# New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Some Basic Concepts

**Definition**

**1.**

_{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}(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 μ

_{A}(x), η

_{A}(x), ν

_{A}(x) satisfy μ

_{A}(x) + η

_{A}(x) + ν

_{A}(x) ≤ 1, for all x ∈ X. Then, ∀ x ∈ X, 1 − (μ

_{A}(x) + η

_{A}(x) + ν

_{A}(x)) 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].

**Definition**

**3.**

_{R}(x, y), η

_{R}(x, y), ν

_{R}(x, y))

**|**x ∈ X, y ∈ Y},

_{R}: X × Y → [0, 1], η

_{R}: X × Y → [0, 1], and ν

_{R}: X × Y → [0, 1] satisfy the condition 0 ≤ μ

_{R}(x, y) + η

_{R}(x, y) + ν

_{R}(x, y) ≤ 1, for every (x, y) ∈ X × Y. PFR(X × Y) the set of all the picture fuzzy relations in X × Y is denoted [29].

**Definition**

**4.**

^{−}

^{1}between Y and X: μ

_{R}

^{−1}(y, x) = μ

_{R}(x, y), η

_{R}

^{−1}(y, x) = η

_{R}(x, y), ν

_{R}

^{−1}(y, x) = ν

_{R}(x, y), ∀ (x, y) ∈ X × Y [29].

**Definition**

**5.**

- (1)
- R ⊆ P iff μ
_{R}(x, y) ≤ μ_{P}(x, y), η_{R}(x, y) ≤ η_{P}(x, y), ν_{R}(x, y) ≥ ν_{P}(x, y); - (2)
- R ∪ P = {((x, y), μ
_{R}(x, y) ∨ μ_{P}(x, y), η_{R}(x, y) ∧ η_{P}(x, y), ν_{R}(x, y) ∧ ν_{P}(x, y))**|**x ∈ X, y ∈ Y}; - (3)
- R ∩ P = {((x, y), μ
_{R}(x, y) ∧ μ_{P}(x, y), η_{R}(x, y) ∧ η_{P}(x, y), ν_{R}(x, y) ∨ ν_{P}(x, y))**|**x ∈ X, y ∈ Y}; - (4)
- R
^{c}= {((x, y), ν_{R}(x, y), η_{R}(x, y), μ_{R}(x, y))**|**x ∈ X, y ∈ Y}.

**Proposition**

**1.**

- (a) (R
^{−1})^{−1}= R; - (b) R ⊆ P ⇒ R
^{−1}⊆ P^{−1}; - (c1) (R ∪ P)
^{−1}= R^{−1}∪ P^{−1}; - (c2) (R ∩ P)
^{−1}= R^{−1}∩ P^{−1}; - (d1) R ∩ (P ∪ Q) = (R ∩ P) ∪ (R ∩ Q);
- (d2) R ∪ (P ∩ Q) = (R ∪ P) ∩ (R ∪ Q);
- (e) R ∩ P ⊆ R, R ∩ P ⊆ P;
- (f1) If (R ⊇ P) and (R ⊇ Q), then R ⊇ P ∪ Q;
- (f2) If (R ⊆ P) and (R ⊆ Q), then R ⊆ P ∩ Q.

**Definition**

**6.**

_{PCE}(x, z), η

_{PCE}(x, z), ν

_{PCE}(x, z))

**|**x ∈ X, z ∈ Z},

**Definition**

**7.**

_{α}, η

_{α}, ν

_{α}, ρ

_{α}) be a picture fuzzy number, μ

_{α}+ η

_{α}+ ν

_{α}≤ 1, ρ

_{α}= 1 − μ

_{α}− η

_{α}− ν

_{α}. The score function S can be defined as S(α) = μ

_{α}− ν

_{α}, and the accuracy function H is given by H(α) = μ

_{α}+ η

_{α}+ ν

_{α}, which S(α) ∈ [−1, 1], H(α) ∈ [0, 1]. 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
- (i)
- if H(α) = H(β), implies that α is equivalent to β, denoted by α ~ β;
- (ii)
- if H(α) > H(β), implied that α is superior to β, denoted by α ⊱ β.

_{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.

#### 2.2. On Inclusion Relation of Picture Fuzzy Relations

_{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.

_{1}” is called type-1 order relation and the original inclusion relation of picture fuzzy relations is called a type-1 inclusion relation, and denoted as the following:

_{1}P iff (∀ (x, y) ∈ X × Y, μ

_{R}(x, y) ≤ μ

_{P}(x, y), η

_{R}(x, y) ≤ η

_{P}(x, y), ν

_{R}(x, y) ≥ ν

_{P}(x, y)).

_{1}P = {((x, y), μ

_{R}(x, y) ∨ μ

_{P}(x, y), η

_{R}(x, y) ∧ η

_{P}(x, y), ν

_{R}(x, y) ∧ ν

_{P}(x, y))

**|**x ∈ X, y ∈ Y}

= {((x, y), (μ

_{R}(x, y), η

_{R}(x, y), ν

_{R}(x, y)) ∨

_{1}(μ

_{P}(x, y), η

_{P}(x, y), ν

_{P}(x, y)))

**|**x ∈ X, y ∈ Y};

_{1}P = {((x, y), μ

_{R}(x, y) ∧ μ

_{P}(x, y), η

_{R}(x, y) ∧ η

_{P}(x, y), ν

_{R}(x, y) ∨ ν

_{P}(x, y))}

**|**x ∈ X, y ∈ Y}

= {((x, y), (μ

_{R}(x, y), η

_{R}(x, y), ν

_{R}(x, y)) ∧

_{1}(μ

_{P}(x, y), η

_{P}(x, y), ν

_{P}(x, y)))

**|**x ∈ X, y ∈ Y};

^{c}

^{1}= {((x, y), ν

_{R}(x, y), η

_{R}(x, y), μ

_{R}(x, y))

**|**x ∈ X, y ∈ Y} = {((x, y), (μ

_{R}(x, y), η

_{R}(x, y), ν

_{R}(x, y))

^{c}

^{1})

**|**x ∈ X, y ∈ Y}.

**Definition**

**8.**

_{2}P if and only if ∀ (x, y) ∈ X × Y, (μ

_{R}(x, y) < μ

_{P}(x, y), ν

_{R}(x, y) ≥ ν

_{P}(x, y)), or (μ

_{R}(x, y) = μ

_{P}(x, y), ν

_{R}(x, y) > ν

_{P}(x, y)), or (μ

_{R}(x, y) = μ

_{P}(x, y), ν

_{R}(x, y) = ν

_{P}(x, y) and η

_{R}(x, y) ≤ η

_{P}(x, y)).

