Picture fuzzy Z-linguistic set and its application in multiple attribute group decision-making

Abstract

Picture fuzzy linguistic set is a vital solution to express complex and uncertain information, which has been applied in multi-attribute group decision-making (MAGDM). However, the credibility of decision-making information is unconsidered, which may give rise to the inaccuracy of final result. To solve this problem, the picture fuzzy Z-linguistic set (PFZLS) composed of linguistic term, picture fuzzy number, and credibility is proposed, which could express more complete decision-making information. Subsequently, operation rules, comparison methods, and distance measures of PFZLS are introduced. In addition, the weighted geometric average operator and the classical VIKOR method are extended and combined to solve the MAGDM problem under the picture fuzzy Z-linguistic environment. Finally, an illustrative example about the emergency decision-making (EDM) problem of forest fire accident is proposed, and a series of comparative analyses are presented to verify the rationality and superiority of the PFZLS.

Keywords

Multi-attribute group decision-making picture fuzzy Z-linguistic set weighted geometric average operator VIKOR method

1 Introduction

As a vital branch of decision theory, MAGDM has been broadly used in various practical fields, including EDM problem [1], consensus process [2] and supply chain management [3]. However, because of the uncertainty of information, decision-makers (DMs) could not accurately express decision preferences. To solve this problem, Zadeh [4] firstly developed the fuzzy set (FS) to express complex information. Furthermore, Atannassov [5] introduced the intuitionistic fuzzy set (IFS), which depicts information from two perspectives of membership degree and non-membership degree. IFS has become the most classic fuzzy set theory because it conforms to human intuitive feelings. However, IFS does not fully consider the hesitation of information, so Cuong [6] put forward the picture fuzzy set (PFS), which completely describes information through three perspectives of positive membership, negative membership, and neutral membership. In recent years, PFS theory has gained widespread attention, scholars have mainly studied information fusion [7 –12], decision-making methods [13 –16].

In some qualitative questions, DMs are more accustomed to expressing opinions in natural language such as “very good” and “perfect”. Therefore, Zadeh [17] innovatively put forward the concept of linguistic term set (LTS), which uses linguistic variables to describe subjective information. Inspired by the idea of LTS, scholars have introduced LTS into fuzzy sets to express information more completely, a series of extended fuzzy sets have been put forward [18 –20]. Among them, the picture fuzzy linguistic set (PFLS) [21] proposed by Ashraf provides a solution for linguistic representation in the case of PFS. Because of its good information representation ability, PFLS has become an novel research direction and has produced a series of research results [22 –24].

When describing decision-making information, DMs must think about the complexity of information as well as the credibility of information. Therefore, Zadeh [25] proposed Z-number to solve this problem. Z-number is a generalized constraint of fuzzy events, which consists of two parts, the first part is the description of information ambiguity, and the second part is the credibility measure of information ambiguity. Because of the great property of Z-number, some extended LTS based on Z-number have been put forward. Chai et al. [26, 27] combined the probabilistic linguistic with Z-number to propose the probabilistic Z-linguistic set and the uncertain probabilistic Z-linguistic set. Tao et al. [28] combined the fuzzy soft set with Z-number to propose the linguistic Z-number fuzzy soft set. Yang et al. [29] combined the intuitive fuzzy linguistic set (IFLS) with Z-number to propose the intuitive fuzzy Z-linguistic set (IFZLS). In short, there are few studies regarding Z-number and the PFLS. Besides, in the MAGDM problems of PFLS, the credibility difference of decision-making information is usually unconsidered. DMs’ experiences and preferences are different, thus their credibility will also be different. For example, DMs’ evaluations of their fields are generally accurate, while evaluations of other fields may be inaccurate. For another example, some DMs are accustomed to giving high-score evaluations, while others are accustomed to giving low-score evaluations. Therefore, PFLS literally can not deal with the credibility of information. To solve this problem, this paper introduces Z-number into PFLS and proposes the PFZLS, so that it has the common advantages of both Z-number and PFLS. As a promotion of PFLS, PFZLS describes the ambiguity of information as well as the reliability of ambiguity, which has a wider range of applicability.

After obtaining the fuzzy information, MAGDM requires to integrate decision information and select the optimal alternatives, the decision-making process roughly includes the following three aspects:

(1) Measure theory. Measures are the cornerstones of decision theory and have been broadly used in various practical problems, including decision analysis, pattern recognition, and machine learning. Common measures mainly include distance measures, similarity measures, and correlation measures. In PFS theory, Wei et al. [30] proposed some similarity measures, Son et al. [31] studied the distance measures, and Ganie et al. [32] studied the correlation coefficients.

(2) Information fusion. Whether integrating different experts’ evaluations or different alternatives’ attribute values, aggregation operator is a vital method of information fusion, which has been broadly used in fuzzy sets. At present, the mainstream aggregation operators include weighted aggregation operators, hybrid weighted aggregation operators and generalized hybrid aggregation operators, etc. In PFS theory, Wei [33] proposed the picture fuzzy Hamacher aggregation operator. Xian et al. [34] proposed the picture fuzzy linguistic Muirhead mean aggregation operator. Mahmood et al. [35] put forward the picture fuzzy Frank aggregation operator.

(3) Decision-making methods. In the development of decision theory, many excellent models are put forward to deal with alternative optimization. Among the most famous methods are TOPSIS method, VIKOR method, ELECTRE method, PROMETHEE method, etc. There is no doubt that these models have been also introduced into various fuzzy sets, in PFS theory, Arya et al. [36] put forward an integrated VIKOR-TODIM approach for opinion polls. Liang et al. [37] presented an integrated EDAS-ELECTRE method to estimate production performance. Tian et al. [38] proposed the AHP and PROMETHEE II method for tourism environmental impact assessment.

In short, these measure theories, aggregation operators, and decision-making methods are the issues that need to be studied urgently in the novel PFZLS, and the contributions of this paper could be summarized as follows: (romannumeral1) Combining the PFLS with Z-number, PFZLS is proposed, operation rules, distance measure, and related properties of PFZLS are studied. (romannumeral2) According to the operation rules and distance measure, the geometric average operator and the classical VIKOR method are extended. (romannumeral3) A novel MAGDM model based on PFZLS is constructed, and an illustrative example and comparative analyses are presented to verify the effectiveness and superiority of the proposed method.

In order to complete the above methods, the rest of paper is constructed as follows. Section 2 introduces the basic concepts that used below. Section 3 proposes the concept of PFZLS, introduces its operation rules, scores and accuracy functions, distance measures, and extended the weighted geometric average operator. Section 4 establishes the PFZLWGA-VIKOR method for MAGDM. Section 5 applies the method to a forest fire EDM case. Section 6 presents the comparative analysis. Section 7 summarizes this paper.

2 Preliminaries

In this section, the basic concepts including the PFS, the LTS, the PFLS, the VIKOR method and Z-number are reviewed.

2.1 The picture fuzzy set

Definition 1. Let X be a nonempty set, then $A = {〈 x | μ_{A} (x), η_{A} (x), ν_{A} (x) 〉 | x \in X},$ is called the picture fuzzy set in X, where μ_A (x), η_A (x) and ν_A (x) are defined as the positive, neutral and negative membership degree of x and satisfied $0 \leq μ_{A} (x) + η_{A} (x) + ν_{A} (x) \leq 1 .$

Besides, γ_A (x) =1 - μ_A (x) - η_A (x) - ν_A (x) could be defined as the refusal membership degree of x, which means that DMs have no inclination but refuse to comment.

2.2 The linguistic term set

Definition 2. Suppose S = {s_t|t = 0, 1, ⋯ , l - 1} is an ordered discrete set with odd cardinality, S could be defined as a LTS, where s_t represents the possible value in the linguistic terms.

Besides, let s_i and s_j be any two linguistic terms in S, the LTS must satisfy the properties as follows.

(1) S is ordered, that is, if i > j, we have s_i > s_j;

(2) Negation operator of S, that is, neg (s_i) = s_l-i-1;

(3) Max operator of S, that is, if s_i > s_j, we have max(s_i, s_j) = s_i;

(4) Min operator of S, that is, if s_i < s_j, we have min(s_i, s_j) = s_i.

Then, the operation rules of LTS could be summarized as follows.

