Study on Chaotic Multi-Attribute Group Decision Making Based on Weighted Neutrosophic Fuzzy Soft Rough Sets

: In this article, we have proposed a multi-attribute group decision making (MAGDM) with a new scenario or new condition named Chaotic MAGDM, in which not only the weights of the decision makers (DMs) and the weights of the decision attributes are considered, but also the familiarity of the DMs with the attributes are considered. Then we applied the weighted neutrosophic fuzzy soft rough set theory to Chaotic MAGDM and proposed a new algorithm for MAGDM. Moreover, we provide a case study to demonstrate the application of the algorithm. Our contributions to the literature are as follows: (1) familiarity is rubbed into MAGDM for the ﬁrst time in the context of neutrosophic fuzzy soft rough sets; (2) a new MAGDM model based on neutrosophic fuzzy soft rough sets has been designed; (3) a sorting/ranking algorithm based on a neutrosophic fuzzy soft rough set is constructed.


Introduction
Multi-attribute decision making (MADM) is an important branch of modern decision theory and methodology with a wide range of practical contexts, such as human resource performance assessment, economic performance assessment, political election assessment, military performance assessment, etc.However, due to the limitations of human knowledge and the specialization of professions, as well as the diversity and complexity of real-world decision making, a single decision maker (DM) cannot make the optimal option.As a result, in most MADMs, decision makers (DMs) from diverse sectors, areas of expertise, or knowledge backgrounds are frequently required to collaborate in order to make more scientifically sound conclusions.That is, multi-attribute group decision making (MAGDM).In addition, there is a lot of uncertainty and ambiguity in practical MAGDM.Therefore, the study of MAGDM under fuzzy scenarios has become a popular research direction in recent years.Considering that different DMs have different professional backgrounds, areas of knowledge, expertise, etc.Therefore, in a MAGDM, how to engage DMs to evaluate the attributes of the alternatives in their areas of expertise and familiarity is an issue that must be considered in the decision making.In the existing literature or research results, there are two common approaches to address this concern.One is to assign weights to DMs, the other is to group DMs according to certain rules.

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Assigning weights to DMs.
Liu et al. [1] proposed a variable weighting approach by considering DMs and attributes weights together for MAGDM problems under interval-valued intuitionistic fuzzy sets.Yu et al. [2] developed a novel consensus reaching process for MAGDM based on hesitant fuzzy linguistic term sets (HFLTSs) which not only can deal with multi-granular HFLTSs, but also considers the weight vectors of DMs and attributes in the proposed consensus model.Liu et al. [1] presented a hybrid approach based on variable weights for multi-attribute group decision making, and so on [3][4][5][6][7][8][9][10][11][12].

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Grouping DMs according to certain rules.
For example, Su et al. [13] proposed a MAGDM approach for evaluation and selfconfidence in online learning platforms based on probabilistic linguistic term sets.Sun et al. [14] provided diverse fuzzy multi-branch rough sets based on binary relations for MAGDM.Sun et al. [15] analyzed the diversified MAGDM problem with the personal preference parameters, etc. [16][17][18][19].

Comparison and Motivation
To date, research on MAGDM in fuzzy scenarios has produced a very large number of theoretical and practical results.All these studies have investigated the relationship between DM and attributes from different perspectives either by assigning weights or by grouping; however, the following issues still need to be further explored.

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Most of the existing studies are empowered weight according to the level of DMs.In other words, DMs with high weight will have a higher proportion of the evaluation of all attributes, even if it is an attribute that he or she is not familiar with.

