Consistency Analysis and Priorities Deriving for Pythagorean Fuzzy Preference Relation in the “Computing in Memory”

Pythagorean fuzzy set, characterized by membership function and non-membership function, has received increasing attention in recent years. In this paper, a new approach to decision making is proposed based on Pythagorean fuzzy preference relation and its additive consistency. Firstly, the concepts of Pythagorean fuzzy preference relations and its additive consistency are introduced, and followed by a discussion of their desirable properties. Then, a linear goal programming model is proposed to determine the consistency of PFPRs. For the PFPRs that does not satisfy the consistency, the consistency index is defined to measure the degree of consistency, and a consistency adjustment algorithm is proposed. Finally, based on the additive consistency, a new algorithm for decision making is presented. An example of CIM(Computing In Memory) is provided, and in comparison with other methods, the validity and rationality of the proposed method are verified.


I. INTRODUCTION
In multiple criteria decision making (MCDM) process, a decision maker (DM) need to make a pairwise comparison of alternatives or criteria, so as to propose his/her preference in a set of n alternatives or criteria, and establish a preference relation to reflect the DM's judgment. Analytic hierarchy process (AHP) [1] is one of the most commonly used and most powerful methods for solving MCDM. It chooses the optimal solution from multiple alternatives according to the preferences provided by decision makers. The AHP provides a convenient framework for the derivation of multiplicative preference relations based on pairwise comparisons. In recent years, with the introduction of fuzzy logic and fuzzy methods into AHP, fuzzy preference relations have received more and more attention [2]- [4].
Due to the complexity of objective things and the uncertainty of actual problems, it is often difficult for DM to express his/her judgments with precise numerical values. In order to express this ambiguity and uncertainty, different uncertain preference relations are proposed. such as interval The associate editor coordinating the review of this manuscript and approving it for publication was Francisco J. Garcia-Penalvo . fuzzy preference relations [5]- [7], interval multiplicative preference relations [8], reciprocal fuzzy preference relations [9], and triangular fuzzy preference relations [10], trapezoidal fuzzy preference relations [54]- [57], hesitant fuzzy linguistic preference relation [61]. Saaty [8] introduce interval multiplicative preference relations and propose a Monte Carlo simulation method to generate priority weights from the interval multiplicative preference relations. Many methods have been proposed to derive priority weights from interval multiplicative preference relations, such as goal programming models [11], [12] and convex combination method [13]. Xu and Chen [14] give additive and multiplicative transitivity conditions for interval fuzzy preference relations based on normalized crisp weights and present some linear programming models for deriving priority weights. hou [60] propose an optimal group continuous logarithm compatibility measure for interval multiplicative preference relations based on the COWGA operator. By using interval arithmetic, Wang and Li [15] introduce new definitions of additive consistent, multiplicative consistent and weakly transitive interval fuzzy preference relations. Xu [16] made a survey on different kinds of preference relations and discussed their properties.
As an extension of the fuzzy set [17], Atanassov [18], [19] introduced the concept of intuitionistic fuzzy sets (IFSs), the sum of its membership degree and non-membership degree is less than or equal to 1. Up to now, IFSs have been widely applied in real-life MCDM problems, the studies of methods of MCDM problems with IFSs have received extensive attentions [20]- [24], and the applications of MCDM problems based on IFSs have attracted widespread attention of researchers. Szmidt [25] generalize fuzzy preference relations to intuitionistic fuzzy preference relations and discuss how to reach consensus with intuitionistic fuzzy preference relations in group decision making. Xu and Yager [26] introduce a new similarity measure between IFSs and apply it to consensus analysis in group decision making with intuitionistic fuzzy preference relations. Xu [27] defines multiplicative consistent intuitionistic fuzzy preference relations based on intuitionistic fuzzy number operations, and develops a new group decision making method by using intuitionistic fuzzy