A Method to Multi-Attribute Group Decision-Making Problem with Complex q-Rung Orthopair Linguistic Information Based on Heronian Mean Operators

The notions of complex q-rung orthopair fuzzy sets (Cq-ROFSs) and linguistic sets (LSs) are two different concepts to deal with uncertain information in multi-attribute group decision-making (MAGDM) problems. The Heronain mean (HM) and geometric Heronain mean (GHM) operators are an effective tool used to aggregate some q-rung orthopair linguistic fuzzy numbers (q-ROLFNs) into a single element. The purpose of this manuscript is to propose a new concept called complex q-rung orthopair linguistic sets (Cq-ROLSs) to cope with complex uncertain information in real decision-making problems. Then the fundamental laws and their examples of the Cq-ROLSs are also given. Furthermore, the notions of complex q-rung orthopair linguistic Heronian mean (Cq-ROLHM) operator, complex q-rung orthopair linguistic weighted Heronian mean (Cq-ROLWHM) operator,complexq-rungorthopairlinguisticgeometricHeronianmean(Cq-ROLGHM)operator,complexq-rungorthopairlinguis-ticweightedgeometricHeronianmean(Cq-ROLWGHM)operatorareproposedandtheirbasicpropertiesarealsodiscussed. Moreover,wedevelopanovelapproachtoMAGDMusingproposedoperatorsandanumericalexampleisusedtodescribethe flexibilityandexplicitlyoftheinitiatedoperators.Inlast,thecomparisonbetweenproposedmethodandexistingworkisalso discussedindetail.