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The scope of DMs' expertise is ambiguous, so in practice it is difficult to group DMs according to their expertise only.For example, if each DM only examines attributes that relate to their own expertise, who should evaluate these combined attributes that encompass a variety of expertise?• Grouping DMs based only on their professional background would inevitably lose many very valuable evaluations.Because most DMs have a great deal of practical experience, they are well-placed to evaluate attributes even in non-specialist areas.• Most studies did not consider the relationship between DMs and decision attributes.That is, the familiarity of DMs with the decision attributes.
The following situations are often encountered in decision making.For example, in a large-scale fire rescue, an important attribute of the rescue plan is the ability to quickly rescue trapped people.This attribute is not only related to the cause of the fire, but also to the structure of the building, construction materials, etc.Therefore, both fire experts and building experts should be important evaluators of this attribute.Hence, it is obviously unreasonable to have only one of the expert group (e.g., the fire expert group) to evaluate it.In other words, simply grouping experts would result in a lack of many valuable evaluations.Furthermore, fires are related to weather conditions, and a meteorologist is a very well-known expert whose evaluation of this property will be less important if he or she is not familiar with the fire scene.On the contrary, the person who built the building, even if he is just a worker, will have a very important evaluation because he is more familiar with the structure of the building at the fire site.
This suggests that the weights of DM's evaluations are related to familiarity with the attributes, rather than just considering the weights of the DM.In other words, in MAGDM, not only the weight of the DM and the weight of the attribute are considered, but also the familiarity of the DM with the attributes.
There will be a lot of uncertainty and fuzziness in the actual MAGDM problem, and also the familiarity of the DM with the decision attributes will change in different scenarios.Therefore, in the absence of an explicit method to determine the familiarity between DM and decision attributes, using fuzzy theory to describe the familiarity between DM and attributes is a good choice.In order to allow DMs to focus their attention on evaluation scoring without considering the limitations of scoring values, we chose NFN (Neutrosophic Fuzzy Number) for evaluation scoring.
In summary, we have proposed a MAGDM with a new scenario or new condition in which not only the weights of the DMs and the weights of the decision attributes are considered, but also the familiarity of the DMs with the attributes needs to be considered.
That is Chaotic MAGDM (CMAGDM) which was proposed by Zhang et al. [20].Then we applied neutrosophic fuzzy soft rough set theory to CMAGDM and proposed a new approach for CMAGDM.Our contributions are mainly as follows.

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Familiarity is rubbed into MAGDM for the first time in the context of neutrosophic fuzzy soft rough sets; • A new Chaotic MAGDM model based on neutrosophic fuzzy soft rough sets has been designed; • A sorting/ranking algorithm based on a neutrosophic fuzzy soft rough set is constructed.
The remainder of this paper is structured as follows: Section 3 briefly introduces the basic concepts and framework of MAGDM, provides a brief overview of fuzzy theory and several concepts in fuzzy theory.In Section 4, combining the neutrosophic fuzzy soft rough set and CMAGDM, we provide a new Chaotic MAGDM model based on neutrosophic fuzzy soft rough sets.A case study and the numerical analysis of the proposed model is illustrated in Section 5.At last, conclusions are given in Section 6.

Theoretical Background
In this section, first, we will review the basic concepts and framework of MAGDM.Second, we will provide a brief overview of fuzzy theory.Finally, we will review several important concepts in fuzzy theory, as well as their basic rules and properties.

The Basic Concepts and Framework of MAGDM
The problem of selecting the best answer from a list of potential solutions, based on a set of attributes or criteria, can be summarized as a decision problem.In real-world decision making, there will always be a group of DMs.The corresponding MADM translates into multi-attribute group decision making (MAGDM).The problem of MAGDM is represented by the following notation [20]: |e k ∈ E} is the decision matrix set, and X(e k ) is the decision matrix of the kth DM: In order to describe MAGDM more clearly and concisely, a MAGDM problem can usually be represented by a sextuple A, C, E, w, τ, X , that is: (2)

