Application of decision method based on aggregation operators of <i>n</i>PIVFNs and graph structure in supplier selection

Chunfeng Suo; Lili Zhang; Shuang Guo; Chunfeng Suo; Lili Zhang; Shuang Guo

doi:10.3934/math.2026393

AIMS Mathematics

2026, Volume 11, Issue 4: 9492-9528. doi: 10.3934/math.2026393

Previous Article Next Article

Research article

Application of decision method based on aggregation operators of nPIVFNs and graph structure in supplier selection

School of Mathematics and Statistics, Beihua University, Jilin, 132000, China

Received: 08 January 2026 Revised: 17 March 2026 Accepted: 24 March 2026 Published: 09 April 2026
MSC : 90B50, 91A35

The $ n $-polygonal interval-valued fuzzy set is an effective tool to deal with multi-attribute group decision-making problems, which can represent fuzzy information. Aggregating multiple evaluation values in the decision-making process is a key challenge. In this study, new operations are proposed for $ n $-polygonal interval-valued fuzzy numbers based on the algebraic sum and the Einstein sum. The arithmetic mean operators and geometric mean operators are constructed, and the related properties of the constructed operator are explored. The newly proposed aggregation operators have clear fuzzy logic semantics, which can accurately express the "or" logic and compensation between attributes, and reflects the decision intention by which "any attribute can improve the overall evaluation". In view of the fact that graph theory can intuitively depict various attributes, it has significant practical value in promoting more accurate evaluation of alternatives. The path analysis can abstract a complex system into a graph structure and simplify the problem. We develop a decision-making method that combines path and graph strategies. This method is relatively simple to operate and can represent the path between the scheme and all attributes. In order to illustrate the effectiveness of the method, it is applied to supplier selection. Finally, the practicability and stability of the method are verified by comparative experiments and a sensitivity analysis.
Citation: Chunfeng Suo, Lili Zhang, Shuang Guo. Application of decision method based on aggregation operators of nPIVFNs and graph structure in supplier selection[J]. AIMS Mathematics, 2026, 11(4): 9492-9528. doi: 10.3934/math.2026393

Related Papers:

Abstract

The $ n $-polygonal interval-valued fuzzy set is an effective tool to deal with multi-attribute group decision-making problems, which can represent fuzzy information. Aggregating multiple evaluation values in the decision-making process is a key challenge. In this study, new operations are proposed for $ n $-polygonal interval-valued fuzzy numbers based on the algebraic sum and the Einstein sum. The arithmetic mean operators and geometric mean operators are constructed, and the related properties of the constructed operator are explored. The newly proposed aggregation operators have clear fuzzy logic semantics, which can accurately express the "or" logic and compensation between attributes, and reflects the decision intention by which "any attribute can improve the overall evaluation". In view of the fact that graph theory can intuitively depict various attributes, it has significant practical value in promoting more accurate evaluation of alternatives. The path analysis can abstract a complex system into a graph structure and simplify the problem. We develop a decision-making method that combines path and graph strategies. This method is relatively simple to operate and can represent the path between the scheme and all attributes. In order to illustrate the effectiveness of the method, it is applied to supplier selection. Finally, the practicability and stability of the method are verified by comparative experiments and a sensitivity analysis.

