q-Rung orthopair fuzzy N-soft aggregation operators and corresponding applications to multiple-attribute group decision making

In this paper, by integrating the q-rung orthopair fuzzy set (q-ROFS) with the N-soft set, we first propose a q-rung orthopair fuzzy N-soft set (q-ROFNSS). Based on the q-ROFNSS, we explore the q-rung orthopair fuzzy N-soft weighted average (q-ROFNSWA) operator and q-rung orthopair fuzzy N-soft weighted geometric (q-ROFNSWG) operator, and investigate some properties of the q-ROFNSWG operator and q-ROFNSWG operator including idempotency, monotonicity and boundedness. Finally, two kinds of multiple-attribute group decision-making (MAGDM) methods based on q-rung orthopair fuzzy N-soft aggregation operators are established. In addition, a practical example is provided to illustrate the effectiveness and correctness of the new decision-making approaches. Through comparison with existing methods, the advantages of our method are also elaborated.


Introduction
In order to solve various types of uncertainties and complex MAGDM problems, the theory of fuzzy sets is proposed by Zadeh (1965). Later on, Atanassov (1986) introduced the intuitionistic fuzzy set (IFS) theory to extend the concept of fuzzy set. Because decision makers consider both membership degree and non-membership degree in decisionmaking process, IFS theory is more accurate to deal with the uncertainties and MAGDM problems than fuzzy set. However, this theory needs to satisfy the restricted condition that the sum of the degrees of membership and the degrees of non-membership is less than or equal to 1. Under this restricted condition, the range of applications of IFSs is very narrow. Therefore, Yager (2014) proposed the Pythagorean fuzzy set (PFS) in which the restricted condition is that the square sum of the membership degree and non-membership degree is less than or equal to 1. However, both IFS and PFS have limitations to describe uncertainty and fuzziness problems. For example, when decision makers adopt 0.6 and 0.9 as membership degree and non-membership degree to express their opinions, it is obvious beyond the range of applications of IFSs or PFSs. For this, Yager (2017) again proposed a new theory which is referred to as the q-ROFS, in which the restricted condition is that the sum of the qth power of membership degree and non-membership degree is less than or equal to 1. Obviously, the range of its application is more accurate and sufficient than IFSs and PFSs. Since then, many scholars have studied the q-ROFS in many fields including the aggregation operators (Du 2018;Liu et al. 2020;Xing 2020;Liu and Wang 2017;Wei et al. 2018;Riaz and Tehrim 2020;Riaz and Hashmi 2019) and the combination with other theories (Wang 2019b;Hussain et al. 2020;Joshi 2018). On the one hand, in terms of aggregation operators there exist plenty of research results. For example, Liu and Wang (2017) gave the concept of q-rung orthopair fuzzy numbers (q-ROFNs), proposed the q-rung orthopair fuzzy weighted average (q-ROFWA) operator and applied it to MAGDM problems. In , the authors introduced the classic Bonferroni average operator into the q-ROFS, and proposed a cluster of q-rung orthopair fuzzy Bonferroni means (q-ROFBM) operators. Wang (2019a) presented a series of q-rung orthopair fuzzy Muirhead means (q-ROFMM) operators and applied them to decision-making problems. Xing (2019) defined a q-rung orthopair fuzzy point weighted aggregation operator and explored its application to MAGDM problems. In , Liu discussed the MAGDM problems based on q-rung orthopair fuzzy Heronian mean operators. Peng et al. (2018) proposed the exponential operation of q-ROFNs, and also studied the exponential aggregation operator of q-ROFSs.  presented some q-rung orthopair fuzzy hybrid aggregation operators and TOPSIS method for MAGDM problems. Liu and Wang (2019) gave a new q-ROFBM operator based on Archimedean t-norm and tconorm, and explored its applications to MAGDM problems. On the other hand, in terms of the combination with other theories, a series of fusion models have been produced in order to extend q-ROFSs. For example, by combining q-ROFSs and rough sets, Hussain (2019) proposed q-rung orthopair fuzzy rough sets, and studied their applications to MAGDM problems based on the classic decision-making method TOP-SIS. In Hussain et al. (2020), the authors defined the concept of q-rung orthopair hesitant fuzzy sets through integrating hesitant fuzzy sets with q-ROFSs. Hussain (2020) presented a q-ROFSS model by combining soft sets and q-ROFSs. On the basis of the model, he gave the q-rung orthopair fuzzy soft average aggregation operators and considered their applications. In addition, Joshi (2018) proposed the interval-value q-ROFSs and some related concepts. Wang et al. (2019) initiated a generalized interval-valued orthopair fuzzy set and applied it to decision-making problems.
