Models for multiple attribute decision making with picture fuzzy information

Abstract

In this paper, we investigate the picture fuzzy multiple attribute decision making problems where the information about attribute weights is partly known or completely unknown. We introduce some notions, such as picture fuzzy ideal point, the normalized Hamming distance of picture fuzzy numbers. We also introduce the grey relational coefficient between the attribute value vectors of each alternative and the picture fuzzy ideal point. Then we establish the grey relational analysis models to measure the grey relational degree between each alternative and the picture fuzzy ideal point. Based on the grey relational analysis models, we can rank the given alternatives and then select the most desirable one. Finally, we illustrate the developed grey relational analysis models with a numerical example for potential evaluation of emerging technology commercialization.

Keywords

Multiple attribute decision making picture fuzzy set picture fuzzy ideal point projection model included angle

1. Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the concept of fuzzy set [3]. Atanassov and Gargov [4] and Atanassov [5] proposed the concept of interval-valued intuitionistic fuzzy sets, which are characterized by a membership function, a non-membership function, and a hesitancy function whose values are intervals. The intuitionistic fuzzy set and interval-valued intuitionistic fuzzy sets have received more and more attention since its appearance [6 –36]. Recently, Cuong [37] proposed picture fuzzy set (PFS) and investigated the some basic operations and properties of PFS. The picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of non-membership. The only constraint is that the sum of the three degrees must not exceed 1. Basically, PFS based models can be applied to situations requiring human opinions involving more answers of types: yes, abstain, no, refusal, which can’t be accurately expressed in the traditional FS and IFS. Until now, some progress has been made in the research of the PFS theory. Singh [38] investigated the correlation coefficients for picture fuzzy set and apply the correlation coefficient to clustering analysis with picture fuzzy information. Son etc. [39, 40] introduce several novel fuzzy clustering algorithms on the basis of picture fuzzy sets and applications to time series forecasting and weather forecasting. Thong [41] developed a novel hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis and application to health care support systems.Wei [42] proposed the picture fuzzy cross-entropy for multiple attribute decision making problems.

Although, Atanassov’s intuitionistic fuzzy set theory has been successfully applied in different areas, but there are situations in real life which can’t be represented by Atanassov’s intuitionistic fuzzy sets. Voting can be a good example of such situation as the human voters may be divided into four groups of those who: vote for, abstain, refusal of voting. Basically, picture fuzzy sets [37] based models may be adequate in situations when we face human opinions involving more answers of the type: yes, abstain, no, refusal. However, all the above approaches are unsuitable to deal with the situations where the information about attribute weights are partly known or completely unknown. To solve this issue, in this paper, we shall establish some grey relational analysis models for picture fuzzy MADM, which can handle the cases where the attribute weights are partly known or completely unknown. In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy set and picture fuzzy sets. In Section 3, we shall propose some grey relational analysis models for picture fuzzy MADM, which can handle the cases where the attribute weights are completely unknown or completely known. In Section 4, we shall present a numerical example for potential evaluation of emerging technology commercialization with picture fuzzy information in order to illustrate the method proposed in this paper. Section 5 concludes the paper with some remarks.

2. Preliminaries

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets and picture fuzzy sets.

Definition 1 [1, 2]. An IFS A in X is given by $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in X}$ (1) where μ_A : X → [0, 1] and ν_A : X → [0, 1], where, 0 ⩽ μ_A (x) + ν_A (x) ⩽ 1, ∀ x ∈ X. The number μ_A (x) and ν_A (x) represents, respectively, the membership degree and non-membership degree of the element x to the set A.

Although, Atanassov’s intuitionistic fuzzy set theory [1, 2] has been successfully applied in different areas, but there are situations in real life which can’t be represented by Atanassov’s intuitionistic fuzzy sets. Picture fuzzy sets are extension of Atanassov’s intuitionistic fuzzy sets. Picture fuzzy set [37] based models may be adequate in situations when we face human opinions involving more answers of types: yes, abstain, no, refusal. It can be considered as a powerful tool represent the uncertain information in the process of patterns recognition and cluster analysis.

