Logarithmic Aggregation Operators of Picture Fuzzy Numbers for Multi-Attribute Decision Making Problems

: The objective of this study was to create a logarithmic decision-making approach to deal with uncertainty in the form of a picture fuzzy set. Firstly, we deﬁne the logarithmic picture fuzzy number and deﬁne the basic operations. As a generalization of the sets, the picture fuzzy set provides a more proﬁtable method to express the uncertainties in the data to deal with decision making problems. Picture fuzzy aggregation operators have a vital role in fuzzy decision-making problems. In this study, we propose a series of logarithmic aggregation operators: logarithmic picture fuzzy weighted averaging / geometric and logarithmic picture fuzzy ordered weighted averaging / geometric aggregation operators and characterized their desirable properties. Finally, a novel algorithm technique was developed to solve multi-attribute decision making (MADM) problems with picture fuzzy information. To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method, and concluded from the comparison and sensitivity analysis that our proposed technique is more e ﬀ ective and reliable.


Introduction
Multiple attribute decision making (MADM) is a process for selecting the optimal alternative from a set of given alternatives according to some attributes.Due to the complexity of the socio-economic environment and decision makers' insufficient knowledge and judgment, it is difficult for decision makers to provide accurate information about alternatives.For this, Atanassov [1] proposed the intuitionistic fuzzy set (IFS).IFS is an extension of fuzzy set (FS) [2], which is characterized by a positive membership degree and a negative membership degree, satisfying the condition that the sum of these two degrees is equal to or less than one; therefore, IFS is a useful tool in processing fuzzy and uncertainty information.
IFS has been practically applied in many fields, such as clustering analysis [3], decision making (DM) problems [4], medical diagnosis [5], and pattern recognition [6].Many researchers have contributed to IFS.Chen et al. [7] presented a transformation method of intuitionistic fuzzy numbers (IFNs) and interval valued intuitionistic fuzzy (IVIF) numbers to aggregation operations to solve DM problems.Garg [8] introduce a robust improved geometric aggregation operation.Huang [9] introduced the intuitionistic fuzzy Hamacher aggregation operations and discussed their applications in decision making problems.
To aggregate the intuitionistic fuzzy information and make decisions, many intuitionistic fuzzy aggregation operators have been proposed for aggregating different alternatives [10,11].
However, in some special conditions, the sum of the positive membership degree and the negative membership degree provided by the decision maker may be greater than one.For instance, if a decision maker determines the positive membership as 0.3 and the negative membership degree may be 0.9, 0.3 + 0.9 = 1.2 > 1.This situation cannot be handled by IFS.To overcome this problem, Yager [12,13] developed the concept of the Pythagorean fuzzy set (PyFS), which is also characterized by the positive membership degree and the negative membership degree, but the square sum of these two degrees is equal to or less than one.Corresponding to the above considered example, 0.3 2 + 0.9 2 = 0.9 < 1. PyFS can solve many problems that IFS cannot.In other words, PyFS is a generalization of IFS; therefore, PyFS is more capable than IFS of modeling the imprecise and imperfect information in practical MADM problems.Since PyFS's introduction, numerous aggregation operators have been proposed for aggregating Pythagorean fuzzy information [14,15].
In real life, some problems cannot be symbolized in IFS and PyFS.For example, for a voting system, human opinions include many answers of such types: yes, no, abstain, and refusal.Therefore, Cuong [16] introduced the core idea of the picture fuzzy set (PFS).PFS is an extension of IFS and PyFS.In the concept of the PFS, Cuong added the neutral term along with the positive membership and negative membership degrees and the sum of their membership degrees is equal to or less than one.Many researchers have attempted to contribute to the application of PFS.In 2014, Phong et al. [17] developed some composition of picture fuzzy relations.Singh [18] developed correlation coefficients for PFS and discussed their applications in decision making problems.In 2015, Cuong et al. [19] introduced the core idea about some fuzzy logic operations for picture fuzzy numbers.Thong et al. [20] developed a policy for multi-variable fuzzy forecasting using picture fuzzy (PF) clustering and a PF rules interpolation system.Son [21] presented the generalized picture distance measure and discussed its application.Wei [22] introduced PF cross-entropy and discussed its application in MADM problems.Ashraf et al. [23] proposed geometric aggregation operators and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach to deal with the uncertainty in decision making problems in the form of picture fuzzy sets.Garg [24] proposed some aggregation operators for picture fuzzy information and discussed their application in decision making problems.In 2017, Wei [25] introduced the PF aggregation operator based on t-norm and t-conorm and provided some discussion on its applications in decision making.In 2019, Zeng et al. [26] proposed the exponential Jensen picture fuzzy divergence measure and discussed its applications in multi-criteria decision making (MCDM).
Many aggregation operators have been developed based on the algebraic t-norm and t-conorm to deal with the aggregate uncertainty in the form of picture fuzzy sets.Logarithmic operational rules are important mathematical operations that conveniently use to aggregate uncertain and inaccurate data.Motivated by these ideas, we developed a picture fuzzy MCDM method based on the logarithmic aggregation operators and the logarithmic operations of PFSs to handle picture fuzzy MADM within PFSs.
The contributions of our method are as follows: (1) We developed some novel logarithmic operations for picture fuzzy sets, which can overcome the weaknesses of algebraic operations and capture the relationship between various PFSs.(2) We extended logarithmic operators to logarithmic picture fuzzy operators: logarithmic picture fuzzy weighted averaging (log-PFWA), logarithmic picture fuzzy ordered weighted averaging (log-PFOWA), logarithmic picture fuzzy weighted geometric (log-PFWG), and logarithmic picture fuzzy ordered weighted geometric (log-PFOWG) aggregation operators for picture fuzzy information, which can overcome the algebraic operators' drawbacks.(3) We developed an algorithm to deal with multi-attribute decision making problems using picture fuzzy information.(4) To show the effectiveness and reliability of the proposed picture fuzzy logarithmic aggregation operators, we applied the proposed operator to a circulation center evaluation problem.(5) The results indicate that the proposed technique is more effective and provides a more accurate output compared to existing methods.
The remainder of this paper is organized as follows: Some basic knowledge related to IFS, PyFS, PFS, score, and accuracy function are defined in Section 2. In Section 3, we discuss the logarithmic operation laws and the properties of picture fuzzy numbers (PFNs), as well as logarithmic picture fuzzy averaging/geometric aggregation operators.In Section 4, we address the MADM problem, using logarithmic picture fuzzy aggregation operators.A descriptive example is outlined in Section 5; Section 5.1 provides a comparative analysis of the existing aggregation operators with the novel logarithmic aggregation operators defined in this study, and the conclusion of this study is draw in Section 6.

