A Choquet-based multi-expert decision-making methodology with N-soft sets

Alcantud, José Carlos R.; Akram, Muhammad; Santos-García, Gustavo; Ding, Weiping

doi:10.22111/ijfs.2026.52870.9342

Document Type : Research Paper

Authors

¹ Unidad de Excelencia Gestión Económica para la Sostenibilidad GECOS and IME, University of Salamanca, 37007 Salamanca, Spain

² Institute of Mathematics, University of the Punjab, New Campus, Lahore 4590, Pakistan

³ BORDA Research Unit and IME, University of Salamanca, 37007 Salamanca, Spain

⁴ School of Artificial Intelligence and Computer Science, Nantong University, Nantong 226019, China, and Faculty of Data Science, City University of Macau, Macau 999078, China

https://doi.org/10.22111/ijfs.2026.52870.9342

Abstract

The objective of this paper is to provide advanced multi-expert decision-making techniques using N-soft sets as a referential framework. For the first time, the primary analytical tool for achieving this goal is the Choquet integral. First, the application of this aggregation operator within the context of a set {0, 1, 2, . . . , N }, representing the available ratings, is investigated. A straightforward formulation of the Choquet integral tailored to this specific set, followed by a detailed presentation of its computational implementation, is presented. Then, practical implications of these constructions in the realm of N-soft set theory are shown. They encompass the computation of new scores for the assessment of alternatives in N-soft sets (both in individual and multi-agent cases), and aggregation of data that come in the form of N-soft sets. Ultimately, we demonstrate how these innovative tools enhance multi-expert decision-making methodologies within the framework of N-soft sets. Three different approaches are discussed. Examples and comparisons with existing methodologies are provided too.

Keywords

Main Subjects

Decision Making

References

[1] M. E. M. Abdalla, A. Uzair, A. Ishtiaq, M. Tahir, M. Kamran, Algebraic structures and practical implications of
interval-valued fermatean neutrosophic super hypersoft sets in healthcare, Spectrum of Operational Research, 2(1)
(2025), 199-218. https://doi.org/10.31181/sor21202523

[2] M. Akram, A. Adeel, J. C. R. Alcantud, Group decision-making methods based on hesitant N-soft sets, Expert
Systems with Applications, 115 (2019), 95-105. https://doi.org/10.1016/j.eswa.2018.07.060

[3] M. Akram, A. Adeel, A. N. Al-Kenani, J. C. R. Alcantud, Hesitant fuzzy N-soft ELECTRE-II model: A new
framework for decision-making, Neural Computing and Applications, 33(13) (2021), 7505-7520. https://doi.
org/10.1007/s00521-020-05498-y

[4] M. Akram, G. Ali, J. C. R. Alcantud, F. Fatimah, Parameter reductions in N-soft sets and their applications in
decision-making, Expert Systems, 38(1) (2021), e12601. https://doi.org/10.1111/exsy.12601

[5] M. Akram, M. Sultan, J. C. R. Alcantud, An integrated ELECTRE method for selection of rehabilitation center
with m-polar fuzzy N-soft information, Artificial Intelligence in Medicine, 135 (2023), 102449. https://doi.org/
10.1016/j.artmed.2022.102449

[6] J. C. R. Alcantud, A. Z. Khameneh, G. Santos-Garc´ıa, M. Akram, A systematic literature review of soft set theory,
Neural Computing and Applications, 36 (2024), 8951-8975. https://doi.org/10.1007/s00521-024-09552-x

[7] J. C. R. Alcantud, G. Santos-Garc´ıa, M. Akram, OWA aggregation operators and multi-agent decisions with N-soft
sets, Expert Systems with Applications, 203 (2022), 117430. https://doi.org/10.1016/j.eswa.2022.117430

[8] J. C. R. Alcantud, G. Santos-Garc´ıa, M. Akram, A novel methodology for multi-agent decision-making based on
N-soft sets, Soft Computing, (2023). https://doi.org/10.1007/s00500-023-08522-0

[9] O. Aristondo, J. L. Garc´ıa-Lapresta, C. Lasso de la Vega, R. A. Marques Pereira, Classical inequality indices,
welfare and illfare functions, and the dual decomposition, Fuzzy Sets and Systems, 228 (2013), 114-136. https:
//doi.org/10.1016/j.fss.2013.02.001

[10] G. Beliakov, On random generation of supermodular capacities, IEEE Transactions on Fuzzy Systems, 30(1) (2022),
293-296. https://doi.org/10.1109/TFUZZ.2020.3036699