Definition 3 [37]. A picture fuzzy set (PFS) A on the universe X is an object of the form $A = {〈 x, μ_{A} (x), η_{A} (x), ν_{A} (x) 〉 | x \in X}$ (2) where μ_A (x) ∈ [0, 1] is called the “degree of positive membership of A ”, η_A (x) ∈ [0, 1] is called the “degree of neutral membership of A ”and ν_A (x) ∈ [0, 1] is called the “degree of negative membership of A”, and μ_A (x), η_A (x), ν_A (x) satisfy the following condition:0 ⩽ μ_A (x) + η_A (x) + ν_A (x) ⩽ 1, ∀ x ∈ X. Then for x ∈ X, π_A (x) = 1 - (μ_A (x) + η_A (x) + ν_A (x)) could be called the degree of refusal membership of x in A. For convenience, we call $\tilde{α} = (μ_{α}, η_{α}, ν_{α})$ a picture fuzzy number.

Definition 4. Let ${\tilde{a}}_{1} = (μ_{1}, η_{1}, ν_{1})$ and ${\tilde{a}}_{2} = (μ_{2}, η_{2}, ν_{2})$ be two picture fuzzy numbers, then the normalized Hamming distance between ${\tilde{a}}_{1} = (μ_{1}, η_{1}, ν_{1})$ and ${\tilde{a}}_{2} = (μ_{2}, η_{2}, ν_{2})$ is defined as follows: $d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) = \frac{1}{2} (| μ_{1} - μ_{2} | + | η_{1} - η_{2} | + | ν_{1} - ν_{2} |)$ (3)

3. Grey relational analysis models for picture fuzzy multiple attribute decision making

Grey relational analysis (GRA) method was originally developed by Deng [43] and has been successfully applied in solving a variety of MADM problems [44 –59]. The main procedure of GRA is firstly translating the performance of all alternatives into a comparability sequence. This step is called grey relational generating. According to these sequences, an ideal target sequence is defined. Then, the grey relational coefficient between all comparability sequences and ideal target sequence is calculated. Finally, base on these grey relational coefficients, the grey relational degree between ideal target sequence and every comparability sequences is calculated. If a comparability sequence translated from an alternative has the highest grey relational degree between the ideal target sequence and itself, that alternative will be the best choice. Based the traditional idea of grey relational analysis methods [43], in this section, we shall propose the grey relational analysis methods for multiple attribute decision making with picture fuzzy information.

Let A ={ A₁, A₂, ⋯, A_m } be a discrete set of alternatives, and G ={ G₁, G₂, ⋯, G_n } be the set of attributes, w = (w₁, w₂, ⋯, w_n) is the weighting vector of the attribute G_j (j = 1, 2, ⋯, n), where w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ . H is a set of the known weight information, which can be constructed by the following forms [60 –63], for i ≠ j: Form 1. A weak ranking: w_i ⩾ w_j; Form 2. A strict ranking:w_i - w_j ⩾ α_i, α_i > 0; Form 3. A ranking of differences: w_i - w_j ⩾w_k - w_l, for j ≠ k ≠ l; Form 4. A ranking with multiples: w_i ⩾ β_iw_j, 0 ⩽ β_i ⩽ 1; Form 5. An interval form: α_i ⩽ w_i ⩽ α_i + ɛ_i, 0 ⩽ α_i < α_i + ɛ_i ⩽ 1.

Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = {(μ_{ij}, η_{ij}, ν_{ij})}_{m \times n}$ is the picture fuzzy decision matrix, where μ_ij indicates the degree of positive membership that the alternative A_i satisfies the attribute G_j given by the decision maker, η_ij indicates the degree of neutral membership that the alternative A_i doesn’t satisfy the attribute G_j, ν_ij indicates the degree that the alternative A_i doesn’t satisfy the attribute G_j given by the decision maker, μ_ij ∈ [0, 1], η_ij ∈ [0, 1] ν_ij ∈ [0, 1], μ_ij + η_ij + ν_ij ⩽ 1, i = 1, 2, ⋯, m, j = 1, 2, ⋯, n.

