New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation

In this paper, some new operations and basic properties of picture fuzzy relations are intensively studied. First, a new inclusion relation (called type-2 inclusion relation) of picture fuzzy relations is introduced, as well as the corresponding type-2 union, type-2 intersection and type-2 complement operations. Second, the notions of anti-reflexive kernel, symmetric kernel, reflexive closure and symmetric closure of a picture fuzzy relation are introduced and their properties are explored. Moreover, a new method to solve picture fuzzy comprehensive evaluation problems is proposed by defining the new composition operation of picture fuzzy relations, and the picture fuzzy comprehensive evaluation model is built. Finally, an application example (about investment risk) of picture fuzzy comprehensive evaluation is given, and the effective experiment results are obtained.

Although an intuitionistic fuzzy set has been successfully applied in different areas, but there are situations that cannot be represented by it in real life, such as voting, we may face human opinions involving more answers of the type: yes, abstain, no and refusal.Thus, in 2013, B. C. Cuong proposed a new concept named picture fuzzy sets (PFSs) [28][29][30], which is an extension of fuzzy sets and intuitionistic fuzzy sets.Picture fuzzy sets give three degrees of an element named degree of positive membership, degree of neutral membership and degree of negative membership, respectively.The picture fuzzy set solved the voting problem successfully, and is applied to clustering [31], fuzzy inference [32], and decision-making [33,34].
Relations are a suitable tool for describing correspondences between objects.Crisp relations have served well in mathematical theories.However, there are some problems that can't be solved through classic relationships, such as the relationship of two objects being vague.Therefore, after the fuzzy set was defined, the definition of fuzzy relations was also proposed by Zadeh in paper [1] as an extension of classic relationship.Then, some scholars study it and used it widely in many fields, such as decision making, clustering analysis [35][36][37][38], fuzzy comprehensive evaluation [39][40][41] and so on.Fuzzy relations can model vagueness; however, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees.Intuitionistic fuzzy sets, as defined by Atanassov [4], gave us a way to incorporate uncertainty in an additional degree.Burillo and Bustince gave the definition of intuitionistic fuzzy relations [42,43] and discussed some properties of them.Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a Cartesian product of universes.In 2005, Lei et al. further researched intuitionistic fuzzy relations and composition operation of intuitionistic fuzzy relations [44].Yang proposed the definition of kernels and closures of intuitionistic fuzzy relations and proved fourteen theorems of intuitionistic fuzzy relations [45].B. C. Cuong proposed the notion of picture fuzzy relations and studied their operations and properties [28,29].
The rest of this paper is structured as follows.In Section 2, some basic notions and operations of picture fuzzy sets and picture fuzzy relations are provided.In Section 3, type-2 union, type-2 intersection, and type-2 complement operations of picture fuzzy relations are well described and their properties are studied.In Sections 4 and 5, kernels and closures of a picture fuzzy relation are discussed.Their computational formulas and some properties are also obtained.Meanwhile, some examples are given.In Section 6, a new composition operation of picture fuzzy relations is investigated.Furthermore, according to the new composition operation, a new method to solve picture fuzzy comprehensive evaluation problems is proposed, and we also prove that this method is doable by an application example.The last section summarizes the conclusions.

Some Basic Concepts
In this section, several basic concepts and operations about picture fuzzy sets and picture fuzzy relations are provided.Definition 1.An intuitionistic fuzzy set (IFS) A on the universe X is an object of the form A = {(x, µ A (x), ν A (x)) | x ∈ X}, where µ A (x) ∈ [0, 1] is called the "degree of membership of x in A", ν A (x) ∈ [0, 1] is called the "degree of non-membership of x in A", and where µ A (x) and ν A (x) satisfy µ A (x) + ν A (x) ≤ 1 for all x ∈ X.In this paper, let IFS(X) denote the sets of all the intuitionistic fuzzy sets on X [4].Definition 2. A picture fuzzy set A on the universe X is an object of the form A = {(x, µ A (x), η A (x), ν A (x))|x ∈ X }, where µ A (x) ∈ [0, 1] is called the "degree of positive membership of x in A", η A (x) ∈ [0, 1] is called the "degree of neutral membership of x in A", and ν A (x) ∈ [0, 1] is called the "degree of negative membership of x in A", and ) is called the "degree of refusal membership of x in A".Let PFS(X) denote the set of all the picture fuzzy sets on a universe X [28].
the set of all the picture fuzzy relations in X × Y is denoted [29].
Symmetry 2017, 9, 268 3 of 17 Definition 5. Let R and P be two picture fuzzy relations between X and Y, for every (x, y) ∈ X × Y we define [29]: is called to the one defined by [29]: where ∀ (x, z) ∈ X × Z: The score function S can be defined as S(α) = µ α − ν α , and the accuracy function H is given by H Then, for two picture fuzzy numbers α and β [33], (1) if S(α) > S(β), then α is superior to β, denoted by α β; (2) if S(α) = S(β), then We also use voting as a good example to explain the above definition, where S(α) = µ α − ν α represents goal difference and H(α) = µ α + η α + ν α can be interpreted as the effective degree of voting.When S(α) increases, we can know that there are more people who vote for α and people who vote against α become less.When H(α) increases, we can know that there are more people who vote for or against α and people who refuse to vote become less.Therefore, H(α) depicts the effective degree of voting.