**Remark**

**1.**

_{2}”, it is different from [30].

_{2}y nor y ≤

_{2}x, then x and y are incomparable, denoted as $x|{|}_{{\le}_{2}}y.$

## 3. New Operations and Properties of Picture Fuzzy Relations

_{2}P if and only if ∀ (x, y) ∈ X × Y, (μ

_{R}(x, y), η

_{R}(x, y), ν

_{R}(x, y)) ≤

_{2}(μ

_{P}(x, y), η

_{P}(x, y), ν

_{P}(x, y)).

**Proposition**

**2.**

- (1)
- R ⊆
_{2}R; - (2)
- (R ⊆
_{2}P, P ⊆_{2}R) ⇒ R = P; - (3)
- (R ⊆
_{2}P, P ⊆_{2}Q) ⇒ R ⊆_{2}Q.

**Definition**

**9.**

- (1)
- R ∪
_{2}P =$$\{\begin{array}{l}\left\{\left(\left(x,y\right),{\mu}_{R}\left(x,y\right),{\eta}_{R}\left(x,y\right),{\nu}_{R}\left(x,y\right)\right)|\left(x,y\right)\in X\times Y\right\},\text{}if\text{}P{\subseteq}_{2}R\hfill \\ \left\{\left(\left(x,y\right),{\mu}_{P}\left(x,y\right),{\eta}_{P}\left(x,y\right),{\nu}_{P}\left(x,y\right)\right)|\left(x,y\right)\in X\times Y\right\}\text{},\text{}if\text{}R{\subseteq}_{2}P\hfill \\ \left\{\left(\left(x,y\right),{\mu}_{R}\left(x,y\right)\vee {\mu}_{P}\left(x,y\right),0,{\nu}_{R}\left(x,y\right)\wedge {\nu}_{P}\left(x,y\right)\right)|\left(x,y\right)\in X\times Y\right\},\text{}else;\hfill \end{array}$$ - (2)
- R ∩
_{2}P =$$\{\begin{array}{l}\left\{\left(\left(x,y\right),{\mu}_{R}\left(x,y\right),{\eta}_{R}\left(x,y\right),{\nu}_{R}\left(x,y\right)\right)|\left(x,y\right)\in X\times Y\right\},\text{}if\text{}R{\subseteq}_{2}\text{}P\text{}\\ \left\{\left(\left(x,y\right),{\mu}_{P}\left(x,y\right),{\eta}_{P}\left(x,y\right),{\nu}_{P}\left(x,y\right)\right)|\left(x,y\right)\in X\times Y\right\}\text{},\text{}if\text{}P{\subseteq}_{2}R\\ \{(\left(x,y\right),{\mu}_{R}\left(x,y\right)\wedge {\mu}_{P}\left(x,y\right),1-\left({\mu}_{R}\left(x,y\right)\wedge {\mu}_{P}\left(x,y\right)\right)-\\ \text{}\left({\nu}_{R}\left(x,y\right)\vee {\nu}_{P}\left(x,y\right)\right),{\nu}_{R}\left(x,y\right)\vee {\nu}_{P}\left(x,y\right))|\left(x,y\right)\in X\times Y\},\text{}else;\end{array}$$ - (3)
- co(R) = R
^{c}^{2}=$$\left\{\right(\left(x,y\right),{\nu}_{R}\left(x,y\right),1-{\mu}_{R}\left(x,y\right)-{\eta}_{R}\left(x,y\right)-{\nu}_{R}\left(x,y\right),{\mu}_{R}\left(x,y\right))|\left(x,y\right)\in X\times Y\}.$$

**Example**

**1.**

**Definition**

**10.**

- (1)
- If ∀ (x, y) ∈ X × Y, μ
_{R}(x, y) = η_{R}(x, y) = 0 and ν_{R}(x, y) = 1, then R is called a null PFR, denoted by ∅_{N}. - (2)
- If ∀ (x, y) ∈ X × Y, μ
_{R}(x, y) = 1 and η_{R}(x, y) = ν_{R}(x, y) = 0, then R is called an absolute PFR, denoted by U_{N}. - (3)
- If ∀ (x, y) ∈ X × Y, μ
_{R}(x, y) = $\{\begin{array}{c}1,\text{}x=y\\ 0,\text{}x\ne y\end{array}$, η_{R}(x, y) = 0 and ν_{R}(x, y) = $\{\begin{array}{c}0,\text{}x=y\\ 1,\text{}x\ne y\end{array}$, then R is called an identity PFR, denoted by Id_{N}.

_{N}denoted by (Id

_{N})

^{c}

^{2}is a PFR satisfying: ∀ (x, y) ∈ X × Y,

**Definition**

**11.**

- (1)
- If ∀ x ∈ X, μ
_{R}(x, x) = 1 and η_{R}(x, x) = ν_{R}(x, x) = 0, then R is called a reflexive PFR. - (2)
- If ∀ (x, y) ∈ X × Y, μ
_{R}(x, y) = μ_{R}(y, x), η_{R}(x, y) = η_{R}(y, x), ν_{R}(x, y) = ν_{R}(y, x), then R is called a symmetric PFR. - (3)
- If ∀ x ∈ X, μ
_{R}(x, x) = η_{R}(x, x) = 0 and ν_{R}(x, x) = 1, then R is called an anti-reflexive PFR.

**Proposition**

**3.**

- (1)
- (R ∩
_{2}P) ∪_{2}Q ≠ (R ∪_{2}Q) ∩_{2}(P ∪_{2}Q), - (2)
- (R ∪
_{2}P) ∩_{2}Q ≠ (R ∩_{2}Q) ∪_{2}(P ∩_{2}Q).

**Example**

**2.**

_{1}, x

_{2}}, Y = {y

_{1}, y

_{2}}. Picture fuzzy relations R, P, Q in X × Y are given in Table 1, Table 4 and Table 5. Then (R ∩

_{2}P) ∪

_{2}Q, (R ∪

_{2}Q) ∩

_{2}(P ∪

_{2}Q), (R ∪

_{2}P) ∩

_{2}Q, (R ∩

_{2}Q) ∪

_{2}(P ∩

_{2}Q) in X × Y are given in Table 6, Table 7, Table 8 and Table 9. Furthermore, according to Table 6, Table 7, Table 8 and Table 9, we can get the conclusion of Propositions 3 (1) and (2).