In the following, we apply GRA method to solve picture fuzzy MADM with incompletely known weight information. The method involves the following steps:

Step 1. Determine the positive ideal alternative with picture fuzzy information.

${\tilde{r}}^{+} = ((μ_{1}^{+}, η_{1}^{+}, ν_{1}^{+}), (μ_{2}^{+}, η_{2}^{+}, ν_{2}^{+}), \dots, (μ_{n}^{+}, η_{n}^{+}, ν_{n}^{+}))$ (4) where

${\tilde{r}}_{j}^{+} = (μ_{j}^{+}, η_{j}^{+}, ν_{j}^{+}) = (max_{i} μ_{ij}, min_{i} η_{ij}, min_{i} ν_{ij})$ ,j ∈ 1, 2, ⋯, n.

Step 2. Calculate the grey relational coefficient of each alternative from PIS using the following equation: $\begin{matrix} ξ_{ij}^{+} & = & \frac{min_{1 ⩽ i ⩽ m} min_{1 ⩽ j ⩽ n} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+}) + ρ max_{1 ⩽ i ⩽ m} max_{1 ⩽ j ⩽ n} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+})}{d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+}) + ρ max_{1 ⩽ i ⩽ m} max_{1 ⩽ j ⩽ n} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+})} \\ i = 1, 2, \dots, m, j \in 1, 2, \dots, n . \end{matrix}$ (5) where the identification coefficient ρ = 0.5.

Step 3. Calculating the grey relational degree of each alternative from PIS using the following equation, respectively: $ξ_{i}^{+} = \sum_{j = 1}^{n} w_{j} ξ_{ij}^{+}, i = 1, 2, \dots, m .$ (6)

The basic principle of the GRA method is that the chosen alternative should have the “largest degree of grey relation” from the positive ideal solution. Obviously, for the weight vector given, the larger $ξ_{i}^{+}$ , the better alternative A_i is.

If the information about attribute weights is incompletely known, in order to get the $ξ_{i}^{+}$ , firstly, we must calculate the weight information. The grey relational coefficient between PIS and itself is(1, 1, ⋯, 1), so the comprehensive grey relational coefficient deviation sum is $d_{i} (w) = \sum_{j = 1}^{n} (1 - ξ_{ij}^{+}) w_{j}$ (7)

So, we can establish the following multiple objective optimization models to calculate the weight information: $(M . 1) {\begin{matrix} min d_{i} (w) = \sum_{j = 1}^{n} (1 - ξ_{ij}^{+}) w_{j}, i = 1, \dots, m . \\ subject to : w \in H \end{matrix}$

Since each alternative is non-inferior, so there exists no preference relation on the all the alternatives. Then, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model: $(M . 2) {\begin{matrix} min d (w) = \sum_{i = 1}^{m} d_{i} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} (1 - ξ_{ij}^{+}) w_{j} \\ s . t . w \in H \end{matrix}$

By solving the model (M.2), we get the optimal solution w = (w₁, w₂, ⋯, w_n), which can be used as the weight vector of attributes. Then, we can get $ξ_{i}^{+} (i = 1, \dots, m)$ by Equations (6).

If the information about attribute weights is completely unknown, we can establish another multiple objective programming model as follows: $(M . 3) {\begin{matrix} min d_{i} (w) = \sum_{j = 1}^{n} {[(1 - ξ_{ij}^{+}) w_{j}]}^{2} \\ s . t . \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$

Similarly, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model: $(M . 4) {\begin{matrix} min d (w) = \sum_{i = 1}^{m} d_{i} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {[(1 - ξ_{ij}^{+}) w_{j}]}^{2} \\ s . t . \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$

To solve this model, we construct the Lagrange function: $L (w, λ) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {[(1 - ξ_{ij}^{+}) w_{j}]}^{2} + 2 λ (\sum_{j = 1}^{n} w_{j} - 1)$ (8) where λ is the Lagrange multiplier.

