Abstract
Fuzzy sets, soft sets and N-soft set are essential concepts for fuzzy modeling, decision making under uncertainty and computational intelligence. Nevertheless, all these models have some limitations imposed on the alternatives and attributes. These mathematical models were unable to handle the situations whenever decision makers need to assign non-binary evaluations to both attributes and alternatives. To overcome these problems, Riaz et al. (Symmetry 13(5):1–31, 2021) proposed the idea of M-parameterized N-soft set (MPNSS), through examining independent non-binary evaluations to both parameters and alternatives. MPNSSs are highly convenient in representation of ambiguous and uncertain data in decision analysis during ranking, rating and grading the objects. The proposed model has greater efficiency to handle the imprecision rather than existing mathematical models. Additionally, we develop multi-attribute decision-making (MADM) techniques on account of M-parameterized N-soft sets (MPNSSs), M-parameterized N-soft aggregation operators (MPNSAO) and M-parameterized N-soft weighted aggregation operators (MPNSWAO). For these objectives, we define fundamental operations on MPNSSs and their properties. Meanwhile, several related results are well proven and some algorithms for MADM are also developed. Moreover, the applications of MADM techniques corresponding to proposed algorithms are explained by illustrative material. To explain the presumption, authenticity and effectiveness of the suggested course of actions, we compared the techniques with few existing MADM methods.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Akram M, Adeel A, Alcantud JCR (2018) Fuzzy N-soft sets: a novel model with applications. J Intell Fuzzy Syst 35(4):4757–4771
Akram M, Adeel A, Alcantud JCR (2019) Group decision-making methods based on hesitant N-soft sets. Expert Syst Appl 115:95–105
Aktas H, Çağman N (2007) Soft sets and soft group. Inf Sci 1(77):2726–2735
Ali MI (2011) A note on soft sets, rough soft sets and fuzzy soft sets. Appl Soft Comput 11:3329–3332
Ali MI, Feng F, Liu XY, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57:1547–1553
Ashraf S, Abdullah S, Mahmood T (2019) Spherical fuzzy Dombi aggregation operators and their application in group decision making problems. J Ambient Intell Human Comput. https://doi.org/10.1007/s12652-019-01333-y
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Çağman N, Enginoglu S (2010) Soft matrix theory and its decision making. Comput Math Appl 59:3308–3314
Çağman N, Çitak F, Enginoglu S (2010) Fuzzy parameterized fuzzy soft set theory and its applications. Turk J Fuzzy Syst 1(1):21–35
Çağman N, Karataş S, Enginoglu S (2011) Soft topology. Comput Math Appl 62:351–358
Çağman N, Enginoglu S, Çitak F (2011) Fuzzy soft set theory and its applications. Iran J Fuzzy Syst 8(8):137–147
Çağman N, Çitak F, Enginoglu S (2011) FP-soft set theory and its applications. Ann Fuzzy Math Inform 2(2):219–226
Chen CT (2000) Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 1(114):1–9
Chen TY, Tsao CY (2008) The interval-valued fuzzy TOPSIS method and experimental analysis. Fuzzy Sets Syst 159(11):1410–1428
Dey PP, Pramanik S, Giri BC (2016) TOPSIS for solving multi-attribute decision making problems under bipolar neutrosophic environment. New trends in neutrosophic theory and applications. Pons Editions, Brussels, pp 65–77
Eraslan S, Karaaslan F (2015) A group decision making method based on TOPSIS under fuzzy soft environment. J New Theory 3:30–40
Fatimah F, Rosadi D, Hakim RBF, Alcantud JCR (2018) N-soft sets and their decision-making algoritms. Soft Comput 22:3829–3842
Feng F, Jun YB, Liu X, Li L (2010) An adjustable approach to fuzzy soft set based decision making. J Comput Appl Math 234(1):10–20
Feng F, Li C, Davvaz B, Ali MI (2010) Soft sets combined with fuzzy sets and rough sets, a tentative approach. Soft Comput 14(9):899–911
Garg H, Arora R (2018) Dual hesitant fuzzy soft aggregation operators and their application in decision-making. Cogn Comput 10(5):769–789
Garg H, Arora R (2019) Generalized intuitionistic fuzzy soft power aggregation operator based on t-norm and their application in multicriteria decision-making. Int J Intell Syst 34(2):215–246
Garg H, Rani D (2019) A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making. Appl Intell 49:496–512
