Complex fermatean fuzzy N-soft sets: a new hybrid model with applications

Decision-making methods play an important role in the real-life of human beings and consist of choosing the best options from a set of possible choices. This paper proposes the notion of complex Fermatean fuzzy N-soft set (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSfS) which, by means of ranking parameters, is capable of handling two-dimensional information related to the degree of satisfaction and dissatisfaction implicit in the nature of human decisions. We define the fundamental set-theoretic operations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSfS and elaborate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFS}_f$$\end{document}CFFSfS associated with threshold. The algebraic and Yager operations on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSf numbers are also defined. Several algorithms are proposed to demonstrate the applicability of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSfS to multi-attribute decision making. The advanced algorithms are described and accomplished by several numerical examples. Then, a comparative study manifests the validity, feasibility, and reliability of the proposed model. This method is compared with the Fermatean fuzzy Yager weighted geometric (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {FFY}_w$$\end{document}FFYwG) and the Fermatean fuzzy Yager weighted average (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {FFY}_w$$\end{document}FFYwA) operators. Further, we developed a remarkable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSf-TOPSIS approach by applying innovative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSf weighted average operator and distance measure. The presented technique is fantastically designed for the classification of the most favorable alternative by examining the closeness of all available choices from particular ideal solutions. Afterward, we demonstrate the amenability of the initiated approach by analyzing its tremendous potential to select the best city in the USA for farming. An integrated comparative analysis with existing Fermatean fuzzy TOPSIS technique is rendered to certify the terrific capability of the established approach. Further, we decisively investigate the rationality and reliability of the presented \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSfS and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CFFNS}_f$$\end{document}CFFNSf-TOPSIS approach by highlighting its advantages over the existent models and TOPSIS approaches. Finally, we holistically describe the conclusion of the whole work.


Introduction
Multi-attribute decision-making (MADM) methods play an important role in the real life of human beings. The process of choosing the best option among a set of possible options is present in all human activities. Decision making in the domain of crisp sets to handle exact and precise data has been a growing field of research for mathematicians.

