Scholar 

  • Osuna-Enciso V, Cuevas E, Castañeda BM (2022) A diversity metric for population-based metaheuristic algorithms. Inf Sci 586:192–208

    Google Scholar 

  • Ouertani MW, Manita G, Korbaa O (2022) Improved antlion algorithm for electric vehicle charging station placement. In: 2022 IEEE 9th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT), pp. 265–271. IEEE

  • Pardalos PM, Romeijn HE, Tuy H (2000) Recent developments and trends in global optimization. J Comput Appl Math 124(1–2):209–228

    MathSciNet  MATH  Google Scholar 

  • Hughes M, Goerigk M, Dokka T (2020) Particle swarm metaheuristics for robust optimisation with implementation uncertainty, Elsevier. Comput Operat Res 122:104998

  • Pozna C, Precup R-E, Horváth E, Petriu EM (2022) Hybrid particle filter-particle swarm optimization algorithm and application to fuzzy controlled servo systems. IEEE Trans Fuzzy Syst 30(10):4286–4297

    Google Scholar 

  • Ramadas M, Abraham A (2019) Metaheuristics for data clustering and image segmentation. Intelligent systems reference library. Springer

    MATH  Google Scholar 

  • Rani D, Garg H (2022) Multiple attributes group decision-making based on trigonometric operators, particle swarm optimization and complex intuitionistic fuzzy values. Artif Intell Rev 56(2):1787–1831

    Google Scholar 

  • Razmjooy N, Ashourian M, Foroozandeh Z (eds.) (2021) Metaheuristics and Optimization in Computer and Electrical Engineering. Lecture Notes in Electrical Engineering, vol. 696. Springer, Cham

  • Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4(3):284–294

    Google Scholar 

  • Sahoo SK, Saha AK, Nama S, Masdari M (2023) An improved moth flame optimization algorithm based on modified dynamic opposite learning strategy. Artif Intell Rev 56(4):2811–2869

    Google Scholar 

  • Sahoo SK, Sharma S, Saha AK (2023) A novel variant of moth flame optimizer for higher dimensional optimization problems. J Bionic Eng. https://doi.org/10.1007/s42235-023-00357-7

    Article  Google Scholar 

  • Sallam KM, Elsayed SM, Chakrabortty RK, Ryan MJ (2020) Improved multi-operator differential evolution algorithm for solving unconstrained problems. In: 2020 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE

  • Samadi Bonab M, Ghaffari A, Soleimanian Gharehchopogh F, Alemi P (2020) A wrapper-based feature selection for improving performance of intrusion detection systems. Int J Commun Syst 33(12):4434

    Google Scholar 

  • Sarker R, Mohammadian M, Yao X, Runarsson T, Yao X (2002) Constrained evolutionary optimization: The penalty function approach. Evolutionary Optimization, 87–113

  • Sharma S, Chakraborty S, Saha AK, Nama S, Sahoo SK (2022) mlboa: a modified butterfly optimization algorithm with LaGrange interpolation for global optimization. J Bionic Eng 19(4):1161–1176

    Google Scholar 

  • Sharma S, Saha AK, Roy S, Mirjalili S, Nama S (2022) A mixed sine cosine butterfly optimization algorithm for global optimization and its application. Clust Comput 25(6):4573–4600

    Google Scholar 

  • Shishavan ST, Gharehchopogh FS (2022) An improved cuckoo search optimization algorithm with genetic algorithm for community detection in complex networks. Multimed Tools Appl 81(18):25205–25231

    Google Scholar 

  • Sugeno M (1974) Theory of fuzzy integrals and its applications. Doct. Thesis, Tokyo Institute of technology

  • Tessema B, Yen GG (2006) A self adaptive penalty function based algorithm for constrained optimization. In: 2006 IEEE International Conference on Evolutionary Computation, pp. 246–253. IEEE

  • Venter G, Sobieszczanski-Sobieski J (2003) Particle swarm optimization. AIAA J 41(8):1583–1589

    Google Scholar 

  • Wang S, Zhao Q (2022) Probabilistic Tabu search algorithm for container liner shipping problem with speed optimisation. Int J Prod Res 60(12):3651–3668

    Google Scholar 

  • Wang Y, Zhang Y, Zhang C, Zhou J, Hu D, Yi F, Fan Z, Zeng T (2023) Genetic algorithm-based fuzzy optimization of energy management strategy for fuel cell vehicles considering driving cycles recognition. Energy 263:126112

    Google Scholar 

  • Xiao Y, Guo Y, Cui H, Wang Y, Li J, Zhang Y (2022) Ihaoavoa: an improved hybrid Aquila optimizer and African vultures optimization algorithm for global optimization problems. Math Biosci Eng 19(11):10963–11017

    MATH  Google Scholar 

  • Yang X-S (2012) Flower pollination algorithm for global optimization. In: International Conference on Unconventional Computing and Natural Computation, pp. 240–249 . Springer

  • Zaman HRR, Gharehchopogh FS (2022) An improved particle swarm optimization with backtracking search optimization algorithm for solving continuous optimization problems. Eng Comput 38(Suppl 4):2797–2831

    Google Scholar 

  • Zhang J, Ma Z (2020) Hybrid fuzzy clustering method based on fcm and enhanced logarithmical pso (elpso). Comput Intell Neurosci. https://doi.org/10.1155/2020/1386839

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory MARS, LR17ES05, ISITCom, University of Sousse, Sousse, Tunisia

    Maha Nssibi & Ghaith Manita

  2. ENSI, University of Manouba, Manouba, Tunisia

    Maha Nssibi

  3. ESEN, University of Manouba, Manouba, Tunisia

    Ghaith Manita

  4. IRIT, CNRS, University of Toulouse, Toulouse, France

    Francis Faux

  5. ISITCom, University of Sousse, Sousse, Tunisia

    Ouajdi Korbaa

  6. Industrial Engineering Center, IMT Mines Albi, 81000, Albi, France

    Elyes Lamine

Authors
  1. Maha Nssibi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ghaith Manita
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francis Faux
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ouajdi Korbaa
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Elyes Lamine
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conception and design of the study: MN, GM, and FF were responsible for the conception and design of the study. Data collection, analysis, and manuscript preparation: MN and GM acquired the data for the study, while MN, GM, and FF analyzed and interpreted the data. MN, GM, and FF drafted the manuscript, and FF and OK revised the manuscript critically for important intellectual content. Approval of the final manuscript: MN, GM, OK, FF, and EL all approved the final version of the manuscript to be published. In summary, MN, GM, and FF played a crucial role in designing the study, collecting and analyzing data, and drafting the manuscript. FF and OK provided critical input during the revision process, while all authors approved the final version of the manuscript.

Corresponding author

Correspondence to Maha Nssibi.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

Appendix

Appendix

See Tables 17, 18.

Table 17 A comparison of the fitness values over 30 experiments obtained by the proposed ci-AVOA and other competitor algorithms over CEC2020 test suite with \(Dim=20\)
Full size table
Table 18 Wilcoxon rank sum test between the proposed ci-AVOA and other competitors algorithms on CEC2020 test suit functions with \(Dim=20\)
Full size table

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nssibi, M., Manita, G., Faux, F. et al. African vultures optimization algorithm based Choquet fuzzy integral for global optimization and engineering design problems. Artif Intell Rev 56 (Suppl 3), 3205–3271 (2023). https://doi.org/10.1007/s10462-023-10602-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10462-023-10602-4

Keywords

Search

Navigation