Development of the hesitant fuzzy weighted averaging–geometricaggregation operator (HFW-AG) under hesitant fuzzy information

Authors

  • Rafi Raza
    Faculty of Computer Science and Mathematics, University of Malaysia Terengganu, Kuala Nerus 210300, Malaysia
  • Ahmad Termimi Bin Ab Ghani
    Faculty of Computer Science and Mathematics, University of Malaysia Terengganu, Kuala Nerus 210300, Malaysia
  • Iraq T. Abbas
    College of Science, Department of Mathematics, University of Baghdad, Baghdad 10071, Iraq
  • Lazim Abdullah
    Faculty of Computer Science and Mathematics, University of Malaysia Terengganu, Kuala Nerus 210300, Malaysia

Keywords:

Hesitant fuzzy sets, Multi-criteria group decision-making, Aggregation operator, HFW-AG operator

Abstract

The hesitant fuzzy set (HFS) framework has been widely used in multi-criteria group decision-making (MCGDM) to represent uncertainty and hesitation in decision-makers' assessments. However, existing hesitant fuzzy aggregation operators often rely solely on either averaging or geometric aggregation mechanisms, which may limit their ability to capture different characteristics of hesitant fuzzy information. To address this issue, this study proposes a new hesitant fuzzy weighted averaging--geometric aggregation (HFW-AG) operator that combines the features of weighted averaging and geometric aggregation within a unified framework. Several mathematical properties of the proposed operator, including boundedness, monotonicity, and idempotency, are established to demonstrate its theoretical validity. Furthermore, an MCGDM procedure based on the HFW-AG operator is developed for decision-making under hesitant fuzzy environments. A numerical example is presented to illustrate the implementation of the proposed approach, and comparative analyses are conducted with the existing hesitant fuzzy weighted averaging (HFWA) and hesitant fuzzy weighted geometric (HFWG) operators. The results show that the HFW-AG operator exhibits a distinct aggregation behavior and produces a different ranking pattern of alternatives, reflecting its ability to capture hesitant fuzzy information from both additive and multiplicative perspectives. These findings demonstrate the applicability of the proposed operator and its potential usefulness for MCGDM problems involving uncertainty and ambiguity..

Dimensions

[1] M. Harvey, R. Fisher, R. McPhail & M. Moeller, ``Globalization and its impact on global managers' decision processes'', Human Resource Development International 12 (2009) 353. https://doi.org/10.1080/13678860903135730.

[2] T. Blumentritt, ``The big picture: decision making and globalization'', Journal of Emerging Knowledge on Emerging Markets 3 (2011) 426. https://doi.org/10.7885/1946-651X.1057.

[3] Y. Rong, W. Niu, H. Garg, Y. Liu & L. Yu, ``A hybrid group decision approach based on MARCOS and regret theory for pharmaceutical enterprises assessment under a single-valued neutrosophic scenario'', Systems 10 (2022) 106. https://doi.org/10.3390/systems10040106.

[4] M. Akram, W. A. Dudek & F. Ilyas, ``Group decision-making based on Pythagorean fuzzy TOPSIS method'', International Journal of Intelligent Systems 34 (2019) 1455. https://doi.org/10.1002/int.22103.

[5] Y. Wu, Y. Gao, B. Zhang & W. Pedrycz, ``Minimum information-loss transformations to support heterogeneous group decision making in a distributed linguistic context'', Information Fusion 89 (2023) 437. https://doi.org/10.1016/j.inffus.2022.07.009.

[6] L. D. D. R. Calache, V. C. B. Camargo, L. Osiro & L. C. R. Carpinetti, ``A genetic algorithm based on dual hesitant fuzzy preference relations for consensus group decision making'', Applied Soft Computing 121 (2022) 108778. https://doi.org/10.1016/j.asoc.2022.108778.

[7] I. M. Sharaf, ``The differential measure for Pythagorean fuzzy multiple criteria group decision-making'', Complex & Intelligent Systems 9 (2023) 3333. https://doi.org/10.1007/s40747-022-00913-4.