INTRODUCTION
The framework of the complex fuzzy set (CFS) was proposed by Ramot et al. [1], which is a generaeronian mean lization of a fuzzy set (FS) [2]. The difference between CFS and FS is that the range of CFS is not restricted to [0, 1], but is extended into a unit disc in a complex plane. The CFS has received more attention in the environment of FS theory. While, Alkouri and Salleh [3] proposed the notions of linguistic variable, hedges and several distances on CFS. Yazdanbakhsh and Dick [4] proposed time-series forecasting via complex fuzzy logic and a systematic review of CFS. Recently, Bi et al. [5] proposed complex fuzzy geometric aggregation operators. Because of its merits and advantages, CFS has been extensively applied to decision-making problems and other fields [6][7]. Because FS and CFS can only describe the membership degree and complex-valued membership degree, and cannot express the non-membership degree and complex-valued non-membership degree. Then the framework of intuitionistic fuzzy set (IFS) is introduced by Atanassov [8] as a generalization of FS by including nonmembership degree. The IFS is characterized by two different degrees such as membership and non-membership grades, and their sum is limited to [0,1]. IFS has been extensively used in different fields [9][10]. Further, Alkouri et al. [11] proposed the framework of complex intuitionistic fuzzy set (CIFS) as a generalization of FS to deal with uncertain and unpredictable information in real-life problems. The CIFS is characterized by complex-valued membership grade and complex-valued non-membership grade in the form of polar coordinates. Because there is a restrict condition in IFS, further, Yager [12] initiated the idea of Pythagorean fuzzy set (PFS) as an effective tool to describe the uncertainty for the multi-attribute group decision making (MAGDM) problems. The notion of PFS is more general than IFS and FS to cope with difficult information in real decision problems. When a decision maker provides (0.6,0.7) for membership and non-membership grades, i.e., 0.6 + 0.7 = 1.3 ≥ 1, the IFS cannot describe it effectively, but the PFS can describe such kinds of information effectively, i.e., 0.6 2 + 0.7 2 = 0.36 + 0.49 = 0.85 < 1. Based on PFS, Garg [13,14] proposed a novel correlation coefficient between PFSs, and a new generalized Pythagorean fuzzy Einstein aggregation operators and their application to decision-making were developed. Dick et al. [15] introduced Pythagorean and complex fuzzy operations. Garg [16] further proposed a novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multi-attribute decision-making (MADM) problem. Further, some new MADM methods and *Corresponding author. Email: peide.liu@gmail.com operators about PFSs were developed. Ren et al. [17] developed the notion of Pythagorean fuzzy TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision-Making) approach to MADM. Garg [18] proposed a new improved score function of an interval-valued PFSs based TOPSIS method. Wei and Wei [19] proposed the concept of similarity measures for PFSs based on the cosine function and their applications. Garg [20], proposed generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for the multi-criteria decision-making process. Wei [21] proposed the Pythagorean fuzzy interaction aggregation operators and their application to MADM problems.
In some particular cases, the IFS and PFS are failed, if a decision maker provides (0.9, 0.7) for membership and non-membership degrees, i.e., 0.9 + 0.7 = 1.6 ≥ 1 and 0.9 2 + 0.7 2 = 0.81 + 0.49 = 1.30 ≥ 1, the IFS and PFS cannot describe effectively such kinds of information. To precisely cope with such kind of problems, Yager [22] proposed the framework of q-rung orthopair fuzzy set (q-ROFS) whose restriction is that the sum of q-power of membership and q-power of non-membership grade is belonging to [0,1]. Obviously, the q-ROFS can describe effectively such kinds of information, i.e., 0.9 4 + 0.7 4 = 0.7 + 0.24 = 0.94 ≤ 1. The FS, IFS, and PFS all are the special cases of q-ROFS, this characteristic makes q-ROFS more general than existing FSs. For example, if q=1 and non-membership equals to zero then, the q-ROFS is converted to FS. If q=1, then the q-ROFS is converted to IFS. If q=2, then the q-ROFS is converted to PFS. To better understand the relationship among q-ROFS, PFS, and IFS, please see Figure 1.
Further, Liu and Liu [23] initiated the concept of some q-rung orthopair fuzzy Bonferroni mean operators and their application to MAGDM. Wei et al. [24] proposed some q-rung orthopair fuzzy Heronian mean (HM) operators. Liu and Wang [25,26] proposed some q-rung orthopair fuzzy aggregation operators and their application to MADM based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers. Further, the power maclaurin symmetric mean [27], partitioned maclaurin symmetric mean [28], as a powerful operator to aggregate the interrelation among q-ROFNs, were developed. Li et al. [29] proposed q-rung orthopair linguistic HM operators with their application to MAGDM.
Moreover, the linguistic variable (LV), proposed by Zadeh [30], can easily express the qualitative information. Many researchers combined the notion of LV with IFS, PFS, q-ROFS, and proposed the novel concepts of intuitionistic linguistic fuzzy numbers [31], linguistic Pythagorean FS [32] and q-rung orthopair linguistic HM operators [29]. Obviously, these combinations can easily describe the complex fuzzy information.
Consequently, motivated by the idea from IFS to CIFS, it is necessary to extend q-ROFS to complex q-ROFS (Cq-ROFS) because Cq-ROFS is a powerful idea to cope with uncertain and unpredictable information, and it is also a generalization of CFS and FS, whose constraint is like q-ROFS, but the range of membership and non-membership grades are bounded to unit disc in a complex plane instead of [0,1]. The complex-valued membership and complex-valued non-membership grades are represented in the polar form. The q-ROFS copes with onedimension information at a time in a single elements, which results in data loss sometimes. But, the Cq-ROFS is a powerful tool to deal with uncertain information as compared to q-ROFS, because it contains two-dimension information in a single elements. So by introducing the second dimension to the grade of membership and non-membership, loss of data can be avoided. At the same time, motivated by combining the LV with q-ROFS, it is meaningful to combine the LV with Cq-ROFS, and propose complex q-rung orthopair linguistic number (Cq-ROLN) which is more general than existing fuzzy sets, such as complex Pythagorean linguistic set (CPYLS) and complex intuitionistic linguistic set (CILS). If we take the imaginary part is zero, in the terms of membership grade and non-membership grade, then the Cq-ROLS is convert into q-rung orthopair linguistic set (q-ROLS), i.e., q-ROLS is its special case. If we set the value of parameter q = 1 in the environment of q-ROLS, then the q-ROLS is converted for intuitionistic linguistic set (ILS). Similarly, if we take the value of parameter q = 2 in the environment of q-ROLS, then the q-ROLS is converted for Pythagorean linguistic set (PYLS). The ILS and PYLS are the particular Figure 1 Geometrical interpretation of q-rung orthopair fuzzy set. Pdf_Folio:2