aggregation operators. Xu et al. [28] by directly employing the membership and non-membership degrees in intuitionistic fuzzy judgments, they propose a new definition of multiplicative consistent intuitionistic fuzzy preference relations and develop two algorithms to estimate missing values for incomplete intuitionistic preference relations. Xu [29] presents an error-analysis-based approach to determine priority interval weights for both consistent and inconsistent intuitionistic fuzzy preference relations. Gong et al. [30] give another multiplicative consistency definition for intuitionistic fuzzy preference relations based on the corresponding membership degree interval fuzzy relations with multiplicative consistency and propose goal programming approaches to obtain priority weights. Gong et al. [31] further define additive consistent intuitionistic fuzzy preference relations and establish some optimization models to obtain intuitionistic fuzzy weights from intuitionistic fuzzy preference relations.
In recent years, Pythagorean fuzzy sets (PFSs) proposed by Yager [32], [33] is an useful extension of the concept of Atanassov's intuitionistic fuzzy sets (IFSs) [18]. PFSs have more powerful abilities than IFSs do in modeling the uncertainty of practical decision making problems, because it satisfies the condition that the square sum of its membership degree and non-membership degree is equal to or less than 1. Yager [33] gave an example to illustrate this situation: the membership degree and the non-membership degree of one alternative in a criterion are √ 3 2 and 1 2 , it is easily seen that √ 3 2 + 1 2 ≥ 1, thus this situation cannot be described by using the IFSs, but ( √ 3 2 ) 2 +( 1 2 ) 2 ≤ 1 holds. Obviously, PFSs have more capability than IFSs do in modeling the vagueness of practical multiple attribute decision making (MADM) problems. Yager [32], [33] proposed a series of aggregation operators: Pythagorean fuzzy weighted average (PFWA) operator, Pythagorean fuzzy weighted geometric average (PFWG) operator, Pythagorean fuzzy weighted power average (PFWPA) operator, Pythagorean fuzzy weighted power geometric (PFWPG) operator, and applied them to MADM problems. Peng and Yang [34] discussed their relationships, and also proposed a method called superiority and inferiority ranking (SIR) multiple attribute group decision making (MAGDM). At the same time, inspired by soft set theory [35] and linguistic set theory [36], they proposed Pythagorean fuzzy soft sets [37] and Pythagorean fuzzy linguistic sets [38], respectively. Yager and Abbasov [32] studied the relationship between the Pythagorean fuzzy numbers (PFNs) and the complex numbers and concluded that Pythagorean degrees are a subclass of complex numbers. Wang and Li [39] presented Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making. Xu [40] defined the algorithms to detect and rectify multiplicative and ordinal inconsistencies of fuzzy preference relations. Zhang and Xu [42] presented a technique for finding the best alternative based on its ideal solution under the Pythagorean fuzzy environment. Xu [41] propose the algorithms to identify and rectify ordinal inconsistencies for incomplete fuzzy linguistic preference relations. Chen [43] defined an extended ELECTRE approach in Pythagorean fuzzy sets. Later on, Garg [44] presented a novel accuracy function for intervalvalued PFSs and apply it to solve the decision-making problem. A correlation coefficient between the two PFSs has been proposed by Garg [45] by showing the advantages as compared to the existing correlation coefficients under IFSs environment.
This paper focuses on PFPRs and its additive consistency, and a linear goal programming model is developed to determine whether the PFPR has additive consistency. For PFPRs that do not meet the consistency, a consistency index is defined to measure its consistency degree, and an algorithm for consistency adjustment is presented to adjust the PFPR until its consistency reaches an acceptable range. The remainder of this paper is organized as follows. In Section 2, we will briefly review some basic concepts and operations, including IFSs, IFPRs, PFSs, and so on. In section 3, a linear goal programming model is defined to determine the additive consistency of PFPRs, for the PFPRs that does not meet the consistency, a consistency adjustment algorithm is proposed. In section 4, we develop an approach to decision making based on Pythagorean fuzzy preference relation (PFPR) and its additive consistency. And in Section 5, we will provide two practical examples to illustrate the developed approaches respectively. Section 6 ends this paper with some concluding remarks.

