The q-Rung Orthopair Hesitant Fuzzy Uncertain Linguistic Aggregation Operators and Their Application in Multi-Attribute Decision Making

This paper combines the q-rung orthopair hesitant fuzzy sets (q-ROHFSs) with the uncertain linguistic variables, and proposes the q-rung orthopair hesitant fuzzy uncertain linguistic sets (q-ROHFULSs). In addition, the Schweizer-Sklar T-norm is introduced, and a multi-attribute decision-making method based on the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar aggregation operators is established. Firstly, based on the Schweizer-Sklar T-norm, the operational properties of q-rung orthopair hesitant fuzzy uncertain linguistic elements are defined, and the score function, accuracy function and ranking method of the q-rung orthopair hesitant fuzzy uncertain linguistic elements are proposed. Secondly, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar Bonferroni mean (BM) operator and geometric Bonferroni mean (GBM) operator, Maclaurin symmetric mean (MSM) operator and dual Maclaurin symmetric mean (DMSM) operator, Maclaurin mean (MM) operator and dual Maclaurin mean (DMM) operator are defined. The calculation formulas of the operators are given, the related properties are studied, and the special forms of the operators are discussed. Finally, a multi-attribute decision-making model based on the q-rung orthopair hesitant fuzzy uncertain linguistic aggregation operators is established, and the feasibility and effectiveness of the decision-making method are demonstrated through calculation examples and comparative analyses.