**Proposition**

**4.**

- (1)
- R is symmetric iff R = R
^{−1}; - (2)
- (R
^{c}^{2})^{−1}= (R^{−1})^{c}^{2}; - (3)
- (R
^{c}^{2})^{c}^{2}= R, (R^{−1})^{−1}= R; - (4)
- R ⊆
_{2}R ∪_{2}P, P ⊆_{2}R ∪_{2}P; - (5)
- R ∩
_{2}P ⊆_{2}R, R ∩_{2}P ⊆_{2}P; - (6)
- If R ⊆
_{2}P, then R^{−1}⊆_{2}P^{−1}; - (7)
- If R ⊆
_{2}P and Q ⊆_{2}P, then R ∪_{2}Q ⊆_{2}P; - (8)
- If P ⊆
_{2}R and P ⊆_{2}Q, then P ⊆_{2}R ∩_{2}Q; - (9)
- If R ⊆
_{2}P, then R ∪_{2}P = P, R ∩_{2}P = R; - (10)
- (R ∪
_{2}P)^{−1}= R^{−1}∪_{2}P^{−1}, (R ∩_{2}P)^{−1}= R^{−1}∩_{2}P^{−1}; - (11)
- (R ∪
_{2}P)^{c}^{2}= R^{c}^{2}∩_{2}P^{c}^{2}, (R ∩_{2}P)^{c}^{2}= R^{c}^{2}∪_{2}P^{c}^{2}.

**Proof.**

_{(R}

^{c}

^{2}

_{)}

^{−1}(x, y) = μ

_{R}

^{c}

^{2}(y, x) = ν

_{R}(y, x) = ν

_{R}

^{−1}(x, y) = μ

_{(R}

^{−1}

_{)}

^{c}

^{2}(x, y); ν

_{(R}

^{c}

^{2}

_{)}

^{−1}(x, y) = ν

_{R}

^{c}

^{2}(y, x) = μ

_{R}(y, x) = μ

_{R}

^{−1}(x, y) = ν

_{(R}

^{−1}

_{)}

^{c}

^{2}(x, y); η

_{(R}

^{c}

^{2}

_{)}

^{−1}(x, y) = η

_{R}

^{c}

^{2}(y, x) = 1 − μ

_{R}(y, x) − η

_{R}(y, x) − ν

_{R}(y, x) = 1 − μ

_{R}

^{−1}(x, y) − η

_{R}

^{−1}(x, y) − ν

_{R}

^{−1}(x, y) = η

_{(R}

^{−1}

_{)}

^{c}

^{2}(x, y).

^{c}

^{2})

^{−1}= (R

^{−1})

^{c}

^{2}.

_{2}P, then R

^{−1}⊆

_{2}P

^{−1}, so (R ∪

_{2}P)

^{−1}= P

^{−1}= R

^{−1}∪

_{2}P

^{−1}; If P ⊆

_{2}R, then P

^{−1}⊆

_{2}R

^{−1}, so (R ∪

_{2}P)

^{−1}= R

^{−1}= R

^{−1}∪

_{2}P

^{−1}; If neither R ⊆

_{2}P nor P ⊆

_{2}R, then (R ∪

_{2}P)

^{−1}= {((x, y), μ

_{R}(x, y) ∨ μ

_{P}(x, y), 0, ν

_{R}(x, y) ∧ ν

_{P}(x, y))

**|**(x, y) ∈ X × Y }

^{−1}= {((y, x), μ

_{R}

^{−1}(x, y) ∨ μ

_{P}

^{−1}(x, y), 0, ν

_{R}

^{−1}(x, y) ∧ ν

_{P}

^{−1}(x, y))

**|**(x, y) ∈ X × Y }, R

^{−1}∪

_{2}P

^{−1}= {((y, x), μ

_{R}

^{−1}(x, y), η

_{R}

^{−1}(x, y), ν

_{R}

^{−1}(x, y))

**|**(x, y) ∈ X × Y } ∪

_{2}{((y, x), μ

_{P}

^{−1}(x, y), η

_{P}

^{−1}(x, y), ν

_{P}

^{−1}(x, y))

**|**(x, y) ∈ X × Y } = {((y, x), μ

_{R}

^{−1}(x, y) ∨ μ

_{P}

^{−1}(x, y), 0, ν

_{R}

^{−1}(x, y) ∧ ν

_{P}

^{−1}(x, y))

**|**(x, y) ∈ X × Y } = (R ∪

_{2}P)

^{−1}. Hence, (R ∪

_{2}P)

^{−1}= R

^{−1}∪

_{2}P

^{−1}. Similarly, we can show (R ∩

_{2}P)

^{−1}= R

^{−1}∩

_{2}P

^{−1}.

_{2}P, then P

^{c}

^{2}⊆

_{2}R

^{c}

^{2}, so (R ∪

_{2}P)

^{c}

^{2}= P

^{c}

^{2}= R

^{c}

^{2}∩

_{2}P

^{c}

^{2}; If P ⊆

_{2}R, then R

^{c}

^{2}⊆

_{2}P

^{c}

^{2}, so (R ∪

_{2}P)

^{c}

^{2}= R

^{c}

^{2}= R

^{c}

^{2}∩

_{2}P

^{c}

^{2}; If neither R ⊆

_{2}P nor P ⊆

_{2}R, then neither R

^{c}

^{2}⊆

_{2}P

^{c}

^{2}nor P

^{c}

^{2}⊆

_{2}R

^{c}

^{2}and (R ∪

_{2}P)

^{c}

^{2}= {((x, y), ν

_{R}(x, y) ∧ ν

_{P}(x, y), 1 – (μ

_{R}(x, y) ∨ μ

_{P}(x, y)) – (ν

_{R}(x, y) ∧ ν

_{P}(x, y)), μ

_{R}(x, y) ∨ μ

_{P}(x, y))

**|**(x, y) ∈ X × Y }, R

^{c}

^{2}∩

_{2}P

^{c}

^{2}= {((x, y), ν

_{R}(x, y), 1 – μ

_{R}(x, y) – η

_{R}(x, y) – ν

_{R}(x, y), μ

_{R}(x, y))

**|**(x, y) ∈ X × Y } ∩

_{2}{((x, y), ν

_{P}(x, y), 1 – μ

_{P}(x, y) – η

_{P}(x, y) – ν

_{P}(x, y), μ

_{P}(x, y))

**|**(x, y) ∈ X × Y } = {((x, y), ν

_{R}(x, y) ∧ ν

_{P}(x, y), 1 – (μ

_{R}(x, y) ∨ μ

_{P}(x, y)) – (ν

_{R}(x, y) ∧ ν

_{P}(x, y)), μ

_{R}(x, y) ∨ μ

_{P}(x, y))

**|**(x, y) ∈ X × Y }. Hence, (R ∪

_{2}P)

^{c}

^{2}= R

^{c}

^{2}∩

_{2}P

^{c}

^{2}. Similarly, we can get (R ∩

_{2}P)

^{c}

^{2}= R

^{c}

^{2}∪

_{2}P

^{c}

^{2}. ☐

## 4. Kernels of Picture Fuzzy Relations

**Definition**

**12.**

- (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).