On Inclusion Relation of Picture Fuzzy Relations
Similar with [30], we consider the set D* defined by: From now on, we will assume that if x, y ∈ D*, then x 1 , x 2 and x 3 denote, respectively, the first, the second and the third component of x, i.e., x = (x 1 , x 2 , x 3 ).We denote the units of D* by 1 D* = (1, 0, 0) and 0 D* = (0, 0, 1), respectively.
In fact, type-2 inclusion relation is based on the following order relation on D* (see [5], and it is called a type-2 order relation in this paper): The above "∧" denote "and", "∨" denote "or".
Note that, if for any x, y ∈ D* that neither x ≤ 2 y nor y ≤ 2 x, then x and y are incomparable, denoted as x|| ≤ 2 y.

New Operations and Properties of Picture Fuzzy Relations
In this section, we introduce some new operations named type-2 inclusion, type-2 union, type-2 intersection and type-2 complement operation of picture fuzzy relations and study their properties.
For any picture fuzzy relations R and P on X × Y, by Definition 8, we have: From this, we can get the following proposition.
According the Definitions 9 and 10, we can get the Type-2 complement of Proposition 3. Type-2 union and type-2 intersection of PFRs don't satisfy distributive law, which means that 1, 4 and 5 6-9.Furthermore, according to Tables 6-9, we can get the conclusion of Propositions 3 (1) and (2).

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Discussion
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Kernels of Picture Fuzzy Relations
In this section, we will give the definition of anti-reflexive kernel and symmetric kernel about a PFR, and then study their properties.
(1) The maximal anti-reflexive PFR contained in R is called anti-reflexive kernel of R, denoted by ar(R).
(2) The maximal symmetric PFR contained in R is called symmetric kernel of R, denoted by s(R).
Suppose P is an anti-reflexive PFR and P ⊆ 2 R.

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Authors should discuss the results and how they can be interpreted in perspective of pre studies and of the working hypotheses.The findings and their implications should be disc in the broadest context possible.Future research directions may also be highlighted, Table 1.10.By Proposition 5, we can obtain ar(R) and s(R), which are given in Tables 11 and 12, respectively.Proposition 6.The anti-reflexive kernel operator ar of the PFR has the following properties: Proof.

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Authors should discuss the results and how they can be interpreted in perspect studies and of the working hypotheses.The findings and their implications shoul in the broadest context possible.Future research directions may also be highlighted, T Example 4. Consider X, Y and R given in Example 3. By Proposition 8, we can obtain r (R) and s(R), which are given in Tables 13 and 14, respectively.Proposition 9.The reflexive closure operator r has the following properties: Proof.
(1) By Definition 10, we can get then by the definition of Id N , we can get R ⊆ 2 Id N or R = Id N , also we can get P ⊆ 2 Id N or P = Id N , when x = y, then we can get Id N and Id N ⊆ 2 P, then by Proposition 3 (8), we have Id N ⊆ 2 R ∩ 2 P, so r(R , by Label (3) and Proposition 3 (4) and Label ( 9), we have r (R) Diseases 2017, xx, x

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Authors should discuss the results and how they can be interpreted in perspective studies and of the working hypotheses.The findings and their implications should be in the broadest context possible.Future research directions may also be highlighted, Table Proposition 10.The symmetric closure operator s has the following properties: (3) ∀ R, P ∈ PFR(X × Y), by Proposition 8 (2) and Proposition 3 (10), we have , by Label (3) and Proposition 3 ( 4) and ( 9), s(R) ⊆ 2 s(R) ∪ 2 s(P) = s(R ∪ 2 P) = s(P). ( Diseases 2017, xx, x

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Authors should discuss the results and how they can be interpreted in pe studies and of the working hypotheses.The findings and their implications in the broadest context possible.Future research directions may also be highligh Proof.(1) By Proposition 8 (1), r(R c 2 ) = R c 2 ∪ 2 Id N .By Proposition 3 (11) and Proposition 5 (1), (2) By Propositions 5 and 8, ∀ (x, y) ∈ X × Y, Hence, ar (r(R)) = ar(R).