Definition 3. Let s_i,s_j are any two linguistic term, and λ > 0, we have

(1) λs_i = s_λi;

(2) s_i ⊕ s_j = s_i+j;

(3) s_i ⊗ s_j = s_ij;

(4) (s_i) ^λ = s_{i
^λ}.

2.3 The picture fuzzy linguisitc set

Definition 4. Let X be a nonempty set, then $A = {〈 x | s_{θ (x_{i})}, (μ_{A} (x), η_{A} (x), ν_{A} (x)) 〉 | x \in X},$ is called the picture fuzzy linguistic set in X, where μ_A (x), η_A (x), and ν_A (x) respectively represent the positive membership degree, neutral-membership degree and negative-membership degree of x to the linguistic variable s_θ(x) and satisfied $0 \leq μ_{A} (x) + η_{A} (x) + ν_{A} (x) \leq 1 .$

Besides, γ_A (x) =1 - μ_A (x) - η_A (x) - ν_A (x) is defined as the refusal degree of x to the linguistic variable s_θ(x).

2.4 VIKOR method

As a compromise solution, VIKOR method [39] comprehensively considers the maximum group utility and the minimum individual loss, which has the great interpretability. Let D = [r_ij] _m×n be a decision matrix, A_i (i = 1, 2, ⋯ , m) represents alternatives, and C_j (j = 1, 2, ⋯ , n) represents attributes, attribute weights vector is w = (w₁, w₂, ⋯ , w_n) ^T and satisfied $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1$ , r_ij is the evaluation value of ith alternative under jth attribute. Then the VIKOR method has the following steps:

Step 1. Select positive-ideal solution r⁺ and negative-ideal solution r^- of D. $\begin{matrix} r^{+} = {r_{1}^{+}, r_{2}^{+}, \dots, r_{n}^{+}} \\ = {max_{i} r_{ij} | j = 1, 2, \dots, n}, \end{matrix}$ (1) $\begin{matrix} r^{-} = {r_{1}^{-}, r_{2}^{-}, \dots, r_{n}^{-}} \\ = {min_{i} r_{ij} | j = 1, 2, \dots, n} . \end{matrix}$ (2)

Step 2. Determine the maximum group benefit value S_i and the minimum individual loss value R_i of alternative. $S_{i} = \sum_{j = 1}^{n} ω_{j} \frac{r_{j}^{+} - r_{ij}}{r_{j}^{+} - r_{j}^{-}},$ (3) $R_{i} = max_{j} {ω_{j} \frac{r_{j}^{+} - r_{ij}}{r_{j}^{+} - r_{j}^{-}}},$ (4) where i = 1, 2, ⋯ , m .

Step 3. Calculate the benefit ratio value Q_i of alternative. $Q_{i} = v \frac{S_{i} - S^{-}}{S^{+} - S^{-}} + (1 - v) \frac{R_{i} - R^{-}}{R^{+} - R^{-}} .$ (5) where i = 1, 2, ⋯ , m, and S^- = min{ S_i }, S⁺ = max{ S_i }, R⁺ = max{ R_i }, R^- = min{ R_i }. Value v is defined as the maximum group utility coefficient, usually set to 0.5.

Step 4. Determine alternative rankings, the smaller the values R_i, S_i, Q_i are, the higher the alternative ranks.

Step 5. Determine optimal or compromise solution, the solution set must meet the following two conditions:

I1: $Q (A^{2}) - Q (A^{1}) \geq \frac{1}{n - 1}$ , where A¹ and A² are the best and second-best alternatives.

I2: A¹ should be also the alternative corresponding to the minimum value of S_i and R_i.

If I1 is not satisfied, find the maximum value of m by the equation as follow: $Q (A^{m}) - Q (A^{1}) \leq \frac{1}{m - 1} .$ (6)

Then set Aⁱ (i = 1, 2, ⋯ , m) will be the compromise alternatives. If I2 is not satisfied, then the alternatives A¹ and A² will be the compromise alternatives.

2.5 Z-number

Z-number is denoted as Z = (X, A, B), if X is a random variable, A is the fuzzy restriction of X, and B is the credibility measure of A. The ordered triple (X, A, B) could be viewed as a generalized constraint on X, which satisfied $Prob (XisA) isB .$

According to the note by Zadeh, the Z-valuation is more likely be an assignment statement, so that it can be summarized in various ways. For the convenience of calculation, when X is a random variable, B could be equated to probability briefly, which also could be interpreted as the confidence, reliability, possibility and so on.

3 The picture Z-linguistic set

In this section, the concept of the PFZLS is put forward, its operation rules, comparison functions, distance measures, and extended operator are proposed.

3.1 The concept of picture Z-linguistic set

When describing decision-making information, DMs should consider not only the complexity of information, but also the credibility of information. The Z-valuation can solve this problem well, the idea of Z-number is introduced into the PFLS.

Definition 5. Let X be a nonempty set and S be a continuous LTS, then $A = {〈 x | s_{θ (x_{i})}, (μ_{A} (x), η_{A} (x), ν_{A} (x)); ρ (x) 〉 | x \in X},$ called the picture fuzzy Z-linguistic set in X, where s_θ(x) ∈ S represents the possible values in the linguisitc terms. μ_A (x) : X → [0, 1] , x ∈ X → μ_A (x) ∈ [0, 1]; η_A (x) : X → [0, 1] , x ∈ X → η_A (x) ∈ [0, 1]; ν_A (x) : X → [0, 1] , x ∈ X → ν_A (x) ∈ [0, 1] respectively represent the positive membership degree, neutral-membership degree and negative-membership degree of x to the linguistic variable s_θ(x) and satisfied $0 \leq μ_{A} (x) + η_{A} (x) + ν_{A} (x) \leq 1 .$

Besides, γ_A (x) =1 - μ_A (x) - η_A (x) - ν_A (x) is defined as the refusal degree of x to the linguistic variable s_θ(x). ρ (x) is defined to describe the credibility of quadruple and satisfied 0 ≤ ρ (x) ≤1. For convenience, is usually called the picture fuzzy Z-linguistic variable (PFZLV).

PFZLS can express three-dimensional information and better deal with MAGDM problems. For example, in a market survey, DMs’ degree of support, neutrality, and opposition to the prediction result of “extremely good” are 0.6, 0.3, and 0.1 respectively, and they have 80% confidence in this prediction.

Remark 1. If ρ (x) =1, then the PFZLS will be simplified to the PFLS.

Remark 2. If η_A (x) =0, then the PFZLS will be simplified to the IFZLS.

Remark 3. If ρ (x) =1 and η_A (x) =0, then the PFZLS will be simplified to the IFLS.

Remark 4. If μ_A (x) =1, then the PFZLS will be simplified to the Z-number.

Definition 6. Let A, A₁, A₂ be any three PFZLVs and λ ≥ 0, then the operation rules are defined as follows: (1) $A_{1} \oplus A_{2} = 〈 \begin{matrix} s_{θ (x_{1}) + θ (x_{2})}, \\ 1 - (1 - μ_{A} (x_{1})) (1 - μ_{A} (x_{2})), \\ η_{A} (x_{1}) η_{A} (x_{2}), ν_{A} (x_{1}) ν_{A} (x_{2}), \\ \frac{ρ^{2} (x_{1}) + ρ^{2} (x_{2})}{ρ (x_{1}) + ρ (x_{2})} \end{matrix} 〉;$ (2) $A_{1} \otimes A_{2} = 〈 \begin{matrix} s_{θ (x_{1}) θ (x_{2})}, \\ μ_{A} (x_{1}) μ_{A} (x_{2}), η_{A} (x_{1}) η_{A} (x_{2}), \\ 1 - (1 - ν_{A} (x_{1})) (1 - ν_{A} (x_{2})) \\ , ρ (x_{1}) ρ (x_{2}) \end{matrix} 〉$ (3) $λ A = 〈 \begin{matrix} s_{λ θ (x)}, 1 - {(1 - μ_{A} (x))}^{λ}, \\ η_{A}^{λ} (x), ν_{A}^{λ} (x), ρ (x) \end{matrix} 〉;$ (4) $A^{λ} = 〈 \begin{matrix} s_{θ^{λ} (x)}, μ_{A}^{λ} (x), η_{A}^{λ} (x), \\ 1 - {(1 - ν_{A} (x))}^{λ}, ρ^{λ} (x) \end{matrix} 〉 .$

Obviously, the operational results above still belong to the PFZLV.