Background
The concepts and basic operations of existing extensions of fuzzy sets are recalled in this section, and they were the foundation of this study.

For all
. a ∈ A, the refusal degree is a , £ J 2 .a be any three PFNs, then: a ∈ A be a PFN.The score function of J is defined as: The accuracy function of J is defined as: a be two PFNs.Then, the comparison rules are defined as:

Picture Fuzzy Logarithmic Aggregation Operators
In this section, we develop some novel logarithmic operational laws for picture fuzzy numbers and discuss their properties.The main purpose of this section is to propose novel logarithmic aggregation operators based on picture fuzzy information.
Note that Log τ 0 is meaningless in the real number system and Log τ 1 is not well defined.Therefore, we assume that J 0, where J is a picture fuzzy number (PFN).Also, the base of log τ 1, τ is in the open interval (0, 1) and always a positive real number.a ≤ 1.Then, the positive membership is defined as: the neutral membership is defined as: the negative membership is defined as: and the refusal degree is defined as:

a , and J
a be any three PFNs.Then, the logarithmic operations are defined as: Proof.(1) According to the Definitions 6 and 7, we have: (2) Similarly, we have: a be any two PFNs with min(β 1 , Proof.(1) According to the Definitions 6 and 7, we have Therefore, we have: (2) Similarly, we have: .
Therefore, we have:

Logarithmic Picture Fuzzy Averaging Aggregation Operators
In this subsection, we define some logarithmic aggregation operators: picture fuzzy logarithmic weighted averaging and picture fuzzy logarithmic order weighted averaging aggregation operators.We also discuss their properties in detail.
Then, using Definitions 6 and 7, we have: Proof.Now, we proof using mathematical induction.So, for n = 2, we have where Using Definition 7, we have Suppose that Theorem 3 is true for n = k: Now, we must prove that Theorem 3 is true for n = k + 1: Using Definition 7, we have: Hence, the result holds for n = k + 1.Thus, the result is true for all n, i.e.: Then we have: where any collection of PFNs, and we have (3) Monotonicity: , holds for any h, then:

Proof of Theorem 4 (3).
Let and However, if β h ≤ β * h , γ h ≥ γ * h , and £ h ≥ £ * h , for any h, then: Therefore, using the definition of the score function and the accuracy function, we conclude that: Then, the mapping Log − PFOWA : Φ n → Φ is defined as: Then, Log-PFOWA is called the logarithmic picture fuzzy ordered weighted averaging operator of dimension n such that the weighted vectors are any collection of PFNs with 0 < τ h ≤ β h ≤ 1, and τ h 1.Then, using Definitions 6 and 7, we have: Proof.The proof is similar to Theorem 3, so the procedure is not provided here.

Theorem 6. (1) Idempotency: Let
Then, we have: where Then we show that: (3) Monotonicity: , holds for any h, then we have: where Proof.The proof is similar to Theorem 4, so the procedure is not provided here.

Logarithmic Picture Fuzzy Geometric Aggregation Operator
Definition 10.Let the family of the PFNs be J and τ h 1,.Let the function be defined as Log − PFWG : then the function Log-PFWG is the logarithmic PF-weighted geometric operator, where the weighted a family of PFNs, and 0 < τ h ≤ β h ≤ 1, and τ h 1, that is, h = 1, 2, • • • n .Then, using the operator Log-PFWG, the aggregated value is also a picture fuzzy number; hence, we know from Definition 5 that: where the weighted vectors are υ Proof.The proof is similar to Theorem 3, so the procedure is not provided here.

Theorem 8. (1) Idempotency: If
(2) Boundedness: ) is a family of PFNs, and also let (3) Monotonicity: , holds for any h, then: where Proof.The proof is similar to Theorem 3, so the procedure is eliminated here.
then the above define function Log-PFOWG is known as the log-PF-ordered weighted geometric operator of dimension n such that the weighted vectors are υ = (υ Then, by using the Log-PFOWG operator, the aggregated value is also a log-PFN; hence: and their sum, i.e., n h=1 υ h = 1,.We also take Proof.The proof is similar to Theorem 3, so the procedure is eliminated here.

Definition 12. Let the logarithmic PFNs be Log
, then S is the score function of Log τ J, which is defined as follows: Let H be the accuracy function of Log τ J, which is defined as follows:

Proposed Method for Solving the MADM Problem
In this section, we establish the introduced method for solving MADM problems using the logarithmic PF aggregation operators.For a finite collection of m alternatives, i.e., U = {u 1 , u 2 , . . ., u m }, and any finite of n attributes, i.e.; G = g 1 , g 2 , . . ., g n , let: be the decision matrices (DMs), where β jk , γ jk , £ jk are the collections of log-PFNs.We also take the weighed vectors of the attribute as υ = (υ 1 , υ 2 , • • • , υ n ) T , with υ h ∈ [0, 1], and n h=1 υ h = 1.The detail of the propsoed method is given in Figure 1.Step 2.
In this step, we use the proposed method to aggregate the different preference types of the DMs.