I. INTRODUCTION
Multi-attribute decision-making refers to the decisionmaking problem of sorting alternatives and choosing the best while considering multiple attributes. It is an important part of modern decision-making science and is widely used in performance evaluation, factory site selection, and supplier selection and other fields. In recent years, due to the increasingly prominent uncertainty and ambiguity of the decision-making environment, the corresponding fuzzy multi-attribute decision-making theory has emerged as the times require, and has gradually become a key issue in The associate editor coordinating the review of this manuscript and approving it for publication was Mu-Yen Chen . the field of decision science. Since Zadeh [1] proposed the concept of fuzzy set in 1965, fuzzy set theory has been widely used in marginal detection [2], pattern recognition [3], image reconstruction [4], decision-making [5] and other fields. In addition, many scholars have carried out research and development on fuzzy set theory. On the one hand, on the basis of fuzzy sets, Atanassov considers the membership degree and non-membership degree of elements belonging to the set at the same time, and proposed the concept of intuitionistic fuzzy sets (IFSs) [6]. Yager proposed the Pythagorean fuzzy sets (PFSs) [7], [8], which can describe the situation where the sum of the membership degree and non-membership degree exceeds 1, and the sum of the squares does not exceed 1. However, there are some situations that Pythagorean fuzzy sets cannot describe. Yager [9] further generalized the Pythagorean fuzzy sets and proposed the concept of q-rung orthopair fuzzy sets (q-ROFSs). The q-rung orthopair fuzzy sets is the generalization of intuitionistic fuzzy sets and Pythagorean fuzzy sets, which can describe fuzzy phenomena more widely. On the other hand, Torra considered the hesitation of decision makers on the basis of fuzzy sets, and proposed the concept of hesitant fuzzy sets(HFSs) [10]. Khan et al. [11] combined the hesitant fuzzy set with the Pythagorean fuzzy set, proposed the Pythagorean hesitant fuzzy sets(PHFSs), gave the distance formula of the Pythagorean hesitant fuzzy element, and a multi-attribute decision-making method based on Pythagorean hesitant fuzzy weighted average operator is constructed. In addition, Liu et al. [12] combined hesitant fuzzy sets and the q-rung fuzzy sets, and proposed the concept of the q-rung orthopair hesitant fuzzy sets (q-ROHFSs), which can better describe the uncertainty and fuzzy phenomena in reality.
However, both Pythagorean hesitant fuzzy sets and q-rung orthopair hesitant fuzzy sets can only quantitatively describe the membership degree and non-membership degree of a fuzzy concept [13]. In practical problems, experts prefer to give a qualitative evaluation of the attributes of the scheme [14]. For example, in the selection of investment options, decision makers often use linguistic variables such as ''general'' and ''higher'' to express evaluation opinions regarding the economic benefits of the options.
When linguistic variables express the expert's decisionmaking opinions, they can only express their qualitative decision-making opinions and ignore the quantitative decision-making opinions. For this reason, Wang et al. [15] combined linguistic variables with intuitionistic fuzzy sets and proposed the intuitionistic linguistic sets (IFLSs). Since the intuitionistic linguistic term set was put forward, it has attracted many scholars to study and discuss it [16]- [18]. Similarly, Peng and Yang [19] combined linguistic variables with the Pythagorean fuzzy set, and proposed the Pythagorean fuzzy linguistic sets (PFLSs). In addition, Wang et al. [20] combined linguistic variables with q-rung orthopair fuzzy sets and proposed q-rung orthopair fuzzy linguistic sets (q-ROFLSs), and studied the q-rung orthopair fuzzy linguistic aggregation operators and their applications in decision-making. The q-rung orthopair fuzzy linguistic set is a generalization of the intuitionistic linguistic set, which can use linguistic variables to express qualitative opinions, and can also describe the membership degree and non-membership degree of decision attributes to linguistic variables.
In practical problems, uncertain linguistic variables can more fully represent fuzzy data than linguistic variables. Therefore, Liu and Jin [21] combined uncertain linguistic variables and intuitionistic fuzzy sets, and proposed the concept of intuitionistic uncertain linguistic sets (IFULSs). Since the introduction of the intuitionistic uncertain linguistic set, many scholars have carried out research and development on the theory of intuitionistic uncertain linguistic set [22]- [24]. Liu and Liu [25] studied the intuitionistic uncertain linguistic Bonferroni mean aggregation operator and its application. Liu and Wang [26] studied the intuitionistic uncertain linguistic power geometric aggregation operator and its application in project evaluation. Liu [27] combined uncertain linguistic variables and interval intuitionistic fuzzy sets, and proposed the concept of interval intuitionistic uncertain linguistic sets (IVFULSs). Meng and Chen [28] studied the interval intuitionistic uncertain linguistic hybrid weighted average operator and its application in multi-attribute decisionmaking. In addition, Lu and Wei [29] combined uncertain linguistic variables with Pythagorean fuzzy sets and proposed the Pythagorean uncertain linguistic sets (PFULSs). Geng et al. [30] proposed a multi-attribute group decisionmaking model of Pythagoras uncertain linguistic set. Wang et al. [31] combined uncertain linguistic variables with q-rung orthopair fuzzy sets and proposed the q-rung orthopair fuzzy uncertain linguistic sets (q-ROFULSs). Liu et al. [32] studied the aggregation operators of the q-rung orthopair fuzzy uncertain linguistic sets and their applications in multi-attribute decision-making.
However, the intuitionistic uncertain linguistic set cannot describe the degree of hesitant of the decision maker. For this reason, drawing on the ideas of Pythagorean uncertain linguistic sets, this paper combines the q-rung orthopair hesitant fuzzy sets with uncertain linguistic sets, and proposes the q-rung orthopair hesitant fuzzy uncertain linguistic sets (q-ROHFULSs). In addition, the Schweizer-Sklar T-norm is introduced into the q-rung orthopair hesitant fuzzy uncertain linguistic sets, and the Schweizer-Sklar Bonferroni mean (BM) operator, Maclaurin symmetric mean (MSM) operator and Maclaurin mean (MM) operator of are defined, and the multiple attributes decision-making methods of the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar aggregation operator are established. In summary, the innovations of this article are: 1) The q-rung orthopair hesitant fuzzy uncertain linguistic set is an extension and generalization of the intuitionistic uncertain linguistic set, which makes up for the lack of q-rung orthopair hesitant fuzzy set and uncertain linguistic set in describing decision information, and it can fully express the true opinions of decision makers. And the q-rung orthopair hesitant fuzzy uncertain linguistic set further considers the hesitation of the decision maker, its application range is wider, and it can describe the decision information of the decision maker more widely. 2) The q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar aggregation operator can flexibly select different parameter values according to the decision-making situation to meet the requirements of different decision-making problems in practice. In addition, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar MM operators can describe the correlation between decision attributes, fully reflect the decision information, reduce the information loss in the decision, and make the decision result more reasonable and reliable. VOLUME 8, 2020 The structure of this paper is as follows: In Chapter 2, the definitions and related knowledge of q-rung hesitant fuzzy sets and uncertain linguistic variables are briefly introduced. In Chapter 3, the definition of the q-rung orthopair hesitant fuzzy uncertain linguistic set is given, the score function and the accuracy function of the q-rung orthopair hesitant fuzzy uncertain linguistic element are proposed, and based on Schweizer-Sklar T-norm, the operation properties of q-rung orthopair hesitant fuzzy uncertain linguistic elements are defined. In Chapter 4, Chapter 5, and Chapter 6, the Schweizer-Sklar BM operator, MSM operator, and MM operator of the q-rung orthopair hesitant fuzzy uncertain linguistic sets are defined, respectively. The calculation formulas of the operators are given, the related properties are studied, and the special forms of the operators are discussed. In Chapter 7, a multi-attribute decision-making method based on the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar MM operator is established. Chapter 8 introduces an example of PhD admissions selection evaluation and uses the proposed decision-making model to get the final decision result. The sensitivity analysis of different parameters in the operators is carried out, and other methods are further introduced to carry out comparative analysis, which shows the validity and rationality of the decision-making method in this paper. Chapter 9 summarizes the research content.