Example 1. Let S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆}, , . According to the operation rules, we have , , 2A₁ = 〈s₂, (0.84, 0.09, 0.01) , 0.9〉, and $A_{1}^{2} = 〈 s_{1}, (0.36, 0.09, 0.19), 0.81 〉$ .

Theorem 1. Let A, A₁, A₂ be any three PFZLVs, then the operation rules have properties as follows:

(1) A₁ ⊕ A₂ = A₂ ⊕ A₁;

(2) A₁ ⊗ A₂ = A₂ ⊗ A₁;

(3) λ (A₁ ⊕ A₂) = λA₁ ⊕ λA₂;

(4) $A_{1}^{λ} \otimes A_{2}^{λ} = (A_{1} \otimes A_{2})^{λ}$ ;

(5) λ₁A ⊕ λ₂A = (λ₁ + λ₂) ⊗ A;

(6) A^λ₁ ⊗ A^λ₂ = A^λ₁+λ₂.

Proof: According to the operation rules, property(1), (2) can be simply proved. As for (3), we separately have $\begin{matrix} λ (A_{1} \oplus A_{2}) \\ 〈 \begin{matrix} s_{θ (x_{1}) + θ (x_{2})}, \\ 1 - (1 - μ_{A} (x_{1})) (1 - μ_{A} (x_{2})), \\ η_{A} (x_{1}) η_{A} (x_{2}), ν_{A} (x_{1}) ν_{A} (x_{2}), \\ \frac{ρ^{2} (x_{1}) + ρ^{2} (x_{2})}{ρ (x_{1}) ρ (x_{2})} \end{matrix} 〉 \\ 〈 \begin{matrix} s_{λ (θ (x_{1}) + θ (x_{2}))}, \\ 1 - (1 - μ_{A} (x_{1}))^{λ} (1 - μ_{A} (x_{2}))^{λ} \\ η_{A}^{λ} (x_{1}) η_{A}^{λ} (x_{2}), ν_{A}^{λ} (x_{1}) ν_{A}^{λ} (x_{2}), \\ \frac{ρ^{2} (x_{1}) + ρ^{2} (x_{2})}{ρ (x_{1}) + ρ (x_{2})} \end{matrix} 〉 \end{matrix}$

$\begin{matrix} λ A_{1} \oplus λ A_{2} = \begin{matrix} 〈 \begin{matrix} s_{λ θ (x_{1})}, 1 - {(1 - μ_{A} (x_{1}))}^{λ} \\ η_{A}^{λ} (x_{1}), ν_{A}^{λ} (x_{1}), ρ (x_{1}) \end{matrix} 〉 \oplus \\ 〈 \begin{matrix} s_{λ θ (x_{2})}, 1 - {(1 - μ_{A} (x_{2}))}^{λ} \\ η_{A}^{λ} (x_{2}), ν_{A}^{λ} (x_{2}), ρ (x_{2}) \end{matrix} 〉 \end{matrix} \\ = 〈 \begin{matrix} s_{λ θ (x_{1}) + λ θ (x_{2})}, \\ 1 - {(1 - μ_{A} (x_{1}))}^{λ} {(1 - μ_{A} (x_{2}))}^{λ}, \\ η_{A}^{λ} (x_{1}) η_{A}^{λ} (x_{2}), ν_{A}^{λ} (x_{1}) ν_{A}^{λ} (x_{2}), \\ \frac{ρ^{2} (x_{1}) + ρ^{2} (x_{2})}{ρ (x_{1}) ρ (x_{2})} \end{matrix} 〉 \end{matrix}$

So λ (A₁ ⊕ A₂) = λA₁ ⊕ λA₂, proof completed.

As for (4), we separately have $\begin{matrix} A_{1}^{λ} \otimes A_{2}^{λ} = \begin{matrix} 〈 \begin{matrix} s_{θ (x_{1})}^{λ}, μ_{A}^{λ} (x_{1}), η_{A}^{λ} (x_{1}), \\ 1 - {(1 - ν_{A} (x_{1}))}^{λ}, ρ^{λ} (x_{1}) \end{matrix} 〉 \oplus \\ 〈 \begin{matrix} s_{θ (x_{2})}^{λ}, μ_{A}^{λ} (x_{2}), η_{A}^{λ} (x_{2}), \\ 1 - {(1 - ν_{A} (x_{2}))}^{λ}, ρ^{λ} (x_{2}) \end{matrix} 〉 \end{matrix} \\ = 〈 \begin{matrix} s_{θ^{λ} (x_{1}) θ^{λ} (x_{2})}, μ_{A}^{λ} (x_{1}) μ_{A}^{λ} (x_{2}), \\ η_{A}^{λ} (x_{1}) η_{A}^{λ} (x_{2}), \\ (1 - {(1 - ν_{A} (x_{1}))}^{λ}) + (1 - {(1 - ν_{(A_{2})})}^{λ}) - \\ (1 - {(1 - ν_{A} (x_{1}))}^{λ}) (1 - {(1 - ν_{A} (x_{2}))}^{λ}), \\ ρ^{λ} (x_{1}) ρ^{λ} (x_{2}) \end{matrix} 〉 \end{matrix}$

$\begin{matrix} (A_{1} \otimes A_{2})^{λ} \\ = 〈 \begin{matrix} s_{(θ (x_{1}) \times θ (x_{2}))^{λ}}, (μ_{A} (x_{1}) μ_{A} (x_{2}))^{λ}, \\ (η_{A} (x_{1}) η_{A} (x_{2}))^{λ}, \\ (1 - (1 - ν_{A} (x_{1}))^{λ}) + (1 - (1 - ν_{A} (x_{2}))^{λ}) \\ - (1 - (1 - ν_{A} (x_{1}))^{λ}) (1 - (1 - ν_{A} (x_{2}))^{λ}), \\ ρ^{λ} (x_{1}) ρ^{λ} (x_{2}) \end{matrix} 〉 \end{matrix}$ So $A_{1}^{λ} \otimes A_{2}^{λ} = (A_{1} \otimes A_{2})^{λ}$ , proof completed.

As for (5), we separately have $\begin{array}{l} λ_{1} A \oplus λ_{2} A \\ = \begin{matrix} λ_{1} 〈 s_{θ (x)}, μ_{A} (x), η_{A} (x), ν_{A} (x), ρ (x) 〉 \oplus \\ λ_{2} 〈 s_{θ (x)}, μ_{A} (x), η_{A} (x), ν_{A} (x), ρ (x) 〉 \end{matrix} \\ \begin{matrix} 〈 \begin{matrix} s_{λ_{1} θ (x)}, 1 - {(1 - μ_{A} (x))}^{λ_{1}}, \\ η_{A}^{λ_{1}} (x), ν_{A}^{λ_{1}} (x), ρ (x) \end{matrix} 〉 \oplus \\ 〈 \begin{matrix} s_{λ_{2} θ (x)}, 1 - {(1 - μ_{A} (x))}^{λ_{3}}, \\ η_{A}^{λ_{3}} (x), ν_{A}^{λ_{1}} (x), ρ (x) \end{matrix} 〉 \end{matrix} \\ = 〈 \begin{matrix} s_{λ_{1} θ (x) + λ_{2} θ (x)}, \\ 1 - {(1 - μ_{A} (x))}^{λ_{1}} {(1 - μ_{A} (x))}^{λ_{2}}, η_{A}^{λ_{1}} (x) \\ η_{A}^{λ_{2}} (x), ν_{A}^{λ_{1}} (x) ν_{A}^{λ_{2}} (x), \\ \frac{ρ^{2} (x) + ρ^{2} (x)}{ρ (x) + ρ (x)} \end{matrix} 〉 \end{array}$

$\begin{array}{l} (λ_{1} + λ_{2}) \otimes A \\ = 〈 \begin{matrix} s_{(λ_{1} + λ_{2}) θ (x)}, 1 - {(1 - μ_{A} (x))}^{λ_{1} + λ_{2}}, \\ η_{A}^{λ_{1} + λ_{2}} (x), ν_{A}^{λ_{1} + λ_{2}} (x), ρ (x) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{λ_{1} θ (x_{1}) + λ_{2} θ (x_{2})}, \\ 1 - {(1 - μ_{A} (x))}^{λ_{1}} {(1 - μ_{A} (x))}^{λ_{2}}, \\ η_{A}^{λ_{1}} (x) η_{A}^{λ_{2}} (x), ν_{A}^{λ_{1}} (x) ν_{A}^{λ_{2}} (x), ρ (x) \end{matrix} 〉 \end{array}$ Since $\frac{ρ^{2} (x) + ρ^{2} (x)}{ρ (x) + ρ (x)} = ρ (x)$ , so λ₁A ⊕ λ₂A = (λ₁ + λ₂) ⊗ A, proof completed.