References

[1]	P. Angelov, Optimization in an intuitionistic fuzzy environment, Fuzzy Set. Syst., 86 (1997), 299–306. https://doi.org/10.1016/S0165-0114(96)00009-7 doi: 10.1016/S0165-0114(96)00009-7
[2]	K. Atanassov, Intuitionistic fuzzy sets, Int. J. Bioautomation, 20 (2016), 1. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[3]	E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Set. Syst., 114 (2000), 505–518. https://doi.org/10.1016/S0165-0114(98)00244-9 doi: 10.1016/S0165-0114(98)00244-9
[4]	I. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Set. Syst., 20 (1986), 191–210. https://doi.org/10.1016/0165-0114(86)90077-1 doi: 10.1016/0165-0114(86)90077-1
[5]	W. Zeng, H. Li, Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Set. Syst., (2006), 1477–1484. https://doi.org/10.1016/j.fss.2005.11.011 doi: 10.1016/j.fss.2005.11.011
[6]	V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst., 25 (2010), 529–539. https://doi.org/10.1002/int.20418 doi: 10.1002/int.20418
[7]	R. Rodríguez, L. Martínez, V. Torra, Z. S. Xu, F. Herrera, Hesitant fuzzy sets: state of the art and future directions, Int. J. Intell. Syst., 29 (2014), 495–524. https://doi.org/10.1002/int.21654 doi: 10.1002/int.21654
[8]	P. Liu, A new fuzzy neural network and its approximation capability, Sci. China Inf. Sci., 32 (2002), 76–86. https://doi.org/10.1007/BF02714569 doi: 10.1007/BF02714569
[9]	X. Li, D. Li, The structure and realization of a polygonal fuzzy neural network, Int. J. Mach. Learn. Cybern., 7 (2016), 375–389. https://doi.org/10.1007/s13042-015-0391-0 doi: 10.1007/s13042-015-0391-0
[10]	G. Wang, Y. Duan, TOPSIS approach for multi-attribute decision making problems based on n-intuitionistic polygonal fuzzy sets description, Comput. Ind. Eng., 124 (2018), 573–581. https://doi.org/10.1016/j.cie.2018.07.038 doi: 10.1016/j.cie.2018.07.038
[11]	D. Li, G. Wang, Complete separability and approximation of the intuitionistic polygonal fuzzy number space, J. Zhejiang Univ. Sci. Ed., 49 (2022), 522–539. https://doi.org/10.3785/j.issn.1008-9497.2022.05.003 doi: 10.3785/j.issn.1008-9497.2022.05.003
[12]	C. Suo, Y. Li, L. Guo, On n-polygonal interval-valued fuzzy numbers and application in e-commerce risk assessment, J. Intell. Fuzzy Syst., (2023), 10739–10755. https://doi.org/10.3233/jifs-230040 doi: 10.3233/jifs-230040
[13]	C. Suo, Y. Li, Z. Li, On n-polygonal interval-valued fuzzy sets, Fuzzy Set. Syst., 417 (2021), 46–70. https://doi.org/10.1016/j.fss.2020.10.014 doi: 10.1016/j.fss.2020.10.014
[14]	G. Beliakov, A. Pradera, T. Calvo, Averaging Functions, in Aggregation Functions: A Guide for Practitioners, Springer-Verlag Berlin, (2007), 39–122. https://doi.org/10.1007/978-3-540-73721-6_2
[15]	Z. Xu, R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, Int. J. Gen. Syst., 35 (2006), 417–433. https://doi.org/10.1080/03081070600574353 doi: 10.1080/03081070600574353
[16]	M. Boczek, L. Jin, M. Kaluszka, Interval-valued seminormed fuzzy operators based on admissible orders, Inf. Sci., 574 (2021), 96–110. https://doi.org/10.1016/j.ins.2021.05.065 doi: 10.1016/j.ins.2021.05.065
[17]	S. Zeng, M. Munir, T. Mahmood, M. Naeem, Some T-spherical fuzzy Einstein interactive aggregation operators and their application to selection of photovoltaic cells, Math. Probl. Eng., 2020 (2020), 1904362. https://doi.org/10.1155/2020/1904362 doi: 10.1155/2020/1904362
[18]	Z. Liu, S. Wang, P. Liu, Multiple attribute group decision making based on q-rung orthopair fuzzy Heronian mean operators, Int. J. Intell. Syst., 33 (2018), 2341–2363. https://doi.org/10.1002/int.22032 doi: 10.1002/int.22032
[19]	L. Ma, K. Jabeen, W. Karamti, K. Ullah, Q. Khan, H. Garg, et al., Aczel-Alsina power bonferroni aggregation operators for picture fuzzy information and decision analysis, Complex Intell. Syst., 10 (2024), 3329–3352. https://doi.org/10.1007/s40747-023-01287-x doi: 10.1007/s40747-023-01287-x
[20]	S. Chen, C. Chang, Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Inf. Sci., 352 (2016), 133–149. https://doi.org/10.1016/j.ins.2016.02.049 doi: 10.1016/j.ins.2016.02.049
[21]	W. Wang, X. Liu, Intuitionistic fuzzy geometric aggregation operators based on Einstein operations, Int. J. Intell. Syst., 26 (2011), 1049–1075. https://doi.org/10.1002/int.20498 doi: 10.1002/int.20498
