Abstract
Multiple criteria decision-making (MCDM) based on interval intuitionistic fuzzy value (IVIFV) is a process of aggregating decision criteria represented by multiple interval-valued intuitionistic fuzzy numbers to select the optimal alternative. Among them, an aggregation operator is an indispensable tool, and the properties of an aggregation operator directly affect the decision results. Existing aggregation operators based on IVIFV have satisfactory results in eliminating the correlation between criteria and removing the influence of outliers on the results. However, there are some unreasonable results due to some undesired properties of IVIFVs. In this paper, IVIFV operation under the Dempster-Shafer theory (DST) framework is applied to combine the power average and Muirhead mean operators and interval intuitionistic fuzzy power Muirhead mean operators under DST framework are presented. Then a method based on the presented operators for MCDM problems is proposed. Finally, a set of numerical experiments are conducted to demonstrate the proposed method. The experimental results suggest that the proposed method not only retains the robustness of the power average operator and the capability of the Muirhead mean operator, but also eliminates a shortcoming that existing interval intuitionistic fuzzy operators cannot handle the case where the weights are in IVIFVs.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Alyami H, Ansari MTJ, Alharbi A, Alosaimi W, Alshammari M, Pandey D, Khan RA (2022) Effectiveness evaluation of different IDSs using integrated fuzzy MCDM model. Electronics 11(6):859
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Atanassov KT (1999) Interval valued intuitionistic fuzzy sets. In: Intuitionistic fuzzy sets. Studies in fuzziness and soft computing. Physica, Heidelberg, vol 35. https://doi.org/10.1007/978-3-7908-1870-3_2
Blanco-Mesa F, Merigó JM, Gil-Lafuente AM (2017) Fuzzy decision making: a bibliometric-based review. J Intell Fuzzy Syst 32(3):2033–2050
Chen TY (2015) The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making. Appl Soft Comput 26:57–73
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339
Dempster AP (2008) Upper and lower probabilities induced by a multivalued mapping. In: Classic works of the Dempster-Shafer theory of belief functions. Springer, Berlin, Heidelberg, pp 57–72
Devaraj A, Aldring J (2021) Tangent similarity measure of cubic spherical fuzzy sets and its application to MCDM. In: International conference on intelligent and fuzzy systems. Springer, Cham, pp 802–810
Du Y, Liu D (2021) A novel approach for probabilistic linguistic multiple attribute decision making based on dual Muirhead mean operators and VIKOR. Int J Fuzzy Syst 23(1):243–261
Dymova L, Sevastjanov P (2010) An interpretation of intuitionistic fuzzy sets in terms of evidence theory: decision making aspect. Knowl-Based Syst 23(8):772–782
Dymova L, Sevastjanov P (2012) The operations on intuitionistic fuzzy values in the framework of Dempster-Shafer theory. Knowl-Based Syst 35:132–143
Dymova L, Sevastjanov P (2016) The operations on interval-valued intuitionistic fuzzy values in the framework of Dempster-Shafer theory. Inf Sci 360:256–272
Einstein A, Podolsky B, Rosen N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 47(10):777
Gao H, Zhang H, Liu P (2019) Multi-attribute decision making based on intuitionistic fuzzy power Maclaurin symmetric mean operators in the framework of Dempster-Shafer theory. Symmetry 11(6):807
Greco S, Figueira J, Ehrgott M (2016) Multiple criteria decision analysis: state of the art surveys. Springer, New York
He Y, Chen H, Zhou L, Liu J, Tao Z (2013) Generalized interval-valued Atanassov's intuitionistic fuzzy power operators and their application to group decision making. Int J Fuzzy Syst 15(4):401–411
Hung WL, Wu JW (2002) Correlation of intuitionistic fuzzy sets by centroid method. Inf Sci 144(1–4):219–225
Jianfang FU, Tao MIAO, Keqin WU (2021) A modified comprehensive evaluation system of ground-water pollution based on fuzzy set theory. Meteorol Environ Res 5(1):75–86
Khan Q, Hassan N, Mahmood T (2018) Neutrosophic cubic power Muirhead mean operators with uncertain data for multi-attribute decision-making. Symmetry 10(10):444
Li C, Jiang H (2011) Extension of VIKOR method with interval-valued intuitionistic fuzzy sets. In: 2011 international conference on management and service science. IEEE, pp 1–4
Liu P (2017) Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Comput Ind Eng 108:199–212
Liu P, Li H (2017) Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making. Cogn Comput 9(4):494–512
Liu B, Guo S, Yan K, Li L, Wang X (2017) Double weight determination method for experts of complex multi-attribute large-group decision-making in interval-valued intuitionistic fuzzy environment. J Syst Eng Electron 28(1):88–96