A Brief Overview of Fuzzy Theory
In order to better combine uncertainty in MAGDM with fuzzy theory, we will make a brief review of the development of fuzzy set (FS).
Zadeh [21] (1965) first proposed fuzzy theory.It breaks through the limitations of classical set theory by introducing a membership function to represent uncertainty.Atanassov [22] (1983) suggested a generalization of fuzzy sets making the degrees of membership (µ) and non-membership (ν) intervene to describe the vinculation of an element to a set, and the sum of these degrees is less or equal to 1 (µ + ν ≤ 1), that is, an intuitionistic fuzzy set (IFS).However, an IFS fails when the sum of these degrees is more then 1. So, Yager [23] (2016) developed the concept of q-rung orthopair fuzzy sets (q-ROFS) and considered an efficient method to explain the vagueness of MADM problems.In q-ROFS, the sum of two degrees can be more then 1, it just needs to satisfy the condition µ q + η q ≤ 1, (q ≥ 1).It is clear that, for q = 1, it is an IFS, it is a Pythagorean fuzzy set (Yager and Abbasov [24] 2013) (PyFS) if q = 2, and it is a Fermatean fuzzy set (Senapati and Yager [25] 2020) (FFS) if q = 3, thus, q-ROFS generalize the IFS, PyFS, FFS.When we face human opinions involving more types of answers: yes, abstain, no, or refusal.Voting can be a good example of such a situation as the human voters may be divided into four group of those who : vote for, abstain, vote against, or refuse to vote.In order to solve this problem, Cuong and Kreinovich [26,27] in 2013 introduced a new notion of picture fuzzy set (PFS), which are directly extensions of FS and of intuitionistic fuzzy set (IFS).In PFS, the following three dimensions are considered simultaneously: degree of positive membership (µ), degree of neutral membership (η), and degree of negative membership (ν), and satisfy the following condition µ + η + ν ≤ 1.The structure of PFS is of great importance as it has the ability to deal with human opinion efficiently.It is observed that the constraint on PFS makes us unable to assign values by own choice.In simple words, one can say that the domain of PFS is restricted.The concept of spherical fuzzy set (SFS) and T-spherical fuzzy set (T-SFS) is introduced as a generalization of FS, IFS, and PFS by Mahmood et al. [28] in 2019.In T-SFS, the sum of three degrees can more then 1, instead, need to satisfy condition µ t + η t + ν t ≤ 1, (t ≥ 1).Obviously, when t = 1, T-SFS degenerates to PFS, and if t = 2, T-SFS is a SFS.Neutrosophic set (NS) was introduced by Smarandache [29].In NS, there is a need to satisfy the condition Although there are many other fuzzy sets, such as fuzzy multi-set (FMS) [30], interval-valued fuzzy set (IVFS) [31], hesitant fuzzy set (HFS) [32], hybrid fuzzy set, and so on.These theories play a very important role in practice and theory.However, due to limited space and the focus of our article, we will not repeat them here.
According to the above review and analysis of FSs, we can clearly draw the relationship between different FSs, as shown in Figure 1.There are two ideas for the promotion of the FS, one is from the dimension of the variable, the other is from the domain of the variable.Researchers have expanded the fuzzy set from the perspective of variable dimension and its domain.These works not only played a very important role in the development of fuzzy set theory, but also played a very important role in practical applications.However, these ideas for generalizing fuzzy sets only consider the dimension and domain of fuzzy variables and do not consider the ambiguity of the attribute.Here we still use [26]'s voting example to illustrate our opinions.Suppose there are two candidates p 1 and p 2 participating in the campaign.A voter is very familiar with p 1 , but only knows about p 2 based on his campaign speech.Now, the voter evaluates the two candidates using the PFS method and receives the same score.Clearly, it is unreasonable to consider these two evaluations as identical due to the difference in familiarity with the candidates.Phenomena such as the above will be frequently encountered in MAGDM problems.Fortunately, in addition to fuzzy sets, there are soft sets [33] and rough sets [34] that can describe unclear and fuzzy relations.Especially, the combination of these uncertainty theories can describe more details in MAGDM.These include fuzzy soft set (FSS) [35], fuzzy soft rough set (FSRS), picture fuzzy soft rough set (PFSRF), spherical fuzzy soft rough set (SFSRS), T-spherical fuzzy soft rough set (T-SFSRS) [36] and so on.

Concepts of Fuzzy Set
Classes and sets in the traditional mathematical sense do not include things such as "the class of all real numbers which are significantly bigger than 1," "the class of attractive ladies," or "the class of tall men."However, it is still true that such loosely defined "classes" are crucial to human thought.In essence, rather than the existence of random variables, the source of imprecision is the absence of well specified criteria for class membership.Zadeh [21] explored a concept which may be of use in dealing with "classes" of the type cited above.That is the fuzzy set (FS), a "class" with a continuum of grades of membership.It is defined as follows: Definition 1 (Zadeh [21]).A fuzzy set A on a universe X is an object of the form where µ A (x) ∈ [0, 1] is called the "degree of membership of x in A".The variable µ A (x) is called a Fuzzy Number (FN).
Intuitionistic fuzzy set (IFS) was developed by Atanassov [22] and is suitable for situations in which there is uncertainty about the degree of membership of an element in a defined set: each element in an IFS has a membership degree and a nonmembership degree between 0 and 1 [37].
Definition 2 (Atanassov [22]).A intuitionistic fuzzy set A on a universe X is an object of the form where is called the "degree of non-membership of x in A", and where µ A (x) and ν A (x) satisfy the following condition: The pair (µ A (x), ν A (x)) is called an Intuitionistic Fuzzy Number (IFN).
When we face human opinions involving more types of answers such as yes, abstain, and refusal.Cuong and Kreinovich [26] introduced the concept of picture fuzzy set (PFS), and Mahmood et al. [28] provided the concept of T-spherical fuzzy set (T-SFS), both of which are direct extensions of the fuzzy set (FS) and the intuitonistic fuzzy set (IFS).[26]).A picture fuzzy set A on a universe X is an object of the form