In real life, there are a lot of ambiguities. In order to solve such problems, many uncertain theories have been produced, such as the fuzzy set theory (Zadeh 1965), IFS theory (Atanassov 1986), PFS theory (Yager 2014) and other theories based on soft set (Maji et al. 2001;Maji and Samanta 2010;Yang et al. 2009Yang et al. , 2013Yager et al. 2018;Riaz et al. 2021). It is noted that most of the works related on the theories focus on binary estimation (either 0 or 1), or else real numbers between 0 and 1. However, we often find that the data structure is not all binary evaluation structure in practical life, such as evaluation systems, ranking or voting situations. In reality, we usually use the number of points and stars to represent the ranking of evaluation objects. For example, one point means bad; one star means better; two stars mean good; three stars mean well; four stars mean best. To solve this issue, Fatimah et al. (2018) extended the concept of soft set theory, proposed a new model called N-soft set, and explained the importance of ordered grades in practical problems under non-binary evaluation environment.  introduced some fundamental operations on N-soft set and the notion of N-soft topology, and presented their applications by using N-soft set and N-soft topology to deal with uncertainties problems. Afterward, by combining fuzzy sets and N-soft sets, Akram and Adeel (2018) proposed the fuzzy N-soft set (FNSS). It is noted that the FNSS only considers the membership degree of the parameterized objects without considering non-membership degree. In order to remedy this defect, Akram and Ali (2019) presented the intuitionistic fuzzy N-soft set (IFNSS) theory by integrating IFSs with N-soft sets. However, in practical problems, when the decision maker evaluates the decision object from two perspectives of negative and positive, the sum of membership degree and non-membership degree provided by the decision maker may be greater than 1. In other words, IFNSSs cannot fully express decision information, which will result in the loss of decision information. In light of that, through combining PFS and N-soft set, Zhang et al. (2020) initiated the theory of Pythagorean fuzzy N-soft set (PFNSS) and applied it to MAGDM problems. However, as previously stated, both IFS and PFS have limitations to describe uncertainty and fuzziness problems. Therefore, whether it is IFNSSs or Pythagorean fuzzy N-soft set, they have fatal flaws in handling uncertain information. As a result, considering that membership degree and non-membership degree provided by the decision maker may express knowledge and information in a narrow range, it is natural to extend IFNSSs and Pythagorean fuzzy N-soft set into generalized orthopair fuzzy environment. In this paper, we attempt to establish the q-ROFNSSs through extending IFNSSs and Pythagorean fuzzy N-soft set into generalized orthopair fuzzy environment. In addition, we focus on the weighted average (WA) operators and weighted geometric (WG) operators to handle MAGDM problems based on the q-ROFNSSs.
The structure of this paper is arranged as follows. The next section reviews some basic definitions on q-ROFSs and N-soft sets. Section 3 gives the concept of q-ROFNSS. In Sect. 4, we propose the q-ROFNSWA operator and q-ROFNSWG operator based on the q-ROFNSS, and explore some properties of the q-ROFNSWG operator and q-ROFNS WG operator, such as idempotency, monotonicity and boundedness. Section 5 establishes two algorithms related to the q-ROFNSWA operator and q-ROFNSWG operator to handle MAGDM problems. In Sect. 6, a practical example is provided to illustrate the effectiveness and practicality of our decision-making method. Comparative analysis with the other methods is also conducted. We conclude in Sect. 7.

