An Approach to Multicriteria Group Decision-Making with Unknown Weight Information Based on Pythagorean Fuzzy Uncertain Linguistic Aggregation Operators

With respect to multicriteria group decision-making (MCGDM) problems in which the experts have different priority levels, the criteria values are in the form of Pythagorean fuzzy uncertain linguistic variables (PFULVs), and the information about weights of experts and criteria is completely unknown, a novel decision-making method is developed. Firstly, the concept of PFULV is defined, and some operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs are presented. Then, to aggregate information given by all experts, the Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation (PFULPWAA) operator and the Pythagorean fuzzy uncertain linguistic prioritized weighted geometric aggregation (PFULPWGA) operator are proposed. Furthermore, in order to get a comprehensive evaluation value for each alternative, the Pythagorean fuzzy uncertain linguistic Maclaurin symmetric mean aggregation (PFULMSMA) operator and the weighted PFULMSMA (WPFULMSMA) operator are proposed. Moreover, to obtain the information about the weights of criteria, themodel based on grey relational analysis (GRA)method is established. Finally, a method ofMCGDMwith PFULVs is developed, and an application example is given to illustrate the validity and feasibility of the provided procedure.


Introduction
The notion of intuitionistic fuzzy set (IFS) was presented by Atanassov [1], which consists of a membership degree and a nonmembership degree meeting the restriction that the sum of two degrees is equal to or less than 1.Thus, it is an effective tool to handle uncertainty and vagueness and MCGDM problems with intuitionistic fuzzy numbers (IFNs) have received more and more attention [2,3].Chen et al. [4] defined a new similarity measure between IFSs so as to solve pattern recognition problems.Based on GRA method and evidence theory, Qiu et al. [5] proposed a novel approach to the MCGDM problems in which both the criteria weights and criteria values take the form of IFNs.Montajabiha [6] developed a new version of the PROMETHE II method to solve intuitionistic fuzzy MCGDM problem.He et al. [7] extended power averaging operator, which can reflect the relationship between the arguments being aggregated, to IFS.He et al. [8] proposed some neutral aggregation operators for IFS, which reflect the interactions between IFNs and the attitude of the experts, and applied them to the MCGDM problem.
However, in some MCGDM problems, the sum of the membership degree and the nonmembership degree to which an alternative satisfies a criterion is bigger than 1, but their square sum is equal to or less than 1.Thus, the notion of intuitionistic fuzzy set of second type was presented by Atanassov [9,10].Later, the notion of Pythagorean fuzzy set (PFS) was presented by Yager [11,12], which consists of a membership degree and a nonmembership degree, whose sum of squares is equal to or less than 1.Zhang [13] developed a novel decision method based on similarity measure to deal with selection problem of photovoltaic cells with Pythagorean fuzzy numbers (PFNs).Zhang and Xu [14] gave an extension of TOPSIS method to solve the MCGDM problems under Pythagorean fuzzy environment.Based on 2 Mathematical Problems in Engineering prospect theory, Ren et al. [15] extended TODIM method to PFS and developed an extended TODIM method.Peng and Yang [16] extended Choquet integral, which can consider the interactions among the criteria, to PFS, and proposed several Pythagorean fuzzy Choquet integral operators.
Under many conditions, it is difficult to handle the fuzziness and uncertainty in real MCGDM problems by numerical numbers, especially for qualitative aspects, while it is easy to express the evaluation values by means of linguistic variables (LVs) or uncertain linguistic variables (ULVs).For example, when the moral character of students, the computer performance, and so on are evaluated, they are easy to be described by the LVs, such as poor, fair, and very good.On the basis of given functions satisfying certain characteristics and distance measures, Tao et al. [17] defined two groups of entropies for LVs and ULVs, respectively.Liu et al. [18] extended Heronian mean (HM) operator, which can consider the interrelationship of the aggregated arguments, to uncertain linguistic set (ULS).Wei et al. [19] extended Bonferroni mean (BM) operator, which can also reflect the interrelationship of the aggregated arguments, to ULS, and applied them to the MCGDM problem with ULVs.