Related work
dissatisfaction with human decisions. To salvage these shortcomings, Atanassov (1986) extended the FS with intuitionistic fuzzy sets (IFS) and added the non-membership function which is limited to the interval [0,1] in order to express the level of discontent with human decisions. In his model the sum of satisfaction and dissatisfaction degrees is in the unit interval.
In 2013, Yager (2013aYager ( , 2013b adapted the conditions of IFS, to present the novel concept of Pythagorean fuzzy set (PFS) with relaxing conditions that the sum of square of belongingness degree and non-membership degree should enclose in unit interval. Due to the constraints in PFS, Yager (2016) introduced the model of q-rung orthopair fuzzy set (q-ROFS) with conditions that sum of q th power of belongingness degree and non-membership degree should not exceed from 1. Later on, Senapati and Yager (2020) developed the theory of Fermatean fuzzy set (FFS) that is more general model than IFS and PFS in which the cubic sum of membership degree and non-membership degree should lie in unit interval. FFS as an extension of IFS and PFS can support more amount of inexactness and vagueness that provide more precise results in decision making framework.
Aforementioned models were not applicable in 2-dimensional problems. Thus, Ramot et al. (2002) introduced the complex fuzzy set (CFS) which was proposed by the emerging relationship of complex and FS theory having complex unit circle as the range of membership function that enables the CFS to handle the 2-dimensional information along with amplitude and phase terms. The amplitude part and phase part both are real-valued functions which can take values from the unit interval to show the vagueness of both dimensions. Later, Alkouri and Salleh (2012) put forward the idea of complex intuitionistic fuzzy set (CIFS), in order to describe the non-membership degree along with membership degree within the complex unit circle, where the sum of phase terms and amplitude terms of belongingness degree and falseness degree should not exceed from 1. Further, Akram and Naz (2019) & Ullah et al. (2020) presented the new model of complex Pythagorean fuzzy set (CPFS), as an extension of CIFS, which has more generalized structure than CFS and CIFS as it possesses more relaxed conditions on the phase and amplitude terms.
The idea of soft sets (S f Ss) theory was proposed by Molodtsov (1999), who also presented its relevancy and remarkable significance in the fields of operational research, probability theory, game theory and smoothness of functions (Molodtsov 1999(Molodtsov , 2004. Alcantud and Santos-García (2017) proposed a totally revised approach for S f S based decision-making issues under imperfect information. Many researchers brought up many models to enhance the literature of S f S, inclusive of fuzzy S f Ss (FS f Ss) (Maji et al. 2001b), Intuitionistic FS f Ss (IFS f Ss) (Maji et al. 2001a), Pythagorean FS f Ss (PFS f Ss) (Peng et al. 2015), Fermatean FS f Ss (FFS f Ss) (Sivadas and John 2020), et cetera. The idea of a new perspective for the selection of best alternatives problems based on FS f Ss was given by Alcantud (2016). Fatimah et al. (2019) worked on a new structure of S f Ss, namely, probabilistic S f S.  proposed a new hybrid model named as valuation fuzzy S f S and used it for real case study that uses data from the Spanish real estate market.
From latest studies of hybrid S f S models, it can be concluded that primarily work of the researchers was based on real numbers between [0,1] or binary evaluation in S f S models (Ma et al. 2017). But nowadays, objects are evaluated by non-binary structures such as voting system and rating or ranking objects. Due to that, numerous researchers for instance Alcantud and Laruelle (2014), Chen et al. (2013), and Herawan and Deris (2009) have worked in formal models for non-binary evaluations. Stimulated by these concerns, Fatimah et al. (2018) proposed the model of N-soft set (NS f S) which is an extension of S f S and encapsulate the idea of parameterized characterization of the alternatives that depend on the finite number of ordered grades. Fatimah and Alcantud (2021) introduced the idea of multi-fuzzy NS f S. Later on, Akram et al. (2018Akram et al. ( , 2021bAkram et al. ( , 2021d combined the concept of NS f S with FS and explored the new hybrid model, namely, fuzzy Nsoft set (FNS f S). This novel concept involves the finite number of ordered grades along with the vagueness in the conception of the attributes that are used for decision making. Another hybrid model called the hesitant N-soft set was introduced by Akram et al. (2019a). Akram et al. (2019b) extended the idea of FNS f S and presented the hybrid model of intuitionistic fuzzy N-soft set (IFNS f S) that can also capture the non-membership grades. Moreover, Zhang et al. (1965) extended IFNS f S to Pythagorean fuzzy N-soft set (PFNS f S) that possesses more relaxed conditions than existing models. Recently, Akram and his contributors set forth the hybrid models of bipolar FNS f Ss (Akram et al. 2021a), complex spherical FNS f Ss (Akram et al. 2021c) and complex Pythagorean FNS f Ss (CPFNS f Ss) (Akram et al. 2021e).
The characteristic comparison of proposed and existing models is organized in Table 1 that present a broad view concerning the superiority of the manifested model.
In recent years, a technique for order preference by similarity to the ideal solution (TOPSIS) was proposed by Hwang and Yoon (1981) to solve the MADM problems. The basic idea of TOPSIS technique is to find out the best opt which is closest to the positive ideal solution (PIS) and farthest away from the negative ideal solution (NIS). Chen (2000) utilized the TOPSIS technique for multi-attribute group decision-making (MAGDM) under a fuzzy environment. Li et al. (2019) applied the fuzzy TOPSIS approach for the case-study of the Beijing rail transportation system. Boran et al. (2009Boran et al. ( , 2011Boran et al. ( , 2012 built up the theory of intuitionistic fuzzy TOPSIS (IF-TOPSIS) and presented various real applications related to technology and business. Akram and his collaborators proposed the methodologies of interval-valued hesitant fuzzy TOPSI-S (Akram and Adeel 2019), Pythagorean Fuzzy TOPSIS (PF-TOPSIS) (Akram et al. 2019c), and complex Pythagorean fuzzy TOPSIS (CPF-TOPSIS) (Akram et al. 2020) to address the tricky MAGDM problems. Senapati and Yager (2020) put forward the Fermatean fuzzy TOPSIS (FF-TOPSIS) to capture the MADM problems. Eraslan (2015) redesigned the TOPSIS approach under the environment of S f S (S f -TOPSIS) and illustrated the methodology by means of its potential application. Eraslan and Karaaslan (2015) adapted the approach of TOPSIS under the framework of FS f Ss (FS f -TOPSIS) and demonstrated its cogent applications to select the suitable house. Han et al. (2019) extended the technique of TOPSIS under entropy on PFS f Ss environment and implemented it for the selection of missile position. Salsabeela and John presented the TOPSIS method based on FFS f Ss (FFS f -TOPSIS) (-Salsabeela and John 2021) and elaborated it with the practical application for the selection of supplier for fivestar hotel.
The comparison of proposed and existing techniques based on TOPSIS method, according to their characteristics, is arranged in Table 2 which provide an extensive view about the dominance of the presented methodology.

Motivation
The motivation of the proposed hybrid model is given by the following facts: • The idea of NS f S captures the graded parameterized information but it has no potential to handle the fuzziness and vagueness of the provided data. • The brilliant models of CIFS and CPFS are competitive frameworks for capturing the 2-dimensional vague data simultaneously. But they also have some restrictions due to the inadequacy of ranking based criteria. • Moreover, the FFS f S theory outstandingly renders the binary parameterized mechanism that handles ambiguity and vagueness of information with fantastic universality. But still, it is a 1-dimensional model that cannot present the uncertain periodic information as well as unable to cope with the ordered graded parameters of tricky practical problems. • The decision-making technique based models FNS f S, IFNS f S and PFNS f S can only deal with 1-dimensional data. None of the described models can handle 2-dimensional problems. • Further, the novel idea of CPFNS f S is an efficacious model with splendid characteristics to handle the obscurity of parameterized fuzzy information. Despite that, it has some flaws that spring up due to its restricted space. • Classical TOPSIS technique is specifically devised to determine the optimal solution based on the assessed closeness of the preferences choices from the ideal solution. But this hypothetical technique must be altered to tackle the ordered graded obscurity and vagueness of inexact information.  Akram et al. 2019b) Because of all these constraints motivated us to put forward the idea of a ground-breaking hybrid model called CFFNS f Ss along with CFFNS f -TOPSIS approach which competently handles two-dimensional information with relaxed conditions that cubic sum of amplitude and phase terms belongs to the interval [0,1]. Moreover, CFFNS f S efficiently deals with the finite order grades of the alternatives according to the attributes. Therefore, the proposed model is the extension of FNS f S (Akram et al. 2018), IFNS f S (Akram et al. 2019b), PFNS f S (Zhang et al. 1965), and CPFNS f S (Akram et al. 2021e) models and in fact dominates overall traditional models of literature as it has comparatively wide range.