[11] G. Beliakov, T. Cao, V. Mak-Hau, Aggregation of interacting criteria in land combat vehicle selection by using
fuzzy measures, IEEE Transactions on Fuzzy Systems, 30(9) (2022), 3979–3989, https://doi.org/10.1109/
TFUZZ.2021.3135972

[12] G. Beliakov, S. James, Choquet integral-based measures of economic welfare and species diversity, International
Journal of Intelligent Systems, 37(4) (2022), 2849–2867, https://doi.org/10.1002/int.22609

[13] G. Beliakov, S. James, J. Wu, Choquet capacities and fuzzy integrals, Springer Nature, Switzerland, (2026). https:
//doi.org/10.1007/978-3-031-97070-2

[14] A. Bilbao-Terol, V. Ca˜nal Fern´andez, C. Gonz´alez-P´erez, Evaluating energy security using choquet integral: Analysis
in the southern E.U. countries, Annals of Operations Research, 355 (2025), 721-763. https://doi.org/10.
1007/s10479-023-05748-x

[15] S. Biswas, S. Bhattacharjee, B. Biswas, K. Mitra, N. Khawas, An expert opinion-based soft computing framework
for comparing nanotechnologies used in agriculture, Spectrum of Operational Research, 4(1) (2025), 1-39. https:
//doi.org/10.31181/sor4156

[16] H. Bustince, R. Mesiar, J. Fern´andez, M. Galar, D. Paternain, A. Altalhi, G. Dimuro, B. Bedregal, Z. Tak´aˇc,
d-Choquet integrals: Choquet integrals based on dissimilarities, Fuzzy Sets and Systems, 414 (2021), 1-27. https:
//doi.org/10.1016/j.fss.2020.03.019

[17] N. C¸ a˘gman, F. C¸ ıtak, S. Engino˘glu, Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal
of Fuzzy Systems, 1(1) (2010), 21-35.

[18] G. Choquet, Theory of capacities, Annales de l’institut Fourier, 5 (1954), 131-295. https://doi.org/10.5802/
aif.53

[19] A. K. Das, C. Granados, IFP-intuitionistic multi fuzzy N-soft set and its induced IFP-hesitant N-soft set in
decision-making, Journal of Ambient Intelligence and Humanized Computing, 14 (2023), 10143-10152. https:
//doi.org/10.1007/s12652-021-03677-w

[20] I. Demir, N-soft mappings with application in medical diagnosis, Mathematical Methods in the Applied Sciences,
44(8) (2021), 7343-7358. urlhttps://doi.org/10.1002/mma.7266

[21] G. P. Dimuro, J. Fern´andez, B. Bedregal, R. Mesiar, J. A. Sanz, G. Lucca, H. Bustince, The state-of-art of the
generalizations of the choquet integral: From aggregation and pre-aggregation to ordered directionally monotone
functions, Information Fusion, 57 (2020), 27-43. https://doi.org/10.1016/j.inffus.2019.10.005

[22] Y. Even, E. Lehrer, Decomposition-integral: Unifying choquet and the concave integrals, Economic Theory, 56
(2014), 33-58. https://doi.org/10.1007/s00199-013-0780-0

[23] F. Fatimah, D. Rosadi, R. B. F. Hakim, J. C. R. Alcantud, N-soft sets and their decision making algorithms, Soft
Computing, 22 (2018), 3829-3842. https://doi.org/10.1007/s00500-017-2838-6

[24] M. Grabisch, A new algorithm for identifying fuzzy measures and its application to pattern recognition, In Proceedings of 1995 IEEE International Conference on Fuzzy Systems, 1 (1995), 145-150. https://doi.org/10.1109/
FUZZY.1995.409673

[25] M. Grabisch, C. Labreuche, A decade of application of the Choquet and Sugeno integrals in multi-criteria decision
aid, Annals of Operations Research, 6 (2008), 144. https://doi.org/10.1007/s10288-007-0064-2

[26] H. Kamacı, Introduction to N-soft algebraic structures, Turkish Journal of Mathematics, 44(6) (2020), 2356-2379.
https://doi.org/10.3906/mat-1907-99

[27] H. Kamacı, S. Petchimuthu, Bipolar N-soft set theory with applications, Soft Computing, 24 (2020), 16727-16743.
https://doi.org/10.1007/s00500-020-04968-8

[28] E. Korkmaz, M. Riaz, M. Deveci, S. Kadry, A novel approach to fuzzy N-soft sets and its application for identifying
and sanctioning cyber harassment on social media platforms, Artificial Intelligence Review, 57 (2024), 1-14. https:
//doi.org/10.1007/s10462-023-10640-y