As for (6), we separately have

$\begin{array}{l} A^{λ_{1}} \otimes A^{λ_{2}} \begin{matrix} = 〈 \begin{matrix} s_{θ^{λ_{1}} (x)}, μ_{A}^{λ_{1}} (x), η_{A}^{λ_{1}} (x), \\ 1 - {(1 - ν_{A} (x))}^{λ_{1}}, ρ^{λ_{1}} (x) \end{matrix} 〉 \otimes \\ 〈 \begin{matrix} s_{θ^{λ_{2}} (x)}, μ_{A}^{λ_{2}} (x), η_{A}^{λ_{2}} (x), \\ 1 - {(1 - ν_{A} (x))}^{λ_{2}}, ρ^{λ_{2}} (x) \end{matrix} 〉 \end{matrix} \\ = 〈 \begin{matrix} s_{θ^{λ_{1}} (x) θ^{λ_{2}} (x)}, μ_{A}^{λ_{1}} (x) μ_{A}^{λ_{2}} (x), \\ η_{A}^{λ_{1}} (x) η_{A}^{λ_{2}} (x), \\ (1 - {(1 - ν_{A} (x))}^{λ_{1}}) + (1 - {(1 - ν_{A} (x))}^{λ_{2}}) \\ - (1 - {(1 - ν_{A} (x))}^{λ_{1}}) (1 - {(1 - ν_{A} (x))}^{λ_{2}}), \\ ρ^{λ_{1}} (x) ρ^{λ_{2}} (x) \end{matrix} 〉 \end{array}$ $\begin{matrix} A^{λ_{1} + λ_{2}} = 〈 \begin{matrix} s_{θ^{λ_{1} + λ_{2}} (x)}, μ_{A}^{λ_{1} + λ_{2}} (x), η_{A}^{λ_{1} + λ_{2}} (x), \\ 1 - {(1 - ν_{A} (x))}^{λ_{1} + λ_{2}}, ρ^{λ_{1} + λ_{2}} (x) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ^{λ_{1}} (x) θ^{λ_{2}} (x)}, μ_{A}^{λ_{1}} (x) μ_{A}^{λ_{2}} (x), \\ η_{A}^{λ_{1}} (x) η_{A}^{λ_{2}} (x), \\ (1 - {(1 - ν_{A} (x))}^{λ_{1}}) + (1 - {(1 - ν_{A} (x))}^{λ_{2}}) - \\ (1 - {(1 - ν_{A} (x))}^{λ_{1}}) (1 - {(1 - ν_{A} (x))}^{λ_{2}}), \\ ρ^{λ_{1}} (x) ρ^{λ_{2}} (x) \end{matrix} 〉 \end{matrix}$

So A^λ₁ ⊗ A^λ₂ = A^λ₁+λ₂, proof completed.

3.2 Score and accuracy functions of PFZLVs

To calculate the preference relationship of PFZLVs, the score function and accuracy function of PFZLS are proposed.

According to the work of Ashraf [24], the score function and the accuracy function of PFLS are denoted as: $\begin{matrix} E (A) = \frac{1}{6} θ (x) (μ_{A} (x) + 2 - η_{A} (x) - ν_{A} (x)), \\ H (A) = \frac{1}{2} θ (x) (μ_{A} (x) - ν_{A} (x)) . \end{matrix}$

It is proved that adding the value of reliability in a linear way will not change the monotonicity of function. Therefore, based on the functions of PFLS, the novel score function and accuracy function of PFZLS could be obtained as follows.

Definition 7. Suppose A be a PFZLV, then the score function E (A) of A could be defined as: $E (A) = \frac{1}{12} θ (x) ρ (x) (μ_{A} (x) + 2 - η_{A} (x) - ν_{A} (x)) .$ (7)

Definition 8. Suppose A be a PFZLV, then the accuracy function H (A) of A could be defined as: $H (A) = \frac{1}{4} θ (x) ρ (x) (μ_{A} (x) - ν_{A} (x)) .$ (8)

Using score function and accuracy function, any two PFZLV A₁ and A₂ could be compared as follows:

(1) If E (A₁) > E (A₂), then A₁ ≻ A₂;

(2) If E (A₁) < E (A₂), then A₁ ≺ A₂;

(3) If E (A₁) = E (A₂), then

If H (A₁) > H (A₂), then A₁ ≻ A₂;

If H (A₁) < H (A₂), then A₁ ≺ A₂;

If H (A₁) = H (A₂), then A₁ ∼ A₂.

Example 2. Let S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆}, , . Firstly, using score function, we have E (A₁) =0.15 and E (A₂) =0.15. Then, using accuracy function, we have H (A₁) =0.1 and H (A₂) =0.08, so we can get the comparison result A₁ > A₂.

3.3 Distance measure of PFZLVs and Propertie

To study the relationship between two PFZLVs, we propose the distance measure of PFZLS.

Similar to the construction method of comparison function, the distance function of PFZLS could be obtained in a linear way.

Definition 9. Let A₁ and A₂ be any two PFZLVs, then the Minkowski distance between A₁ and A₂ could be defined as:

$\begin{matrix} d (A_{1}, A_{2}) \\ = \frac{1}{4 (l - 1)} {| \begin{matrix} {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) - \\ ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) \end{matrix} ‖}^{r} \\ + {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - η_{A} (x_{2})) \end{matrix} ‖}^{r} \\ + {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖}^{r} \end{matrix} |}^{\frac{1}{r}} . \end{matrix}$ (9)

Remark 5. If r = 2, the distance will be simplified to the Euclidean distance. $\begin{matrix} d (A_{1}, A_{2}) \\ = \frac{1}{4 (l - 1)} {| \begin{matrix} {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) \\ - ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) \end{matrix} ‖}^{2} \\ + {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - η_{A} (x_{2})) \end{matrix} ‖}^{2} \\ + {‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) \\ - ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖}^{2} \end{matrix} |}^{\frac{1}{2}} . \end{matrix}$ (10)

Remark 6. If r = 1, the distance will be simplified to the Hamming distance. $\begin{matrix} d (A_{1}, A_{2}) \\ = \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) - \\ ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - η_{A} (x_{2})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖ \end{matrix} | . \end{matrix}$ (11)

Theorem 2. Let A₁ and A₂ be any two PFZLVs, the distance d (A₁, A₂) could meet the following conditions:

(1) 0≤ d (A₁, A₂) ≤1 ;

(2) d (A₁, A₁) =0 ;

(3) d (A₁, A₂) = d (A₂, A₁) ;

(4) d (A₁, A₂) + d (A₁, A₃) ≥ d (A₂, A₃) .

Proof: We only prove the case of n = 1, and other cases can be obtained by the same principle.

Since ρ (x), μ_A (x), η_A (x), ν_A (x) ∈[0, 1], (1) can be simply proved.