A. THE Q-RUNG ORTHOPAIR HESITANT FUZZY SET
Based on the HFSs and q-ROFs, Khan et al. [11] proposed the q-rung orthopair hesitant fuzzy sets (q-ROHFSs). Some fundamental theorems regarding the q-ROHFSs are presented as follows.
Definition 1 [11]: Let X = {x 1 , x 2 , . . . , x n } be a fixed set, then a q-ROHFS on X can be represented as: where A (x i ) and A (x i ) are two sets of values in [0, 1], denoting all the possible membership degrees and non-membership degrees of the element x i ∈ X , and they satisfies the following conditions: is called the q-rung orthopair hesitant fuzzy element (q-ROHFE), and for convenience, the q-rung orthopair hesitant fuzzy element is recorded as h = < h , h > q . The overall q-rung orthopair hesitant fuzzy elements are denoted as q − ROHFE(X ).
In particular, when | h | = | h | = 1, the q-rung orthopair hesitant fuzzy set reduces to the q-rung orthopair fuzzy set; when q = 1, the q-rung orthopair hesitant fuzzy set reduces to the dual hesitant fuzzy set.
Definition 2 [11]: Let h = < h , h > q , h 1 = < h 1 , h 1 > q and h 2 = < h 2 , h 2 > q be three q-ROHFEs, for λ>0, q ≥ 1, the operational laws between them are described as follows: Definition 3 [11]: , the numbers of set h and h are denoted as | h | and | h |, respectively, then a score function S(h) can be defined as: The accuracy function H (h) can be defined as: Definition 4 [11]:

B. UNCERTAIN LINGUISTIC VARIABLES
The linguistic term set S = {s 0 , s 1 , s 2 , . . . , s γ } is composed of an odd number of elements, that is, γ should be an even number. For any linguistic term set S, the following conditions should be met [33]: 1) If i>j, then s i >s j , it means that s i is superior to s j ; 2) There is a negative operator neg(s i ) = s j , making j = γ − i; 3) If s i ≥ s j , it means that s i is not inferior to s j , then max(s i , s j ) = s i ; 4) If s i ≤ s j , it means that s i is not better than s j , then min(s i , s j ) = s i . For any one linguistic term set S = {s 0 , s 1 , s 2 , . . . , s γ }, there is a strictly monotonically increasing relationship between element s i and its subscript i. Therefore, the function f : s i = f (i) can be defined, and the function f (i) is a strictly monotonically increasing function with the subscript i.
Extending the discrete linguistic term set S = {s 0 , s 1 , s 2 , . . . , s γ } into a continuous linguistic term set S = {s t |t ∈ [0,p],p ≥ γ }, the extended linguistic term set still satisfies a strict monotonic increasing relationship.
Definition 4 [34]: Let S = {s 0 , s 1 , s 2 , . . . , s γ } be a linguistic term set, and S = {s t |t ∈ [0, p], p ≥ γ } be an extended linguistic term set. Assume s = [s θ , s τ ], s θ , s τ ∈ S, and θ ≤ τ , where s θ and s τ are the lower and upper limits of s, respectively, then s is called the uncertain linguistic variable, and S is called the set of all uncertain linguistic variables.

III. THE Q-RUNG ORTHOPAIR HESITANT FUZZY UNCERTAIN LINGUISTIC SETS
By combing the uncertain linguistic sets with q-ROHFSs, we propose the concept of q-ROHFULSs. Definition 6: Let [s θ (x) , s τ (x) ] ∈ s and X is a given domain, then a q-ROHFULS is defined as follows where A (x i ) and A (x i ) are two sets of values in [0, 1], denoting all the possible membership degrees and nonmembership degrees of the element x to the uncertain linguistic variable [s θ (x) , s τ (x) ], respectively. And they satisfies the following conditions: is the linguistic part of the q-rung orthopair hesitant fuzzy uncertain linguistic element, and ( A (x), A (x)) is the q-rung orthopair hesitant fuzzy part of q-rung orthopair hesitant fuzzy uncertain linguistic element.
For convenience, the q-rung orthopair hesitant fuzzy uncertain linguistic element is recorded as h = <[s θ , s τ ], h , h > q . The overall q-rung orthopair hesitant fuzzy uncertain linguistic elements is recorded as q − ROHFULE(X ).
To compare two q-ROHFULEs,we provide the concepts of score function and accuracy function of a q-ROHFULE.
, h , h > be any a q-ROHFULE, the numbers of set h and h are denoted as | h | and | h |, respectively, then the score function S(h) can be defined as: The accuracy function H (h) can be defined as: Based on the two above concepts, a comparison law for q-ROHFULEs can be provided.
, h 2 , h 2 > be any two q-ROHFULEs, S(h 1 ) and S(h 2 ) represent the score function of h 1 and h 2 respectively, H (h 1 ) and H (h 2 ) represent the accuracy function of h 1 and h 2 respectively, then , then h 1 = h 2 . Schweizer-Sklar operations involve the Schweizer-Sklar T-norm and Schweizer-Sklar T-conorm, which are special cases of T-norm and T-conorm, respectively.
Based on the Schweizer-Sklar T-norm and T-conorm, we can give the operational rules of q-ROHFULEs as follows.
In the same way, we can get Therefore, it is proved that λ(h 1 ⊕ h 2 ) = λh 1 ⊕ λh 2 is established.
In the same way, we can get