A. FPRS AND IFPRS
In this section, we briefly review some basic concepts, including fuzzy preference relation (FPR) and its additive consistency, intuitionistic fuzzy preference relation (IFPR) and its additive consistency and Pythagorean fuzzy set (PFS).
For a decision-making problem, let X = {x 1 , x 2 , . . . , x n } be a finite set of alternatives, where x i (i = 1, 2, . . . , n) denote the ith alternatives. In the decision-making process, a DM need to provide his/her preferences for each pair of alternatives, and then construct a preference relation matrix, which can be defined as follows.
Definition 2.1 [5]: A preference relation P on set X is characterized by a function µ P : X × X → D, where D is the domain of representation of preference degrees. These preference relations can be mainly classed into three categories: multiplicative preference relations, fuzzy preference relations, linguistic preference relations.
Definition 2.2 [46]: A fuzzy preference relation R on set X is represented by a complementary matrix where r ij denotes the degree that the alternatives x i is preferred to x j . In particular, r ij = 0.5 indicates that there is no difference between alternative x i and x j ; r ij > 0.5 indicates that the alternative x i is preferred to x j , especially, r ij = 1 means that the alternative x i is absolutely preferred to x j ; and r ij < 0.5 indicates that the alternative x j is preferred to x i , especially, r ij = 0 means that the alternative x j is absolutely preferred to x i . Definition 2.3 [47]: A fuzzy preference relation R = (r ij ) n×n is called an additive consistency FPR, if it satisfies the following condition: for all i, j, k = 1, 2, · · · , n.
For a fuzzy preference relation R = (r ij ) n×n , if there exists a normalized crisp weight vector ω=(ω 1 , ω 2 , . . . , ω n ) T such that where n i=1 ω i = 1 and ω i ≥ 0 for i = 1, 2, · · · , n, then R is additive consistent [22], [23], [48]. Due to the complexity, ambiguity and uncertainty of decision-making problems, it is difficult to be convinced to express the DM's preference information with exact numbers. For this case, Atanassov [7] proposed intuitionistic fuzzy sets (IFSs) composed of membership function and nonmembership function.
Definition 2.4. [18], [19]: Let a set X be fixed. An IFS A in X is shown as follows: Which is characterized by a membership function µ A : A → [0, 1] and a non-membership function v A : A → [0, 1] with the condition: where µ A (x) and v A (x) represent the membership degree and the non-membership degree of the element x ∈ X to the set A, is called the indeterminacy degree of the membership of the element x ∈ X to the set A.
According to the concept of IFS, the concept of IFPR is defined as follows.
Definition 2.5 [27]: An intuitionistic fuzzy preference relation (IFPR) B on set X is represented by a matrix where b ij is an intuitionistic fuzzy number composed by the certainty degree µ ij to which x i is preferred to x j and the certain degree v ij to which x i is non-preferred to x j , and π ij = 1 − µ ij − v ij is interpreted as the uncertainty degree to which x i is preferred to x j .
Furthermore, for all i, j = 1, 2, . . . , n, µ ij and v ij satisfy the following characteristics: Consistency is a very important issue for all kinds of preference relations, and the lack of consistency in a preference relation may result in unreasonable conclusions. For IFPR, several different consistency have been proposed, of which there are two main types: the additive consistency and the multiplicative consistency. Xu [49], Gong et al. [31], and Wang [48] proposed some different definitions of additive consistent IFPR, respectively. Definition 2.6 [50]: Let B = (b ij ) n×n be an IFPR with b ij = (µ ij , v ij ), (i, j = 1, 2, · · · , n), if there exists a vector ω=(ω 1 , ω 2 , . . . , ω n ) T such that where Definition 2.7 [48]: An intuitionistic fuzzy preference rela- is called additive consistent, if it satisfies the following transitivity: for all i, j, k = 1, 2, · · · , n.