PRELIMINARIES
In this section, we will review the existing concepts and initiate the idea of Cq-ROFSs. The operational laws of Cq-ROFSs are also discussed in detail.

The q-ROFS
In this sub-section, we review some basic concepts of q-ROFS, LV, HM, GHM and their operations. Definition 1. [22] For ordinary fixed set X, the q-ROFS is given by where ′ (x) , ′ (x) ∶ X → [0, 1] denoted the membership and non-membership degrees respectively, satisfying the condition 0 ≤ . The hesitancy degree is defined by is called q-ROFN. The geometrical interpretation of Cq-ROFS is shown in Figure 1.
, the score and accuracy functions are defined by For any two q-ROFNs and ) , then we have 2. If S (P 1 ) = S (P 2 ), then 2. If H (P 1 ) = H (P 2 ), then If P 1 = P 2 .  HM and geometric Heronian mean (GHM) are a more generalized operators than existing operators like averaging mean operator, geometric mean operator, weighted averaging mean operator, weighted geometric mean operator and more others. The operators which are discussed in [5,20,24,25,29,31,33,34] are all the special cases of the proposed operators. In this article, we will use HM and GHM operators to propose the complex q-rung orthopair linguistic Heronian mean and complex q-rung orthopair linguistic geometric Heronian mean operators. Definition 5. [29] For a set of crisp numbers P i (i = 1, 2, .., n) withs, t > 0, Heronian mean (HM) is given by

The Complex q-Rung Orthopair Fuzzy Set
In this section, we will propose the notion of complex q-rung orthopair fuzzy set (Cq-ROFS) and their operations.

Definition 7.
For ordinary fixed set X, the Cq-ROFS is given by . The hesitancy degree is defined by ) is called complex q-rung orthopair fuzzy number (Cq-ROFN). Simply we write ( e i2 , e i2 ) .
Next, we will examine a numerical example for Cq-ROFNs as follows:

Example 1
We consider the two Cq-ROFNs for q = 2 and = 3 such that P 1 = and , then we have , the score and accuracy functions are defined by Definition 10. For any two Cq-ROFNs P 1 = ( 1 e i2 1 , 1 e i2 1 ) and P 2 = ( 2 e i2 2 , 2 e i2 2

Example 2
We consider the two Cq-ROFNs for q = 2 and such that P 1 = and . Then the score values of the P 1 and P 2 are calculated as follows: So it is clear that S (P 2 ) > S (P 1 ), then we say that P 2 > P 1 . If (P 2 ) = S (P 1 ), the we will use the accuracy function of the Cq-ROFNs.

COMPLEX q-RUNG ORTHOPAIR LINGUISTIC SET
Motivated by the notion of Cq-ROFS and LV, we will initiate the novelty of Cq-ROLS by combing the two different concepts. Throughout this article, is represented the continuous linguistic term set of = { i /i = 1, 2, .., z}.
Next, we defined some operations for Cq-ROLNs.
First, we give the numerical example for Cq-ROLNs and the four points of Definition 8. Then, we will proposed the ideas of score function and accuracy function for compression between two Cq-ROFLNs.

Example 3
We will consider the two Cq-ROLNs P 1 =  )) with q = 2 and = 3, then we can get 1.