IV. THE Q-RUNG ORTHOPAIR HESITANT FUZZY UNCERTAIN LINGUISTIC SCHWEIZER-SKLAR BONFERRONI MEAN OPERATOR
A. q-ROHFULSSBM OPERATOR Bonferroni mean (BM) operator [37] can describe the relationship between any two decision attributes. Therefore, BM operator can meet the practical needs of multi-attribute decision making, and is widely used in information aggregation and multi-attribute decision making [38]- [40]. In this article, the Bonferroni mean operator is introduced into the q-rung orthopair hesitant fuzzy uncertain linguistic set, and the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar Bonferroni mean operator and geometric Bonferroni mean operator are defined based on the Schweizer-Sklar T-norm.
. . , n), then the aggregate results of Definition 11 is still a q-ROHFULE, and (19), as shown at the bottom of the page.
Proof: 1) when n = 2, by the operational law in Definition 10, we have , Thus, result is true for n = 2.
2) If Equation (19) holds for n = k, that is Thus, Equation (19) holds for n = k + 1. Therefore, Equation (19) holds for all n, which completes the proof.
It can be proved by mathematical induction that Then, it can be proved that Theorem 3 is true.
Thus, from the definition of the score function, it can be proved that Proof: Based on Theorem 4 and 5, we have By giving different values of the parameters r, s and t, we get the following special cases.
Case 1: When r = 0, then the q-ROHFULSSBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Bonferroni mean (q-ROHFULBM) operator which can be presented as: Case 2: When s → 0, then the q-ROHFULSSBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar generalized arithmetic average operator (q-ROHFULSSGAA) operator which can be presented (20), as shown at the bottom of the next page.
Case 3: When t = 1 and s → 0, then the q-ROHFULSSBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar arithmetic average operator (q-ROHFULSSAA) operator which can be presented (21), as shown at the bottom of the next page.
The q-ROHFULSSBM operator can describe the correlation between any two input variables, but it ignores the importance of the q-rung orthopair hesitant fuzzy uncertain linguistic element itself, which is the weight of the q-rung orthopair hesitant fuzzy uncertain linguistic element. Therefore, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar weighted Bonferroni mean operator is proposed. VOLUME 8, 2020 . . , n), then we can defined the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar weighted Bonferroni mean (q-ROHFULSSWBM) operator as follow: where s, t ≥ 0 and s + t>0. w = (w 1 , w 2 , . . . , w n ) T is the weight vector of h j (j = 1, 2, . . . , n), which satisfies w j ∈ [0, 1] and n j=1 w j = 1.
. . , n), and their weight vector w = (w 1 , w 2 , . . . , w n ) T satisfies w j ∈ [0, 1] and n j=1 w j = 1, then the aggregate results of Definition 12 is still a q-ROHFULE, and (23), as shown at the bottom of the next page.
Proof: The proof process of Theorem 8 is similar to that of Theorem 3. Replace h i and h j of Theorem 3 with nw i h i and Thus, it can be proved that Theorem 8 is valid. In addition, similar to the q-ROHFULSSBM operator, it can be proved that the q-ROHFULSSWBM operator satisfies monotonicity and boundedness.

B. q-ROHFULSSGBM OPERATOR
Next, we introduce the geometric Bonferroni mean operator [41] into the q-rung orthopair hesitant fuzzy uncertain linguistic set, and define the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric Bonferroni mean operator.
Definition 13: . . , n), then we can defined the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric Bonferroni mean (q-ROHFULSSGBM) operator as follow: where s, t ≥ 0 and s + t>0. n), then the aggregate results of Definition 13 is still a q-ROHFULE, and (24), as shown at the bottom of the next page.
Proof: The proof process of Theorem 9 is similar to that of Theorem 3.
Moreover, it can be proved that the q-ROHFULSSGBM operator satisfies commutativity, boundedness, monotonicity and idempotence.
By giving different values of the parameters r, s and t, we get the following special cases.
Case 1: When r = 0, then the q-ROHFULSSGBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic geometric Bonferroni mean (q-ROHFULGBM) operator which can be presented (25), as shown at the bottom of the next page.
Case 2: When s → 0, then the q-ROHFULSSGBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar generalized geometric average operator (q-ROHFULSSGGA) operator which can be presented (26), as shown at the bottom of the next page.
Case 3: When t = 1 and s → 0, then the q-ROHFULSSGBM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric average operator (q-ROHFULSSGA) operator which can be presented as: Next, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar weighted geometric Bonferroni mean operator is proposed.
Thus, it can be proved that Theorem 10 is valid.
In addition, similar to the q-ROHFULSSGBM operator, it can be proved that the q-ROHFULSSWGBM operator satisfies monotonicity and boundedness.