[29] R. Krishankumar, D. Pamucar, M. Deveci, M. Aggarwal, K. S. Ravichandran, Assessment of renewable energy
sources for smart cities’ demand satisfaction using multi-hesitant fuzzy linguistic based choquet integral approach,
Renewable Energy, 189 (2022), 1428-1442. https://doi.org/10.1016/j.renene.2022.03.081

[30] G. Lucca, J. A. Sanz, G. P. Dimuro, B. Bedregal, M. J. Asiain, M. Elkano, H. Bustince, CC-integrals: Choquet-like
Copula-based aggregation functions and its application in fuzzy rule-based classification systems, Knowledge-Based
Systems, 119 (2017), 32-43. https://doi.org/10.1016/j.knosys.2016.12.004

[31] E. A. Ok, L. Zhou, The choquet bargaining solutions, Games and Economic Behavior, 33(2) (2000), 249-264.
https://doi.org/10.1006/game.1999.0778

[32] Z. Ontkoviˇcov´a, V. Torra, Computation of choquet integrals: Analytical approach for continuous functions, Information Sciences, 679 (2024), 121105. https://doi.org/10.1016/j.ins.2024.121105

[33] J. Sartori, G. Lucca, T. Asmus, H. Santos, E. Borges, B. Bedregal, H. Bustince, G. P. Dimuro, d-CC integrals:
Generalizing CC-integrals by restricted dissimilarity functions with applications to fuzzy-rule based systems, In
Naldi, M. C. Bianchi, R. A. C., editors, Intelligent Systems, volume 14195, Springer Nature, Switzerland, (2023),
243-258. https://doi.org/10.1007/978-3-031-45368-7_16

[34] D. Schmeidler, Integral representation without additivity, Proceedings of the American Mathematical Society, 97(2)
(1986), 255-261. https://doi.org/10.2307/2046508

[35] Z. Tak´aˇc, M. Uriz, M. Galar, D. Paternain, H. Bustince, Discrete IV dG-Choquet integrals with respect to admissible
orders, Fuzzy Sets and Systems, 441 (2022), 169-195. https://doi.org/10.1016/j.fss.2021.09.013

[36] A. Theerens, O. U. Lenz, C. Cornelis, Choquet-based fuzzy rough sets, International Journal of Approximate
Reasoning, 146 (2022), 62-78. https://doi.org/10.1016/j.ijar.2022.04.006

[37] V. Torra, Υ-values: Power indices `a la orness for non-additive measures, IEEE Transactions on Fuzzy Systems,
32(7) (2024), 4099-4108. https://doi.org/10.1109/TFUZZ.2024.3392268

[38] V. Torra, Differentially private choquet integral: Extending mean, median, and order statistics, International Journal
of Information Security, 24 (2025), 68. https://doi.org/10.1007/s10207-025-00984-7

[39] V. Torra, Y. Narukawa, Numerical integration for the choquet integral, Information Fusion, 31 (2016), 137-145.
https://doi.org/10.1016/j.inffus.2016.02.007

[40] E. T¨urkarslan, V. Torra, Measure identification for the choquet integral: A python module, International Journal
of Computational Intelligence Systems, 15(1) (2022), 89. https://doi.org/10.1007/s44196-022-00146-w

[41] J. Wang, X. Zhang, J. Dai, J. Zhan, TI-fuzzy neighborhood measures and generalized choquet integrals for granular
structure reduction and decision making, Fuzzy Sets and Systems, 465 (2023), 108512. https://doi.org/10.
1016/j.fss.2023.03.015

[42] Z. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences, 180(5) (2010), 726-736.
https://doi.org/10.1016/j.ins.2009.11.011

[43] R. R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems, Man and Cybernetics, 18(1) (1988), 183-190. https://doi.org/10.1109/21.87068

[44] H. Zhang, T. Nan, Y. He, q-Rung orthopair fuzzy N-soft aggregation operators and corresponding applications to
multiple-attribute group decision making, Soft Computing, 26(13) (2022), 6087-6099. https://doi.org/10.1007/
s00500-022-07126-4

[45] X. Zhang, J. Wang, J. Zhan, J. Dai, Fuzzy measures and choquet integrals based on fuzzy covering rough sets, IEEE
Transactions on Fuzzy Systems, 30(7) (2022), 2360-2374. https://doi.org/10.1109/TFUZZ.2021.3081916

Iranian Journal of Fuzzy Systems

A Choquet-based multi-expert decision-making methodology with N-soft sets

References

References

Volume 23, Issue 1
January and February 2026
Pages 163-181

A Choquet-based multi-expert decision-making methodology with N-soft sets

References

References

Volume 23, Issue 1January and February 2026Pages 163-181

Volume 23, Issue 1
January and February 2026
Pages 163-181