Preliminaries
In this section, we recall some fundamental notions concerning q-ROFSs and N-soft sets that are useful for discussions in the next sections.
Definition 1 (Yager 2017) Let C be an universal set, then a q-rung orthopair fuzzy set Z on C is expressed as follows: where μ Z (c) and η Z (c) denote the membership degree and non-membership degree, respectively, with the condition that for all μ Z (c), η Z (c) ∈ [0, 1] and q ≥ 1, 0 ≤ μ Z (c) q + η Z (c) q ≤ 1.
In the following, Fatimah et al. (2018) introduced the concept of N -soft sets.
Definition 4 (Fatimah et al. 2018) Let S be an universal set of objects, A be a set of attributes, and L ⊆ A. Given that R = {0, 1, 2, . . . , N − 1} is a family of ordered grades with N = {2, 3, . . .}. A triple (Q, L, N ) is referred to as an N-soft set over S if Q is a mapping Q : L → 2 S×R , where for each l ∈ L, there exists a (s, r l ) ∈ S × R such that (s, r l ) ∈ Q(l) with s ∈ S and r l ∈ R.
Example 1 Let S = {s 1 , s 2 , s 3 } be a collection of three electronic products and L = {l 1 , l 2 , l 3 } be a set of three consumers. Suppose that the consumer l 1 chooses the product s 1 , the consumer l 2 chooses the product s 2 , and the consumer l 3 chooses the product s 3 . Take R = {0, 1}. Thus, a 2-soft set Q : L → 2 S×R can be used to explain this situation as follows:

q-Rung orthopair fuzzy N-soft set
In this current section, we shall propose the some concepts of q-ROFNSSs, and an example is provided to elaborate the concept.
Definition 5 Suppose that S is an universal set of objects and A is a collection of attributes.
can be expressed as where r (k) denotes the level of the element attribute; μ(k) and η(k) denote the membership and non-membership degrees, respectively, satisfying the condition 0 ≤ μ q (k) + η q (k) ≤ 1, for all k ∈ K and q ≥ 1.
Example 2 Assume that S = {s 1 , s 2 , s 3 , s 4 , s 5 } is a collection of five electronic products and K = {k 1 =Shape, k 2 =Performance, k 3 =Price, k 4 =Lifetime} is a set of attributes. The evaluation information provided by experts can be expressed as Table 1 For being convenient to study, we use numbers to replace the symbols in Table 1. Therefore, the evaluation data of Table 1 can be converted into a 5-soft set which is shown in Table 2, where • 0 stands for '•,' • 1 stands for ' ,' • 2 stands for ' ,' • 3 stands for ' ,' • 4 stands for ' .' Due to the ambiguity and complexity of the data information, the evaluation data for these electronic products are  characterized by data in the range from 0 to 1. Therefore, when the expert evaluates an electronic product to determine its grade, we generally determine the grade of the product by using the degree of membership. Therefore, we introduce q-ROFNSSs to establish how grades are scaled. The grade standard of the evaluation provided by experts meets the following requirements: If r (k) = 3, based on the above requirements, we can take the value of μ(k) as 0.71 or any a value that satisfies the condition 0.6 ≤ μ(k) < 0.8. Further, by Definition 5, when the value μ(k) is 0.71 or any a value that satisfies the condition 0.6 ≤ μ(k) < 0.8, we can take the value of η(k) as 0.32 or any a value that satisfies the condition 0 ≤ μ 3 (k) + η 3 (k) ≤ 1. Similarly, when r (k) = 0, 1, 2 or 4, the values of μ(k) and η(k) can also be obtained. Therefore, the q-rung orthopair fuzzy 5-soft set can be expressed as follows: Remark 2 In Example 2, we just consider five assessment grades. However, in practical problems, the assessment grades do not necessarily the five grades and may be arbitrary. In this situation, the range concerning the membership value and non-membership value of q-rung orthopair fuzzy numbers can vary with actual grade requirements. When the set of alternatives and the set of attributes are both finite, by Definition 5, the q-rung orthopair fuzzy N-soft set can be expressed as follows: In what follows, we shall explore the basic operations of qrung orthopair fuzzy N -soft set, such as 'weak complement,' 'top weak complement' and 'bottom weak complement.' Definition 6 Let ( f q , K , N ) be a q-rung orthopair fuzzy Nsoft set over S. Then, the weak complement ( f q , K , N ) of ( f q , K , N ) is given as follows: Consider ( f q , K , 5) in Example 2, then its weak complement ( f q , K , 5) is given in Table 3.
Definition 7 Let ( f q , K , N ) be a q-rung orthopair fuzzy Nsoft set over S. Then, the top weak complement ( f q , K , N ) of ( f q , K , N ) is given as follows: Consider ( f q , K , 5) in Example 2, then its top weak complement ( f q , K , 5) is given in Table 4.