By combining ULVs with IFS, the concept of intuitionistic uncertain linguistic set (IULS) was introduced by Liu and Jin [20], which gives the information about the membership and nonmembership of an element to an ULV.Then, the research on the MCGDM problems with intuitionistic uncertain linguistic numbers (IULNs) has made many achievements [21][22][23].However, the IULS cannot handle the situation; the sum of membership degree and nonmembership degree belonging to uncertain linguistic variable is bigger than 1.To deal with this situation, based on ULVs and PFS, the concept of Pythagorean fuzzy uncertain linguistic set (PFULS), which is only required to meet the restriction that the square sum of the two degrees is less than or equal to 1, is defined in this paper.To understand the PFULS better, we provide an instance: computer performance is perhaps felt to be lower than "very good" ( 6 ) but higher than "fair" ( 3 ), the membership degree to [ 3 ,  6 ] is 1/2, and nonmembership degree is √ 3/2.The evaluation result can be denoted as ⟨[ 3 ,  6 ], (1/2, √ 3/2)⟩.Due to the fact that the sum of two degrees is ( √ 3/2 + 1/2) > 1, the evaluation value is not available for IULS but is available for PFULS since ( √ 3/2) 2 + (1/2) 2 = 1.Clearly, the PFULS has more powerful ability than the IULS to depict the uncertainty in the real-world MCGDM problems.It should be noted that when the upper and lower limits of the uncertain linguistic part of PFULS are identical, PFULS reduces to the Pythagorean fuzzy linguistic set (PFLS) introduced by Peng and Yang [24], which indicates that the former is an extension of the latter.Compared with PFLS, PFULS is defined by utilizing ULVs, whose membership degree and nonmembership degree are no longer with respect to a LV, but to an ULV, which makes the experts express uncertain information more easily and precisely.
Information aggregation is a pervasive activity in our daily life; many operators have been provided on this issue.Among them, the prioritized averaging (PA) operator and the Maclaurin symmetric mean (MSM) are two of the most common operators for aggregating information.The PA operator was initially presented by Yager [25], which can capture the prioritization phenomenon of the aggregated arguments.Then, it was extended to hesitant fuzzy set (HFS) [26], triangular fuzzy set (TFS) [27], IFS [28], trapezoidal intuitionistic fuzzy set (TIFS) [29], linguistic set (LS) [30], 2-tuple linguistic set (2TLS) [31], multigranular uncertain linguistic set (MULS) [32], and so on.The MSM was initially given by Maclaurin [33], which can consider the interdependent characteristics among the multi-input arguments.The MSM is different from Choquet integral or power average operator.The MSM pays attention to the input arguments while the Choquet integral or power average operator pays attention to the weights information.The MSM is also different form BM or HM.The MSM operator considers the interdependent characteristics among the multi-input arguments and should take one parameter from finite integer set, while the BM or HM captures the interrelationship between two input arguments and should take two parameters both from infinite set.Due to the advantages of the MSM, it was extended to HFS [34], IFS [3], 2TLS [35], ULS [36], intuitionistic linguistic set (ILS) [21], IULS [21], and so on.However, both PA and MSM operators fail to aggregate Pythagorean fuzzy uncertain linguistic information.Therefore, we shall propose the PFULPWAA, PFULPWGA, PFULMSMA, and WPFULMSMA operators.The significant features of these operators are that not only can they handle PFULVs, but also the PFULPWAA and PFULPWGA operators can capture prioritization among the criteria or experts, and the PFULMSMA and WPFULMSMA operators can consider the interrelationship among the multi-input arguments.
The aim of this paper is to develop a novel method based on proposed Pythagorean fuzzy uncertain linguistic aggregation operators to solve the MCGDM problems in which the experts have different priority levels, the criteria values are in the form of PFULVs, and the information about weights of experts and criteria is completely unknown.To do so, the remainder of this paper is constructed as follows: Section 2 reviews some concepts of PFS, PA operator, and MSM.Section 3 defines the concept of PFULS and presents some operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs.Section 4 proposes the PFULPWAA and PFULPWGA operators and investigates their corresponding properties.Section 5 proposes the PFULMSMA and WPFULMSMA operators.Section 6 presents an approach to Pythagorean fuzzy linguistic MCGDM based on GRA model and the proposed new operators.Section 7 provides an example to demonstrate the decision-making application.Section 8 gives the concluding remarks.
With respect to the operational rules and characteristics of PFNs, please refer to [11,14].