III. ADDITIVE CONSISTENCY IMPROVING APPROACH
In this section, PFPR and its additive consistency are defined. Based on the additive consistency of PFPR, a model for determining Pythagorean weights is presented. Based on Pythagorean fuzzy weight information, a PFPR with additive consistency is constructed. In addition, this section also proposes a consistency index to measure the Pythagorean fuzzy preference relation matrix, in order to judge whether the PFPRs has acceptable consistency. For the PFPRs with unacceptable consistency, the algorithm of consistency adjustment is proposed.
for all i, j = 1, 2, · · · , n, where p ij is a Pythagorean fuzzy value composed by the certainty degree µ ij to which x i is preferred to x j and the certainty degree v ij to which x i is nonpreferred to x j . Furthermore, µ ij and v ij satisfy the following conditions: } be a set of alternatives, suppose that there is a PFPR on X , which is shown as follows: According to the PFPR P, it is obvious that p is ≥ p it for all i ∈ {1, 2, · · · , n}, where s ∈ {1, 2, · · · , n} and t ∈ {1, 2, · · · , n}. Since p i2 ≥ p i1 , p i3 ≥ p i2 and p i4 ≥ p i3 for all i ∈ {1, 2, · · · , n}, which denote the alternative x 2 is better than x 1 , the alternative x 3 is better than x 2 , the alternative x 4 is better than x 3 , then PFPR P indicating that the ranking is that We can see that intuitionistic fuzzy preference relations can be viewed as a degeneration of Pythagorean fuzzy preference relation. Based on the additive consistency of intuitionistic fuzzy preference relation, a new definition of additive consistency is introduced by directly employing the membership and non-membership degrees in a Pythagorean fuzzy preference relation. Definition 3.3: For a Pythagorean fuzzy preference relation if there exists a s vector ω=(ω 1 , ω 2 , . . . , ω n ) T such that where ω i ∈ [0, 1](i = 1, 2, · · · , n) that n i=1 ω i = 1. Then, P is called an additive consistent PFPR.
And we can also get the following equation
Definition 3.6: A Pythagorean fuzzy weight vector n is said to be normalized if it satisfies the following conditions: Let Then the following result is obtained. Theorem 3.3: Assume that the elements of the matrix P = (p ij ) n×n are defined by Eq. (17), thenP is a Pythagorean fuzzy preference relation.
Proof: It is obvious thatp Since According to Definition 3.1,P = (p ij ) n×n is an PFPR, which completes the proof.
Proof: From Eq. (17), ki ) 2 , According to Definition 3.4, it is certified that the PFPRP = (p ij ) n×n is additive consistent, which completes the proof.
From Theorem 3.3, one can easily obtain the following corollary.
Corollary 3.1: For a PFPR P = (p ij ) n×n in which p ij = (µ ij , v ij ), if there exists a normalized Pythagorean fuzzy weight vectorω = (ω 1 ,ω 2 , · · · ,ω n ) T such that for i = 1, 2, · · · , n, then P = (p ij ) n×n is an additive consistent Pythagorean fuzzy preference relation. Inspired by Corollary 3.1, we can develop a method to derive the priority weight vector from PFPRs.