Example 4
We will consider the two Cq-ROLNs P 1 =  So it is clear that S (P 1 ) > S (P 2 ), then we say that P 1 > P 2 . If S (P 2 ) = S (P 1 ) , then we will use the accuracy function of the Cq-ROFLNs. Pdf_Folio:8

COMPLEX q-RUNG ORTHOPAIR LINGUISTIC HM OPERATORS
In this section, we generalize the HM operator to Cq-ROLS and propose the concepts of Cq-ROLHM, Cq-ROLWHM, Cq-ROLGHM, Cq-ROLWGHM operators and discuss their properties in detailed, where s, t.
According to the operational laws of Cq-ROLNs, we can get the following results.
Proof: Using the Definition 12, we get Further, we discuss the properties of Cq-ROLNs as follows.
for all i = 1, 2, … , n and j = i, i + 1, .., n. Then it is clear that for linguistic number Similarly procedure for imaginary-valued membership grades, we get Combined both values, we have Nest we will describe the real-valued non-membership grade such that Similarly procedure for imaginary-valued non-membership grades, we get Combined both values, we have So by combining the values of complex-valued membership and complex-valued non-membership grades, then we get Hence proved the result.
Hence proved the result.

Special Cases
In this sub-section, the particular cases of Cq-ROLHM operator is discuss about the parameters s and t.
According to the operational laws of Cq-ROLNs, we can get the following result.

))
, i = 1, 2, … , n be a family of Cq-ROLNs, then the Cq -ROLGHM s,t is defined: where n is denoted the family of all Cq-ROLNs.
According to the operational laws of Cq-ROLNs, we can get the following result.

Special Cases
In this sub-section, the particular cases of Cq-ROLGHM operator is discussed based on the parameters s and t.
According to the operational laws of Cq-ROLNs, we can get the following result.

A NEW MULTI-ATTRIBUTE GROUP DECISION-MAKING (MAGDM) METHOD
In this section, we would propose a new decision-making method with complex q-rung orthopair linguistic information. Consider the set of alternatives and the set of attributes with respect to weight vectors, i.e., X = {x 1 , x 2 , .., x m } isset of alternatives, Y = {y 1 , y 2 , .., y n } is set of attributes, and ω = (ω 1 , ω 2 , .., ω n ) T is the weight vector of the attributes such that ∑ , and the complex q-rung orthopair linguistic decision matrices is represented by . Then we will use two different operators to solve this problem. The procedure of the MAGDM is shown as follows: 1. Construct the decision matrices, it is necessary to consider two kinds of attribute like cost and benefits. The decision matrices is obtained by The symbol I 1 and I 2 represent the benefits and cost attributes.

Use the Cq-ROLWHM operator
Or the Cq-ROLWHM operator To aggregate the decision matrices A k = 3. Use the Cq-ROLWHM operator Or the Cq-ROLWHM operator To aggregate the decision matrices A k = ( P k ij ) m×n into a single value Cq-ROLN.
4. Calculate the score function and accuracy function of Cq-ROLNs.
5. Rank to all Cq-ROLNs and choose the best alternative.

Example 6
In this sub-section, we adopted a numerical example from [29] to show the application of the proposed method. The saving enterprise wants to invest its share with another enterprise. After search, there are four possible enterprises in the list of applicants which are

Decision-Making Process
The steps of this decision-making problem are given as 1. The four attributes are all benefits types, so we cannot normalize the decision matrix. Table 1 Complex q-rung orthopair linguistic decision matrix R 1 by D 1 )) ( 4 ,

))
Pdf_Folio:24 2. We will consider the Cq-ROLWHM operator To aggregate the decision matrices A k = which is shown in Table 4 for q = 3.
3. We use the Cq-ROLWGHM operator To aggregate the decision matrix (in Table 4) and get the comprehensive value of four alternatives which is listed in Table 5.
4. Calculate the score functions of four alternatives which is listed in Table 6.
5. Rank all Cq-ROLNs and choose the best alternative.
So, A 1 is the best alternative.