Outline of the article
The essence of the first part of this article is to propose the hybrid model of CFFNS f Ss and the related concepts including score function and accuracy function. Further, we investigate the remarkable properties and basic operations of CFFNS f Ss. We have also constructed the CFFNS f S derived by the threshold. Furthermore, algebraic and Yager operations for CFFNS f numbers (CFFNS f Ns) are also defined. The proposed model is supported by the construction of three algorithms of decision-making and the applications are presented in contemplation of comparing the results of our algorithms. The comparative results of the model with existing FFY w A ) and FFY w G  operators are given in the paper. On the other hand, we revamp the TOPSIS approach for the environment of CFFNS f to account for MAGDM problems. The innovative CFFNS f weighted average operator and the distance measure of alternatives from positive and negative ideal solutions are employed to examine the contiguity of optimal variables from ideal solutions. The accountability of the presented technique is illustrated by implementing its magnificent procedure to select the suitable city in the USA for farming. A comparative analysis with the existing FF-TOPSIS (Senapati and Yager 2020) approach has been demonstrated to endorse the phenomenal feasibility and viability of the set forth strategy. The merits of the developed model and TOPSIS approach are also narrated for the appropriate manifestation of its marvelous and incredible feasibility over the existing models and approaches.
We summarize the main contributions of our research work as follows:

Layout of the paper
From this point on, the paper is organized as follows. Section 2 contains some definitions of existing models. In Sect. 3, we introduce the novel concept of CFFNS f S followed by operations on CFFNS f Ss. Section 4 scrutinizes the algebraic and Yager operations on CFFNS f Ns. Section 5 describes the three proposed algorithms of the decision-making process and also provides some applications of multi-variable decision-making procedures. Section 6 carries out a comparative analysis with existing models and offers experimental results that illustrate the effectiveness of the proposed algorithms. Then, Sect. 7 introduces the CFFNS f -TOPSIS method for MAGDM problems. A real example and a comparative study of its usefulness is shown in Sects. 8 and 9 . Finally, merits of the proposed model and conclusions are drawn in Sects. 10 and 11 .