The Prioritized Averaging
Operator.In many real and practical MCGDM problems, the criteria or the experts usually have different priority levels.For instance, regarding decision-making in a company, general manager usually has a higher priority than vice manager.To deal with this issue, the prioritized averaging (PA) operator was proposed by Yager [25], which is shown as follows.

The Maclaurin Symmetric Mean.
The MSM can consider the interdependent characteristics among the multi-input arguments, which is shown as follows.

Pythagorean Fuzzy Uncertain
Linguistic Variable (2 Clearly, the above operational results are still PFULVs.

Comparison of Two Pythagorean Fuzzy Uncertain
Linguistic Variables Definition 6. Suppose that p1 = ⟨[  1 ,   1 ], ( 1 , V 1 )⟩ is a PFULV; then the score function of p1 is shown as follows: It should be mentioned that the score function   (p 1 ) is between −1 and 1.In order to facilitate the following study, we provide another score function   (p 1 ) whose range is between 0 and 1.
If (p 1 , p2 ) satisfies the three restrictions ) is called the distance between p1 and p2 .
In what follows, we shall prove that the normalized Hamming distance (p 1 , p2 ) defined in (19) can also satisfy restriction (3) in Definition 9.
and therefore

Pythagorean Fuzzy Uncertain Linguistic Prioritized Aggregation Operators
In this section, we shall extend the PA operator to PFULS and propose the PPFULPWAA and PFULPWGA operators.
Proof.Formula ( 23) can be proved by mathematical introduction on  as follows.
(i) For  = 2, according to the operational laws of PFULVs, we obtain So, formula ( 23) is right for  = 2.
In addition, since formula ( 23) is still a PFULV, and the proof of Theorem 12 is finished.
In what follows, some properties of the PFULPWAA operator shall be explored.
Proof.Similar to Theorem 12, the proof of Theorem 15 is omitted.
Obviously, the PFULPWGA operator also has idempotency and boundness, and the proofs are omitted here.

Pythagorean Fuzzy Uncertain Linguistic Maclaurin Symmetric Mean Aggregation Operators
In this section, we will extend the MSM to PFULS and propose the PFULMSMA and WPFULMSMA operators.
is a collection of PFULVs, and  = 1, 2, . . ., ; then the aggregating value by PFULMSMA operator is still a PFULV, and

Mathematical Problems in Engineering
Proof.According to the operational laws of PFULVs, we have then (43) that is, formula (40) is right.
In addition, since formula ( 40) is still a PFULV, and the proof of Theorem 17 is finished.
In what follows, some properties of the PFULMSMA operator will be explored. Proof.
Now, the monotonicity of the PFULMSMA operator about the parameter  shall be analyzed.In the first place, we give two lemmas which shall be utilized in the following investigation.
According to Lemma 19, we know that functions  1 () and  2 () are both monotonically decreasing about the parameter .In addition, according to formulas (65), we know that   () and   () are monotonically decreasing and increasing about the parameter , respectively.
Now, several special cases of the WPFULMSMA operator about the parameter  shall be discussed.
Mathematical Problems in Engineering 13 Next, the monotonicity of the WPFULMSMA operator about the parameter  shall be analyzed.
Proof.Similar to Theorem 21, the proof of Theorem 24 is omitted.

A Group Decision-Making Method with Unknown Weight Based on Pythagorean Fuzzy Uncertain Linguistic Aggregation Operators
In this section, we will develop a method to solve the MCGDM problems with unknown weight under Pythagorean fuzzy uncertain linguistic environment by utilizing the Pythagorean fuzzy uncertain linguistic aggregation operators and grey relational analysis (GRA) method.

The Model Based on GRA Method to Obtain the Weight
Vector of Criteria.The GRA method as an important approach to obtain the criteria weights has been researched by many scholars [39,40].In what follows, the model based on GRA method shall be established to obtain the optimal weight vector of criteria.
Assume that  = [  ] × is the collective Pythagorean fuzzy uncertain linguistic matrix, where The grey relational coefficient of each solution from positive ideal alternative and negative ideal alternative is obtained by utilizing the following equations, respectively: for all  = 1, 2, . . .,  and  = 1, 2, . . ., , where the identification coefficient  = 0.5 and the distance is normalized Hamming distance between PFULVs.
If the information about criteria weight is completely unknown, the single objective programming model is established to obtain the weight information.
By solving the above model, we can obtain the optimal solution  = ( 1 ,  2 , . . .,   )  , which can be utilized as the weight vector of criteria.