Definition 3 (Cuong and Kreinovich
where µ A (x) ∈ [0, 1] is called the "degree of positive membership of x in A", η A (x) ∈ [0, 1] is called the "degree of neutral membership of x in A" and ν A (x) ∈ [0, 1] is called the "degree of negative membership of x in A", and where µ A (x), η A (x) and ν A (x) satisfy the following condition: ), π A (x) could be called the "degree of refusal membership of x in A".Let PFS(X) denote the set of all the picture fuzzy sets on a universe X.A triplet (µ A (x), η A (x), ν A (x)) can be referred to as a Picture Fuzzy Number (PFN).Definition 4 (Mahmood et al. [28]).A T-spherical fuzzy set A on a universe X is an object of the form where the "degree of neutral membership of x in A", and ν A (x) ∈ [0, 1] is called the "degree of negative membership of x in A", and where µ A (x), η A (x) and ν A (x) satisfy the following condition: ), π A (x) could be called the "degree of refusal membership of x in A".Let T-SFS(X) denote the set of all the T-spherical fuzzy sets (T-SFS)on a universe X.A triplet (µ A (x), η A (x), ν A (x)) can be identified as a spherical fuzzy number (T-SFN).If t = 2 the T-spherical fuzzy set is called spherical fuzzy set (SFS), and corresponding SFS(X) denote the set of all the spherical fuzzy sets on a universe X.A triplet (µ A (x), η A (x), ν A (x)) can be referred to as a spherical fuzzy number (SFN).
Smarandache [29] generalized intuitionistic fuzzy sets (IFSs) to neutrosophic sets (NSs).A neutrosophic set (NS) contains three parameters: truth membership function, indeterminacy membership function, and falsity membership function.Unlike the PFS and T-SFS, the NS has a broader definition domain, giving DMs more options for evaluating scores in MAGDM.
Definition 5 (Smarandache [29]).A neutrosophic set A on a universe X is an object of the form where µ A (x) ∈ [0, 1] is called the "truth membership function of x in A", η A (x) ∈ [0, 1] is called the "indeterminacy membership function of x in A" and ν A (x) ∈ [0, 1] is called the "falsity membership function of x in A", and where µ A (x), η A (x) and ν A (x) satisfy the following condition: ) can be referred to as a Neutrosophic Fuzzy Number (NFN).
By comparing the above concepts, it is easy to conclude that NS has a broader field of definition, thus allowing the DM to focus more on scoring the options without having to think too much about the constraints that need to be met for the evaluation scores.It is with this in mind that the Neutrosophic Fuzzy Number (NSN) will be chosen for scoring in this paper.

The Concept of Rough Set
Pawlak [34] introduced the concept of rough sets (RS) in 1982, which can handle uncertainty, imprecision, and ambiguity in sets.It is defined as follows.
Definition 6 (Pawlak [34]).Let R be an equivalence relation on the universe X (X = ∅), (X, R) be a Pawlak approximation space.A subset A ⊆ X is called definable if R(x) = R(x); in the opposite case, i.e., if R(x) − R(x) = ∅, A is said to be a rough set, where the two operations are defined as: As an illustration, let us consider the following example (Example 1).

The Concept of Soft Set
Molodtsov [33] proposed in 1999 a mathematical approach to dealing with uncertain information with the core idea of emphasizing the study of uncertainty and ambiguity of information from a parametric perspective, which is known as soft set (SS) theory.The concept of SS is as follows.
Definition 7 (Molodtsov [33]).Let X be the universe.C is a set of parameters (attributes) about objects in X, (X, C) is called a soft space, and C ⊆ C, ϕ is a mapping given by ϕ : C → 2 X ; here 2 X is the power set of X, then a pair (ϕ, C) is named a soft set (SS) over the universe X.
To illustrate the point, let us consider the following example (Example 2).
According to the Definition of soft set, we could easily come to the results as follows: Take ϕ(c 3 ) as an example, it means that the objects with attribute c 3 are a 1 , a 2 and a 3 .