Preliminaries
Definition 2.1 (Molodtsov (1999)) Let U be a universe of discourse under consideration and A be the set of all attributes, B A: A pair ðq; BÞ is called soft set over U if q : B À! PðUÞ where q is a set-valued function.
Definition 2.2 (Fatimah et al. (2018)) Let U be a universe of discourse and A be the set of all attributes, B A: Consider R ¼ f0; 1; . . .; N À 1g be a set of ordered grades where N 2 f2; 3; . . .g: A triple ðF ; B; NÞ is an NS f S on U if F : B À! 2 UÂR ; with the property that for each b t 2 B there exists a unique ðu g ; r a Þ 2 U Â R such that ðu g ; r a Þ 2 Fðb t Þ; u g 2 U; r a gt 2 R: Definition 2.3 (Senapati and Yager (2020)) Consider U be a universe of discourse. An FFS E on U is defined as an object of the form where the functions . E : U À! ½0; 1 and -E : U À! ½0; 1 denote the degree of membership (namely . E ðu g Þ) and the degree of non-membership (namely -E ðu g Þ) of the element u g 2 U, respectively, and for all u g 2 q is called degree of uncertainty of the elements u g 2 U to the FFS E.
Definition 2.4 A complex Fermatean fuzzy set (CFFS, in short) B, defined on the universal set U, is characterized by the membership and non-membership functions l B ðu g Þ and m B ðu g Þ, respectively, which assign to each element u g 2 U a complex-valued grade of membership and nonmembership functions in B: The CFFS may be represented as the set of triples: where l B ðu g Þ : U À! fu g j u g 2 C; j u g j 1g, m B ðu g Þ : and s B ðu g Þ; k B ðu g Þ; x B ðu g Þ; w B ðu g Þ are real-valued functions such that s B ðu g Þ; k B ðu g Þ 2 ½0; 1; x B ðu g Þ; w B ðu g Þ 2 ½0; 2p: s B ðu g Þ; k B ðu g Þ are called the amplitude terms and x B ðu g Þ; w B ðu g Þ are called the phase terms with 0 ðs B ðu g ÞÞ 3 þ ðk B ðu g ÞÞ 3 1; and 0 ð For convenience, Hðb t Þ ¼ hðu g ; r a gt Þ; sðu g ; r a gt Þe ixðu g ;r agt Þ ; kðu g ; r a gt Þe iwðu g ;r a gt Þ i is denoted by a a gt ¼ hr a gt ; ðs a gt e ix agt ; k a gt e iw agt Þi which represents CFFNS f number (CFFNS f N).
Definition 3.2 Let a a gt ¼ hr a gt ; ðs a gt e ix agt ; k a gt e iw agt Þi be a CFFNS f N then is called the degree of hesitancy/indeterminacy of CFFNS f N.
Definition 3.3 Let a a gt ¼ hr a gt ; ðs a gt e ix a gt ; k a gt ; e iw a gt Þi be any CFFNS f N over U. The score function and accuracy function of a a gt are defined as follows: Aða a gt Þ ¼ð r a gt N À 1 respectively, where Sða a gt Þ 2 ½À2; 3 and Aða a gt Þ 2 ½0; 3: For a better understanding of the concept of our new model, we present the following example: Example 3.1 Consider that an auto broker decides to purchase the car from auto company. The best car is chosen by spade ratings endowed by an expert. These rankings are on the basis of launched cars in the last 5 years and their performances. Before purchasing the car, auto broker obtained some rating and ranking based information from an expert about four different models of vehicles having different manufacturing dates. Let X ¼ fx 1 ; x 2 ; x 3 ; x 4 g be the set of vehicles and B ¼ fb 1 ¼ Reliability; b 2 ¼ Maximumpayload; b 3 ¼ Purchasingcostg A be the set of attributes, that are used to set grades for each vehicle with respect to each attribute. The expert assigned the rating of the cars according to the above-mentioned conflicting criteria and the initial review recapped in Table 3, where: • four spades represent 'excellent', • three spades represent 'very good', • two spades represent 'good', • one spade represents 'regular', and • a bullet represents 'bad'.
Based on the overall qualities of the cars, the auto broker gives evaluation scores of the cars which is shown as Table 3 Information extracted from the expert x 4 €€ €€€€ Hðb t Þ ¼ fhðu g ; r a gt Þ; sðu g ; r a gt Þe ixðu g ;r a gt Þ ; kðu g ; r a gt Þe iwðu g ;r a gt Þ ijb t 2 B; ðu g ; r a gt Þ 2 U Â Rg: Table 3 and the tabular representation of its associated 5-soft set is given in Table 4. The grade data in the actual information can be easily extracted. However, according to the Definition 3.1 when the data possess fuzzy uncertainty characteristics, we need CFFNS f N. It provides us information in which the auto brokers evaluate the cars and specify their rankings based on the same multiple fuzzy characteristics from the perspective of the two-dimensional membership degree and non-membership degree. This assessment of cars by auto brokers complies with the guidelines as follows: À2:0 SðXÞ\ À 1:2 when grade 0; À1:2 SðXÞ\ À 0:4 when grade 1; À0:4 SðXÞ\0:4 when grade 2; 0:4 SðXÞ\1:2 when grade 3; 1:2 SðXÞ\2:0 when grade 4: According to above criteria, we can obtain Table 5.
Therefore, by Definition 3.1, the CFF5S f S ðH; Q; 5Þ can be defined as follows: The CFF5S f S ðH; Q; 5Þ can be represented more clearly in tabular form shown as in Table 6 as follows: Remark 1 The following observations are in order: 1. In Example 3.1, we consider the five assessment grades, but the assessment grades in practical problems do not necessarily utilize the 5 grades, it can be arbitrary. Generally, the range concerning the score function of CFF numbers can vary with actual grade requirements. 2. Any CFF2S f S ðH; Q; 2Þ can be naturally associated with a CFFS f S. We identify a CFF2S f S H : B À! CFF ðUÂf0;1gÞ with a CFFS f S ð}; BÞ; which is given by: for every b t 2 B; where CFF ðUÂf0;1gÞ is the collection of all CFF subsets of U Â f0; 1g: 3. An arbitrary CFFNS f S over a universe U can be identified as a CFF ðN þ 1Þ-soft set. For example, from Table 6, a CFF5S f S ðH; Q; 5Þ can be identified as a CFF6S f S over U. In a CFF6S f S, we consider that there is a 5 grade, which is never used in Example 3.1. 4. In Definition 3.1, grade 0 describes the lowest score. It does not mean that there is incomplete information or no assessment.
Definition 3.5 A CFFNS f S ðH; Q; NÞ over universe of discourse U, where Q ¼ ðF; B; NÞ is an NS f S, is said to be efficient if Hðb t Þ ¼ hðu g ; N À 1Þ; 1e i2p ; 0e i0p i for some b t 2 B; u g 2 U: Example 3.2 By inspection, it can be checked that the CFF5S f S defined in Example 3.1 is not efficient. However, CFF5S f S ðH; Q; 5Þ in Table 7 is efficient.
Definition 3.6 Let ðH 1 ; Q 1 ; N 1 Þ and ðH 2 ; Q 2 ; N 2 Þ be two CFFNS f Ss over universe of discourse U, where Q 1 ¼ ðF 1 ; B 1 ; N 1 Þ; Q 2 ¼ ðF 2 ; B 2 ; N 2 Þ are NS f Ss, then x 1 2 3 1 x 2 1 3 2 x 3 3 4 0 x 4 0 2 4 We now define the concept of complementarity CFFNS f S: Definition 3.7 Let ðH; Q; NÞ be a CFFNS f S over universe of discourse U, where Q ¼ ðF; B; NÞ is an NS f S, then ðH; Q c ; NÞ is said to be weak complement if Q ¼ ðF c ; B; NÞ is a weak complement of Q ¼ ðF ; B; NÞ: By this mean that F c ðb t Þ \ F ðb t Þ ¼ ; for all b t 2 B: The term weak complement is used because this complement is not unique.
Definition 3.8 Let ðH; Q; NÞ be a CFFNS f S over universe of discourse U, where Q ¼ ðF; B; NÞ is an NS f S, then a CFF complement is denoted by ðH c ; Q; NÞ; such that H c is defined as H c : B À! CFF ðUÂRÞ ; which is given by: H c ðb t Þ ¼ fhðu g ; r a gt Þ; kðu g ; r a gt Þe ixðu g ;r agt Þ ; sðu g ; r a gt Þe iwðu g ;r agt Þ ijb t 2 B; ðu g ; r a gt Þ 2 U Â Rg: In CFF complement, the grades are same as in the original NS f S, however all their membership and nonmembership degrees are complementary.
Definition 3.9 Let ðH; Q; NÞ be a CFFNS f S over universe of discourse U, where Q ¼ ðF; B; NÞ is an NS f S, then ðH c ; Q c ; NÞ is said to be weak CFF complement when ðH; Q c ; NÞ is a weak complement and ðH c ; Q; NÞ is a CFF complement.
In other words, a weak CFF complement of CFFNS f S is the CFF complement of any of its weak complement.
NÞ is the bottom weak complement of Q ¼ ðF ; B; NÞ and defined as follows: if r a gt [ 0; hN À 1; ðs a gt e ix agt ; k a gt e iw agt Þi; if r a gt ¼ 0: ( ðH c ; Q \ ; NÞ ¼ h0; ðk a gt e iw a gt ; s a gt e ix a gt Þi; if r a gt [ 0; hN À 1; ðk a gt e iw agt ; s a gt e ix agt Þi; if r a gt ¼ 0: ( Example 3.5 The bottom weak complement and the bottom weak CFF complement of the CFF5S f S Table 6 in Example 3.1 are given by Tables 13 and 14 .
Definition 3.12 Let U be a universe of discourse and ðH 1 ; Q 1 ; N 1 Þ and ðH 2 ; Q 2 ; N 2 Þ be two CFFNS f Ss over non- a gt ÞÞ ; if a gt Þ i 2 B 1 ðb t Þ and hðu g ; r 2 a gt Þ; s D ðu g ; r 2 a gt Þe ix D ðu g ;r 2 agt Þ ; k D ðu g ; r 2 a gt Þe iw D ðu;r 2 agt Þ i 2 B 2 ðb t Þ; C and D are CFFSs on F 1 ðb t Þ and F 2 ðb t Þ; respectively.
Definition 3.16 Suppose that U be a universe of discourse and ðH; Q; NÞ be a CFFNS f S over non-empty set U, where Q ¼ ðF ; B; NÞ is an NS f S on U. Let 0\L\N be a threshold. A CFFS f S related with ðH; Q; NÞ and L, denoted by ðH L ; BÞ; is given as follows: Example 3.10 Consider the CFF5S f S in Example 3.1, represented by Table 6. From Definition 3.16, we can find the associated CFFS f Ss with CFF5S f S. Let 0\L\5 be threshold. Then the possible CFFS f S associated with thresholds 1, 2, 3 and 4 are shown by Tables 21-24.