B. LINEAR GOAL PROGRAM MING MODEL FOR GENERATING PYTHAGOREA FUZZY WEIGHT
This section develops a linear goal programming model for deriving Pythagorean fuzzy weights from PFPR.
As presented above, for the purpose of obtaining a reasonable result, the PFPR given by the DM should satisfy additive consistency which can be expressed as Eq. (18) according to Corollary 1. However, in practical situations of decision making, it is too difficult for a DM to construct such an additive consistent PFPR. Hence, it is expected that the deviation between the given PFPR and its corresponding additive consistent PFPR should be as small as possible. As a result, we introduce the deviation variables as follows: As can be seen, the smaller the absolute deviations are, the more exact the results are. Thus, the following fractional programming model can be established to derive the Pythagorean fuzzy weights (M1), as shown at the bottom of the next page.
Theorem 3.5: M1 is equivalent to the following M2, as shown at the bottom of the next page.
Proof: Because of µ ij = v ji and µ ji = v ij , the deviation of the upper diagonal elements is equal to the deviation of the lower diagonal elements. Hence, we only need to consider the deviation of the upper (or lower) diagonal elements. That is to say, the objective functions of M1 and M2 are equivalent, which completes the proof.
In order to simplify the calculation, in the following, we will use M2 for further discussion. If can be further expressed and simplified as follows: Theorem 3.6: The PFPR P = (p ij ) n×n is an additive consistent preference relation if and only if J * = 0, where J * is the optimal value of the objective function.
Proof: If P = (p ij ) n×n is an additive consistent PFPR, then the deviation of both the membership and non-membership degrees should be equal to 0, which indicates that J * = 0. If J * = 0, i.e., . . , n, VOLUME 8, 2020 then we can derive that ε + ij = ε − ij = η + ij = η − ij = 0 for all i, j = 1, 2, . . . , n. That is to say, the PFPR P = (p ij ) n×n is equal to its corresponding additive consistent PFPR, which completes the proof.
M3, as shown at the bottom of the page, can be solved by using some optimization computer packages, such as MATLAB and LINGO. Thus, the optimal objective function value J * and the optimal Pythagorean fuzzy weight vector can be yielded. If J * = 0, the given PFPR P = (p ij ) n×n is additive consistent in which case the derived Pythagorean fuzzy weight vector is reasonable. If J * = 0, P does not have additive consistency, so we need to consider whether P have acceptable consistency. If P has acceptable additive consistency, the next calculation can be performed; if P does not have acceptable additive consistency, we will adjust it with an additive consistency adjustment algorithm until it reaches acceptable additive consistency. Once the result is consistent with decision maker's preferences, the calculation process ends. Otherwise, the decision maker should reevaluate the alternatives to construct a more acceptable consistent PFPR P = (p ij ) n×n , or the process will stop as the repetition times reach the maximum number which we specified previously. The adjustment algorithm for additive consistency will be introduced as below.

C. ALGORITHM FOR IMPROVING ADDITIVE CONSISTENCY
In this section, we define the consistency index of PFPR and develop a feasible algorithm for improving consistency degree of PFPR without achieving acceptable consistency. Definition 3.7: Assume two PFPRs is called the distance between P 1 and P 2 . Definition 3.8: Assume a PFPR P = (p ij ) n×n with p ij = (µ ij , v ij ), and its additive consistent PFPRP = (p ij ) n×n with p ij = (μ ij ,ṽ ij ), to make P approximateP as much as possible, we define CI (P) as a consistency index (CI) of the PFPR P as follows.
According to Definition 3.7, the CI (P) can be used to measure the distance between P andP. Theorem 3.7: The consistency index CI (P) between two PFPRs P and its additive consistent PFPRP satisfies the following properties: 1. 0 ≤ CI (P) ≤ 1; 2. CI (P) = 0 if and only if P =P.
Additionally, according to Theorem 3.7, the smaller the CI (P), the more consistent the PFPR P. Especially, CI (P) = 0 if and only if P is an additive consistent PFPR.
In most cases, it is unrealistic to construct an additive consistent PFPR due to the reason that decision makers must be affected by many factors in the decision-making process. Based on this, a definition of acceptably additive consistent PFPR will be further developed to allow a certain of level of acceptable deviation. Definition 3.9: Let P = (p ij ) n×n be a PFPR. Given a threshold value CI , if the additive consistency index satisfies the following, then we call a PFPR P with acceptably additive consistency. The value of CI can be determined according to the preferences of the decision maker or the actual situation of the problem. Which is a question worthy of further discussion in the future.
Due to the complexity of objective things and the limitations of human cognition, the PFPR P constructed by DMs often has unacceptable additive consistency, i.e., CI (P) ≥ CI . In order to obtain more reasonable results, DMs need to construct a new PFPR based on additive consistency. To help the DMs to obtain an additive consistent PFPR, we provide the following formula to adjust or repair the inconsistent PFPR P (t) = (p (t) ij ) n×n until it has acceptably additive consistency.
Theorem 3.8: ) n×n be a PFPR defined by Eq. (24), as shown at the bottom of the next page. Then, we have CI (P (t+1) ) ≤ CI (P (t) ).
Consistency means that decision makers do not have conflicts when expressing their preferences. If PFPR P = (p ij ) n×n does not have acceptable consistency, we should adjust it to achieve acceptable consistency before using it to resolve decision problems. We propose the following algorithm to modify P = (p ij ) n×n that do not have acceptable consistency to meet the consistency requirements.