As for (2), we have $\begin{matrix} d (A_{1}, A_{1}) = \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) - \\ ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η A (x_{1})) - \\ ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) - \\ ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) \end{matrix} ‖ \end{matrix} | \\ = 0 \end{matrix}$ As for (3), we have $\begin{matrix} d (A_{1}, A_{2}) \\ = \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) \\ - ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - η_{A} (x_{2})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖ \end{matrix} | \end{matrix}$ $= \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) - \\ ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{2}) θ (x_{2}) (1 - η_{A} (x_{2})) - \\ ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{2}) θ (x_{2} (1 - ν_{A} (x_{2})) - \\ ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) \end{matrix} ‖ \end{matrix} | = d (A_{2}, A_{1})$ As for (4), using transform scaling, we have $\begin{matrix} d (A_{1}, A_{3}) \\ = \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) - \\ ρ (x_{3}) θ (x_{3}) μ_{A} (x_{3}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{3}) θ (x_{3}) (1 - η_{A} (x_{3})) \end{matrix} ‖ \\ + ‖ \begin{matrix} r h o (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) - \\ ρ (x_{3}) θ (x_{3}) (1 - ν_{A} (x_{3})) \end{matrix} ‖ \end{matrix} | \\ \leq \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) μ_{A} (x_{1}) \\ - ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - η_{A} (x_{1})) - \\ ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{1}) θ (x_{1}) (1 - ν_{A} (x_{1})) \\ - ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) \end{matrix} ‖ \end{matrix} | + \\ \frac{1}{4 (l - 1)} | \begin{matrix} ‖ \begin{matrix} ρ (x_{2}) θ (x_{2}) μ_{A} (x_{2}) - \\ ρ (x_{3}) θ (x_{3}) μ_{A} (x_{3}) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2}) - \\ ρ (x_{3}) θ (x_{3}) (1 - η_{A} (x_{3})) \end{matrix} ‖ \\ + ‖ \begin{matrix} ρ (x_{2}) θ (x_{2}) (1 - ν_{A} (x_{2})) - \\ ρ (x_{3}) θ (x_{3}) (1 - ν_{A} (x_{3})) \end{matrix} ‖ \end{matrix} | \\ = d (A_{1}, A_{2}) + d (A_{2}, A_{3}) \end{matrix}$

3.4 PFZLWGA operator and its properties

In decision theory, the weighted geometric average operator is one of the most common aggregation operators, thus the extended weighted geometric average operator of PFZLS is put forward.

Definition 10. Let A = (A₁, A₂, ⋯ , A_n) is a set of PFZLVs, then the picture fuzzy Z-linguistic weighted geometric average operator (PFZLWGA) is defined as ^varOmegaN → varOmega, ${PFZLWGA}_{ω} (A_{1}, A_{2}, \dots, A_{n}) = \prod_{i = 1}^{n} A_{i}^{ω_{i}},$ (12) where i = 1, 2, ⋯ , n, and ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the corresponding weight vector of A, and satisfied $ω_{i} \in [0, 1], \sum_{i = 1}^{n} ω_{i} = 1$ .

Remark 7. If $ω = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then the PFZLWGA operator will be simplified to the picture fuzzy Z-linguistic geometric average operator.

Remark 8. If ρ (x_i) =1 (i = 1, 2, ⋯ , n), then the PFZLWGA operator will be simplified to the picture fuzzy linguistic weighted geometric average operator.

Remark 9. If ρ (x_i) =1 (i = 1, 2, ⋯ , n) and $ω = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})$ , then the PFZLWGA operator will be simplified to the picture fuzzy linguistic weighted geometric average operator.

Theorem 3. Let A = (A₁, A₂, ⋯ , A_n) is a set of PFZLVs, then the PFZLWGA aggregate result is also the PFZLV. $\begin{matrix} \begin{matrix} {PFZLGA}_{ω} (A_{1}, A_{2}, \dots, A_{n}) \end{matrix} \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - η_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉, \end{matrix}$ (13)

where i = 1, 2, ⋯ , n, and ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight vector of A, and satisfied ω_i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ .

Proof: 1. Suppose n=1, Equation (13) is obviously established.

2. Suppose n=2, since

$\begin{array}{l} A_{1}^{ω_{1}} = 〈 \begin{matrix} s_{θ^{ω_{1}} (x_{1})}, 1 - {(1 - μ_{A} (x_{1}))}^{ω_{1}}, \\ η_{A}^{ω_{1}} (x_{1}), ν_{A}^{ω_{1}} (x_{1}), ρ^{ω_{1}} (x_{1}) \end{matrix} 〉 \\ A_{2}^{ω_{2}} = 〈 \begin{matrix} s_{θ^{ω_{2}} (x_{2})}, 1 - {(1 - μ_{A} (x_{2}))}^{ω_{2}}, \\ η_{A}^{ω_{2}} (x_{2}), ν_{A}^{ω_{2}} (x_{2}), ρ^{ω_{2}} (x_{2}) \end{matrix} 〉 \end{array}$

3. Suppose n = k, we have $\begin{matrix} \begin{matrix} PFLWGA (A_{1}, A_{2}, \dots, A_{n}) \end{matrix} \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉 . \end{matrix}$

4. Now suppose n = i + 1, we have $\begin{matrix} \begin{matrix} PFZLWGA (A_{1}, A_{2}, \dots, A_{n}, A_{n + 1}) \\ = PFZLWGA (A_{1}, A_{2}, \dots, A_{n}) \otimes A_{i + 1}^{ω_{i + 1}} \end{matrix} \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i + 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ^{ω_{n + 1}} (x_{n + 1})}, \\ 1 - (1 - μ_{A} (x_{n + 1}))^{ω_{n = 1}}, \\ η_{A}^{ω_{n + 1}} (x_{n + 1}), ν_{A}^{ω_{n + 1}} (x_{n + 1}), \\ ρ^{ω_{n + 1}} (x_{n + 1}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{\prod_{i = n + 1}^{n + 1} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n + 1} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n + 1} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n + 1} (1 - ν_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i = 1}^{n + 1} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉, \end{matrix}$

the result is satisfied for the value of n in any cases, proof completed.

The PFZLWGA operator has the following properties:

Property 1 (Idempotency). Let $A^{1} = (A_{1}^{1}, A_{2}^{1}, \dots, A_{n}^{1})$ and $A^{2} = (A_{1}^{2}, A_{2}^{2}, \dots, A_{n}^{2})$ be two sets of PFZLVs, if $A_{i}^{1} \leq A_{i}^{2}$ , then $\begin{matrix} PFZLWGA (A_{1}^{1}, A_{2}^{1}, \dots, A_{n}^{1}) \\ \leq PFZLWGA (A_{1}^{2}, A_{2}^{2}, \dots, A_{n}^{2}) . \end{matrix}$

Proof: f (x) = x^a (0 < a < 1) is a monotonically increasing function when x > 0, so each component of PFZLV is monotonically increasing too.

Then, if $A_{i}^{1} \leq A_{i}^{2}$ , we have $PFZLWGA (A_{1}^{1}, A_{2}^{1}, \dots, A_{n}^{1}) \leq PFZLWGA (A_{1}^{2}, A_{2}^{2}, \dots, A_{n}^{2})$ , proof completed.

Property 2 (Boundedness). Let A_i be any PFZLV, and A = (A₁, A₂, ⋯ , A_n) be any set of PFZLVs, if A = A_i for any i (i = 1, 2, ⋯ , n), then $PFZLWGA (A_{1}, A_{2}, \dots, x_{n}) = A .$

Proof: Since we have A = (A₁, A₂, ⋯ , A_n),

$\begin{matrix} P F Z L W G A (A_{1}, A_{2}, \dots, A_{n}) = 〈 \begin{matrix} s_{θ (x)}, μ_{A} (x), η_{A} (x), \\ 1 - (1 - ν_{A} (x)), ρ (x) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} {(1 - ν_{A} (x_{i}))}^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ^{\sum_{i = 1}^{n} ω_{i}} (x_{i})}, μ_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{i}), \\ η_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{i}), \\ 1 - {(1 - ν_{A} (x_{i}))}^{\sum_{i = 1}^{n} ω_{i}}, \\ ρ^{\sum_{i = 1}^{n} ω_{i}} (x_{i}) \end{matrix} 〉 \end{matrix}$

Property 3 (Monotonicity). Let A = (A₁, A₂, ⋯ , A_n) be any set of PFZLVs, assume that A_min = min {A₁, A₂, ⋯ , A_n} and A_max = max {A₁, A₂, ⋯ , A_n} be the upper and lower bounders of A, then $\begin{matrix} A_{min} \leq PFZLWGA (A_{1}, A_{2}, \dots, x_{n}) \leq A_{max} . \end{matrix}$