q-Rung orthopair fuzzy N-soft aggregation operators
In Liu and Wang (2017), Liu and Wang proposed the q-ROFWA operator and q-ROFWG operator based on the operational rules of q-ROFNs. The disadvantage of these methods is that they only focus on the degree of membership and the degree of non-membership without considering the evaluation level of attributes, which leads us to make decisions only in the binary evaluation system. However, in practical problems there exist lots of decision-making problems under non-binary evaluation environment. Faced with this situation, these methods are powerless. As a result, in order to overcome the insufficiencies and flaw of these methods, we should put emphasis on the ordered grades under non-binary evaluation environment.
In the following, we improve the q-ROFWA operator and q-ROFWG operator in Liu and Wang (2017), extend them into non-binary evaluation environment, and initiate some new aggregation operators, which are called the q-ROFNSWG operator and q-ROFNSWA operator, respectively. 1, 2, 3 . . . , n) is a collection of q-rung orthopair fuzzy numbers. Given a mapping q − ROFNSWG : H n → H such that

Definition 9 Suppose that
where H is the set of all q-rung orthopair fuzzy numbers, r (k) = (r (k 1 ), r (k 2 ), . . . , r (k n )) denotes the level of the element attribute, and E = {e 1 , e 2 , . . . , e n } is weight vector Then, the mapping q-ROFNSWG is referred to as a q-rung orthopair fuzzy N-soft weighted geometric operator.
Remark 3 In Definition 9, according to the operational rules of q-rung orthopair fuzzy numbers, we have Proof Firstly, this theorem can be proved by using the mathematical inductive method.
(1) When n = 2, we have (2) When n = j, we suppose that the equation is valid, and it is shown as follows: Then, when n = j + 1, we have Therefore, when n = j + 1, the equation is valid.
(3) According to the steps (1) and (2), we can obtain the conclusion that Eq. (2) holds for any i.
Secondly, we prove that the aggregation result obtained by Eq. (2) is still a q-ROFN. Due to μ q + η q ≤ 1, we have Hence, the aggregation result obtained by Eq. (2) is still a q-ROFN.
In the following, we shall explore some properties of the q-ROFNSWG operator, like idempotency, monotonicity and boundedness.

Theorem 4 (Boundedness) Given that
Proof By Theorem 2, we have In the light of Theorem 3, we obtain Thus, it is easy to obtain Remark 4 (1) If q = 1, then the q-ROFNSWG operator degrades into the intuitionistic fuzzy N-soft weighted geometric (IFNSWG) operator, and it is shown as follows: (2) If q = 2, then the q-ROFNSWG operator converts into the Pythagorean fuzzy N-soft weighted geometric (PFN-SWG) operator, and it is shown as follows: In what follows, a q-ROFNSWA operator will be established, and we shall investigate some of interesting properties concerning the q-ROFNSWA operator.

. , n) be two q-rung orthopair fuzzy numbers. For any i, if
Proof It is similar to the proof of Theorem 3.

Theorem 8 (Boundedness) Given that
Proof It is similar to the proof of Theorem 4.