**Proposition**

**5.**

- (1)
- ar(R) = R ∩
_{2}(Id_{N})^{c2}. - (2)
- s(R) = R ∩
_{2}R^{−1}.

**Proof.**

_{2}(Id

_{N})

^{c2}⊆

_{2}R. According the definition of Id

_{N}, ∀ x ∈ X, we have μ

_{Id}

_{N}(x, x) = 1 and η

_{Id}

_{N}(x, x) = ν

_{Id}

_{N}(x, x) = 0, then μ

_{(Id}

_{N}

_{)}

^{c2}(x, x) = η

_{(Id}

_{N}

_{)}

^{c2}(x, x) = 0 and ν

_{(Id}

_{N}

_{)}

^{c2}(x, x) = 1. Hence, μ

_{(Id}

_{N}

_{)}

^{c2}(x, x) ≤ μ

_{R}(x, x), ν

_{(Id}

_{N}

_{)}

^{c2}(x, x) ≥ ν

_{R}(x, x), η

_{(Id}

_{N}

_{)}

^{c2}(x, x) ≤ η

_{R}(x, x). Therefore, (Id

_{N})

^{c2}⊆

_{2}R, μ

_{R}

_{∩2(Id}

_{N}

_{)}

^{c2}(x, x) = η

_{R}

_{∩2(Id}

_{N}

_{)}

^{c2}(x, x) = 0 and ν

_{R}

_{∩2(Id}

_{N}

_{)}

^{c2}(x, x) = 1. According to Definition 11 (3), R ∩

_{2}(Id

_{N})

^{c2}is an anti-reflexive PFR.

_{2}R. Obviously, P ⊆

_{2}(Id

_{N})

^{c}

^{2}. Hence, P ⊆

_{2}R ∩

_{2}(Id

_{N})

^{c}

^{2}. Therefore, ar(R) = R ∩

_{2}(Id

_{N})

^{c}

^{2}.

_{2}R

^{−1})

^{−1}= R

^{−1}∩

_{2}(R

^{−1})

^{−1}= R

^{−1}∩

_{2}R = R ∩

_{2}R

^{−1}, which implies that R ∩

_{2}R

^{−1}is a symmetric PFR. According to Proposition 3 (5), R ∩

_{2}R

^{−1}⊆

_{2}R.

_{2}R. By Proposition 3 (6), P

^{−1}⊆

_{2}R

^{−1}. Then, by Proposition 3 (1) and (5), P = P

^{−1}⊆

_{2}R ∩

_{2}R

^{−1}. Therefore, s(R) = R ∩

_{2}R

^{−1}. ☐

**Example**

**3.**

**Proposition**

**6.**

- (1)
- ar(∅
_{N}) = ∅_{N}, ar((Id_{N})^{c}^{2}) = (Id_{N})^{c}^{2}; - (2)
- ∀ R ∈ PFR(X × Y), ar(R) ⊆
_{2}R; - (3)
- ∀ R, P ∈ PFR(X × Y), ar(R ∪
_{2}P) = ar(R) ∪_{2}ar(P), ar(R ∩_{2}P) = ar(R) ∩_{2}ar(P); - (4)
- ∀ R, P ∈ PFR(X × Y), if R ⊆
_{2}P, then ar(R) ⊆_{2}ar(P); - (5)
- ∀ R ∈ PFR(X × Y), ar(ar(R)) = ar(R).

**Proof.**

_{N}, we can get ∅

_{N}⊆

_{2}(Id

_{N})

^{c}

^{2}. Therefore, ar(∅

_{N}) = ∅

_{N}∩

_{2}(Id

_{N})

^{c}

^{2}= ∅

_{N}. ar((Id

_{N})

^{c}

^{2}) = (Id

_{N})

^{c}

^{2}∩

_{2}(Id

_{N})

^{c}

^{2}= (Id

_{N})

^{c}

^{2}.

_{2}(Id

_{N})

^{c}

^{2}⊆

_{2}R.

_{2}P) = (R ∪

_{2}P) ∩

_{2}(Id

_{N})

^{c}

^{2}, ar(R) ∪

_{2}ar(P) = (R ∩

_{2}(Id

_{N})

^{c}

^{2}) ∪

_{2}(P ∩

_{2}(Id

_{N})

^{c}

^{2}). ∀ (x, y) ∈ X × Y, when x = y, (Id

_{N})

^{c}

^{2}= {((x, y), 0, 0, 1)

**|**(x, y) ∈ X × Y }, so (Id

_{N})

^{c}

^{2}⊆

_{2}R, (Id

_{N})

^{c}

^{2}⊆

_{2}P, then ar(R ∪

_{2}P) = ar(R) ∪

_{2}ar(P) = (Id

_{N})

^{c}

^{2}; when x ≠ y, (Id

_{N})

^{c}

^{2}= {((x, y), 1, 0, 0)

**|**(x, y) ∈ X × Y }, so R ⊆

_{2}(Id

_{N})

^{c}

^{2}, P ⊆

_{2}(Id

_{N})

^{c}

^{2}, then ar(R ∪

_{2}P) = ar(R) ∪

_{2}ar(P) = R ∪

_{2}P. Hence, ar(R ∪

_{2}P) = ar(R) ∪

_{2}ar(P). ar(R ∩

_{2}P) = (R ∩

_{2}P) ∩

_{2}(Id

_{N})

^{c}

^{2}= ((R ∩

_{2}(Id

_{N})

^{c}

^{2}) ∩

_{2}(P ∩

_{2}(Id

_{N})

^{c}

^{2}) = ar(R) ∩

_{2}ar(P).

_{2}P, by Label (3) and Proposition 3 (4) and Label (9), ar(R) ⊆

_{2}ar(R) ∪

_{2}ar(P) = ar(R ∪

_{2}P) = ar(P).