The Concept of Fuzzy Soft Set
Maji et al. [35] combined the theory of fuzzy sets and soft sets in 2001 and proposed the definition of fuzzy soft sets.A fuzzy soft set can essentially be seen as a parametric fuzzy set for a given universe, which is a representation model that combines parameters and fuzzy information together.It is no longer restricted to the 0 and 1 values of the parameters in the soft set, but is a more flexible form of parameter selection that can be used in a wide range of uncertainty areas [15].The concept is as follows.
Definition 8 (Maji et al. [35]).Let X be a universal set, C be a collection of parameters regarding X, and FS(X) represents the collection of all FSs over the universe X.A pair (ϕ, C) is said to ba a Fuzzy Soft Set (FSS) over X, where C ⊆ C and ϕ : C → FS(X), where FS(X) represents the collection of all the fuzzy sets (FSs) on a universe X.For every x ∈ X, the FSS can be defined as follows: Particularly, when |C| = 1, the fuzzy soft set degenerates to a fuzzy set.
Specially, when the fuzzy set is PFS, the corresponding concept has the following form: Definition 9 (Khan et al. [38]).Let X be a universal set, C be a collection of parameters (attributes) regarding to X and PFS(X) represents the collection of all picture fuzzy set over the universe X.A pair (ϕ, C) is said to ba a Picture Fuzzy Soft Set (PFSS) over X, where C ⊆ C and ϕ : C → PFS(X).For every x ∈ X, the PFSS can be defined as follows: Obviously, when |C| = 1, the picture fuzzy soft set degenerates to a picture fuzzy set.

The Concept of Fuzzy Soft Rough Set
Combining fuzzy sets, soft sets, and rough sets can lead to a more flexible method of describing parameters, namely fuzzy soft rough sets.If the fuzzy set is a picture fuzzy set, the corresponding fuzzy soft rough set is called a picture fuzzy soft rough set and is defined as follows.
Definition 10 (Muahmmad and Martino [36]).Let X be the universe.C be a set of parameters (attributes) about objects in X, PFS{X} be the collection of all picture fuzzy soft sets over the universe X, R be a picture fuzzy soft set relation from universe X to C (That is ∀c ∈ C ⊆ C, R(c) ∈ PFSS(X)), and ψ be a mapping given by ψ : C → PFSS{X}.Then (ψ, C, R) is known as a Picture Fuzzy Soft Rough Approximation Space.For every F ∈ PFS(C), the lower and upper approximation of F can be defined as follows: where and Here, 0 ≤ µ(x) The score function can be defined as: Example 3. In a (MADM), suppose the alternatives set is A = {a 1 , a 2 , a 3 } and the attributes set is C = {c 1 , c 2 , c 3 , c 4 }, as presented in Table 3.Let F be a PFS over C as follows.