Remark 5
(1) If q = 1, then the q-ROFNSWA operator degrades into the intuitionistic fuzzy N-soft weighted average (IFNSWA) operator, and it is shown as follows: (2) If q = 2, then the q-ROFNSWA operator converts into the Pythagorean fuzzy N-soft weighted average (PFNSWA) operator, and it is shown as follows:

The decision-making methods based on q-rung orthopair fuzzy N-soft aggregation operators
Let S = {s 1 , s 2 , . . . , s m } be a family set of m alternatives and A = {a 1 , a 2 , . . . , a n } be a set of n attributes. Given that E = {e 1 , e 2 , . . . , e n } is a weight vector of A with e j ∈ [0, 1] and n j=1 e j = 1. Suppose that (M i j ) m×n = (r (k i j ), (μ i j ,η i j )) m×n is a q-rung orthopair fuzzy N-soft decision matrix, where r (k i j ) denotes the level of the attribute a j (1 ≤ j ≤ n);μ i j andη i j are the membership degree and non-membership degree of alternative s i (1 ≤ i ≤ m) to attribute a j , respectively. Then, the methods for MAGDM based on q-ROFNSWG operator or q-ROFNSWA operator are established as follows: Step 1. Normalize the decision matrix. In practical MAG DM problems, there are two basic types of attributes: one is benefit type; the other is cost type. From Xu (2010), we know that cost type can be converted into benefit type by using the following formula: Step 2. Aggregate all the evaluation values of alternative s i by virtue of the q-ROFNSWG operator or q-ROFNSWA operator to obtain the comprehensive value m i , Step 3. Rank m i using the score function.
Step 4. Choose the best s i based on the sorting results. The higher the value m i is, the better the alternative s i is.

Application example
With the improvement of people's living standards, sports are getting more and more attention. Under the circumstances, a large number of sports clubs have been produced, such as basketball clubs, football clubs and volleyball clubs. Now, suppose that a certain sport club needs to recruit an outstanding athlete to improve the strength of the club. After investigation, the evaluation experts from the club decide to choose the best one from seven candidates s 1 , s 2 , s 3 , s 4 , s 5 , s 6 and s 7 . Assume that the indicators evaluated by the evaluation experts are composed of five attributes: explosiveness (a 1 ), patience (a 2 ), concentration (a 3 ), development potential (a 4 ) and personal quality (a 5 ). The weight vector of attributes is e = {0.25, 0.15, 0.12, 0.27, 0.21}. Now, the experts evaluate the attributes for the candidates by using q-ROFNs. Furthermore, a q-rung orthopair fuzzy 5-soft decision matrix is shown in Table 6 as follows.

Decision-making steps
Firstly, the decision-making steps are performed by using the q-ROFNSWA operator.
Step 1. Because all the attributes we give are benefit types, our decision matrix is a normalized decision matrix. The result is shown in Table 6.
Step 2 Step 4. Now, we make the conclusion that s 2 is the best candidate.
Secondly, the decision-making steps are conducted by using the q-ROFNSWG operator.
Step 1. A normalized decision matrix is shown in Table 6.
Step 2 Step 4. Now, we make the conclusion that s 2 is the best candidate.