Algorithm 1
Input: The original PFPR P = (p ij ) n×n with p ij = (µ ij , v ij ), the parameter σ ∈ (0, 1) that is the trade-off parameter between the inconsistent preference relation and the corresponding consistent preference relation, the maximum number of iterations t * , and the threshold value CI ∈ (0, 1]. Output: The adjusted PFPRP = (p ij ) n×n withp ij = (μ ij ,v ij ), and the consistency index CI (P).
Step 6: End. The proposed algorithm can be described by using FIGURE. 1.

IV. METHOD FOR DECISION MAKING WITH PFPR
In this section, we develop an approach to decision making based on Pythagorean fuzzy preference relation (PFPR) and its additive consistency, which can be described as Algorithm 2.
The proposed algorithm can also be described by using FIGURE. 2.

V. NUMERICAL EXAMPLES
This section presents a numerical example to validate the proposed models. In this paper, since √ 2 2 is irrational number, we make √ 2 2 ≈ 0.7071.

A. EXAMPLE AND COMPARISON ANALYSIS
As the big data, cloud computing, AI and other fields developing at a high speed, the contradiction of the increasing data calculation volume, flexibility demand and the execution is gradually sharp. Computing-in-memory (CIM) based on DRAM integrates Computing and storage closely, which can partially relieve the ''von neumann bottleneck'', reduce data handling between on-chip cache and Memory, and greatly improve Memory access efficiency. During the research of the CIM in recent years, r&d institutions have all taken the following four factors, CI (P (t+1) ) = d(P (t+1) ,P (t+1) ) = d(P (t+1) ,P (t) ) ,P (t) ) ≤ CI (P (t) ).

Algorithm 2
Step 1: Construct the PFPR matrix P = (µ ij , v ij ) n×n based on the decision-making information; and set the values of the predefined consistency threshold CI .
Step 6: Utilize Algorithm 1 to modify the PFPR that do not achieve acceptable consistency degree. After implementation of Algorithm 1, we can get PFPRs P = (p ij ) n×n = (µ ij , v ij ) n×n with acceptable consistency; then, go to Step 7.
Step 8: Rank all p i (i = 1, 2, . . . , n) by means of the score function Eq. (9) and the accuracy function Eq. (10), and then rank all the alternatives x i (i = 1, 2, . . . , n) and select the best one in accordance with the values of p i (i = 1, 2, . . . , n).
access speed, storage capacity, computing speed,and delay time(CAS),into consideration in the design of hardware structure optimization and software algorithm improvement,such as 3D stacked DRAM with CIM [62], approxPIM(A new CIM architecture for solving the problem of bandwidth and power mismatch between processor and memory ) [63] and a mathematical framework integrated with 3D stacked DRAM accelerators [64].
So,An excellent CIM system is affected by many factors, including x 1 : access speed x 2 : storage capacity x 3 : computing speed x 4 : delay time(CAS) Consider the case that a company prepare to know the importance of these factors for the design of CIM, and because decision makers lack the corresponding expertise, many uncertain messages are generated. Then by pairwise comparison of x i and x j (i, j = 1, 2, 3, 4), the decision maker construct a Pythagorean fuzzy preference relation (PFPR) P (0) as follows P (0) , as shown at the bottom of the next page.
Step 2: By calculating M3, the following linear goal program is established Min J , as shown at the bottom of the next page.
Solving this model by an appropriate optimization computer package, it follows that the optimal objective value J = 1.68, which implies that PFPRP is non-additive consistent, and we get the optimal Pythagorean fuzzy weights vector as:ω Utilize Eq. (17), a PFPRP (0) with complete additive consistency is constructed as followsP (0) , as shown at the bottom of the next page.
Step 4: Utilize Algorithm 1 to modify the PFPR P (0) . After implementation of Algorithm 1, we can get CI (P (3) ) = 0.0530. Since CI (P (3) ) < CI , we can get PFPRs P (3) with acceptable consistency is shown as follows P (3) , as shown at the bottom of the next page.
Step 5: Utilize the Pythagorean fuzzy arithmetic averaging operators Eq. (13) to aggregate all p Step 7: The ranking of the score function of four alternatives is s(p