[8] L. A. Zadeh, ``Fuzzy sets'', Information and Control 8 (1965) 338. https://doi.org/10.1016/S0019-9958(65)90241-X.

[9] K. T. Atanassov & S. Stoeva, ``Intuitionistic fuzzy sets'', Fuzzy Sets and Systems 20 (1986) 87. https://doi.org/10.1016/S0165-0114(86)80034-3.

[10] D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, 1st ed., Academic Press, New York, NY, USA, 1980, pp. 149--334. Available online: https://shop.elsevier.com/books/fuzzy-sets-and-systems/dubois/978-0-12-222750-9.

[11] S. Miyamoto, ``Remarks on basics of fuzzy sets and fuzzy multisets'', Fuzzy Sets and Systems 156 (2005) 427. https://doi.org/10.1016/j.fss.2005.05.040.

[12] R. R. Yager, ``On the theory of bags'', International Journal of General Systems 13 (1986) 23. https://doi.org/10.1080/03081078608934952.

[13] S. Miyamoto, Multisets and fuzzy multisets, in Soft Computing and Human-Centered Machines, Springer, Tokyo, Japan, 2000, pp. 9--33. https://doi.org/10.1007/978-4-431-67907-3_2.

[14] V. Torra, ``Hesitant fuzzy sets'', International Journal of Intelligent Systems 25 (2010) 529. https://doi.org/10.1002/int.20418.

[15] V. Torra & Y. Narukawa, ``On hesitant fuzzy sets and decision'', 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378--1382. https://doi.org/10.1109/FUZZY.2009.5276884.

[16] M. Xia & Z. Xu, ``Hesitant fuzzy information aggregation in decision making'', International Journal of Approximate Reasoning 52 (2011) 395. https://doi.org/10.1016/j.ijar.2010.09.002.

[17] U. F. Simo & H. Gwét, ``Fuzzy triangular aggregation operators'', International Journal of Mathematics and Mathematical Sciences 2018 (2018) 9209524. https://doi.org/10.1155/2018/9209524.

[18] J. L. Marichal, Aggregation Operators for Multicriteria Decision Aid, Ph.D. dissertation, University of Liège, Liège, Belgium, 1998. Available online: https://hdl.handle.net/10993/7224.

[19] J. Ye, ``Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making'', Journal of Intelligent & Fuzzy Systems 26 (2014) 165. https://doi.org/10.3233/IFS-120724.

[20] S. Broumi & F. Smarandache, ``Several similarity measures of neutrosophic sets'', Neutrosophic Sets and Systems 1 (2013) 54. Available online: https://digitalrepository.unm.edu/nss_journal/vol1/iss1/10.

[21] N. Chen, Z. Xu & M. Xia, ``Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis'', Applied Mathematical Modelling 37 (2013) 2197. https://doi.org/10.1016/j.apm.2012.04.031.

[22] J. Ye, ``Multi-criteria decision-making method using the correlation coefficient under single-valued neutrosophic environment'', International Journal of General Systems 42 (2013) 386. https://doi.org/10.1080/03081079.2012.761609.

[23] G. c{S}enel, J.-G. Lee & K. Hur, ``Distance and similarity measures for octahedron sets and their application to MCGDM problems'', Mathematics 8 (2020) 1690. https://doi.org/10.3390/math8101690.

[24] J. Lee, G. c{S}enel, K. Hur & J.-G. Lee, ``Octahedron sets'', Annals of Fuzzy Mathematics and Informatics 19 (2020) 211. https://doi.org/10.30948/afmi.2020.19.3.211.

[25] Z. Xu & Q. Chen, ``A multicriteria decision making procedure based on interval-valued intuitionistic fuzzy Bonferroni means'', Journal of Systems Science and Systems Engineering 20 (2011) 217. https://doi.org/10.1007/s11518-011-5163-0.

[26] P. Liu, S.-M. Chen & J. Liu, ``Multiple attribute group decision making based on intuitionistic fuzzy interaction partitioned Bonferroni mean operators'', Information Sciences 411 (2017) 98. https://doi.org/10.1016/j.ins.2017.05.016.

[27] H. Garg & R. Arora, ``Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making'', Journal of the Operational Research Society 69 (2018) 1711. https://doi.org/10.1080/01605682.2017.1409159.