A Novel Neutrosophic FSRS-Based Method for Chaotic MAGDM 4.1. Chaotic Multi-Attribute Group Decision Making
Zhang et al. [20] proposed the concept of Chaotic MAGDM, in which not only the weights of DMs and decision attributes are considered, but also the familiarity of DMs with the decision attributes.With the crossover factor of familiarity, Chaotic MAGDM is brought closer to the real decision problem.The relevant concepts are as follows.
Definition 11 (Zhang et al. [20]).A MAGDM is called Chaotic MAGDM if there exists at least one decision attribute such that at least two DMs have the different familiarity with it.
For convenience, the symbols of the variables used in Chaotic MAGDM are summarized as follows: [20]: . ., a m } is the set of considered alternatives; • C = {c 1 , c 2 , . . ., c n } is the set of attribute which are used for evaluating of alternatives; • E = {e 1 , e 2 , . . ., e l } is the set of the DMs involved in the decision process; is the weight vector of the attribute (w 0, ∑ n j=1 w j = 1);  Since we consider the relationship between DMs and decision attributes in Chaotic MAGDM, we added the familiarity variable F to the MAGDM.That is using septuple A, C, E, w, τ, X, F to represent the Chaotic MAGDM.As shown in Equation ( 22) Clearly, the diversified multi-attribute group decision making proposed by Sun et al. [14] is a special case of chaotic MAGDM.One of the core ideas of diversified MAGDM is that by establishing a pluralistic binary fuzzy relationship between the set of evaluation attribute indicators and different decision makers.
In order to describe a Chaotic MAGDM more visually, it can be represented by a information form.As shown in Table 4.In the Definition 10 proposed by Muahmmad and Martino.If, for example, in MADM, F denotes the familiarity of the DMs against to the attributes rather then the values of evaluation, then the defining functions of the upper and lower bounds of R(F ) must be changed accordingly.So, we give the new definition as follows: Definition 12. Let X be the universe.C be a set of parameters (attributes) about objects in X, NFS{X} be the collection of all neutrosophic fuzzy soft sets over the universe X, R be a neutrosophic fuzzy soft set relation from universe X to C (That is ∀c ∈ C ⊆ C, R(c) ∈ NFSS(X)), ψ be a mapping given by ψ : C → NFSS{X}.Then (ψ, C, R) is known as a Neutrosophic Fuzzy Soft Rough Approximation Space.For every F ∈ NFS(C), the lower and upper approximation of F can be defined as follows: where and where, 0 ≤ µ(x) The score function is as following: Example 4. Still analyzing the data in Example 3, under the new Definition 12, the corresponding results are: R(F ) = {(a 1 , 0.04, 0.85, 0.60), (a 2 , 0.04, 1.02, 0.22), (a 3 , 0.08, 0.24, 0.36)}; R(F ) = {(a 1 , 0.45, 0.06, 0.01), (a 2 , 1.28, 0.03, 0.00), (a 3 , 0.09, 0.00, 0.01)}.
If different attributes c (c ∈ C) have different weights, then the neutrosophic fuzzy soft rough set will become a weighted neutrosophic fuzzy soft rough set.Definition 13.Let X be the universe.C be a set of parameters (attributes) with the weight w C about objects in X, NFS{X} be the collection of all neutrosophic fuzzy soft sets over the universe X, R be a neutrosophic fuzzy soft set relation from universe X to C (That is ∀c ∈ C ⊆ C, R(c) ∈ NFSS(X)), and ψ be a mapping given by ψ : C → NFSS{X}.Then (ψ, C, R, w C ) is known as a weighted neutrosophic Fuzzy Soft Rough Approximation Space.For every F ∈NFS(C), the lower and upper approximation of F can be defined as follows: where and here, 0 ≤ µ(x) The evaluation function is Considering that as the parameters η, η, ν and ν increase, the value of the evaluation function becomes very close to zero or even negative, this does not facilitate numerical calculations and comparisons.Therefore, the evaluation function needs to be improved accordingly.The new evaluation function is as following: Definition 14.The score function is as following: Example 5. Still analyzing the data in Example 3 with the weight vector w = {0.50,0.25, 0.15, 0.10} T , under the new Definition 13 and Equation ( 14), the corresponding upper and lower bounds and score functions are as follows.
In general, in chaotic multi-attribute group decision making (CMAGDM), different DMs have different decision weights, so the total evaluation function of the corresponding CMAGDM is as follows.

The Algorithm for CMAGDM
Through the analysis in the previous subsection, we summarize the algorithm for solving CMAGDM as Algorithm 1.

Algorithm 1
The Algorithm for CMAGDM.

Numerical Analysis
In this section, we will analyze a real-life home purchase problem to explain the specific application of our proposed method.

Problem Statement
A family with three members (husband, wife, and daughter) are planning to buy one of four houses.Suppose the family considers the following factors in purchasing: price, construction materials, decoration, convenience for shopping (e.g., availability of supermarkets, food markets, shops, etc.), and convenience of transportation (e.g., availability of public parking, bus stops, metro stations, etc.).Assume that the purchase decision weight of the husband is 40%, the wife is 35%, and the daughter is 25%.The weights of the purchasing factors are as follows: 35% for price, 20% for construction materials, 15% for decoration, 10% for convenience of shopping, and 20% for convenience of transportation.
Obviously, this is a MAGDM problem, it can be represented by a sextuple A, C, E, w, τ, X , that is: where A = {a 1 , a 2 , a 3 , a 4 }, a 1 , a 2 , a 3 , a 4 denote the first house, the second house, the third house, and the fourth house, respectively.C = {c 1 , c 2 , c 3 , c 4 , c 5 }, c 1 , c 2 , c 3 , c 4 , c 5 denote price, construction materials, decoration, convenience for shopping, and convenience of transportation, respectively, and the corresponding weight vector is w, where w = {0.35,0.20, 0.15, 0.10, 0.20}.E = {e 1 , e 2 , e 3 }, e 1 , e 2 , e 3 denote husband, wife, and daughter, respectively, the corresponding weight vector is τ, where τ = {0.40,0.35, 0.25}.X = {X(e k )|e k ∈ E} is the decision matrix set, and X(e k ) is the decision matrix of the kth DM.Usually, however, the wife and daughter are not familiar with attribute (indicator) construction materials, while the husband is not particularly familiar with attribute (indicator) decoration.Since DMs have different levels of familiarity with attributes, this is not a regular MAGDM problem, but a CMAGDM problem.It can be represented by a septuple A, C, E, w, τ, X, F , that is: Assuming that the DMs select NFSS for scoring evaluation, the evaluation form is shown in Table 5.