).
Step 8: The ranking of the four alternatives is

B. COMPARISON WITH OTHER METHOD
Because of the value ranges of PFN and IFN are different, in the comparative analysis, we first convert PFN into IFN, and the converted preference relation matrix is denoted as P * , as shown at the bottom of the next page.

VI. DISCUSSIONS
Based on the numerical examples and comparative study, the characteristic of the proposed method is summarized as follows.
1) As the proposed model aim to derive the Pythagorean fuzzy priority weights by minimizing derivation of via a linear programming model, it can be observed that the model is built on PFN. As mentioned above, PFN have more powerful abilities than IFN do in modeling the uncertainty of practical decision-making problems. From this perspective, it can be concluded the proposed model has a wider range of applications.
(2) The specific implementation steps of the two methods are different. Wang [48]'s method developed a linear goal programming model to obtain its intuitionistic fuzzy weights, and then the best alternatives is selected. In the proposed models, we develop a linear goal programming model to obtain its Pythagorean fuzzy weights, and for the PFPR that does not satisfy the consistency, the consistency index is defined to measure the degree of consistency, and a consistency adjustment algorithm is proposed, this method makes our results more accurate.
(3) Finally, the proposed adjustment algorithm can be used to improve the additive consistency of a PFPR. By solving this algorithm, not only can the additive consistency of a PFPR be improved, but also can make reference for decision makers before making decisions.
However, the proposed methods still have some limitations. First, it needed to solve a linear programming model to obtain the Pythagorean fuzzy weights with additive consistency, and for PFPR that does not meet consistency, it needs to be adjusted, it may be a bit complex compared with other methods. However, these models can be easily solved by using some optimization packages, such as Lingo, MATLAB and CPLEX. In addition, the threshold of the additive consistency index is assumed to be given by decision makers. We argue that it will be more convincing if the threshold can be determined by using some automatic methods.

VII. CONCLUSION
As a new type of preference relation, PFPR not only expands the application scope of preference relations, but also more fully expresses the views of decision makers. In this paper, we mainly discussed the application of PFPR in decision making. Firstly, we defined PFPR and its additive consistency based on IFPR, and discussed some properties that satisfy the additive consistency of PFPR. Secondly, we developed a linear goal programming model for generating Pythagorean fuzzy weight based on the additive consistency of PFPR.
Then, for PFPRs that do not satisfy additive consistency, we defined the consistency index of PFPR and develop a feasible algorithm for improving consistency degree of PFPR. In the next section, we developed an approach to decision-making based on PFPR and its additive consistency. The proposed decision-making process and models may be used in many real-world applications in which the DMs may be able to provide his/her preferences for alternatives to a PFN, this was confirmed in the final section of this paper.
During the PPFR research process, we can find that there are some new directions that should be considered in future research, such as the application of PFPR in group decision making, the multiplicative consistency of PFPR and the consensus reaching process of groups in Pythagorean fuzzy environment.
LIGANG ZHOU received the Ph.D. degree in operational research from Anhui University, Hefei, China, in 2013. He is currently a Professor with the School of Mathematical Sciences, Anhui University. He has contributed over 100 journal articles to professional journals, such as the IEEE TRANSACTIONS ON FUZZY SYSTEMS, Fuzzy Sets and Systems, Information Sciences, Applied Mathematical Modelling, and Group Decision and Negotiation. He is also a Paper Reviewer of many international journals. His current research interests include information fusion, group decision making, aggregation operators, and combined forecasting.
KAIFANG YANG is currently the Graduate Student of electronic information with Anhui University. His main research interests include control engineering and sensing technology. VOLUME 8, 2020