Hashmi MR, Riaz M (2020) A novel approach to censuses process by using pythagorean \(m\)-polar fuzzy Dombi’s aggregation operators. J Intell Fuzzy Syst 38(2):1977–1995
Hashmi MR, Riaz M, Smarandache F (2020) \(m\)-polar neutrosophic topology with applications to multi-criteria decision-making in medical diagnosis and clustering analysis. Int J Fuzzy Syst 22(1):273–292
Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, New York. https://doi.org/10.1007/978-3-642-48318-9
Karaaslan F, Hunu F (2020) Type-2 single-valued neutrosophic sets and their applications in multi-criteria group decision making based on TOPSIS method. J Ambient Intell Human Comput. https://doi.org/10.1007/s12652-020-01686-9
Kumar K, Garg H (2018) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput Appl Math 37(2):1319–1329
Maji PK, Biswas R, Roy AR (2001) Fuzzy soft sets. J Fuzzy Math 9(3):589–602
Maji PK, Biswas R, Roy AR (2001) Intuitionistic fuzzy soft sets. J Fuzzy Math 9(3):677–691
Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37:19–31
Naeem K, Riaz M, Peng XD, Afzal D (2019) Pythagorean fuzzy soft MCGDM methods based on TOPSIS, VIKOR and aggregation operators. J Intell Fuzzy Syst 37(5):6937–6957
Naeem K, Riaz M, Afzal Deeba (2019) Pythagorean m-polar fuzzy sets and TOPSIS method for the selection of advertisement mode. J Intell Fuzzy Syst 37(6):8441–8458
Pawlak Z (1982) Rough sets. Int J Inf Comput Sci 11:341–356
Peng XD, Liu L (2019) Information measures for q-rung orthopair fuzzy sets. Int J Intell Syst 34(8):1795–1834
Peng XD, Selvachandran G (2019) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev 52:1873–1927
Peng XD, Yang Y (2015) Some results for Pythagorean fuzzy sets. Int J Intell Syst 30(11):1133–1160
Peng XD, Yuan HY, Yang Y (2017) Pythagorean fuzzy information measures and their applications. Int J Intell Syst 32(10):991–1029
Riaz M, Hashmi MR (2019) Linear Diophantine fuzzy set and its applications towards multi-attribute decision making problems. J Intell Fuzzy Syst 37(4):5417–5439
Riaz M, Tehrim ST (2020) On bipolar fuzzy soft topology with decision-making. Soft Comput 24(24):18259–18272
Riaz M, Çağman N, Zareef I, Aslam M (2019) N-soft topology and its applications to multi-criteria group decision making. J Intell Fuzzy Syst 36(6):6521–6536
Riaz M, Davvaz B, Fakhar A, Firdous A (2020) Hesitant fuzzy soft topology and its applications to multi-attribute group decision-making. Soft Comput 24(21):16269–16289
Riaz M, Razzaq MA, Aslam M, Pamucar D (2021) M-parameterized N-soft topology-based TOPSIS approach for multi-attribute decision making. Symmetry 13(5):1–31
Shabir M, Naz M (2011) On soft topological spaces. Comput Math Appl 61:1786–1799
Varol BP, Aygun H (2012) Fuzzy soft topology. Hacet J Math Stat 41(3):407–419
Xu ZS, Zhang X (2013) Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl-Based Syst 52:53–64
Yager RR (2013) Pythagorean fuzzy subsets. In: IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS). Joint. Edmonton, Canada, IEEE 2013, pp 57–61
Yager RR, Abbasov AM (2013) Pythagorean membership grades, complex numbers, and decision making. Int J Intell Syst 28(5):436–452
Zadeh LA (1965) Fuzzy sets. Inf. Control 8:338–353
Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning–I. Inf Sci 8:199–249
Zhan J, Alcantud JCR (2018) A novel type of soft rough covering and its application to multicriteria group decision making. Artif Intell Rev. https://doi.org/10.1007/s10462-018-9617-3
Zhan J, Ali MI, Mehmood N (2017) On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods. Appl Soft Comput 56:446–457
Zhan J, Liu Q, Herawan T (2017) A novel soft rough set: soft rough hemirings and its multicriteria group decision making. Appl Soft Comput 54:393–402
Zhang XL, Xu ZS (2014) Extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078
Zorlutuna I, Atmaca S (2016) Fuzzy parameterized fuzzy soft topology. New Trends Math Sci 4(1):142–152
Funding
The authors have no funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Informed consent
All authors equally contributed in this research work. All authors read and approved the final manuscript. The paper is submitted with the consent of all authors.
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
Razzaq, A., Riaz, M. M-parameterized N-soft set-based aggregation operators for multi-attribute decision making. Soft Comput 27, 13701–13717 (2023). https://doi.org/10.1007/s00500-023-08853-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-023-08853-y