Comparative analysis with the other methods
To expand on the advantages of the developed methods , we compare them with the existing methods by solving the same example, such as the Pythagorean fuzzy weighted geometric (PFWG) (Garg 2017) operator, the q-ROFWA (Liu and Wang 2017) operator and the q-ROFMM(Wang 2019a) operator. Applying the above-mentioned methods, we obtain the comparison results shown in Table 7. The results of Table 7 can also be more intuitively characterized by Fig. 1.
From Table 7 and Fig.1, we observe that the optimal ranking results are essentially the same in the different methods, even though the score functions are different in different methods. Therefore, the novel methods proposed by us are reasonable and valid. Moreover, by comparing the proposed q-ROFNSWA method with the existing methods from Table  7 and Fig. 1, we observe that when there are seven candidates, there is little difference in the score functions obtained from the existing methods including PFWA, q-ROFWA and q-ROFMM. Therefore, we can infer that when the number of candidates is increased, the decision effect of the existing methods will be further weakened, and it is also unable to describe the uncertainty and vagueness problems very well.    That is, the effect of applying the existing methods in such a case is not obvious. On the other hand, these existing methods put emphasis on describing a certain problem by using the membership degree and the non-membership degree without considering the evaluation level of attributes, which leads us to make decisions only in the binary evaluation system. Unfortunately, in real life, the data structure is not all binary evaluation structure, such as evaluation systems, ranking and voting situations. Under the circumstances, just using the membership degree and non-membership degree is insufficient to provide the accurate information about the ranking of the objects in ranking or evaluation systems. Having said all of above, the existing methods still remain in many insufficiencies and flaw. As for the limitations, two decision-making methods we propose based on q-ROFNSs can avoid those defects of the existing decision-making methods, whose the significant advantage is that they not only focus on the degree of membership and the degree of non-membership, but also consider the evaluation level under non-binary evaluation environment. Therefore, decision-making processing based on the novel methods can be performed in a non-binary evaluation system, which implies that the proposed methods contain more information and have more accuracy and flexibility than the existing decision-making methods. In order to further analyze the flexibility and sensitivity of the attributes q, we set the different values q to rank the alternatives s i and the ranking results are shown in Table 8 and Fig. 2.
From Table 8 and Fig. 2, we observe that when applying the q-ROFNSWA operator, the optimal ranking results are always the same with different values q. When the value of the parameter q is relatively small, the score function is relatively large. When the value of the parameter q is large, the score function is relatively small. It is well-known that the value of parameter q stands for the pessimistic or optimistic attitude of the decision maker toward the decision plan. When the decision maker is pessimistic about the decision plan, he or she can assign a larger value to the parameter q in the q-ROFNSWA operator. Conversely, when the decision maker is S(m 1 )=0.2696,S(m 2 ) =0.4577,S(m more optimistic about the decision plan, a smaller value can be assigned to the parameter q in the q-ROFNSWA operator. So this is for the q-ROFNSWG operator. Meanwhile, from Table 8 and Fig. 2, we observe that the optimal ranking results are different with different values q in the q-ROFNSWG operator. Concretely, in the q-ROFNSWG operator the optimal ranking results change from s 2 to s 1 as the parameter q increases. In light of that the method based on q-ROFNSWA operator is relatively more stable than the method based on q-ROFNSWG operator. Furthermore, it is well-known that the intuitionistic fuzzy numbers and Pythagorean fuzzy numbers need to satisfy the restricted condition 0 ≤ μ(k) + η(k) ≤ 1 and 0 ≤ μ 2 (k) + η 2 (k) ≤ 1, respectively. Under the restricted conditions, the range of applications of intuitionistic fuzzy numbers and Pythagorean fuzzy numbers is narrower than that of q-ROFNs, as shown in Fig. 3. This further illustrates the necessity of fusion research between N-soft sets and q-ROFSs. In addition, the decision matrix we provide is beyond the range of applications of IFNSSs. Therefore, the IFN-SWA and IFNSWG methods cannot be used to deal with the decision-making problem in this paper.
In view of the above analysis, the methods we propose in this paper have greater advantages than the existing methods including the PFWG, the q-ROFWA and the q-ROFMM, which are also more flexible and reasonable than the existing methods. Therefore, the novel methods are more suitable for solving MAGDM problems.

Conclusion
In this study, we introduce a q-ROFNSS model, and investigate two operators including the WA operator and WG operator to handle MAGDM problems based on q-ROFNSS. Then, two algorithms are established based on the q-ROFNSWA operator and q-ROFNSWG operator. A practical example is provided to illustrate the effectiveness of two algorithms, and we also compare the novel decision-making methods with the existing methods. Through comparison and analysis, we make the conclusion that the method based on q-ROFNSWA operator possesses the more advantages than other methods to deal with MAGDM problems. In future research, the combination of the q-ROFNSS and other aggregation operators will be explored. Furthermore, the model will be applied to MAGDM problems.
Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions.
Author Contributions HZ contributed to the manuscript preparation and the conception of the study, and made important revisions to the paper; TN wrote the manuscript; and YH performed the experiment and the data analysis.

Data availability
The authors confirm that the data supporting the findings of this study are available within the article.

Conflict of interest
The authors declare that there is no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.