Case 1. The method under the PFN environment is summarized as follows:
( ( ) ( ) ( ) Graphically presentation of the proposed method.
We planned the following algorithm to resolve MADM problems with PF information by using the log-PFWA and log-PFWG operators.

Step 1. Normalized the given decision matrices (DMs).
There are two types of criterion one is the benefit criteria and the other one is the cost type criteria.The following equation is used to modify the cost type criteria into benefit criteria.If the given criteria are of the benefit type, then there is no need for normalization: Step 2.
In this step, we use the proposed method to aggregate the different preference types of the DMs.
Case 1.The method under the PFN environment is summarized as follows: Step 3.
In this step, we use Definition 5 to calculate the score and accuracy functions of the decision matrices.
In this step, we rank the alternatives based on score and accuracy functions to choose the best option.

Descriptive Example
In this section, the proposed method is applied to a circulation center evaluation problem.Consider a committee of DMs to evaluate and select the most suitable circulation center among four circulation centers: G 1 , G 2 , G 3 , and G 4 .The decision maker evaluates the four circulation center alternatives based on four aspects: cost of transportation (u 1 ), load of capacity (u 2 ), satisfying demand with minimum delay (u 3 ), and security (u 4 ).We selected the best option according to the suitability ratings of the four alternatives, G 1 , G 2 , G 3 and G 4 , under four aspects u 1 , u 2 , u 3 , and u 4 whose weight vector is (0.224, 0.236, 0.304, 0.236), which are assumed by DM.In this example, the DM is designed to be G = (J mn ) 4×4 , as shown in Table 1.Step 1. First, we normalized the DM in Table 2. Step 2. Operate the aggregation operator to evaluate the DMs of the logarithm picture fuzzy information.
Case 1.Using the Log-PFWA operator, we have Table 3. Case 2. In this case, we are using the Log-PFWG, we have Table 4. Case 3. In this case, we use the Log-PFOWA.We have Table 5. Case 4. In this case, we use the Log-PFOWG.We have Table 6.Step 3. In this step, we compute the score values using definition 5, S(G m )(m = 1, . . ., 4) of the Log-PFNs, as shown in Table 7.
Step 4. Thus, the best petroleum circulation center is G 3 .
The comparison of the alternative based on score values is shown in Figure 2.

Comparison Analysis with the Existing Methods
This section provides our comparison analysis of the existing aggregation operators and our novel logarithmic aggregation operators.The existing aggregation operators were proposed by Ashraf et al. [23] in 2018 to handle picture fuzzy quantities.Ashraf introduced the picture fuzzy aggregation operators, such as picture fuzzy geometric, order geometric, and hybrid geometric aggregation operators.We provide novel aggregation operators including logarithmic picture averaging and geometric aggregation operators, such as logarithmic picture fuzzy average, ordered average, geometric, and ordered geometric aggregation operators.The logarithmic aggregation operators proposed in this paper are more general, more accurate, and more flexible To evaluate air quality, we take the descriptive example proposed by Ashraf et al. [23].Three stations are used to evaluate the air quality, denoted as ( 1 , O 2 , O 3 O ).Decision maker weights are (0.314, 0.355, 0.331) T .We have three criteria's  ,    .The collected information from these three stations, which is given in the Tables 8-10.