Differentiating Equation. (8) with respect to w_j (j = 1, 2, ⋯, n) and λ, and setting these partial derivatives equal to zero, we get a simple and exact formula for determining the attribute weights as follows: $w_{j} = {[\sum_{j = 1}^{n} (\sum_{i = 1}^{m} (1 - {ξ_{ij}^{+})^{2})}^{- 1}]}^{- 1} / - \sum_{i = 1}^{m} (1 - ξ_{ij}^{+})^{2}$ (9)

Then, we can get $ξ_{i}^{+} (i = 1, \dots, m)$ by Equations (6).

Step 4. Rank all the alternatives A_i (i = 1, 2, ⋯, m) and select the best one(s) in accordance with $ξ_{i}^{+} (i = 1, \dots, m)$ . If any alternative has the highest $ξ_{i}^{+}$ value, then, it is the most important alternative.

Step 5. End.

4. Numerical example

Thus, in this section we shall present a numerical example for potential evaluation of emerging technology commercialization with picture fuzzy information in order to illustrate the method proposed in this paper. Let us suppose there is a problem to deal with the potential evaluation of emerging technology commercialization which is classical multiple attribute decision making problems. There is a panel with five possible emerging technology enterprises A_i (i = 1, 2, 3, 4, 5) to select. The experts selects six attribute to evaluate the five possible emerging technology enterprises: ding172G₁ is the technical advancement; ding173G₂ is the potential market and market risk; ding174G₃ is the industrialization infrastructure; ding175G₄ is the development of science and technology; ding176G₅ is the financial conditions; ding177G₆ is the employment creation. In order to avoid influence each other, the decision makers are required to evaluate the five possible emerging technology enterprises A_i (i = 1, 2, 3, 4, 5) under the above six attributes and the decision matrix $\tilde{R} = {({\tilde{r}}_{ij})}_{5 \times 6}$ is presented in Table 1, where ${\tilde{r}}_{ij} (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4, 5, 6)$ are in the form of PFNs.

Table 1
The picture fuzzy decision matrix

A ₁ A ₂ A ₃ A ₄ A ₅

x ₁ (0.53,0.33,0.09) (0.73,0.12,0.08) (0.91,0.03,0.02) (0.85,0.09,0.05) (0.90,0.05,0.02)

x ₂ (0.89,0.08,0.03) (0.13,0.64,0.21) (0.07,0.09,0.05) (0.74,0.16,0.10) (0.68,0.08,0.21)

$\begin{matrix} x_{3} \end{matrix}$ (0.42,0.35,0.18) (0.03,0.82,0.13) (0.04,0.85,0.10) (0.02,0.89,0.05) (0.05,0.87,0.06)

x ₄ (0.08,0.89,0.02) (0.73,0.15,0.08) (0.68,0.26,0.06) (0.08,0.84,0.06) (0.13,0.75,0.09)

$\begin{matrix} x_{5} \end{matrix}$ (0.33,0.51,0.12) (0.52,0.31,0.16) (0.15,0.76,0.07) (0.16,0.71,0.05) (0.15,0.73,0.08)

x ₆ (0.17,0.53,0.13) (0.51,0.24,0.21) (0.31,0.39,0.25) (1.00,0.00,0.00) (0.91,0.03,0.05)

	A ₁	A ₂	A ₃	A ₄	A ₅
x ₁	(0.53,0.33,0.09)	(0.73,0.12,0.08)	(0.91,0.03,0.02)	(0.85,0.09,0.05)	(0.90,0.05,0.02)
x ₂	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.07,0.09,0.05)	(0.74,0.16,0.10)	(0.68,0.08,0.21)
$\begin{matrix} x_{3} \end{matrix}$	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)	(0.05,0.87,0.06)
x ₄	(0.08,0.89,0.02)	(0.73,0.15,0.08)	(0.68,0.26,0.06)	(0.08,0.84,0.06)	(0.13,0.75,0.09)
$\begin{matrix} x_{5} \end{matrix}$	(0.33,0.51,0.12)	(0.52,0.31,0.16)	(0.15,0.76,0.07)	(0.16,0.71,0.05)	(0.15,0.73,0.08)
x ₆	(0.17,0.53,0.13)	(0.51,0.24,0.21)	(0.31,0.39,0.25)	(1.00,0.00,0.00)	(0.91,0.03,0.05)