_{2}(Id

_{N})

^{c}

^{2}. Hence, ar(ar(R)) = ar(R ∩

_{2}(Id

_{N})

^{c}

^{2}) = (R ∩

_{2}(Id

_{N})

^{c}

^{2}) ∩

_{2}(Id

_{N})

^{c}

^{2}= R ∩

_{2}(Id

_{N})

^{c}

^{2}= ar(R). ☐

**Proposition**

**7.**

- (1)
- s(∅
_{N}) = ∅_{N}, s(U_{N}) = U_{N}, s(Id_{N}) = Id_{N}; - (2)
- ∀ R ∈ PFR(X × Y), s(R) ⊆
_{2}R; - (3)
- ∀ R, P ∈ PFR(X × Y), s(R ∩
_{2}P) = s(R) ∩_{2}s(P); - (4)
- ∀ R, P ∈ PFR(X × Y), if R ⊆
_{2}P, then s(R) ⊆_{2}s(P); - (5)
- ∀ R ∈ PFR(X × Y), s(s(R)) = s(R).

**Proof.**

_{N}) = ∅

_{N}, s(U

_{N}) = U

_{N}, s(Id

_{N}) = Id

_{N}.

_{2}R

^{−1}⊆

_{2}R.

_{2}P) = (R ∩

_{2}P) ∩

_{2}(R ∩

_{2}P)

^{−1}= (R ∩

_{2}P) ∩

_{2}(R

^{−1}∩

_{2}P

^{−1}) = (R ∩

_{2}R

^{−1}) ∩

_{2}(P ∩

_{2}P

^{−1}) = s(R) ∩

_{2}s(P).

_{2}P, by Label (3) and Proposition 3 (5) and (9), s(R) = s(R ∩

_{2}P) = s(R) ∩

_{2}s(P) ⊆

_{2}s(P).

_{2}R

^{−1}) = (R ∩

_{2}R

^{−1}) ∩

_{2}(R ∩

_{2}R

^{−1})

^{−1}= R ∩

_{2}R

^{−1}= s(R). ☐

## 5. Closures of Picture Fuzzy Relations

**Definition**

**13.**

- (1)
- O is reflexive;
- (2)
- R ⊆
_{2}O; - (3)
- ∀ E ∈ PFR(X × Y), if E is reflexive and R ⊆
_{2}E, then O ⊆_{2}E.

**Definition**

**14.**

- (1)
- O is symmetric;
- (2)
- R ⊆
_{2}O; - (3)
- ∀ E ∈ PFR(X × Y), if E is symmetric and R ⊆
_{2}E, then O ⊆_{2}E.

**Proposition**

**8.**

- (1)
- $\overline{r}$(R) = R ∪
_{2}Id_{N}. - (2)
- $\overline{s}$(R) = R ∪
_{2}R^{−1}.

**Proof.**

_{2}R ∪

_{2}Id

_{N}, Id

_{N}⊆

_{2}R ∪

_{2}Id

_{N}. According the definition of Id

_{N}, ∀ x ∈ X, we have μ

_{Id}

_{N}(x, x) = 1 and η

_{Id}

_{N}(x, x) = ν

_{Id}

_{N}(x, x) = 0. Hence, μ

_{R}(x, x) < μ

_{Id}

_{N}(x, x), ν

_{R}(x, x) ≥ ν

_{Id}

_{N}(x, x), η

_{Id}

_{N}(x, x) ≤ η

_{R}(x, x) or μ

_{R}(x, x) = μ

_{Id}

_{N}(x, x) = 1, ν

_{R}(x, x) = ν

_{Id}

_{N}(x, x) = 0, η

_{Id}

_{N}(x, x) = η

_{R}(x, x) = 0. Therefore, R ⊆

_{2}Id

_{N}or R = Id

_{N}, so μ

_{R}

_{∪}

_{2Id}

_{N}(x, x) = 1 and η

_{R}

_{∪}

_{2Id}

_{N}(x, x) = ν

_{R}

_{∪}

_{2Id}

_{N}(x, x) = 0. According to Definition 11 (1), R ∪

_{2}Id

_{N}is a reflexive PFR.

_{2}P. Obviously, Id

_{N}⊆

_{2}P. Hence, R ∪

_{2}Id

_{N}⊆

_{2}P. Therefore, $\overline{r}$(R)= R ∪

_{2}Id

_{N}.

_{2}R

^{−1})

^{−1}= R

^{−1}∪

_{2}(R

^{−1})

^{−1}= R

^{−1}∪

_{2}R = R ∪

_{2}R

^{−1}, which implies that R ∪

_{2}R

^{−1}is a symmetric PFR. According to Proposition 3 (4), R ⊆

_{2}R ∪

_{2}R

^{−1}.

_{2}P. By Proposition 3 (6), R

^{−1}⊆

_{2}P

^{−1}. Then, by Proposition 3 (1) and Label (7), R ∪

_{2}R

^{−1}⊆

_{2}P = P

^{−1}. Therefore, $\overline{s}$(R) = R ∪

_{2}R

^{−1}. ☐

**Example**

**4.**

**Proposition**

**9.**

- (1)
- $\overline{r}$(U
_{N}) = U_{N}, $\overline{r}$(Id_{N}) = Id_{N}; - (2)
- ∀ R ∈ PFR(X × Y), R ⊆
_{2}$\overline{r}$(R); - (3)
- ∀ R, P ∈ PFR(X × Y), $\overline{r}$(R ∪
_{2}P) = $\overline{r}$(R) ∪_{2}$\overline{r}$(P), $\overline{r}$(R ∩_{2}P) = $\overline{r}$(R) ∩_{2}$\overline{r}$(P); - (4)
- ∀ R, P ∈ PFR(X × Y), if R ⊆
_{2}P, then $\overline{r}$(R) ⊆_{2}$\overline{r}$(P); - (5)
- ∀ R ∈ PFR(X × Y), $\overline{r}$($\overline{r}$(R)) = $\overline{r}$(R).

**Proof.**

_{N}⊆

_{2}U

_{N}. Therefore, $\overline{r}$(U

_{N}) = U

_{N}. $\overline{r}$(Id

_{N}) = Id

_{N}∪

_{2}Id

_{N}= Id

_{N}.

_{2}R ∪

_{2}Id

_{N}= $\overline{r}$(R).