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Picture Fuzzy Comprehensive Evaluation
In this section, we defined a new composition operation of picture fuzzy relations, and according to the new composition, we give a picture fuzzy comprehensive evaluation model about risk investment.Definition 15.Let R ∈ PFR(X × Y) and P ∈ PFR(Y × Z).Then, the composition of P and R is defined by Then, the composition of A and R is defined by where r ij and B j are picture fuzzy sets, i = {1, 2, . . ., n}, j = {1, 2, . . ., s}.

Picture Fuzzy Comprehensive Evaluation Model
Next, we will give the procedure of picture fuzzy comprehensive evaluation.
Step 1: Establish evaluation index system.According to the method of APH (Analytic Hierarchy Process), we classify various factors that influence evaluation and establish a hierarchical relationship of evaluation index.In this paper, we used two levels of evaluation indexes, where U i (i = 1, 2, . . ., m) shows the first level evaluation index, u j (i) (j = 1, 2, . . ., n i ) shows the second level evaluation index.Let subscript sets I = {1, 2, . . ., m}, J (I) = {1, 2, . . ., n i }.
Step 2: Determine factor importance degree of evaluation indexes.In the evaluation index system, the importance degree of the various index of target is different.There are many ways to determine the factor importance degree, such as the Delphi method, Expert investigation method and so on.Suppose the importance degree of the first level evaluation index U i relative to the total goal is W = (w 1 , w 2 , . . ., w m ), where . ., m), µ k shows this evaluation index is useful for the total goal, η k shows this evaluation index is dispensable for the total goal, and ν k shows this evaluation index is not useful for the total goal.The importance degree of the second level evaluation index u j (i) (j ∈ J (I) ) relative to the first level evaluation index U i is W (i) = (w 1 (i) , w 2 (i) , . . ., w ni (i) ).
Step 3: Establish evaluation matrix.Let V = {v 1 , v 2 , . . ., v s } be a natural language comment set, S = {1, 2, . . ., s}, and evaluation experts give the membership degree of waiting evaluation schemes relative to each comment according to the evaluation indexes.In this paper, let the five-level language review set V = {big risk, larger risk, general risk, smaller risk, small risk}.Suppose evaluation experts give evaluation matrix R (i) , which represents the picture fuzzy relation of factor sets and comment sets, and , where r pq (i) is a picture fuzzy set, i ∈ I, p ∈ J (I) , q ∈ S.
Step 4: The second level picture fuzzy comprehensive evaluation.According the evaluation matrix R (i) and the importance degree W (i) , we can get evaluation vector of U i (i ∈ I): , .
Step 5: The first level picture fuzzy comprehensive evaluation.Let the subset of factor sets as the element of total factor sets.Then, the picture fuzzy relation matrix of factor sets U and evaluation sets V is A = (A (1) , A (2) , . . ., A (m) ).According to the factor importance degree vector W = (w 1 , w 2 , . . ., w m ) and the picture fuzzy relation matrix A, to calculate the first level comprehensive evaluation vector Step 6: The risk assessment.All elements of B are picture fuzzy sets.Therefore, the evaluation vector B can be regarded as the picture fuzzy set of the evaluation scheme risk size of the evaluation set V, which comprehensively describe the picture fuzzy membership of the evaluation scheme about all the comments.In order to get the final evaluation result, we used the scoring function and accurate function in Definition 7 to compare the size of each element in B, and, according to the principle of maximum membership, the corresponding v i in V of the maximum value b i in the picture fuzzy comprehensive evaluation set B is selected as the final evaluation result.In this paper, the final evaluation result is to determine the risk level of the investment scheme.
Then, according to step 5, we can get the first level comprehensive evaluation vector: Hence, the risk rating for the evaluation scheme is "larger risk".

Conclusions
In this paper, we investigate some new operations of picture fuzzy relations and discussed their properties.The kernels and closures of a picture fuzzy relation are defined and their properties are obtained.Then, we proposed a new composition operation of picture fuzzy relations and found a new method to solve picture fuzzy comprehensive evaluation problems.In addition, we prove it is feasible by an application example.

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Table 1 .
A picture fuzzy relation R.

Table 2 .
The inverse R −1 of R.

Table 3 .
The complement R c 2 of R.

Table 4 .
A picture fuzzy relation P.

Table 5 .
A picture fuzzy relation Q.

Table 10 .
A picture fuzzy relation R.

Table 11 .
The anti-reflexive kernel ar(R) of R.

Table 12 .
The symmetric kernel s(R) of R.

Table 13 .
The reflexive closure r (R) of R.

Table 14 .
The symmetric closure s (R) of R.