To get the most desirable emerging technology enterprises, the following steps are involved:

Case 1. The information about the attribute weights is partly known and the known weight information is given as follows: $\begin{matrix} H = {0.15 ⩽ w_{1} ⩽ 0.25, 0.10 ⩽ w_{2} ⩽ 0.15, \\ 0.15 ⩽ w_{3} ⩽ 0.20, 0.15 ⩽ w_{4} ⩽ 0.20, \\ 0.15 ⩽ w_{3} ⩽ 0.20, 0.10 ⩽ w_{4} ⩽ 0.13, \\ w_{j} ⩾ 0, j = 1, 2, 3, 4, 5, 6, \sum_{j = 1}^{6} w_{j} = 1} \end{matrix}$

Step 1. Based on the Table 1 , we denote the five possible emerging technology enterprisesA_i (i = 1, 2, 3, 4, 5) by $\begin{matrix} A_{1} = {(0 . 53, 0 . 33, 0 . 09), (0 . 89, 0 . 08, 0 . 03), (0 . 42, 0 . 35, 0 . 18) \\ (0 . 08, 0 . 89, 0 . 02), (0 . 33, 0 . 51, 0 . 12), (0 . 17, 0 . 53, 0 . 13)} \end{matrix}$ $\begin{matrix} A_{2} = {(0 . 73, 0 . 12, 0 . 08), (0 . 13, 0 . 64, 0 . 21), (0 . 03, 0 . 82, 0 . 13) \\ (0 . 73, 0 . 15, 0 . 08), (0 . 52, 0 . 31, 0 . 16), (0 . 51, 0 . 24, 0 . 21)} \end{matrix}$ $\begin{matrix} A_{3} = {(0 . 91, 0 . 03, 0 . 02), (0 . 07, 0 . 09, 0 . 05), (0 . 04, 0 . 85, 0 . 10) \\ (0 . 68, 0 . 26, 0 . 06), (0 . 15, 0 . 76, 0 . 07), (0 . 31, 0 . 39, 0 . 25)} \end{matrix}$ $\begin{matrix} A_{4} = {(0 . 85, 0 . 09, 0 . 05), (0 . 74, 0 . 16, 0 . 10), (0 . 02, 0 . 89, 0 . 05) \\ (0 . 08, 0 . 84, 0 . 06), (0 . 16, 0 . 71, 0 . 05), (1 . 00, 0 . 00, 0 . 00)} \end{matrix}$ $\begin{matrix} A_{5} = {(0 . 90, 0 . 05, 0 . 02), (0 . 68, 0 . 08, 0 . 21), (0 . 05, 0 . 87, 0 . 06) \\ (0 . 13, 0 . 75, 0 . 09), (0 . 15, 0 . 73, 0 . 08), (0 . 91, 0 . 03, 0 . 05)} \end{matrix}$

Step 2. Based on the Table 1 and Equation (4), we can get the A⁺: $\begin{matrix} A^{+} = {(0 . 91, 0 . 03, 0 . 02), (0 . 89, 0 . 08, 0 . 03), (0 . 42, 0 . 35, 0 . 05) \\ (0 . 73, 0 . 15, 0 . 02), (0 . 52, 0 . 31, 0 . 05), (1 . 00, 0 . 00, 0 . 00)} \end{matrix}$