Comparison Analysis with the Existing Methods
This section provides our comparison analysis of the existing aggregation operators and our novel logarithmic aggregation operators.The existing aggregation operators were proposed by Ashraf et al. [23] in 2018 to handle picture fuzzy quantities.Ashraf introduced the picture fuzzy aggregation operators, such as picture fuzzy geometric, order geometric, and hybrid geometric aggregation operators.We provide novel aggregation operators including logarithmic picture averaging and geometric aggregation operators, such as logarithmic picture fuzzy average, ordered average, geometric, and ordered geometric aggregation operators.The logarithmic aggregation operators proposed in this paper are more general, more accurate, and more flexible.
To evaluate air quality, we take the descriptive example proposed by Ashraf et al. [23].Three stations are used to evaluate the air quality, denoted as (O 1 , O 2 , O 3 ).Decision maker weights are (0.314, 0.355, 0.331) T .We have three criteria's C 1 , C 2 and C 3 .The collected information from these three stations, which is given in the Tables 8-10.Step 2.
In this step, we use the proposed Log − PFWA aggregation operator to evaluate the collective picture fuzzy matrix, which is in shown in Tables 11 and 12. Next, we operate the Log − PFOWA aggregation operator, to rank the alternatives.
Step 3. In this step, we compute the score values using definition 5. S(G m )(m = 1, . . ., 4) of the Log-PFNs is as shown in Table 13.Step 4. Thus, the best evaluation result for this case is G 4 .
We compared the existing method [23] and our novel logarithmic aggregation operators.From the analysis, we found our proposed aggregation operator (log-PFOWA) and the existing method aggregation operator [23] provide the same result, as shown in Table 14.
Table 14.Comparison analysis with the existing method [23].

Method Ranking Order
Ashraf [23], Table 14 shows that our proposed method logarithmic picture fuzzy aggregation operator provides new method to find the best alternative in this decision-making problem.

Conclusions
In this paper, we outlined the logarithmic operational laws of PFNs, which are a useful supplement to the existing picture fuzzy aggregation techniques.Based on the logarithmic operational laws, we constructed a series of aggregation operators of PFNs, i.e., Log-PFWA, Log-PFWG, Log-PFOWA, and Log-PFOWG.Several characteristics required of these logarithmic aggregation operators were comprehensively explored.To demonstrate the efficiency and consistency of the proposed picture fuzzy logarithmic aggregation operators, we developed an algorithm to address MADM problems for tacking the uncertainty in the form of picture fuzzy information.A descriptive example using a circulation center evaluation problem was provided to validate and demonstrate the applicability of our proposed technique.Consequently, we also demonstrated the superiority and the validity of the proposed aggregation operators using some of the existing tools.Hence, the analysis showed that our proposed aggregation operator is more reliable and accurate than the existing method.Thus, our proposed method logarithmic aggregation operator provides new method to find the best alternative in the MADM decision-making problem.
In future work, we will discuss the generalized form of these operators for the logarithmic operational law of PFSs and define the hybrid averaging and hybrid geometric operators.

Definition 1 .a ≤ 1 . 2 .
[1] Let a fixed set A φ.Then, the set J = β J . a , £ J . a .a ∈ A , is an IFS, where β J . a , £ J . a ∈ [0, 1] are the positive and negative membership degrees of the element .a ∈ A in J, respectively.β J . a , £ J . a satisfy the condition for all .a ∈ A, 0 ≤ β J . a + £ J .Definition [16] Let a fixed set A φ.Then, the set J = β J . is said to be PFS, where β J . a , γ J . a , £ J . a ∈ [0, 1] are the positive, neutral and negative membership degrees of the element .a ∈ A in J, respectively.β J .

Figure 1 .
Figure 1.Graphically presentation of the proposed method.

Table 1 .
Petroleum circulation center information.

Table 2 .
Normalized petroleum circulation center information.

Table 3 .
Petroleum circulation center information.

Table 4 .
Petroleum circulation center information.

Table 5 .
Petroleum circulation center information.

Table 6 .
Petroleum circulation center information.

Table 8 .
Air quality data from station O1.

Table 9 .
Air quality data from station O2.

Table 10 .
Air quality data from station O3.

Table 8 .
Air quality data from station O 1 .

Table 9 .
Air quality data from station O 2 .

Table 10 .
Air quality data from station O 3 .

Table 11 .
Collective picture fuzzy information.

Table 13 .
Evaluation results of each air station.