Numerical Computations
According to the Algorithm 1, we can obtain following results: Step 1: By Equation (42), the NFSRSs vector R(F ) can be found.To illustrate the exact process of calculation, we take the husband's evaluation of alternative a 1 as an example.Similarly, the values of η(a 1 ), η(a 1 ), ν(a 1 ), ν(a 1 ) can be calculated.The same approach could be used to obtain the husband's evaluation of alternative a 2 , a 3 , a 4 .Using the same method, we can obtain all the evaluation of wife and daughter.Finally, the NFSRSs vector R(F ) can be found.As shown in Table 6.Step 2: By Definition (14), the score vector S can be obtained.Here is an example of the calculation process using the husband's scoring of the alternative a 1 .
Using the same method, all evaluation scores can be derived, then the information of the score vector S can be found, as shown in Table 7. Step 3: By Definition 15, we can obtain the corresponding total evaluation score for four houses as shown in Table 8.Using alternative a 1 as an example, the process for calculating its overall evaluation score is as follows.Step 4: Obtain the final ranking.By the calculation in the previous step, obviously, we can obtain: S(a 4 ) > S(a 2 ) > S(a 1 ) > S(a 3 ).That is, a 4 a 2 a 1 a 3 .The optimal alternative a * = a 4 , so the 4th house is the optimal choice.

Conclusions
Most of the current studies on multi-attribute group decision problems mainly give the corresponding solutions in different practical applications or when DMs use different fuzzy sets [1,[9][10][11][39][40][41].Typically, the study of traditional group decision models and methods consists of two main aspects: consensus building and optimal choice.The former refers to how to make the opinions of all experts as consensual as possible among all candidate alternatives, while the latter focuses on how to select the optimal decision alternative from all candidates based on group preference opinions [14,15].However, there are few research results that consider the structure of the multi-attribute group decision problem itself, such as the relationship between DMs and attributes.Based on such considerations, we propose a chaotic multi-attribute group decision model that considers the familiarity of DMs with attributes, which can well avoid the drawbacks arising from grouping or weighting of decision makers by introducing familiarity in multi-attribute group decision making.At the same time, we combine neutrosophic set with a wider definition domain with soft set and rough set to give the concept of weighted neutrosophic fuzzy soft rough set and apply it to chaotic multi-attributes group decision making to obtain the corresponding algorithm.The validity of the model and the flexibility of the algorithm are well illustrated by practical case studies.
Despite our attempts to solve more realistic problems, however, there are still many shortcomings in our work.We have only considered the evaluation scoring of decision makers using neutrosophic fuzzy sets, whereas in practical decision making, decision makers can choose different evaluation methods, such as using different fuzzy sets or precise numbers or linguistic variables, etc.This is a drawback of our work and is certainly a direction for future research.In addition, this paper does not give a scheme for determining familiarity.However, how to determine the familiarity, just like how to assign weights to decision makers or decision attributes, is still the key to multi-attribute cluster decision making, so this will be another popular direction for future research.
l} is the decision matrix set, and X k is the decision matrix of the kth DM.As shown in Equation (1);• f k j is the familiarity of the kth DM against to the jth attribute the vector of the familiarity of DMs with attributes, where F k is the familiarity of kth DM.

Table 3 .
A information system of PFS.

Table 4 .
The Chaotic MAGDM Information Form.

Table 5 .
CMAGDM information of house purchase.

Table 6 .
Information of the NFSRSs vector R(F ).

Table 7 .
Information of the score vector S.