Step 3. Calculate the grey relational coefficient of each emerging technology enterprise from A⁺ $ξ^{+} = {(ξ_{ij}^{+})}_{5 \times 6} = [\begin{matrix} 0 . 4013 0 . 3848 0 . 6189 0 . 7222 0 . 5983 0 . 7452 \\ 0 . 6024 0 . 7427 0 . 6203 0 . 4730 0 . 5718 0 . 4941 \\ 1 . 0000 0 . 8763 0 . 6103 0 . 4823 0 . 6119 0 . 6180 \\ 0 . 7728 0 . 4257 0 . 6080 0 . 7293 0 . 6205 0 . 3333 \\ 0 . 9488 0 . 4473 0 . 6035 0 . 7050 0 . 6186 0 . 3539 \end{matrix}]$

Step 4. Utilize the model (M.2) to establish the following single-objective programming model: ${\begin{matrix} \begin{array}{l} \min ξ (w) = 1.2747 w_{1} + 2.1232 w_{2} + 1.9391 w_{3} \\ + 1.8881 w_{4} + 1.9789 w_{5} + 2.4555 w_{6} \end{array} \\ S u b j e c t t o : w \in H \end{matrix}$

Solve this model, we get the weight vector of attributes: $w = (\begin{matrix} 0.2500, 0.1000, 0.2000, \\ 0.2000, 0.1500, 0.1000 \end{matrix})$

Then, we can get the grey relational degree of each alternative from A⁺ $\begin{matrix} ξ_{1}^{+} = 0.5713, ξ_{2}^{+} = 0.5787, ξ_{3}^{+} = 0.7097 \\ ξ_{4}^{+} = 0.6296, ξ_{5}^{+} = 0.6718 \end{matrix}$

Step 5. According to the relative relational degree, the ranking order of the five emerging technology enterprises is: A₃ > A₅ > A₄ > A₂ > A₁, and thus the most desirable emerging technology enterprise is A₃.

Case 2. If the information about the attribute weights is completely unknown, we utilize another approach developed to get the most desirable emerging technology enterprise(s).

Utilize the Equation (9) to get the weight vector of attributes: $w = {(\begin{matrix} 0 . 2401, 0 . 1251, 0 . 1822, \\ 0 . 1750, 0 . 1746, 0 . 1030 \end{matrix})}^{T}$

Then, we can get the grey relational degree of each alternative from A⁺ $\begin{matrix} ξ_{1}^{+} = 0.5649, ξ_{2}^{+} = 0.5841, ξ_{3}^{+} = 0.7158 \\ ξ_{4}^{+} = 0.6199, ξ_{5}^{+} = 0.6615 \end{matrix}$

According to the relative relational degree, the ranking order of the five emerging technology enterprises is: A₃ > A₅ > A₄ > A₂ > A₁, and thus the most desirable emerging technology enterprises is also A₃.

5. Conclusion

In this paper, we investigate the picture fuzzy multiple attribute decision making problems where the information about attribute weights are partly known or completely unknown. We introduce some notions, such as picture fuzzy ideal point, the normalized Hamming distance of picture fuzzy numbers. We also introduce the grey relational coefficient between the attribute value vectors of each alternative and the picture fuzzy ideal point. Then we establish the grey relational analysis models to measure the grey relational degree between each alternative and the picture fuzzy ideal point. Based on the grey relational analysis models, we can rank the given alternatives and then select the most desirable one. Finally, we illustrate the developed grey relational analysis models with a numerical example for potential evaluation of emerging technology commercialization. In the future, the application of the proposed grey relational analysis models of PFSs needs to be explored in decision making, risk analysis and many other fields under uncertain environment.

Footnotes

Acknowledgment

The work was supported by the National Natural Science Foundation of China under Grant No. U1609218 and 61572286, Shandong Provincial Key Research and Development Plan (No. 2017CXGC1504) and the Fostering Project of Dominant Discipline an Talent Team of Shandong Province Higher Education.

References

Atanassov , Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov , More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989), 37–46.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–356.

Atanassov and

Gargov , Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349.

Atanassov , Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 64(2) (1994), 159–174.

Bustince and

Burillo , Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 74(2) (1995), 237–244.