V. THE Q-RUNG ORTHOPAIR HESITANT FUZZY UNCERTAIN LINGUISTIC SCHWEIZER-SKLAR MACLAURIN SYMMETRIC MEAN OPERATOR A. q-ROHFULSSMSM OPERATOR
The Maclaurin symmetric mean (MSM) operator [42] can describe the correlation between multiple decision attributes. The Maclaurin symmetric mean operator is introduced into the q-rung orthopair hesitant fuzzy uncertain linguistic set. Based on Schweizer-Sklar T-norm, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar Maclaurin symmetric mean (q-ROHFULSSMSM) operator and q-rung orthopair hesitant fuzzy uncertain linguistic dual Maclaurin symmetric mean (q-ROHFULSSDMSM) operator are defined. 1, 2, . . . , n), then we can defined the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar Maclaurin Symmetric Mean (q-ROHFULSSMSM) operator as follow: where k is a parameter and k = 1, 2, . . . , n, i 1 , i 2 , . . . , i n are k integer values taken from the set {1, 2,. . . ,n} of n integer values, C k n represents the binomial coefficient and 1, 2, . . . , n), then the aggregate results of Definition 15 is still a q-ROHFULE, and (29), as shown at the bottom of the next page.
Case 1: When r = 0, then the q-ROHFULSSMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Maclaurin symmetric mean (q-ROHFULMSM) operator which can be presented (30), as shown at the bottom of the page.
Case 2: When k = 1, then the q-ROHFULSSMSM operator reduces to the of the q-rung orthopair hesitant fuzzy uncertain linguisticc Schweizer-Sklar arithmetic average (q-ROHFULSSAA) operator which can be presented as: Case 3: When k = 2, then the q-ROHFULSSMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain lingistic Schweizer-Sklar Bonferroni mean (q-ROHFULSSBM) operator.
Case 4: When k = n, then the q-ROHFULSSMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric average (q-ROHFULSSGA) operator which can be presented (32) and (33), as shown at the bottom of the next page.
In actual decision-making, the weight of each attribute may be different. For this reason, the the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar weighted Maclaurin symmetric mean (q-ROHFULSSWMSM) operator is proposed. 1, 2, . . . , n), then we can defined the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar weighted Maclaurin symmetric mean (q-ROHFULSSWMSM) operator as follow: where k is a parameter and k = 1, 2, . . . , n, i 1  Proof: The proof process of Theorem 16 is similar to that of Theorem 11. Replace h i j of Theorem11 with nw i j h i j , Thus, it can be proved that Theorem 16 is valid.
Moreover, it can be proved that the q-ROHFULSSWMSM operator satisfies monotonicity and boundedness.

B. q-ROHFULSSDMSM OPERATOR
Next, the dual Maclaurin symmetric mean operator is introduced into the q-rung orthopair hesitant fuzzy uncertain linguistic set, and the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar dual Maclaurin symmetric mean (q-ROHFULSSDMSM) operator is defined. 1, 2, . . . , n), then we can defined the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar dual Maclaurin symmetric mean (q-ROHFULSSDMSM) operator as follow: where k is a parameter and k = 1, 2, . . . , n, i 1 , i 2 , . . . , i n are k integer values taken from the set {1, 2, . . . , n} of n integer values, C k n represents the binomial coefficient and 1, 2, . . . , n), then the aggregate results of Definition 17 is still a q-ROHFULE, and (35), as shown at the bottom of the page.
Proof: The proof process of Theorem 17 is similar to that of Theorem 11.
Moreover, it can be proved that the q-ROHFULSSDMSM operator satisfies commutativity, boundedness, monotonicity and idempotence.
By giving different values of the parameters r, k, we get the following special cases.
Case 1: When r = 0, then the q-ROHFULSSDMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic dual Maclaurin symmetric mean (q-ROHFULDMSM) operator which can be presented (36), as shown at the bottom of the page.
Case 2: When k = 1, then the q-ROHFULSSDMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric average (q-ROHFULSSGA) operator which can be presented as: Case 3: When k = 2, then the q-ROHFULSSDMSM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar geometric Bonferroni mean (q-ROHFULSSGBM) operator.
Proof: The proof process of Theorem 18 is similar to that of Theorem 17. Replace h i j of Theorem17 with nw i j h i j , where Thus, it can be proved that Theorem 18 is valid. Moreover, it can be proved that the q-ROHFULSSWDMSM operator satisfies monotonicity and boundedness.
Proof: According to the operational laws of q-ROHFULEs, we get Moreover, similar to Theorem 3, it can be proved by mathematical induction that Thus, this proves that Theorem 19 is valid.
is any permutation of (h 1 , h 2 , . . . , h n ), then By giving different values of the parameters r, k, we get the following special cases.
Case 1: When r = 0, then the q-ROHFULSSMM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic Maclaurin mean (q-ROHFULMM) operator which can be presented (42), as shown at the bottom of the page.
Proof: The proof process of Theorem 24 is similar to that of Theorem 19. Replace h ϑ(j) of Theorem19 with nw ϑ(j) h ϑ(j) . Thus, it can be proved that Theorem 24 is valid.
Moreover, it can be proved that the q-ROHFULSSWMM operator satisfies monotonicity and boundedness.
Proof: The proof process of Theorem 25 is similar to that of Theorem 3.
Moreover, it can be proved that the q-ROHFULSSDMM operator satisfies commutativity, boundedness, monotonicity and idempotence.
By giving different values of the parameters r, k, we get the following special cases.
Case 1: When r = 0, then the q-ROHFULSSDMM operator reduces to the q-rung orthopair hesitant fuzzy uncertain linguistic dual Maclaurin mean (q-ROHFULDMM) operator which can be presented (49), as shown at the bottom of the page.