Algorithms and applications
In this section, we clarify the decision-making (DM) process for the constructed model. Firstly, we construct the procedures as shown in Algorithms 1-3 for problems that are described by CFFNS f Ss. Then, we apply them to real circumstances to get the particular results.  (F , B, N), corresponding to each attribute for alternative ug.
6 Output Any of the alternative for which maximum outcome and maximum grade will be decision.

Selection of buy new car
Selection of a car is a difficult task for an auto broker. Productive selection is possible only when there is an essential matching. By choosing the best car, the auto broker will get quality performance. In Example 3.1, different CFFNS f Ns for the cars have been defined on the basis of their qualities, by the auto broker. Tabulated form of CFF5S f S is represented by Table 25.
Choice value (CV) of CFF5 S f S We can calculate the CV of CFF5S f S of the car's selection by using Algorithm 1 and calculated results are given in Table 26, where L-Choice value (L-CV) of CFF5 S f S Now, we will choose the threshold L and will calculate the CV by using Algorithm 2, where The result is shown by Table 27.
Complex fermatean fuzzy N-soft sets... 8779 In Table 27, we took L ¼ 2 for DM and get the 2-CV of CFF5S f S. It can observe from Table 27 that the car x 3 has highest output value. So, x 3 will be selected by the auto broker.
Comparison table of CFF5 S f S Comparison table is a square table in which rows and columns are represented by the name of objects of universe such as u 1 ; u 2 ; u 3 ; . . .; u n and q gt ¼ the CFFNS f values of the attributes for which the value of score function of u d ! u j : Membership and non-membership values of Table 25 are given in tabular form in Table 28.
The comparison table of Table 28 is given by Table 29. The result will be derived by subtracting the row and column sum of Table 29.
From Table 30, it is concluded that the highest rank and grade sum is 3 and 7, respectively, which is obtained by x 3 : So, x 3 car is selected by the auto broker.