Atanassov ,

Pasi and

Yager , Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making, International Journal of Systems Science 36(14) (2005), 859–868.

R. R.

Yager , A note on measuring fuzziness for intuition-istic and interval-valued fuzzy sets, Int J General Systems 44(7-8) (2015), 889–901.

Z.S.

Xu , Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15 (2007), 1179–1187.

10.

Z.S.

Xu and

R.R.

Yager , Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems 35 (2006), 417–433.

11.

Wu ,

Wei ,

Gao and

Wei , Some Interval-Valued Intuitionistic Fuzzy Dombi Hamy Mean Operators and Their Application for Evaluating the Elderly Tourism Service Quality in Tourism Destination, Mathematics 6(12) (2018), 294.

12.

Z.S.

Xu , Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems 24 (2011), 749–760.

13.

Zhang and

Xu , Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl Soft Comput 26 (2015), 42–56.

14.

Wu and

Chiclana , A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions, Appl Soft Comput 22 (2014), 272–286.

15.

H.W.

Liu and

G.J.

Wang , Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research 179 (2007), 220–233.

16.

T.Y.

Chen , Bivariate models of optimism and pessimism in multi-criteria decision-making based on intuitionistic fuzzy sets, Information Sciences 181 (2011), 2139–2165.

17.

T.Y.

Chen and

C.H.

Li , Determining objective weights with intuitionistic fuzzy entropy measures: A comparative analysis, Information Sciences 180 (2010), 4207–4222.

18.

T-Y.

Chen , Interval-valued intuitionistic fuzzy QUAL-IFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis, InfSci 261 (2014), 149–169.

19.

T-Y.

Chen , An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis, Appl Soft Comput 42 (2016), 390–409.

20.

S.-M.

Chen and

C-H.

Chiou , Multiattribute Decision Making Based on Interval-Valued Intuitionistic Fuzzy Sets, PSO Techniques, and Evidential Reasoning Methodology, IEEE Trans Fuzzy Systems 23(6) (2015), 1905–1916.

21.

J.-Q.

Wang ,

H.-Y.

Zhang and

X.-H.

Chen , Atanassov’s Interval-Valued Intuitionistic Linguistic Multicriteria Group Decision-Making Method Based on the Trapezium Cloud Model, IEEE Trans Fuzzy Systems 23(3) (2015), 542–554.

22.

Garg , A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems, Appl Soft Comput 38 (2016), 988–999.

23.

D.F.

Li , Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Applied Soft Computing 11 (2011), 3402–3418.

24.

D.-F.

Li and

H.-P.

Ren , Multi-attribute decision making method considering the amount and reliability of intuition-istic fuzzy information, Journal of Intelligent and Fuzzy Systems 28(4) (2015), 1877–1883.

25.

G.W.

Wei ,

Garg ,

Gao and

Wei , Interval-Valued Pythagorean Fuzzy Maclaurin Symmetric Mean Operators in Multiple Attribute Decision Making, IEEE Access 6 (2018), 67866–67884.

26.

S-P.

Wan and

D-F.

Li , Fuzzy mathematical programming approach to heterogeneous multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees, Inf Sci 325 (2015), 484–503.

27.

Li ,

Gao and

Wei , Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators, Symmetry 10(11) (2018), 574.

28.

G.W.

Wei ,

Wei and

Gao , Multiple Attribute Decision Making With Interval-Valued Bipolar Fuzzy Information and Their Application to Emerging Technology Commercialization Evaluation, IEEE Access 6 (2018), 60930–60955.

29.

Li ,

Wei and

Gao , Methods for Multiple Attribute Decision Making with Interval-Valued Pythagorean Fuzzy Information, Mathematics 6(11) (2018), 228.

30.

Li ,

Wei and

Lu , Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection, Symmetry 10(10) (2018), 505.

31.

Wang ,

Wei and

Gao , Approaches to Multiple Attribute Decision Making with Interval-Valued 2-Tuple Linguistic Pythagorean Fuzzy Information, Mathematics 6(10) (2018), 201.