Proof: Since we have A_min ≤ A ≤ A_max, according to the idempotency and monotonicity, $\begin{matrix} \begin{matrix} PFZLWGA (A_{1}, A_{2}, \dots, A_{n}) \end{matrix} \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉 \\ \geq 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{min})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{min}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{min}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{min}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{min}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ^{\sum_{i = 1}^{n} ω_{i}} (x_{min})}, μ_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{min}), \\ η_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{min}), \\ 1 - (1 - ν_{A} (x_{min}))^{\sum_{i = 1}^{n} ω_{i}}, \\ ρ^{\sum_{i = 1}^{n} ω_{i}} (x_{min}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ (x_{min})}, μ_{A} (x_{min}), \\ η_{A} (x_{min}), ν_{A} (x_{min}), \\ 1 - (1 - ν_{A} (x_{min})), ρ (x_{min}) \end{matrix} 〉 \\ = A_{min}, \end{matrix}$ $\begin{matrix} \begin{matrix} PFZLWGA (A_{1}, A_{2}, \dots, A_{n}) \end{matrix} \\ = 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{i})}, \prod_{i = 1}^{n} μ_{A}^{ω_{i}} (x_{i}), \\ \prod_{i = 1}^{n} η_{A}^{ω_{i}} (x_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{i}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{i}) \end{matrix} 〉 \\ \leq 〈 \begin{matrix} s_{\prod_{i = 1}^{n} θ^{ω_{i}} (x_{max})}, \prod_{i = 1}^{n} \underset{A}{μ^{ω_{i}}} (x_{max}), \\ \prod_{i = 1}^{n} \underset{A}{η^{ω_{i}}} (x_{max}), \\ 1 - \prod_{i = 1}^{n} (1 - ν_{A} (x_{max}))^{ω_{i}}, \\ \prod_{i = 1}^{n} ρ^{ω_{i}} (x_{max}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ^{\sum_{i = 1}^{n} ω_{i}} (x_{max})}, μ_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{max}), \\ η_{A}^{\sum_{i = 1}^{n} ω_{i}} (x_{max}), \\ 1 - (1 - ν_{A} (x_{max}))^{\sum_{i = 1}^{n} ω_{i}}, \\ ρ^{\sum_{i = 1}^{n} ω_{i}} (x_{max}) \end{matrix} 〉 \\ = 〈 \begin{matrix} s_{θ (x_{max})}, μ_{A} (x_{max}), \\ η_{A} (x_{max}), ν_{A} (x_{max}), \\ 1 - (1 - ν_{A} (x_{max}), ρ (x_{max}) \end{matrix} 〉 \\ = A_{max} . \end{matrix}$

4 MAGDM model based on PFLNWGA-VIKOR method

In this section, a novel MAGDM model based on PFZLWGA operator and VIKOR method under the picture fuzzy Z-linguistic environment is put forward.

For a MAGDM problem, let A_i (i = 1, 2, ⋯ , m) be a set of alternatives, and C_j (j = 1, 2, ⋯ , n) be a set of attributes, attribute weights vector is w = (w₁, w₂, ⋯ , w_i) ^T and satisfied $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1$ . expert weights vector is ω = (ω₁, ω₂, ⋯ , ω_k) ^T and satisfied $ω_{k} \in [0, 1], \sum_{k = 1}^{p} ω_{k} = 1$ . Assume that experts provide the decision-making information by PFZLV and give decision matrices D_k = [r_ij] _m×n (k = 1, 2, ⋯ , p) respectively, where p is the number of experts, r_ij is the evaluation value of ith alternative under jth attribute. Then, the procedure of the PFLNWGA-VIKOR method based on PFZLS in detail could be summarized as follows:

Step 1. Determine the group decision matrix. Aggregate individual decision matrix D_k into a group decision matrix $\tilde{D} = [{\tilde{r}}_{ij}]_{m \times n}$ by Eq.(12)

Step 2. Select the positive and negative ideal solutions. Because of the complex components of PFZLVs, it is difficult to observe r⁺ and r^- in $\tilde{D}$ . Therefore, calculate and compare the score functions of the corresponding elements of $\tilde{D}$ by Equation (7) and then select r⁺ and r^- by Equations (1) and (2).

Step 3. Calculate the value S_i and R_i. Determine the maximum group benefit value S_i and the minimum individual loss value R_i of alternatives by Equations (3) and (4).

Step 4. Calculate the value Q_i. Calculate the benefit ratio value Q_i of alternatives by Equation (5).

Step 5. Rank the alternatives. Obtain the ranking results of alternatives, the smaller the values R_i, S_i, Q_i are, the higher the alternative ranks.

Step 6. Determine the compromise alternative. If both compromise conditions are satisfied, then the alternative with the smallest value in Q_i will be the optimal alternative and go to Step 8. If not met, go to Step 7.

Step 7. Adjust the compromise alternatives. If I1 is not satisfied, adjust the value of m by Eq 1, then the set Aⁱ (i = 1, 2, ⋯ , m) will be the compromise alternatives. If I2 is not satisfied, then A¹, A² will be the compromise alternatives.

step 8. End.

5 Illustrative example

In this section, an EDM problem of forest fire accident is presented to verify the superiority of the proposed method.

As a large-scale natural disaster, forest fire is sudden, destructive and difficult to rescue, which may cause huge loss of lives and properties. Therefore, it is necessary to scientifically evaluate the emergency alternatives of forest fire accidents. However, because of the uncertainty and complexity of the initial feedback information, experts can only give subjective evaluation and confidence level, so that the existing fuzzy set theory could not deal with such kind of EDM problem very well. Therefore, the PFZLWGA-VIKOR method based on PFZLS could become the optimal choice to deal with this problem.

Assume that there are four emergency alternatives to be considered for the forest fire accident: artificial rainfall (A1); fire forces (A2); setting up isolation belts (A3) and social assistance (A4). And experts set four attributes that may affect the rescue progress: fire area (C1); spreading speed (C2); visibility index (C3) and ventilation level (C4), the corresponding weights vector of attribute is w_c = (0.2, 0.3, 0.1, 0.4) ^T. Three experts participated in the EDM formulation, according to different authorities, the weights vector of experts is w_e = (0.2, 0.5, 0.3) ^T. Each expert gives the evaluations on the four alternatives over each attribute Using the PFZLVs, where the linguistic terms are set as $\begin{matrix} \begin{matrix} S = {s_{0} = very bad, s_{1} = bad, \\ s_{2} = generally bad, s_{3} = fair, s_{4} = good, \\ s_{5} = very good, s_{6} = perfect} . \end{matrix} \end{matrix}$

The decision matrices $D_{k} = [r_{ij}^{k}] (k = 1, 2, 3)$ of experts are shown in Tables 1–3.

Table 1
The decision matrix D₁

C1 C2 C3 C4

A1 〈s1, (0.6, 0.2, 0.2), 0.6〉〈s2, (0.7, 0.2, 0.1), 0.5〉〈s2, (0.5, 0.3, 0.1), 0.6〉〈s3, (0.6, 0.3, 0.1), 0.7〉

A2 〈s2, (0.7, 0.2, 0.1), 0.6〉〈s3, (0.6, 0.2, 0.1), 0.6〉〈s2, (0.6, 0.1, 0.2), 0.7〉〈s2, (0.6, 0.2, 0.2), 0.8〉

A3 〈s4, (0.8, 0.1, 0.1), 0.6〉〈s4, (0.5, 0.2, 0.2), 0.7〉〈s4, (0.7, 0.2, 0.1), 0.6〉, 〈s3, (0.7, 0.2, 0.1), 0.6〉

A4 〈s2, (0.5, 0.3, 0.1), 0.5〉〈s4, (0.6, 0.2, 0.2), 0.7〉〈s5, (0.5, 0.2, 0.2), 0.7〉〈s4, (0.6, 0.2, 0.1), 0.5〉

	C1	C2	C3	C4
A1	〈s1, (0.6, 0.2, 0.2), 0.6〉	〈s2, (0.7, 0.2, 0.1), 0.5〉	〈s2, (0.5, 0.3, 0.1), 0.6〉	〈s3, (0.6, 0.3, 0.1), 0.7〉
A2	〈s2, (0.7, 0.2, 0.1), 0.6〉	〈s3, (0.6, 0.2, 0.1), 0.6〉	〈s2, (0.6, 0.1, 0.2), 0.7〉	〈s2, (0.6, 0.2, 0.2), 0.8〉
A3	〈s4, (0.8, 0.1, 0.1), 0.6〉	〈s4, (0.5, 0.2, 0.2), 0.7〉	〈s4, (0.7, 0.2, 0.1), 0.6〉,	〈s3, (0.7, 0.2, 0.1), 0.6〉
A4	〈s2, (0.5, 0.3, 0.1), 0.5〉	〈s4, (0.6, 0.2, 0.2), 0.7〉	〈s5, (0.5, 0.2, 0.2), 0.7〉	〈s4, (0.6, 0.2, 0.1), 0.5〉