Liu Z, Teng F, Liu P, Ge Q (2018) Interval-valued intuitionistic fuzzy power Maclaurin symmetric mean aggregation operators and their application to multiple attribute group decision-making. Int J Uncertain Quantif 8(3):211–232
Liu P, Gao H (2019) Some intuitionistic fuzzy power Bonferroni mean operators in the framework of Dempster-Shafer theory and their application to multicriteria decision making. Appl Soft Comput 85:105790
Liu P, Liu W (2019) Multiple-attribute group decision-making method of linguistic q-rung orthopair fuzzy power Muirhead mean operators based on entropy weight. Int J Intell Syst 34(8):1755–1794
Li L, Mao C, Lei B, Gao Y, Liu Y, Huang GQ (2020) Decision-making of product-service system solution selection based on integrated weight and technique for order preference by similarity to an ideal solution. IET Collab Intell Manuf 2(3):102–108
Muirhead RF (1902) Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc Edinb Math Soc 21:144–162
Pramanik R, Baidya DK, Dhang N (2021) Reliability assessment of three-dimensional bearing capacity of shallow foundation using fuzzy set theory. Front Struct Civ Eng 15:478–489
Qin Y, Qi Q, Shi P, Scott PJ, Jiang X (2020) Novel operational laws and power Muirhead mean operators of picture fuzzy values in the framework of Dempster-Shafer theory for multiple criteria decision making. Comput Ind Eng 149:106853
Ren H, Wang G (2015) An interval-valued intuitionistic fuzzy MADM method based on a new similarity measure. Information 6(4):880–894
Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton
Sun G, Xia WL (2016) Evaluation method for innovation capability and efficiency of high technology enterprises with interval-valued intuitionistic fuzzy information. J Intell Fuzzy Syst 31(3):1419–1425
Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539
Veeramachaneni S, Kandikonda H (2016) An ELECTRE approach for multicriteria interval-valued intuitionistic trapezoidal fuzzy group decision making problems. Adv Fuzzy Syst 2016:1956303. https://doi.org/10.1155/2016/1956303
Wang YM, Elhag TM (2006) On the normalization of interval and fuzzy weights. Fuzzy Sets Syst 157(18):2456–2471
Wang F, Ali Z, Mahmood T, Zeng S (2021) A multi-MOORA decision making method based on muirhead mean operators and complex spherical fuzzy uncertain linguistic setting. J Intell Fuzzy Syst (Preprint), 1–26
Wang J, Xu L, Cai J, Fu Y, Bian X (2022) Offshore wind turbine selection with a novel multi-criteria decision-making method based on Dempster-Shafer evidence theory. Sustain Energy Technol Assess 51:101951
Xian S, Cheng Y, Liu Z (2021) A novel picture fuzzy linguistic Muirhead Mean aggregation operators and their application to multiple attribute decision making. Soft Comput 25(23):14741–14756
Xu Z (2010) A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making. Group Decis Negot 19(1):57–76
Xu Z (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179–1187
Xu Z, Chen Q (2011) A multi-criteria decision making procedure based on interval-valued intuitionistic fuzzy bonferroni means. J Syst Sci Syst Eng 20(2):217–228
Xu Z, Yager RR (2009) Power-geometric operators and their use in group decision making. IEEE Trans Fuzzy Syst 18(1):94–105
Xu W, Shang X, Wang J, Li W (2019) A novel approach to multi-attribute group decision-making based on interval-valued intuitionistic fuzzy power Muirhead mean. Symmetry 11(3):441
Yager RR (2001) The power average operator. IEEE Trans Syst Man Cybern-Part A Syst Hum 31(6):724–731
Yu D, Wu Y (2012) Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making. Afr J Bus Manag 6(11):4158–4168
Zadeh LA, Klir GJ, Yuan B (1996) Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers, vol 6. World Scientific, Singapore
Zeng S, Chen SM, Kuo LW (2019) Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method. Inf Sci 488:76–92
Zeshui X (2009) Intuitionistic fuzzy hierarchical clustering algorithms. J Syst Eng Electron 20(1):90–97
Ze-Shui X (2007) Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis 2:019
Zhang L, Zhan J, Yao Y (2020) Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems. Inf Sci 517:315–339
Zhong Y, Cao L, Zhang H, Qin Y, Huang M, Luo X (2021) Hesitant fuzzy power Maclaurin symmetric mean operators in the framework of Dempster-Shafer theory for multiple criteria decision making. J Ambient Intell Hum Comput 13:1777–1797
Funding
This work was supported by the National Natural Science Foundation of China (No. 62166011), and the Innovation Key Project of Guangxi Province (No. 222068071).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
About this article
Cite this article
Zhong, Y., Zhang, H., Cao, L. et al. Power Muirhead mean operators of interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory for multiple criteria decision-making. Soft Comput 27, 763–782 (2023). https://doi.org/10.1007/s00500-022-07595-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-022-07595-7