VII. MODELS FOR MADM WITH q-ROHFULSs
Step 1: Carry out the standardized processing of data, convert all cost-type data into profit-type data, so that the obtained data is uniformly transformed into a standard q-rung hesitant fuzzy uncertain linguistic decision matrix.
where h c is the complement of q-rung orthopair orthopair hesitant fuzzy uncertain linguistic element h.
Step 2: Calculate the q-rung orthopair hesitant fuzzy uncertain linguistic aggregation value of each scheme.
Use the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Skla WMM operator or the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Skla WDMM operator to aggregate the data values of each scheme under different attributes, and obtain the q-rung orthopair hesitant fuzzy uncertain linguistic value of each scheme q-ROHFULSSWMM(h k1 , h k2 , . . . , h kn ) or q-ROHFULSSWDMM(h k1 , h k2 , . . . , h kn ).
Step 3: The score function S(h k ) and accuracy function H (h k ) of the q-rung orthopair hesitant fuzzy uncertain linguistic value of each scheme are calculated.
Step 4: According to the score function S(h k ) and the accuray function H (h k ) of each solution, the candidate solutions are sorted.
Step 5: Make the final decision based on the ranking results.

VIII. EXAMPLE APPLICATION A. NUMERICAL EXAMPLE
There is a doctoral enrollment quota for a certain major in a university, and there are currently 5 candidate students A 1 , A 2 , A 3 , A 4 , A 5 entering the retest stage. The five students were evaluated from the four attributes of written test M 1 , interview performance M 2 , foreign language level M 3 , and scientific research level M 4 .
This major invites experts with rich professional background, experience and knowledge in related fields to evaluate the candidates. There are many evaluation factors. For comprehensive and accurate evaluation, q-rung orthopair hesitant fuzzy uncertain linguistic elements are used to reflect the evaluation information of experts.
Since written test situation M 1 , interview performance M 2 , foreign language level M 3 , and scientific research level M 4 are all profit data, there is no need to standardize the data.
According to the expert's decision matrix, the q − ROHFULSSWMM operator is used to aggregate the elements  of different schemes under each attribute, and the q-rung hesitant fuzzy uncertain linguistic aggregation value h k of each scheme is obtained.
It is easy to verify that each q-rung orthopair hesitant fuzzy uncertain linguistic element in Table 1 satisfies the q ≥ 3. Without loss of generality, taking P = (1, 1, 1, 1) and q = 3, r = −1, we can get When the q − ROHFULSSWDMM operator is used, take P = (1, 1, 1, 1), q = 3, and r = −1, so that the q − ROHFULSSWDMM operator aggregates the elements of each scheme in the decision matrix under different attributes, and obtains the q-rung orthopair hesitant fuzzy uncertain linguistic value of each scheme.
It can be seen that the most suitable candidate for admission is A 1 .

B. SENSITIVITY ANALYSIS OF PARAMETERS
The following discusses the influence of the parameter vector P in the q − ROHFULSSWMM operator and the q − ROHFULSSWDMM operator on the decision result. The parameter vector P in the q−ROHFULSSWMM operator and the q − ROHFULSSWDMM operator takes different values, and the score functions and ranking results of each scheme obtained are shown in Table 4 and Table 5.
From the results in Table 2 and Table 3, it can be concluded that although the scoring function and ranking result of each scheme will change when the parameter  vector P takes different values, the best candidate for admission is always A 1 , which shows the decision result reliability. Moreover, the q − ROHFULSSWMM operator and the q−ROHFULSSWDMM operator consider the correlation between all decision attributes, which can fully reflect the decision information and reduce the loss of information in the decision.
In addition, when P = (1, 0, 0, 0), the decision attributes in the operator are independent of each other; when P = (1, 1, 0, 0), the operator can describe the correlation between any two decision attributes; when P = (1, 1, 1, 0), the operator can consider the correlation between multiple attributes. Therefore, the decision maker can flexibly select the parameter vector P to deal with a variety of different types of decision-making problems, making it more widely applicable.
Below, the influence of the parameter q and parameter r in the q − ROHFULSSWMM operator on the decision result is further discussed.
It can be seen from Table 5 that with the continuous decrease of the parameters r in the q − ROHFULSSWMM operator, the score function of each scheme is also continuously reduced, that is, the result of the decision is increasingly pessimistic. Therefore, the value of the parameter r can be regarded as the degree of pessimism and optimism when the expert makes a decision. If the expert is pessimistic about the decision result, assign a smaller value to the parameter r in the q − ROHFULSSWMM operator. Conversely, if the expert is optimistic about the decision result, assign a lower value to the parameter r in the q − ROHFULSSWMM operator.
Combining Table 2, Table 3, Table 4 and Table 5, it can be concluded that although the q − ROHFULSSWMM operator and the q − ROHFULSSWDMM operator are used to aggregate the attribute values of each scheme, and with the continuous changes of the parameters q and r in the operator, the scoring function of the proposed programs is different, and the ranking results of each program are slightly changed, the best candidates for doctoral admission are A 1 , which shows that the decision-making method in this article is effective and reasonable.