Selection of the best cellular telecommunication company in Pakistan
Since due to COVID-19, everything has shifted to the online mode. So, the usage of the internet has increased. Due to that, in market, the competition among different network provider companies has been tough day by day. All companies are presenting different internet packages according to the needs   x 1 3 1 1 2 x 2 2 3 1 2 x 3 2 2 3 2 x 4 1 1 1 3 of the customers. So, it's a difficult task for customers to choose the sim card of the best telecommunication company. Suppose that a student, in Pakistan, decides to purchase a new sim card to attend online classes. Before buying the sim, the student has collected some relevant rating based information related to the internet packages and internet speed about five different network companies such as Zong, Ufone, Telenor, Jazz, and Warid. Each network company has different prices of internet packages depending on time as well. Let Y ¼ fy 1 ¼ zong; y 2 ¼ SCOM; y 3 ¼ telenor; y 4 ¼ jazz; y 5 ¼ ufoneg be a universal set and O ¼ fo a set of attributes, which are used to assign grades to network companies. The ratings are on the basis of internet packages and speed provided in last year and users reviews. It may be noted that the ranking of alternatives with respect to parameters may get affected and altered if the time and location is different for a particular network company. The initial survey is organized in Table 31, where: • five Ç represents 'outstanding', • four Ç represents 'very good', • three Ç represents 'good', • two Ç represents 'average', • one Ç represents 'subpar', • AE represents 'poor'. The set of grades R ¼ f0; 1; 2; 3; 4; 5g can be easily associated with Ç and AE as follows: • 0 stands for 'AE', • 1 stands for 'Ç', • 2 stands for 'ÇÇ', • 3 stands for 'ÇÇÇ', • 4 stands for 'ÇÇÇÇ', • 5 stands for 'ÇÇÇÇÇ'.
Based on the overall qualities of the network companies, the student gives evaluation scores to the sim cards which is shown as Table 31 and the tabular representation of its associated 6-soft set is given in Table 32.
Although it is easy to extract the grade data in actual information, the data possess the fuzzy uncertainty characteristics. In order to address the ambiguity of data, we construct CFFNS f S by using a certain grade. This evaluation of sim cards by students complies with the guidelines as follows: The CFF6S f S ðH; Q; 6Þ can be represented more clearly in tabular form shown as in Table 33.
Choice value (CV) of CFF6 S f S We can calculate the CV of CFF6S f S of the sim card's selection by using Algorithm 1 as given by Table 34, where   x 3 7 9 6 3 x 4 6 6 9 À3 Complex fermatean fuzzy N-soft sets... 8781 From Table 34, it is concluded that according to SðW i Þ; y 1 [ y 3 [ y 5 [ y 4 [ y 2 and hence y 1 has maximum value. So, the student will choose the sim card of zong.
L-Choice value (L-CV) of CFF6 S f S Now, by using the second procedure as given by Algorithm 2, L value will be chosen and the results are given by Table 35, where In Table 35, we have chosen L ¼ 3 for DM and get the 3-CV of CFF6S f S. It can observe from Table 35 that the telecommunication company y 1 ¼ zong has highest output value. So, y 1 will be selected by the student.
Comparison table of CFF6 S f S Now to apply the third procedure as shown in Algorithm 3, membership and non-membership values of Table 33 are given by Table 36.  The comparison table of Table 36 is given by Table 37. The result will be derived by subtracting the row and column sum of Table 37.
From Table 38, it is concluded that the highest rank and grade sum is 17 and 16, respectively, which is obtained by y 1 ¼ zong: So, sim card y 1 is selected by the student.

Superb
Poor Acceptable Ordinary Ranking y 1 y 2 y 3 y 4

Telecommunication Companies in Pakistan
Outstanding y 5 CV of CFF6Sf S (proposed)