[22]	X. Zou, S. Chen, K. Fan, Multiple attribute decision making using improved intuitionistic fuzzy weighted geometric operators of intuitionistic fuzzy values, Inf. Sci., 535 (2020), 242–253. https://doi.org/10.1016/j.ins.2020.05.011 doi: 10.1016/j.ins.2020.05.011
[23]	H. Zhang, H. Wang, Q. Cai, G. Wei, Spherical fuzzy hamacher power aggregation operators based on entropy for multiple attribute group decision making, J. Intell. Fuzzy Syst., 44 (2023), 8743–8771. https://doi.org/10.3233/jifs-224468 doi: 10.3233/jifs-224468
[24]	H. Wang, L. Feng, M. Deveci, K. Ullah, H. Garg, A novel CODAS approach based on Heronian Minkowski distance operator for T-spherical fuzzy multiple attribute group decision-making, Expert Syst. Appl., 244 (2024), 122928. https://doi.org/10.1016/j.eswa.2023.122928 doi: 10.1016/j.eswa.2023.122928
[25]	S. H. Gurmani, S. Zhang, F. A. Awwad, E. A. A. Ismail, Combinative distance-based assessment method using linguistic T-spherical fuzzy aggregation operators and its application to multi-attribute group decision-making, Eng. Appl. Artif. Intell., 133 (2024), 108165. https://doi.org/10.1016/j.engappai.2024.108165 doi: 10.1016/j.engappai.2024.108165
[26]	R. Verma, E. Álvarez-Miranda, Multiple-attribute group decision-making approach using power aggregation operators with CRITIC-WASPAS method under 2-dimensional linguistic intuitionistic fuzzy framework, Appl. Soft Comput., 133 (2024), 111466. https://doi.org/10.1016/j.asoc.2024.111466 doi: 10.1016/j.asoc.2024.111466
[27]	S. Debbarma, S. Chakraborty, A. K. Saha, Health care waste recycling concerning circular economy: A Fermatean fuzzy aggregation operator-based SWARA–MABAC approach, Environ. Dev. Sustain., 27 (2025), 14601–14640. https://doi.org/10.1007/s10668-023-04436-x doi: 10.1007/s10668-023-04436-x
[28]	B. B. Meher, S. Jeevaraj, M. Alrasheedi, Dombi weighted geometric aggregation operators on the class of trapezoidal-valued intuitionistic fuzzy numbers and their applications to multi-attribute group decision-making, Artif. Intell. Rev., 58 (2025), 205. https://doi.org/10.1007/s10462-025-11200-2 doi: 10.1007/s10462-025-11200-2
[29]	D. Ren, X. Ma, H. Qin, S. Lei, X. Niu, A multi-criteria decision-making method based on discrete Z-numbers and Aczel-Alsina aggregation operators and its application on early diagnosis of depression, Eng. Appl. Artif. Intell., 139 (2025), 109484. https://doi.org/10.1016/j.engappai.2024.109484 doi: 10.1016/j.engappai.2024.109484
[30]	C. Wei, F. Yan, R. Rodriguez, Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision-making, J. Intell. Fuzzy Syst., 31 (2016), 673–685. https://doi.org/10.3233/IFS-2180 doi: 10.3233/IFS-2180
[31]	L. Yue, Y. Lv, VIKOR optimization decision model based on poset, J. Intell. Fuzzy Syst., 48 (2025), 673–689. https://doi.org/10.3233/JIFS-230680 doi: 10.3233/JIFS-230680
[32]	S. Albagli-Kim, D. Beimel, Knowledge graph-based framework for decision making process with limited interaction, Mathematics, 10 (2022), 3981. https://doi.org/10.3390/math10213981 doi: 10.3390/math10213981
[33]	D. Jin, L. Wang, Y. Zheng, G. Song, F. Jiang, X. Li, et al., Dual intent enhanced graph neural network for session-based new item recommendation, Proc. ACM Web Conf., (2023), 684–693. https://doi.org/10.1145/3543507.3583526 doi: 10.1145/3543507.3583526
[34]	Z. Meng, R. Lin, B. Wu, Multi-criteria group decision making based on graph neural networks in Pythagorean fuzzy environment, Expert Syst. Appl., 242 (2024), 122803. https://doi.org/10.1016/j.eswa.2023.122803 doi: 10.1016/j.eswa.2023.122803
[35]	P. Dutta, G. Borah, S. Borkotokey, Neo arithmetic and ranking techniques for trapezoidal generalized interval-valued fuzzy numbers: Their applications in decision making for medical investigation, Neural Process. Lett., 54 (2022), 3045–3096. https://doi.org/10.1007/s11063-022-10752-6 doi: 10.1007/s11063-022-10752-6
[36]	X. Q. Qu, J. T. Cao, T. Y. Zhao, Z. Kan, A three-dimensional fuzzy dynamic risk assessment model with dynamic cloud weights for human operational errors in oil and gas pipelines, Pet. Sci., (2026). https://doi.org/10.1016/j.petsci.2026.01.044 doi: 10.1016/j.petsci.2026.01.044
[37]	P. Wang, T. Zhao, J. Cao, P. Li, Self-organizing interval type-2 fuzzy neural network based on eigenvalue decomposition, Appl. Soft Comput., (2025), 113741. https://doi.org/10.1016/j.asoc.2025.113741 doi: 10.1016/j.asoc.2025.113741

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)