C. COMPARATIVE ANALYSIS
A comparative analysis is carried out to illustrate the effectiveness and superiority of the proposed method over the method of q-ROFULSs proposed by Liu et al. [32], the method of ULSs developed by Xu [44], the method of q-ROHFSs developed by Xu et al. [45], the method of DHFULSs proposed by Lu and Wei [46], the method of PHFULSs proposed by Shakeel et al. [47], and other methods using the same illustrative example.

1) COMPARATIVE ANALYSIS WITH Q-ROFULS AGGREGATION OPERATOR
The q-rung orthopair fuzzy uncertain linguistic set is a special form of the q-rung orthopair orthopair hesitant fuzzy uncertain linguistic set. When the membership degree and non-membership degree of q-rung orthopair hesitant fuzzy uncertain linguistic set satisfy | h | = | h | = 1, the qrung orthopair hesitant fuzzy uncertain linguistic set is transformed into the q-rung orthopair fuzzy uncertain linguistic set. And the possible membership degree and the possible non-membership degree in the q-rung orthopair orthopair hesitant fuzzy uncertain linguistic decision data are respectively averaged. Thus, the q-rung orthopair fuzzy uncertain linguistic decision data is obtained, as shown in Table 6 below.
The q-rung orthopair fuzzy uncertain linguistic weighted geometric mean (q − ROFWGA) operator proposed by   Liu et al. [32] is used to aggregate the attribute data, and the q-rung orthopair fuzzy uncertain linguistic evaluation value of each candidate object is obtained. Then, the score function S(h k ) of the q-rung orthopair fuzzy uncertain linguistic value of each scheme is calculated, which are S(h 1 ) = 3.1350, S(h 2 ) = 2.1610, S(h 3 ) = 1.9093, S(h 4 ) = 2.3990, and S(h 5 ) = 1.8482 respectively. According to the score function S(h k ) of each scheme, the ranking is It can be seen that the most suitable candidate for admission is A 1 .
The comparative analysis shows that the decision-making method based on the q − ROFWGA operator is consistent with the most suitable candidates obtained by the method in this paper, and the results are both A 1 . However, the score function value of each scheme and the ranking situation of schemes obtained by the q − ROFWGA operator are different from the method in this paper. The reason is that the q-rung orthopair fuzzy uncertain linguistic set does not take into account the experts' hesitation on the membership degree and non-membership degree, and cannot fully reflect the expert's decision information, which will cause the loss of decision information.

2) COMPARATIVE ANALYSIS WITH ULS AGGREGATION OPERATOR
Uncertain linguistic set is a special form of q-rung orthopair hesitant fuzzy uncertain linguistic set. When the membership degree and non-membership degree of the q-rung orthopair hesitant fuzzy uncertain linguistic set satisfy h = {1}, h = {0}, the q-rung orthopair hesitant fuzzy uncertain linguistic set is transformed into an uncertain linguistic set. Taking the membership degree and non-membership degree in the q-rung orthopair hesitant fuzzy uncertain linguistic decision data as h = {1}, h = {0}, the uncertain linguistic decision data is obtained, as shown in Table 7 below.
The uncertain linguistic weighted average (ULWAA) operator proposed by Xu [44] is used to aggregate the attribute data to obtain the uncertain linguistic evaluation value of each candidate object. Then, the score function S(h k ) of the uncertain linguistic value of each scheme is calculated, which are S(h 1 ) = 2.2300, S(h 2 ) = 1.9400, S(h 3 ) = 1.9750, S(h 4 ) = 1.9950, and S(h 5 ) = 1.8150 respectively. According to the score function S(h k ) of each scheme, the ranking is It can be seen that the most suitable candidate for admission is A 1 . The comparison shows that the decision-making method based on the ULWAA operator is consistent with the most suitable candidates obtained by the method in this paper, and the results are both A 1 . However, the uncertain linguistic set only reflects the qualitative decision-making information of the decision maker, ignoring the quantitative decision-making information of the expert, and cannot reflect the membership degree, non-membership degree and hesitation of the expert's qualitative opinions.