3-CV of CFF6Sf S (proposed)
FFYwA operator  FFYwG operator Step 3. The attributes nominated by experts may not be equally important and valuable in a MAGDM problem. Therefore, each decision-maker ranks these attributes and assigns a CFFNS f weightage according to the grading criteria defined by the experts. Let G ¼ ð b G gt Þ nÂm by using CFFNS f DM G and the weight vector j t of attributes, as follows: The AWCFFNS f DM is constructed as follows: Step 5. Let B À and B þ represent the collection of costtype and benefit-type attributes, respectively. Then CFFNS f positive ideal solution (CFFNS f -PIS) e G t ¼ hr a t ; ðl a t ;m a t Þi related to attribute B t can be chosen as follows: The CFFNS f negative ideal solution (CFFNS f -NIS) G t ¼ h r a t ; ð l a t ; m a t Þi with respect to the attribute B t can be determined as follows: ¼ hr a t ; ðl a t ; m a t Þi; ¼ hr a t ; ðs a t e ix at ; k a t e iw a t Þi; ; ¼hr a gt ; ðl a gt ;m a gt Þi; ¼hr a gt ; ðŝ a gt e ix agt ;k a gt e iŵ agt Þi: Complex fermatean fuzzy N-soft sets...
The CFFNS f Ns are compared on the basis of accuracy function and score function to obtain CFFNS f -PIS and CFFNS f -NIS. • Step 6. Now, to find the optimal alternative which is away from CFFNS f -NIS and closest to CFFNS f -PIS, we evaluate the distance of each alternative I g from CFFNS f -PIS and CFFNS f -NIS. The distance between any of the alternative and CFFNS f -PIS can be calculated as follows: Similarly, the distance between any of the alternative and CFFNS f -NIS can be calculated as follows: Step 7. To find the most suitable alternative, we compare the alternative by some ranking index. The revised closeness index (Vencheh and Mirjaberi 2014) corresponding to the alternative I g can be evaluated by utilizing the formula: where g ¼ 1; 2; 3; . . .; n, and g ; e G t Þ: • Step 8. After the evaluated results of closeness index, the alternatives are arranged in an ascending order with respect to revised closeness index. The alternative having maximum value of closeness index will be the optimal solution of MAGDM problem.
The general steps of CFFNS f -TOPSIS method are summarized in Figure 2. The stepwise solution of this MAGDM problem by following CFFNS f -TOPSIS method is given as follows: • Step 1. According to the above-mentioned attributes, each expert assesses the alternatives regarding all attributes using 5-soft, given in Table 42, where: The experts D 1 ; D 2 ; D 3 and D 4 will use the grading criteria given by Step 2 Complex fermatean fuzzy N-soft sets... 8791 • Step 4. The entries of AWCFFNS f DM b G are obtained by Equation 4 by utilizing ACFFNS f DM, given by Table 47, and the weight vector j of attributes in Equation 10. These entries are tabulated, as shown in Table 49.
• Step 5. In the proposed MAGDM problem, the attributes topography, soil and water quality & availability are benefit-type attributes whereas initial cost, environmental destruction, climate of the area and maintenance cost are cost-type attributes. CFFNS f -PIS and CFFNS f -NIS relative to each attribute, opted by Equations 5 and 6 , are arranged in Table 50.
• Step 6. Distance of each alternative from CFFNS f -PIS and CFFNS f -NIS is calculated by employing Equations 7 and 8 , respectively. These distance measures are tabulated in Table 51.
• Step 7. Table 52 represents the revised closeness index corresponding to each alternative, evaluated by using Equation 9.
• Step 8. The ranking of cities on the basis of revised closeness index is shown by Table 53. Since I 1 has maximum index value. Hence the experts will give suggestions to the investor to select Boston, Massachusetts for farming.
Comparative analysis of CFFNS f -TOPSIS technique In this section, we solve the MAGDM problem ''Selection of the most suitable city in the USA for farming'' by Fermatean fuzzy TOPSIS (FF-TOPSIS) method, proposed by Senapati and Yager (2020), to authenticate the importance and validity of proposed model. The step wise solution of MAGDM problem following the Fermatean fuzzy TOPSIS method is given as follows: • Step 1. The linguistic terms along with grades are same as given in Table 42. Since the existing technique only deals with multi-attribute decision-making (MADM) problems. Hence, the aggregated opinion of all experts, given in Table 47 is used by the investor but the grading part is excluded and CFFNs have taken to be zero to apply FF-TOPSIS method. Fermatean fuzzy decision matrix (FFDM) is arranged in Table 54. Moreover, to determine the role of each criteria, the decision-maker sets the weights of attributes as follows: k ¼ ð0:150:20:170:10:160:130:09Þ T Step 2. The score of all FF numbers (FFNs) are determined to identify the Fermatean fuzzy positive and negative ideal solutions. The score of a FFN can be calculated by the following formula (Senapati and Yager 2020): The score values of all entries of FFDM are assembled in Table 55. Table 56 represents the FF-PIS and FF-NIS relative to each attribute. • Step 3. Distance of each alternative I g from FF-PIS B þ and FF-NIS B À is computed by employing the equations as follows (Senapati and Yager 2020): The results are tabulated in Table 57.
• Step 4. To find out the most suitable alternative, Table 58 represents the closeness index corresponding to each alternative which is evaluated by utilizing the Equation 9. • Step 5. The increasing order ranking of cities is shown by Table 59, where 1 is for minimum closeness index value and 4 is for highest index value. Since I 1 has the maximum index value. Hence the investor will select Boston, Massachusetts for farming.

Results
1. Now, we present a comparison of the proposed technique with the existing FF-TOPSIS method (Senapati and Yager 2020) to assess the accuracy of CFFNS f - Complex fermatean fuzzy N-soft sets... 8793 TOPSIS method. Despite the difference in revised closeness index calculated by both techniques, the final ranking of cities is the same. Thus, the same city is proclaimed as the most suitable one for farming in both methods. The results of the proposed and existing methods, including the final ranking and best alternative, are summarized in Table 60 as follows:      Figure 3 to envision the conformity of final results of compared and proposed MAGDM approaches which shows the effectuality and accountability of our proposed technique. 3. It is clear from the figure that both techniques elucidate the same outcome and ranking order that indicates the feasibility and sustain-ability of the presented technique.