Data Analysis
)) Table 3 Complex q-rung orthopair linguistic decision matrix R 3 by D 3 )) ( )) )) Table 6 The score function for four alternatives

Cq -ROLN s Score Function Ranking
CIFWA operator proposed by Garg and Rani [33] S ( WDM for PYFS proposed by Ullah et al. [36] S ( Method based on Cq-ROLS in this paper for q = 1 Method based on Cq-ROLS in this paper for q = 2 Method based on Cq-ROLS in this paper In order to explain the validity of the proposed method, we use the method for CILS and the method for CPYLS, the ranking results are listed in Table 7.
The geometrical interpretation of the proposed method with existing methods are discussed in Figure 3.
From Table 7, we can get the same ranking result, it can explain the validity of the proposed method.

The Influence of Parameters on Ranking Results
The parameters in the proposed operators play a key role on the final ranking results. By example 6, we assign different values to parameters s and t, and discuss the ranking results which are shown in Table 8.
From Table 8, we can see that although the best choice is the same, the ranking order is different, this can explain the parameters s and t can affect the ranking results.
In order to show clearly the ranking results, we consider the values of parameters for s = t, then the score values of alternatives A i (i = 1, 2, 3, 4) are shown in Figure 4.

Advantages of the Proposed Cq-ROLS with the Existing CFSs
The HM operators for CILS and CPYLS are also the special cases of our proposed method.
The following examples can explain the generalization of the proposed Cq-ROLS.

Example 7
In some practical examples, the CILS cannot described effectively, because the restriction of of CILS is that the sum of membership (for real part and imaginary part) and non-membership (for real part and imaginary part) are limited to 1. So we considered the complex Pythagorean linguistic kinds of information, and solved by our proposed methods and then compared with existing methods. The weight vectors are given by ω = {0.34, 0.32, 0.11, 0.23} T . The complex Pythagorean linguistic decision matrix R shown in Table 9.
The aggregation results for different approaches shown in Table 10.
From Table 10, we can get (1) CILS cannot express the information described by CPYLS; (2) the proposed method in this paper can the same ranking results as method in [33], which can show the effectiveness of the proposed method because the Cq-ROLS is reduced into CPYFS when q = 2, s = t = 5.

Example 8
In this example, we consider the information is expressed by Cq-ROLNs which is listed in Table 11, and the weight vectors is taken from example 6.
Then the ranking results are listed in Table 12 ( for q = 5, s = t = 1 ) .
From Table 12, we can know the Cq-ROLS is more generalized than existing CFSs, so we easily find that our proposed method is more superior and more reliable than existing methods.

Figure 4
Scores of alternatives for parameters s and t Table 9 Decision matrix for complex Pythagorean linguistic information's

Example 9
In this example, we consider the information expressed by Pythagorean linguistic sets, which is listed in Table 13, and the weight vectors is taken from example 6. The information discussed in this example is taken from [29].
We will convert the Table 13 into Table 14, and we also clear that about e 0 = 1.
Then the ranking results are listed in Table 15 ( for q = 5, s = t = 1 ) .
From Table 15, we can know the Cq-ROLS is more generalized than existing CPYLS, CILS, q-ROLS, PYLS, ILS and CFSs, so we easily find that our proposed method is more superior and more reliable than existing methods. Therefore, the proposed method is more generalized than existing to cope with uncertain and complicated types of information easily.

The Qualitative Comparison with the Existing Methods
In this sub-section, we give some comparisons with some existing methods from a qualitative point of view. We compare our method with the work proposed by Ullah et al. [36] based on the similarity measures for complex PFS, the method proposed by Rani and Garg [35,37] based on the distance measures and power aggregation operators for CIFS, the method proposed by Garg and Rani [38,33] based on some robust correlation coefficient and generalized CIFS and their aggregation operators. The characteristic comparison of the proposed method with existing works is shown in Table 16. )) ( cq-ROLS, complex q-rung orthopair linguistic set.
Method based on Cq-ROLS in this paper cq-ROLS, complex q-rung orthopair linguistic set.