_{2}P) = (R ∪

_{2}P) ∪

_{2}Id

_{N}= (R ∪

_{2}Id

_{N}) ∪

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∪

_{2}$\overline{r}$(P); $\overline{r}$(R ∩

_{2}P) = (R ∩

_{2}P) ∪

_{2}Id

_{N}, $\overline{r}$(R) ∩

_{2}$\overline{r}$(P) = (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}), ∀ (x, y) ∈ X × Y, when x = y, 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}⊆

_{2}R or R = Id

_{N}, also we can get Id

_{N}⊆

_{2}P or P = Id

_{N}. If R ⊆

_{2}Id

_{N}and P ⊆

_{2}Id

_{N}, then R ∩

_{2}P ⊆

_{2}Id

_{N}, so $\overline{r}$(R ∩

_{2}P) = Id

_{N}= (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If R = Id

_{N}and P ⊆

_{2}Id

_{N}, then $\overline{r}$(R ∩

_{2}P) = Id

_{N}= (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If R ⊆

_{2}Id

_{N}and P = Id

_{N}, then $\overline{r}$(R ∩

_{2}P) = Id

_{N}= (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If R = Id

_{N}and P = Id

_{N}, then $\overline{r}$(R ∩

_{2}P) = R = P = Id

_{N}= (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If Id

_{N}⊆

_{2}R and Id

_{N}⊆

_{2}P, then by Proposition 3 (8), we have Id

_{N}⊆

_{2}R ∩

_{2}P, so $\overline{r}$(R ∩

_{2}P) = R ∩

_{2}P = (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If Id

_{N}= R and Id

_{N}⊆

_{2}P, then $\overline{r}$(R ∩

_{2}P) = Id

_{N}= R = (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P). If Id

_{N}⊆

_{2}R and Id

_{N}= P, then $\overline{r}$(R ∩

_{2}P) = Id

_{N}= P = (R ∪

_{2}Id

_{N}) ∩

_{2}(P ∪

_{2}Id

_{N}) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P).

_{2}P) = $\overline{r}$(R) ∩

_{2}$\overline{r}$(P).

_{2}P, by Label (3) and Proposition 3 (4) and Label (9), we have $\overline{r}$ (R) ⊆

_{2}$\overline{r}$(R) ∪

_{2}$\overline{r}$(P) = $\overline{r}$(R ∪

_{2}P) = $\overline{r}$(P).

_{2}Id

_{N}= (R ∪

_{2}Id

_{N}) ∪

_{2}Id

_{N}= R ∪

_{2}Id

_{N}= $\overline{r}$(R). ☐

**Proposition**

**10.**

- (1)
- $\overline{s}$(∅
_{N}) = ∅_{N}, $\overline{s}$(U_{N}) = U_{N}, $\overline{s}$(Id_{N}) = Id_{N}; - (2)
- ∀ R ∈ PFR(X × Y), R ⊆
_{2}$\overline{s}$(R); - (3)
- ∀ R, P ∈ PFR(X × Y), $\overline{s}$(R ∪
_{2}P) = $\overline{s}$(R) ∪_{2}$\overline{s}$(P); - (4)
- ∀ R, P ∈ PFR(X × Y), if R ⊆
_{2}P, then $\overline{s}$(R) ⊆_{2}$\overline{s}$(P); - (5)
- ∀ R ∈ PFR(X × Y), $\overline{s}$($\overline{s}$(R)) = $\overline{s}$(R);

**Proof.**

_{N}, U

_{N}and Id

_{N}, we have $\overline{s}$(∅

_{N}) = ∅

_{N}, $\overline{s}$(U

_{N}) = U

_{N}, $\overline{s}$(Id

_{N}) = Id

_{N}.

_{2}R ∪

_{2}R

^{−1}= $\overline{s}$(R).

_{2}P) = (R ∪

_{2}P) ∪

_{2}(R ∪

_{2}P)

^{−1}= (R ∪

_{2}R

^{−1}) ∪

_{2}(P ∪

_{2}P

^{−1}) = $\overline{s}$(R) ∪

_{2}$\overline{s}$(P).

_{2}P, by Label (3) and Proposition 3 (4) and (9), $\overline{s}$(R) ⊆

_{2}$\overline{s}$(R) ∪

_{2}$\overline{s}$(P) = $\overline{s}$(R ∪

_{2}P) = $\overline{s}$(P).

_{2}R

^{−1}) = (R ∪

_{2}R

^{−1}) ∪

_{2}(R ∪

_{2}R

^{−1})

^{−1}= R ∪

_{2}R

^{−1}= $\overline{s}$(R). ☐

**Proposition**

**11.**

- (1)
- ($\overline{r}$(R
^{c}^{2}))^{c}^{2}= ar(R); - (2)
- ar ($\overline{r}$(R)) = ar(R).

**Proof.**

^{c}

^{2}) = R

^{c}

^{2}∪

_{2}Id

_{N}. By Proposition 3 (11) and Proposition 5 (1),

- (i)
- If x = y and R = Id
_{N}, then (Id_{N})^{c}^{2}⊆_{2}Id_{N}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R); - (ii)
- If x = y and R = (Id
_{N})^{c}^{2}, then (Id_{N})^{c}^{2}⊆_{2}Id_{N}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= Id_{N}∩_{2}(Id_{N})^{c}^{2}= (Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R); - (iii)
- If x = y and (Id
_{N})^{c}^{2}⊆_{2}R ⊆_{2}Id_{N}, then (Id_{N})^{c}^{2}⊆_{2}Id_{N}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= Id_{N}∩_{2}(Id_{N})^{c}^{2}= (Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R); - (iv)
- If x ≠ y and R = Id
_{N}, then Id_{N}⊆_{2}(Id_{N})^{c}^{2}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R); - (v)
- If x ≠ y and R = (Id
_{N})^{c}^{2}, then Id_{N}⊆_{2}(Id_{N})^{c}^{2}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R); - (vi)
- If x ≠ y and Id
_{N}⊆_{2}R ⊆_{2}(Id_{N})^{c}^{2}, then Id_{N}⊆_{2}(Id_{N})^{c}^{2}, so ar ($\overline{r}$(R)) = (R ∪_{2}Id_{N}) ∩_{2}(Id_{N})^{c}^{2}= R ∩_{2}(Id_{N})^{c}^{2}= ar(R);

**Proposition**

**12.**

- (1)
- ($\overline{s}$(R
^{c}^{2}))^{c}^{2}= s(R); - (2)
- $\overline{s}$(s(R)) = s(R);
- (3)
- s($\overline{s}$(R)) = $\overline{s}$(R).