4.
Our proposed CFFNS f -TOPSIS technique has capability to handle the vagueness and periodicity involve in the data simultaneously, but the compared FF-TOPSIS technique is limited to capture the ambiguity of nonperiodic data that may cause to the inconsistency and specious outcomes. This extraordinary trait of the proposed strategy depicts that it is the more effective and generalized MAGDM strategy. 5. Due to the inadequacy of multi-valued grades and periodic terms, FF-TOPSIS cannot deal with CFFNS f information. On the other hand, CFFNS f -TOPSIS method has potential to handle the FF information by taking phase terms equal to zero and neglecting the grades. Since the results are same in both cases which   depicts the proposed method more adaptable than existing methods.
10 Merits of CFFNS f S model and CFFNS f -TOPSIS approach 1. In the modern era, the performance appraisal system is commonly used for the rating of restaurant management, schools, candidates for job, online services, online applications, products and websites, etc. The proposed model is designed to handle the rating-based assessment framework along with imprecise and vague two-dimensional information. 2. In this article, the robust technique of CFFNS f -TOPSIS is developed for determining the best solution obtaining the closest distance from PIS and far away from NIS. The framework of the presented MAGDM strategy has remarkable aspects: it merges the fascinating advantages of TOPSIS with the hybrid model of CFFNS f S. The advantage of the hybrid model is that it has the potency to handle vagueness and periodicity of parameterized graded information simultaneously. 3. The proposed model shows the same accuracy when applied to the existing models inclusive of FF, CIF, CPF, FFS f , CIFS f , CPFS f , FFNS f , CIFNS f , and CPFNS f by taking either N ¼ 2 or substituting phase terms equal to zero or by applying both strategies. Hence, the developed technique deprives a adaptable tool that skillfully and efficiently accomplishes its decision-making chores with preciseness under traditional as well as two-dimensional vague information along with finely-graded parameters.

Conclusion
Decision-making methods play an important role in the real life of human beings. The process of choosing the best option among a set of possible options is present in all human activities. In this paper, a new theory has been developed that serves as a mathematical tool which deals with the two-dimensional vague information, and which is a generalization of the fuzzy N-soft set. We have advanced a model, CFFNS f S, that assesses the uncertain and vague data which has complex membership and non-membership values, parameterized information, and ordinal ranking systems. To establish a comparison between two CFFNS f Ns, we have developed score and accuracy functions in a CFFNS f environment. We have defined some basic operations for the CFFNS f S model that include: complement (weak complement, CFFNS f S complement, and weak CFFNS f S complement), union (extended union and restricted union), intersection (extended intersection and restricted intersection). We have also included relevant examples for these operations. In addition, we have presented algebraic and Yager operations for CFFNS f Ns. Moreover, we have accomplished three algorithms to resolve multi-attribute decision-making problems. These algorithms have been validated by two real-life examples related to the selection of cars and the selection of the best telecommunication company in Pakistan.
Furthermore, in order to analyze the validity, feasibility, and reliability of the proposed model, we have conducted a comparative study of our approach with two operators: the FFY w G operator and the FFY w A operator.
With respect to the proposed CFFNS f -TOPSIS method, our method possesses the MAGDM potential of TOPSIS along with the adequacy of the proposed CFFNS f model to CFFNSf -TOPSIS method (proposed) FF-TOPSIS method (Senapati and Yager, 2020) Cities in the USA for farming Fig. 3 Comparative analysis improve the exactness of decision-making results. The proposed method's primary dominance is due to its capability to tackle two-dimensional imprecise information along with level of attributes based on alternative with the help of N-soft grading values as well as complex membership and non-membership values. The basic principle of the CFFNS f -TOPSIS method is to find out the best solution possessing the proximity distance from the ideal solutions.
In the presented approach, the primary information has been equipped by ordered grades and their corresponding CFFNS f Ns. The individual opinions have been aggregated by employing CFFNS f WA operator. Further, the AWCFFNS f DM has been acquired by the multiplication of CFFNS f weight vector of criteria and CFFNS f DM. After examining the CFFNS f -PIS and CFFNS f -NIS, distance of each alternative from ideal solutions have been computed. Further, the revised closeness index of each alternative has been calculated by evolving the discrepancy of these variables from the ideal solution. After the evaluated results of the closeness index, the alternatives are arranged in an ascending order. The alternative having maximum value of closeness index will be the optimal solution of the MAGDM problem. The proposed approach has been endorsed by a numerical example related to the selection of the suitable city in the USA for farming.
Along with beneficial characteristics of the proposed technique based on TOPSIS method for MAGDM problems in two-dimensional data, it ensures the same level of authenticity under Fermatean fuzzy environment by eliminating the grades and substituting phase terms equal to zero. On the contrary, the adeptness of the FF-TOPSIS method is restricted to handle one dimensional phenomena, also it is unable to deal with MAGDM problems.
Moreover, the proposed CFFNS f -TOPSIS method has an edge over the extant decision-making approaches as the CFFNS f S model can effectively apply in the environments of FFS, CIFS, CPFS, FFS f S, FNS f , IFNS f S, PFNS f S, CPFNS f and so forth by taking either N ¼ 2 or substituting phase terms equal to zero or by applying both strategies.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

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.
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