[28] G. Kaur & H. Garg, ``Multi-attribute decision-making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment'', Entropy 20 (2018) 65. https://doi.org/10.3390/e20010065.

[29] Z. Zhang, ``Interval-valued intuitionistic hesitant fuzzy aggregation operators and their application in group decision-making'', Journal of Applied Mathematics 2013 (2013) 670285. https://doi.org/10.1155/2013/670285.

[30] H. Liao & Z. Xu, Hesitant fuzzy set and its extensions, in Hesitant Fuzzy Decision Making Methodologies and Applications, Springer, Singapore, 2017, pp. 1--36. https://doi.org/10.1007/978-981-10-3265-3_1.

[31] Z. Xu, ``On consistency of the weighted geometric mean complex judgment matrix in AHP'', European Journal of Operational Research 126 (2000) 683. https://doi.org/10.1016/S0377-2217(99)00082-X.

[32] V. Torra & Y. Narukawa, Modeling Decisions: Information Fusion and Aggregation Operators, Springer, Berlin, Germany, 2007. https://doi.org/10.1007/978-3-540-68791-7.

[33] M. Sarwar, G. Ali, W. Gulzar, M. Akram & D. Pamucar, ``Two-sided matching optimization model for green housing technology selection based on hesitant 2-tuple linguistic rough numbers'', Engineering Applications of Artificial Intelligence 151 (2025) 110559. https://doi.org/10.1016/j.engappai.2025.110559.

[34] M. Akram, X. Peng & A. Sattar, ``Multi-criteria decision-making model using complex Pythagorean fuzzy Yager aggregation operators'', Arabian Journal for Science and Engineering 46 (2021) 1691. https://doi.org/10.1007/s13369-020-04864-1.

[35] A. Sarkar, S. Moslem, D. Esztergár-Kiss, M. Akram, L. Jin & T. Senapati, ``A hybrid approach based on dual hesitant (q)-rung orthopair fuzzy Frank power partitioned Heronian mean aggregation operators for estimating sustainable urban transport solutions'', Engineering Applications of Artificial Intelligence 124 (2023) 106505. https://doi.org/10.1016/j.engappai.2023.106505.

[36] M. Akram, X. Peng & A. Sattar, ``A new decision-making model using complex intuitionistic fuzzy Hamacher aggregation operators'', Soft Computing 25 (2021) 7059. https://doi.org/10.1007/s00500-021-05658-9.

[37] B. Ning, Y. Zhang, C. Wei & G. Wei, ``Some generalized aggregation operators with probabilistic spherical hesitant fuzzy information and applications to green enterprise credit selection'', Informatica 37 (2026) 435. https://doi.org/10.15388/26-INFOR623.

[38] M. Kiric{s}ci, S. Kuzu & A. Kablan, ``The investment portfolio selection with a hesitant fuzzy decision-making method based on aggregation operators'', Borsa Istanbul Review 26 (2026) 100805. https://doi.org/10.1016/j.bir.2026.100805.

[39] R. Ma & W. Zeng, ``Bonferroni mean operators on weighted hesitant fuzzy set and its application in multi-criteria symmetric group decision making'', Symmetry 18 (2026) 414. https://doi.org/10.3390/sym18030414.

fig 1

Published

2026-08-05

How to Cite

Development of the hesitant fuzzy weighted averaging–geometricaggregation operator (HFW-AG) under hesitant fuzzy information. (2026). African Scientific Reports, 5(2), 491. https://doi.org/10.46481/asr.2026.5.2.491

Issue

Volume 5, Issue 2, August 2026

Section

MATHEMATICAL SCIENCES SECTION
Copyright & Licensing

Copyright (c) 2026 Rafi Raza, Ahmad Termimi Bin Ab Ghani, Iraq T. Abbas, Lazim Abdullah

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Development of the hesitant fuzzy weighted averaging–geometricaggregation operator (HFW-AG) under hesitant fuzzy information. (2026). African Scientific Reports, 5(2), 491. https://doi.org/10.46481/asr.2026.5.2.491

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