**Proof.**

^{c}

^{2}) = R

^{c}

^{2}∪

_{2}(R

^{c}

^{2})

^{−1}. By Proposition 3 (10) and Proposition 5 (2), we have

^{c}

^{2}))

^{c}

^{2}= (R

^{c}

^{2}∪

_{2}(R

^{c}

^{2})

^{−}

^{1})

^{c}

^{2}= ((R

^{c}

^{2})

^{c}

^{2}) ∩

_{2}(((R

^{c}

^{2})

^{−}

^{1})

^{c}

^{2}) = R ∩

_{2}R

^{−}

^{1}= s(R).

_{2}R

^{−1}) = (R ∩

_{2}R

^{−1}) ∪

_{2}(R ∩

_{2}R

^{−1})

^{−1}= R ∩

_{2}R

^{−1}= s(R).

_{2}R

^{−}

^{1}) = (R ∪

_{2}R

^{−}

^{1}) ∩

_{2}(R ∪

_{2}R

^{−}

^{1})

^{−}

^{1}= R ∪

_{2}R

^{−}

^{1}= $\overline{s}$(R). ☐

## 6. Picture Fuzzy Comprehensive Evaluation

**Definition**

**15.**

_{R}

_{°P}(x, z), η

_{R}

_{°P}(x, z), ν

_{R}

_{°P}(x, z))

**|**(x, z) ∈ X × Z)},

_{R}

_{°P}(x, z) + η

_{R}

_{°P}(x, z) + ν

_{R}

_{°P}(x, z) ≤ 1, ∀ (x, z) ∈ X × Z.

**Definition**

**16.**

_{1}, A

_{2}, …, A

_{n}} ∈ PFS(X) and R ∈ PFR(X × Y), where A

_{i}(i = 1, 2, …, n) is picture fuzzy set and

_{ij}and B

_{j}are picture fuzzy sets, i = {1, 2, …, n}, j = {1, 2, …, s}.

#### 6.1. Picture Fuzzy Comprehensive Evaluation Model

_{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}}.

_{i}relative to the total goal is W = (w

_{1}, w

_{2}, …, w

_{m}), where w

_{k}= (μ

_{k}, η

_{k}, ν

_{k}), 0 ≤ μ

_{k}≤ 1, 0 ≤ η

_{k}≤ 1, 0 ≤ ν

_{k}≤ 1, 0 ≤ μ

_{k}+ η

_{k}+ ν

_{k}≤ 1(k = 1, 2, …, 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

_{n}

_{i}

^{(i)}).

_{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

_{pq}

^{(i)}is a picture fuzzy set, i ∈ I, p ∈ J

^{(I)}, q ∈ S.

^{(i)}and the importance degree W

^{(i)}, we can get evaluation vector of U

_{i}(i ∈ I):

^{(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

_{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.

#### 6.2. The Application Example

^{(1)}= {(0.3, 0.1, 0.4), (0.4, 0.3, 0.2), (0.2, 0.3, 0.4), (0.4, 0.2, 0.2)}, W

^{(2)}= {(0.2, 0.2, 0.5), (0.5, 0.1, 0.1), (0.2, 0.3, 0.3), (0.3, 0.4, 0.1)}, W

^{(3)}= {5}, W

^{(4)}= {(0.6, 0.1, 0.2), (0.4, 0.3, 0.1), (0.2, 0.6, 0.1), (0.3, 0.5, 0.2)}, W

^{(5)}= {(0.5, 0.3, 0.1), (0.3, 0.2, 0.2), (0.4, 0.1, 0.2), (0.3, 0.4, 0.2)}, decision makers get the picture fuzzy evaluation matrix about a certain item risk investment projects through the information integration:

^{(1)}= W

^{(1)}° R

^{(1)}= {((0.3, 0.1, 0.4) ∧

_{2}(0.3, 0.2, 0.1)) ∨

_{2}((0.4, 0.3, 0.2) ∧

_{2}(0.2, 0.6, 0.1)) ∨

_{2}((0.2, 0.3, 0.4) ∧

_{2}(0.6, 0.1, 0.1)) ∨

_{2}((0.4, 0.2, 0.2) ∧

_{2}(0.5, 0.1, 0.2)), ((0.3, 0.1, 0.4) ∧

_{2}(0.7, 0.1, 0.1)) ∨

_{2}((0.4, 0.3, 0.2) ∧

_{2}(0.5, 0.1, 0.3)) ∨

_{2}((0.2, 0.3, 0.4) ∧

_{2}(0.5, 0.3, 0.1)) ∨

_{2}((0.4, 0.2, 0.2) ∧

_{2}(0.5, 0.1, 0.3)), ((0.3, 0.1, 0.4) ∧

_{2}(0.1, 0.2, 0.6)) ∨

_{2}((0.4, 0.3, 0.2) ∧

_{2}(0.6, 0.1, 0.2)) ∨

_{2}((0.2, 0.3, 0.4) ∧

_{2}(0.2, 0.1, 0.5)) ∨

_{2}((0.4, 0.2, 0.2) ∧

_{2}(0.6, 0.2, 0.1)), ((0.3, 0.1, 0.4) ∧

_{2}(0.4, 0.1, 0.2)) ∨

_{2}((0.4, 0.3, 0.2) ∧

_{2}(0.4, 0.2, 0.3)) ∨

_{2}((0.2, 0.3, 0.4) ∧

_{2}(0.6, 0.1, 0.2)) ∨

_{2}((0.4, 0.2, 0.2) ∧

_{2}(0.3, 0.4, 0.2)), ((0.3, 0.1, 0.4) ∧

_{2}(0.4, 0.1, 0.4)) ∨

_{2}((0.4, 0.3, 0.2) ∧

_{2}(0.1, 0.6, 0.1)) ∨

_{2}((0.2, 0.3, 0.4) ∧

_{2}(0.3, 0.2, 0.4)) ∨

_{2}((0.4, 0.2, 0.2) ∧

_{2}(0.3, 0.1, 0.4))}

= {(0.4, 0.2, 0.2), (0.4, 0.3, 0.3), (0.4, 0.3, 0.2), (0.4, 0, 0.2), (0.3, 0, 0.2)}.

- A
^{(2)}= {(0.3, 0.3, 0.2), (0.5, 0.1, 0.1), (0.4, 0.3, 0.2), (0.3, 0.3, 0.2), (0.4, 0, 0.1)}; - A
^{(3)}= {(0.3, 0, 0.2), (0.3, 0.5, 0.1), (0.4, 0, 0.1), (0.4, 0, 0.2), (0.3, 0.5, 0.1)}; - A
^{(4)}= {(0.6, 0.1, 0.2), (0.3, 0.4, 0.2), (0.3, 0.3, 0.2), (0.5, 0, 0.1), (0.4, 0, 0.2)}; - A
^{(5)}= {(0.3, 0, 0.1), (0.4, 0, 0.2), (0.4, 0.2, 0.2), (0.4, 0.4, 0.2), (0.5, 0.3, 0.1)}.