The Decision Procedure.
Based on established model and presented operators, we propose a procedure to solve the MCGDM problems in which the criteria values are in the form of PFULVs, and the information about the criteria weights is completely unknown.The detailed decisionmaking steps are listed as follows.
Step 2. Utilize PFULPWAA operator in formula (78) or PFULPWGA operator in formula (79) to aggregate the evaluation information of individual expert to collective information.
Step 3. Utilize formulas (80) to determine the positive and negative ideal alternatives, respectively. where where the identification coefficient  = 0.5 and the distance is normalized Hamming distance between PFULVs.
Step 5. Utilize model (83) to get the optimal weight vector of the criteria  = ( 1 ,  2 , . . .,   )  . min Step 6. Utilize WPFULMSMA operator in formula (84) to aggregate the collective evaluation information of each criterion to the comprehensive evaluation information of each alternative.

An Illustrative Example
To demonstrate the application of the developed MCGDM method, we will cite an example from [24] which is about the investment selection.An investment company wanted to invest lots of money to develop a software project, and there are four possible projects { 1 ,  2 ,  3 ,  4 } as alternatives.In order to make a reasonable decision, three experts { 1 ,  2 ,  3 } are designated to evaluate the alternatives with respect to three criteria: the technical feasibility ( 1 ), the economic feasibility ( 2 ), and the operational feasibility ( 3 ).Suppose that there is a prioritization relationship for the experts,  2 ≻  3 ≻  1 , and the weight vector of three criteria is completely unknown.The experts gave the individual Pythagorean fuzzy uncertain linguistic decision matrices ) by utilizing the linguistic term set  = { 0 ,  1 ,  2 ,  3 ,  4 ,  5 ,  6 } = (very poor, poor, slightly poor, fair, slightly good, good, very good) as listed in Tables 1-3.

The Evaluation Steps by Developed Method.
To obtain the most eligible alternative(s), the detailed steps are shown as follows.
Step 3. Utilize formulas (80) to determine the positive and negative ideal alternatives, respectively.) .

R+ = (⟨[𝑠
(88) Step 5. Utilize model (83) to obtain the optimal weight vector of the criteria.min 2.5434  (89) among the experts, are more objective and reasonable than the known weight vector of experts.
(3) The Pythagorean fuzzy linguistic weighted averaging (PFLWA) operator and extension of TOPSIS in [24] are based on the assumption that the input arguments are independent.However, the PFULPWAA and PFULPWGA operators can consider prioritization among the experts or criteria, and the WPFULMSMA operator can consider the interrelationship among the multi-input arguments, which could ensure the reasonableness and effectiveness of the decision-making results.
(4) The WPFILMSMA operator provides more selecting choice for experts by altering the values of the parameter decided by the preferences of the expert, which is more flexible for coping with MCGDM problems with PFULVS.

Conclusion
To solve more complex MCGDM problems, we define a new class of fuzzy set called PFULS as an extension of PFLS, which could more precisely express ULV and retain the original data integrity.The operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs are presented.Then, based on the PA operator, we propose the PFULPWAA and PFULPWGA operators, which can capture the prioritization among the experts or criteria.In addition, we combine MSM with PFULVs and propose the PFULMSMA and WPFULMSMA operators, which can consider the interrelationship among the multiinput arguments.In the meantime, some special cases and properties are also investigated, such as the idempotency, commutativity, monotonicity, and boundedness.Furthermore, a Pythagorean fuzzy uncertain linguistic MCGDM method based on GRA model and proposed operators with interactive condition and completely unknown weight information is developed.The prominent characteristics of developed method are that they are able to effectively aggregate PFULVS, and they can capture the prioritization among the experts and interrelationship among the multiinput arguments, which can avoid losing and distorting the given preference information so as to make the final results more feasible and practical.Finally, an example is provided to illustrate the feasibility and practicality of the developed method.In the succeeding work, the proposed operators shall be applied to the practical problems.