Table 2

The decision matrix D₂

	C1	C2	C3	C4
A1	〈s1, (0.7, 0.1, 0.2), 0.6〉	〈s2, (0.6, 0.1, 0.2), 0.5〉	〈s1, (0.7, 0.1, 0.1), 0.6〉	〈s2, (0.6, 0.1, 0.1), 0.6〉
A2	〈s2, (0.6, 0.2, 0.2), 0.5〉	〈s1, (0.7, 0.2, 0.1), 0.6〉	〈s2, (0.6, 0.1, 0.1), 0.6〉	〈s1, (0.7, 0.1, 0.2), 0.
A3	〈s4, (0.7, 0.1, 0.1), 0.6〉	〈s3, (0.6, 0.2, 0.2), 0.6〉	〈s3, (0.5, 0.2, 0.2), 0.7〉	〈s2, (0.6, 0.1, 0.1), 0.5〉
A4	〈s2, (0.8, 0.1, 0.1), 0.5〉	〈s4, (0.8, 0.1, 0.1), 0.7〉	〈s4, (0.5, 0.3, 0.1), 0.6〉	〈s3, (0.7, 0.1, 0.1), 0.8〉

Table 3

The decision matrix D₃

	C1	C2	C3	C4
A1	〈s1, (0.7, 0.2, 0.1), 0.9〉	〈s3, (0.6, 0.3, 0.1), 0.7〉	〈s5, (0.6, 0.2, 0.1), 0.8〉	〈s1, (0.5, 0.3, 0.2), 0.7〉
A2	〈s4, (0.6, 0.2, 0.2), 0.7〉	〈s4, (0.7, 0.2, 0.1), 0.6〉	〈s4, (0.5, 0.2, 0.2), 0.6〉	〈s4, (0.5, 0.2, 0.2), 0.5〉
A3	〈s3, (0.7, 0.1, 0.1), 0.8〉	〈s3, (0.5, 0.2, 0.2), 0.7〉	〈s4, (0.5, 0.2, 0.2), 0.6〉	〈s4, (0.5, 0.2, 0.2), 0.5〉
A4	〈s3, (0.5, 0.3, 0.1), 0.8〉	〈s5, (0.7, 0.1, 0.1), 0.6〉	〈s5, (0.7, 0.1, 0.1), 0.7〉	〈s5, (0.7, 0.1, 0.1), 0.6〉

Then the procedure of PFZLWGA-VIKOR method based on PFZLS is summarized as follows:

Step 1. Aggregate all D_k by Equation (13), the group decision matrix $\tilde{D}$ is shown in Table 4.

Table 4

Group decision matrix $\tilde{D}$

	C1	C2
A1	〈s1.0000, (0.6787, 0.1414, 0.1712), 0.6777〉	〈s2.2587, (0.6188, 0.1597, 0.1515), 0.5531〉
A2	〈s2.4623, (0.6188, 0.2000, 0.1809), 0.5736〉	〈s1.8882, (0.6787, 0.2000, 0.1000), 0.6000〉
A3	〈s3.6692, (0.7189, 0.1000, 0.1000), 0.6541〉	〈s3.1777, (0.5477, 0.2000, 0.2000), 0.6481〉
A4	〈s2.2587, (0.6325, 0.1732, 0.1000), 0.5757〉	〈s4.2769(, 0.7256, 0.1149, 0.1210), 0.6684〉
	C3	C4
A1	〈s1.8616, (0.6249, 0.1534, 0.1000), 0.6541〉	〈s1.7617, (0.5681, 0.1732, 0.1312), 0.6481〉
A2	〈s2.2587, (0.6000, 0.1390, 0.1210), 0.6481〉	〈s1.5971, (0.6481, 0.1149, 0.2000), 0.6355〉
A3	〈s3.4641, (0.5348, 0.2000, 0.1810), 0.6481〉	〈s2.6703, (0.5859, 0.1414, 0.1312), 0.5186〉
A4	〈s4.4721, (0.5531, 0.1990, 0.1210), 0.6481〉	〈s3.7039, (0.6787, 0.1149, 0.1000), 0.6680〉

Step 2. Calculate and select positive-ideal solution r⁺ and negative-ideal solution r^- of $\tilde{D}$ by Equation (7), Equations (1) and (2).

$r^{+} = (\begin{matrix} 〈 s_{3.6672}, (07234, 0.1000, 01000), 0.6540 〉, \\ 〈 s_{4.2769}, (07405, 0.1149, 01149), 0.6684 〉, \\ 〈 s_{4.4721}, (05710, 0.1990, 01149), 0.6481 〉, \\ 〈 s_{3.7039}, (06822, 0.1149, 01000), 0.6680 〉 \end{matrix})$ $r^{-} = (\begin{matrix} 〈 s_{1.0000}, (06822, 0.1414, 01625), 0.6776 〉, \\ 〈 s_{1.8882}, (06822, 0.2000, 01000), 0.6000 〉, \\ 〈 s_{1.8616}, (06378, 0.1534, 01000), 0.6540 〉, \\ 〈 s_{1.5971}, (06536, 0.1149, 02000), 0.6355 〉 \end{matrix})$

Step 3. Determine the maximum group benefit value S_i and the minimum individual loss value R_i by Equations (3) and (4).

Step 4. Calculate the benefit ratio value Q_i of each alternative by Equation (5).

Step 5. Obtain the rankings of alternatives, which is shown in Table 5.

Table 5

Ranking result

Value type	Value vector	Ranking result
S _i	(0.96,0.91,0.53,0.13)	A₂ ≺ A₁ ≺ A₃ ≺ A₄
R _i	(0.37,0.40,0.30,0.13)	A₂ ≺ A₁ ≺ A₃ ≺ A₄
Q _i	(0.92,0.98,0.62,0.00)	A₂ ≺ A₁ ≺ A₃ ≺ A₄

Step 6. $Q (A_{3}) - Q (A_{4}) = 0.62 \geq \frac{1}{3}$ and A₂ ≺ A₁ ≺ A₃ ≺ A₄, I1 and I2 are both satisfied, so setting up isolation belts (A₄) is the optimal alternative to the forest fire EDM problem.

6 Comparative analysis

In this section, the comparative experiments are presented from three aspects of operator-optimization method, the special form of PFZLS and other degenerate sets, which could strongly verify the necessity and superiority of PFZLS.

6.1 Comparative analysis of operator-optimization method

Aggregation operator is one of the effective methods for information fusion, which could be used for alternative ranking. Therefore, the main idea of the operator-optimization method is to aggregate all attribute values of each alternative into one comprehensive value and calculate the crisp values to obtain the alternative rankings. The process of operator-optimization method is summarized as follows.

Firstly, taking the group decision matrix $\tilde{D}$ as the initial matrix, the comprehensive evaluation values of each alternative are obtained by Equation (13), which is shown as

$A_{i} = (\begin{matrix} 〈 s_{1.7042}, (06097, 0.1603, 01424), 0.6240 〉, \\ 〈 s_{1.8958}, (06460, 0.1544, 01594), 0.6131 〉, \\ 〈 s_{3.0770}, (05927, 0.1515, 01514), 0.5938 〉, \\ 〈 s_{3.5696}, (06689, 0.1317, 01084), 0.6465 〉 \end{matrix})$

By Equation (7), the crisp comprehensive evaluation values of alternatives are calculated as $\begin{matrix} E (A_{i}) = (0.2044, 0.2259, 0.3486, 0.4671) . \end{matrix}$

Then, the alternative rankings is obtained as A₁ ≺ A₂ ≺ A₃ ≺ A₄, so setting up isolation belts (A₄) is the optimal alternative to this EDM problem. The two results obtained by the operator-optimization method and the PFZLWGA-VIKOR method are basically consistent, which verifies the effectiveness of the proposed method.

In the complex MAGDM process, the operator-optimization method gradually loses advantages because of the time-consuming component-level operations and low logic strength. However, this method still is one of the indispensable tools for decision theory because of the simple steps. Therefore, this paper first aggregates the group decision matrix by the operator method to retain as much information as possible, then uses VIKOR method to obtain the alternative rankings. Combining the advanges of two methods, the PFZLWGA-VIKOR method can better deal with MAGDM problems.