_{1}= ((0.5, 0.2, 0.1) ∧

_{2}(0.4, 0.2, 0.2)) ∨

_{2}((0.6, 0.1, 0.2) ∧

_{2}(0.3, 0.3, 0.2)) ∨

_{2}((0.4, 0.3, 0.3) ∧

_{2}(0.3, 0, 0.2)) ∨

_{2}((0.3, 0.1, 0.5) ∧

_{2}(0.6, 0.1, 0.2)) ∨

_{2}((0.3, 0.2, 0.3) ∧

_{2}(0.3, 0, 0.1)) = (0.4, 0.2, 0.2);

_{2}= (0.5, 0.3, 0.2); b

_{3}= (0.4, 0.3, 0.2); b

_{4}= (0.4, 0, 0.2); b

_{5}= (0.4, 0.4, 0.2).

_{1}) = 0.4 − 0.2 = 0.2, S(b

_{2}) = 0.3, S(b

_{3}) = 0.2, S(b

_{4}) = 0.2, S(b

_{5}) = 0.2. H(b

_{1}) = 0.4 + 0.2 + 0.2 = 0.8, H(b

_{2}) = 1, H(b

_{3}) = 0.9, H(b

_{4}) = 0.6, and H(b

_{5}) = 1.

_{2}⊱ b

_{5}⊱ b

_{3}⊱ b

_{1}⊱ b

_{4}.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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R | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.3, 0.2, 0.1) | (0.5, 0.1, 0.3) |

x_{2} | (0.2, 0.6, 0.2) | (0.2, 0.1, 0.5) |

R^{−1} | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.3, 0.2, 0.1) | (0.2, 0.6, 0.2) |

x_{2} | (0.5, 0.1, 0.3) | (0.2, 0.1, 0.5) |

R^{c}^{2} | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.1, 0.6, 0.3) | (0.2, 0, 0.2) |

x_{2} | (0.3, 0.2, 0.5) | (0.5, 0.3, 0.2) |

P | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.5, 0.2, 0.3) | (0.3, 0.2, 0.4) |

x_{2} | (0.6, 0.1, 0.2) | (0.7, 0.1, 0.1) |

Q | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.4, 0.1, 0.2) | (0.2, 0.1, 0.1) |

x_{2} | (0.2, 0.2, 0.5) | (0.1, 0.4, 0.2) |

(R ∩_{2} P) ∪_{2} Q | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.4, 0.1, 0.2) | (0.3, 0, 0.1) |

x_{2} | (0.2, 0.6, 0.2) | (0.2, 0, 0.2) |

(R ∩_{2} Q) ∪_{2} (P ∩_{2} Q) | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.4, 0.4, 0.2) | (0.3, 0, 0.1) |

x_{2} | (0.2, 0.6, 0.2) | (0.2, 0, 0.2) |

(R ∪_{2} P) ∩_{2} Q | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.4, 0.1, 0.2) | (0.2, 0.5, 0.3) |

x_{2} | (0.2, 0.2, 0.5) | (0.1, 0.4, 0.2) |

(R ∩_{2} Q) ∪_{2} (P ∩_{2} Q) | y_{1} | y_{2} |
---|---|---|

x_{1} | (0.4, 0, 0.2) | (0.2, 0.5, 0.3) |

x_{2} | (0.2, 0.2, 0.5) | (0.1, 0.4, 0.2) |

R | z_{1} | z_{2} | z_{3} |
---|---|---|---|

z_{1} | (0.3, 0.2, 0.1) | (0.5, 0.1, 0.3) | (0.3, 0.2, 0.4) |

z_{2} | (0.2, 0.6, 0.2) | (0.2, 0.1, 0.5) | (0.6, 0.1, 0.2) |

z_{3} | (0.7, 0.1, 0.1) | (0.4, 0.1, 0.2) | (0.2, 0.2, 0.5) |

ar(R) | z_{1} | z_{2} | z_{3} |
---|---|---|---|

z_{1} | (0, 0, 1) | (0.5, 0.1, 0.3) | (0.3, 0.2, 0.4) |

z_{2} | (0.2, 0.6, 0.2) | (0, 0, 1) | (0.6, 0.1, 0.2) |

z_{3} | (0.7, 0.1, 0.1) | (0.4, 0.1, 0.2) | (0, 0, 1) |

s(R) | z_{1} | z_{2} | z_{3} |
---|---|---|---|

z_{1} | (0.3, 0.2, 0.1) | (0.2, 0.5, 0.3) | (0.3, 0.2, 0.4) |

z_{2} | (0.2, 0.5, 0.3) | (0.2, 0.1, 0.5) | (0.4, 0.1, 0.2) |

z_{3} | (0.3, 0.2, 0.4) | (0.4, 0.1, 0.2) | (0.2, 0.2, 0.5) |

$\overline{\mathit{r}}$(R) | z_{1} | z_{2} | z_{3} |
---|---|---|---|

z_{1} | (1, 0, 0) | (0.5, 0.1, 0.3) | (0.3, 0.2, 0.4) |

z_{2} | (0.2, 0.6, 0.2) | (1, 0, 0) | (0.6, 0.1, 0.2) |

z_{3} | (0.7, 0.1, 0.1) | (0.4, 0.1, 0.2) | (1, 0, 0) |

$\overline{\mathit{s}}$(R) | z_{1} | z_{2} | z_{3} |
---|---|---|---|

z_{1} | (0.3, 0.2, 0.1) | (0.5, 0, 0.2) | (0.7, 0.1, 0.1) |

z_{2} | (0.5, 0, 0.2) | (0.2, 0.1, 0.5) | (0.6, 0.1, 0.2) |

z_{3} | (0.7, 0.1, 0.1) | (0.6, 0.1, 0.2) | (0.2, 0.2, 0.5) |

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Bo, C.; Zhang, X.
New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation. *Symmetry* **2017**, *9*, 268.
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**AMA Style**

Bo C, Zhang X.
New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation. *Symmetry*. 2017; 9(11):268.
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**Chicago/Turabian Style**

Bo, Chunxin, and Xiaohong Zhang.
2017. "New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation" *Symmetry* 9, no. 11: 268.
https://doi.org/10.3390/sym9110268