3) COMPARATIVE ANALYSIS WITH Q-ROHFS AGGREGATION OPERATOR
When the qualitative decision information of the decision maker is ignored and only the quantitative decision information of the decision maker is considered, the q-rung hesitant fuzzy uncertain language set degenerates the q-rung orthopair hesitant fuzzy set. The q-rung orthopair hesitant fuzzy decision data are shown in Table 8 below.
Using the q-rung orthopair hesitant fuzzy weighted Heronian mean (q − ROHFWH ) operator proposed by Xu et al. [45] to aggregate all attribute data, the q-rung orthopair hesitant fuzzy evaluation value of each candidate object is obtained. The score function of the q-rung orthopair hesitant fuzzy evaluation value of each scheme is calculated, which are S(h 1 ) = 3.5098, S(h 2 ) = 2.5126, S(h 3 ) = 2.1511, S(h 4 ) = 2.7939, S(h 5 ) = 2.3450. According to the score function of each program, each program is ranked as A 1 A 4 A 2 A 5 A 3 . It can be seen that the most suitable candidate for admission is A 1 .
The comparison shows that the decision-making method based on the q − ROHFWH operator is consistent with the most suitable candidates obtained by the method in this paper, and the results are both A 1 . However, the q-rung orthopair hesitant fuzzy set only quantitatively reflects the decision-making information of the decision maker. In actual decision-making, experts may prefer to use qualitative linguistic variables to express the decision information.

4) COMPARATIVE ANALYSIS WITH PHFULS AND DHFULS AGGREGATION OPERATOR
The q-rung orthopair hesitant fuzzy uncertain linguistic set is a generalization of the dual hesitant fuzzy uncertain linguistic set and the Pythagoras hesitant fuzzy uncertain linguistic set. When q = 1, the q-rung orthopair hesitant fuzzy uncertain linguistic set is transformed into the Pythagorean hesitation fuzzy uncertain linguistic set; when q = 2, the q-rung orthopair hesitant fuzzy uncertain linguistic set is transformed into the dual hesitant fuzzy uncertain linguistic set. In order to illustrate the effectiveness and feasibility of the decision-making method in this paper, the Pythagorean hesitant fuzzy uncertain linguistic hybrid weighted average (PHFULHWA) operator of the literature [46] and the dual hesitant fuzzy uncertain linguistic weighted average (DHFULWAA) operator of the literature [47] are introduced for comparative analysis.
From the data in Table 1, we can see that the element h 15 is <[s 4 , s 4 ], {0.5, 0.8}, {0.5, 0.7}>. However, due to 0.8 + 0.7>1 and 0.8 2 + 0.7 2 >1, the membership degree and nonmembership degree of element h 15 do not meet the constraints of DHFULSs and PHFULSs, so that the evaluation data cannot be represented by DHFULE and PHFULE. Therefore, as shown in Table 9, the Pythagoras hesitant fuzzy uncertain linguistic hybrid weighted average (PHFULHWA) operator and the dual hesitant fuzzy uncertain linguistic weighted average (DHFULWAA) operator cannot be applied to the decision-making problem in this paper.
However, the proposed decision-making method can still be applied to this decision-making problem, because by adjusting the value of q, the q-ROFULE can represent the element <[s 4 , s 4 ], {0.5, 0.8}, {0.5, 0.7}>. Therefore, the decision-making method in this paper has a wider application range than DHFULWAA and PHFULWAA operators.
In addition, according to the results in Table 9, it can be seen that the ranking results of the various schemes obtained by different methods may be slightly different, but the optimal scheme is all A 1 , which illustrates the effectiveness and rationality of the decision-making method in this paper.
Based on the above comparison and analysis, it can be concluded that the advantages of the proposed decision-making method are: 1) Compared with DHFULSs and PHFULSs, q-ROHFULSs can express uncertainty more accurately. In today's complex decision-making environment, the q-ROHFULSs proposed in this paper is a very powerful tool in decision-making. 2) The decision-making method in this paper is an optimization and generalization of existing methods. When q = 1, 2, the DHFULWAA and PHFULWAA operators are special cases of the operators proposed in this paper. 3) For the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar aggregation operator, different parameter values can be selected according to the decision-making situation to meet the requirements of different decision-making problems in practice, so it is flexible and general. 4) The q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar MM operator can describe the correlation between the decision attributes, can fully reflect the decision information, reduce the information loss in the decision, and make the decision result more reasonable and reliable. VOLUME 8, 2020  All in all, the proposed decision-making method is effective and reliable.

IX. CONCLUSION
This paper combines the q-rung orthopair hesitant fuzzy sets with uncertain linguistic variables, and proposes the q-rung orthopair hesitant fuzzy uncertain linguistic sets. And the Schweizer-Sklar T-norm is introduced to define the operational properties of the q-rung orthopair hesitant fuzzy uncertain linguistic elements. Then the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar BM operator, MSM operator and MM operators are respectively defined. In addition, a multi-attribute decision-making method based on the Schweizer-Sklar MM operator of the q-rung orthopair hesitant fuzzy uncertain linguistic set is established. The research results of this paper develope the theory of the q-rung orthopair hesitant fuzzy uncertain linguistic aggregation operators, and extend the q-rung orthopair hesitant fuzzy uncertain linguistic multi-attribute decision-making method. In the future, we will study the information measure and multi-attribute decision-making method of the q-rung orthopair hesitant fuzzy uncertain linguistic set. In addition, the theory of the q-rung orthopair hesitant fuzzy uncertain linguistic set can be expanded by combining other related theories [48], [49], and the aggregation operators of the extended q-rung orthopair hesitant fuzzy uncertain linguistic set and their application in decision-making can be studied.