6.2 Comparative analysis of the special PFZLVs

In this part, the effect on information expression by a special form of PFZLV is discussed. Assume that ρ (x) =1 in decision matrices D_k, then the numerical experiment is restarted and the process is summarized as follows.

Firstly, determine the novel group decision matrix ${\tilde{D}}^{'}$ by Equation (12), which is shown in Table 6. Because of the properties of operation rules, the value of ρ (x) in ${\tilde{D}}^{'}$ are still equal to 1,

Table 6
Special form group decision matrix ${\tilde{D}}^{'}$

C1 C2

A1 〈s1.0000, (0.6787, 0.1414, 0.1712), 1〉〈s2.2586, (0.6187, 0.1597, 0.1514), 1〉

A2 〈s2.4622, (0.6187, 0.2000, 0.1809), 1〉〈s1.8881, (0.6787, 0.2000, 0.1000), 1〉

A3 〈s3.6692, (0.7189, 0.1000, 0.1000), 1〉〈s3.1777, (0.5477, 0.2000, 0.2000), 1〉

A4 〈s2.2587, (0.6325, 0.1732, 0.1000), 1〉〈s4.2769(, 0.7256, 0.1149, 0.1210), 1〉

C3 C4

A1 〈s1.8616, (0.6248, 0.15336, 0.1000), 1〉〈s1.7617, (0.5680, 0.1732, 0.1312), 1〉

A2 〈s2.2586, (0.6000, 0.1390, 0.1209), 1〉〈s1.5971, (0.6481, 0.1149, 0.2000), 1〉

A3 〈s3.4641, (0.5348, 0.2000, 0.1810), 1〉〈s2.6703, (0.6480, 0.1148, 0.2000), 1〉

A4 〈s4.4721, (0.5531, 0.1990, 0.1210), 1〉〈s3.7039, (0.6787, 0.1149, 0.1000), 1

	C1	C2
A1	〈s1.0000, (0.6787, 0.1414, 0.1712), 1〉	〈s2.2586, (0.6187, 0.1597, 0.1514), 1〉
A2	〈s2.4622, (0.6187, 0.2000, 0.1809), 1〉	〈s1.8881, (0.6787, 0.2000, 0.1000), 1〉
A3	〈s3.6692, (0.7189, 0.1000, 0.1000), 1〉	〈s3.1777, (0.5477, 0.2000, 0.2000), 1〉
A4	〈s2.2587, (0.6325, 0.1732, 0.1000), 1〉	〈s4.2769(, 0.7256, 0.1149, 0.1210), 1〉
	C3	C4
A1	〈s1.8616, (0.6248, 0.15336, 0.1000), 1〉	〈s1.7617, (0.5680, 0.1732, 0.1312), 1〉
A2	〈s2.2586, (0.6000, 0.1390, 0.1209), 1〉	〈s1.5971, (0.6481, 0.1149, 0.2000), 1〉
A3	〈s3.4641, (0.5348, 0.2000, 0.1810), 1〉	〈s2.6703, (0.6480, 0.1148, 0.2000), 1〉
A4	〈s4.4721, (0.5531, 0.1990, 0.1210), 1〉	〈s3.7039, (0.6787, 0.1149, 0.1000), 1

Secondly, according to the VIKOR method, the values of S_i, R_i, Q_i are calculated by Equations (3)–(5), the total results including the values of S_i, R_i, Q_i and alternative rankings are shown in Table 8.

Table 8

Ranking result of special form PFZLVs

Value type	Value vector	Ranking result
S	(0.94, 0.89, 0.44, 0.11)	A₂ ≺ A₁ ≺ A₃ ≺ A₄
R	(0.37, 0.40, 0.21, 0.11)	A₂ ≺ A₁ ≺ A₃ ≺ A₄
Q	(0.94, 0.98, 0.36, 00..)	A₂ ≺ A₁ ≺ A₃ ≺ A₄

Finally, verify the conditions of compromise alternative, both I1 and I2 are satisfied, so setting up isolation belts (A₄) is the optimal alternative to this EDM problem.

Besides, an equivalent experiment is proposed, that is, remove all credibility values of D_k and retest. The final ranking results are also consistent with Table 8, which shows that when all experts are convinced of their evaluation, the decision process could be completed by the way of degenerate set. In a word, PFZLS not only retains the advantages of the degenerate sets such as PFLS but also has the characteristics that the degenerate sets does not have. More broadly, these characteristics can be confirmed in the next part we discussed to further verify the necessity of PFZLS in the information representation.

6.3 Comparative analysis of different degenerate sets

In this part, the comparative experiments by other degenerate sets are presented, including PFLS, IFLS, IFZLS, and Z-number. The purpose is to discuss the effect of missing information on the reliability of experimental results. In order to control the variables, the same group matrix $\tilde{D}$ and VIKOR method are used, and the process is summarized as follows.

Firstly, different components of PFZLV are deleted respectively, which makes PFZLS degenerate into other language term sets. For example, if we delete the value of η_A (x), PFZLS will degenerate into the IFZLS.

Then, based on operation rules and distance functions of each corresponding degenerated set, we conduct the numerical experiment and collect the compared results of different fuzzy sets, The total results including the characteristics, the component considerations, and the alternative rankings are shown in Table 7.

Table 7
Compare result

Tpye Characteristics Credibility Neutrality Ranks

PFZLS 〈s_θ(x),η_A(x),ν_A(x)ρ(x)〉 ✓ ✓ A₂ ≺ A₁ ≺ A₃ ≺ A₄

PFLS [24] 〈s_θ(x),η_A(x),ν_A(x)〉 ✓ A₂ ≺ A₁ ≺ A₃ ≺ A₄

IFZLS [29] 〈s_θ(x),ν_A(x)ρ(x)〉 ✓ A₂ ≺ A₁ ≺ A₃ ≺ A₄

IFLS [40] 〈s_θ(x),ν_A(x)〉 A₂ ≺ A₁ ≺ A₃ ≺ A₄

Z-number 〈x,A,B〉

Tpye	Characteristics	Credibility	Neutrality	Ranks
PFZLS	〈s_θ(x),η_A(x),ν_A(x)ρ(x)〉	✓	✓	A₂ ≺ A₁ ≺ A₃ ≺ A₄
PFLS [24]	〈s_θ(x),η_A(x),ν_A(x)〉	✓	A₂ ≺ A₁ ≺ A₃ ≺ A₄
IFZLS [29]	〈s_θ(x),ν_A(x)ρ(x)〉	✓		A₂ ≺ A₁ ≺ A₃ ≺ A₄
IFLS [40]	〈s_θ(x),ν_A(x)〉			A₂ ≺ A₁ ≺ A₃ ≺ A₄
Z-number	〈x,A,B〉

The formula of Z-number is not suitable for the data in this case, so the compared result is not obtained. The ranking results by PFZLS and other degenerate sets are consistent, which indicates that PFZLS has the great scalability and can be compatible with the decision-making information form given by DMs to the greatest extent.

Table 7 shows that other fuzzy sets are more or less lacking in information components, which would inevitably give rise to errors in decision-making results. Therefore, PFZLS does increase the information dimension while ensuring accuracy, which has wider applicability in MAGDM process.

7 Conclusion

In the MAGDM problems of PFLS, the credibility of information is unconsidered, which may case information loss. To solve this problem, this paper proposes the PFZLS consisted of linguistic term, picture fuzzy number, and credibility, which could represent information more comprehensively. Meanwhile, operation rules, measure theories, and comparison methods of PFZLS are proposed and their properties are studied. Then, the PFZLWGA operator and extended VIKOR method based on PFZLS are put forward. Finally, a MAGDM model based on PFZLWGA operator and VIKOR method is established. A numerical experiment and a series of comparative analyses of the forest fire EDM problem are presented, which deeply shows that PFZLS has the greatest advantage of ductility and compatibility.

Two main research directions need to be considered in the future. The first is to apply the proposed PFZLS to other fields and complete the decision-making process in real cases. An interesting assumption is the data labeling work of uncertain information, by using a more compatible fuzzy set such as PFZLS, more available annotation results could be obtained to improve the efficiency of data mining for decision-making process. In addition, with the increase of the expression dimension of fuzzy sets, the operation scale will become larger and larger, and even information distortion caused by the zero-trending phenomenon will occur. How to balance uncertain information and computational efficiency has become a new challenge, processing uncertain information without distortion needs to be further studied.

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