32.

Ye , Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionis-tic fuzzy environment, Expert Systems with Applications 36(2) (2009), 899–6902.

33.

Ye , Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets, Applied Mathematical Modelling 34 (2010), 3864–3870.

34.

Gao , Pythagorean Fuzzy Hamacher Prioritized Aggregation Operators in Multiple Attribute Decision Making, Journal of Intelligent and Fuzzy Systems 35(2) (2018), 2229–2245.

35.

Y.H.

Huang and

G.W.

Wei , TODIM Method for Pythagorean 2-tuple Linguistic Multiple Attribute Decision Making, Journal of Intelligent and Fuzzy Systems 35(1) (2018), 901–915.

36.

J.H.

Park ,

C.K.

Young and

Xue , Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems, Mathematical and Computer Modeling 50 (2009), 1279–1293.

37.

Cuong , Picture fuzzy sets-first results. part 1, in: Seminar “Neuro-Fuzzy Systems with Applications”, Institute of Mathematics, Hanoi, 2013.

38.

Singh , Correlation coefficients for picture fuzzy sets, Journal of Intelligent & Fuzzy Systems 27 (2014), 2857–2868.

39.

Son , DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets, Expert System with Applications 2 (2015), 51–66.

40.

P.H.

Thong and

L.H.

Son , A new approach to multi-variables fuzzy forecasting using picture fuzzy clustering and picture fuzzy rules interpolation method, in: 6th International Conference on Knowledge and Systems Engineering, Hanoi, Vietnam, 2015, pp. 679–690.

41.

N.T.

Thong , HIFCF: An effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis, Expert Systems with Applications 42(7) (2015), 3682–3701.

42.

G.W.

Wei , Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 174 (2016), 491–502.

43.

J.L.

Deng , Introduction to Grey System, The Journal of Grey System (UK) 1(1) (1989), 1–24.

44.

Liu ,

Xie and

Forrest , Novel models of grey relational analysis based on visual angle of similarity and nearness, Grey Systems: T&A 1(1) (2011), 8–18.

45.

Liu ,

Zhou and

Zhao , A new decision support model in multi-criteria decision making with intu-itionistic fuzzy sets based on risk preferences and criteria reduction, JORS 648 (2013), 1205–1220.

46.

Li-Feng

Wu ,

Liu ,

Yao ,

Xu and

Lei , Using fractional order accumulation to reduce errors from inverse accumulated generating operator of grey model, Soft Comput 19(2) (2015), 483–488.

47.

Yan ,

Liu and

Liu , Dynamic grey target decision making method with three-parameter grey numbers, Grey Systems: T&A 6(2) (2016), 169–179.

48.

Mi and

Xia , Prioritizing technical requirements in QFD by integrating grey relational analysis method and analytic network process approach, Grey Systems: T&A 5(1) (2015), 117–126.

49.

J.J.

Zhang ,

D.S.

Wu and

D.L.

Olson , The method of grey related analysis to multiple attribute decision making problems with interval numbers, Mathematical and Computer Modeling 42 (2005), 991–998.

50.

Wei , Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making, Expert Systems with Applications 38(9) (2011), 11671–11677.

51.

Wei , Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Expert Systems with Applications 38(5) (2011), 4824–4828.

52.

Wei , GRA-based linear-programming methodology for multiple attribute group decision making with 2-tuple linguistic assessment information, Information: An International Interdisciplinary Journal 14(4) (2011), 1105–1110.

53.

Wei , A novel efficient approach for interval-valued intu-itionistic fuzzy multiple attribute decision making with incomplete weight information, Information: An International Interdisciplinary Journal 14(1) (2011), 97–102.

54.

Wei , Grey relational analysis model for dynamic hybrid multiple attribute decision making, Knowledge-based Systems 24(5) (2011), 672–679.

55.

G.W.

Wei , GRA method for multiple attribute decision making with incomplete weight information in intuition-istic fuzzy setting, Knowledge-based Systems 23(3) (2010), 243–247.

56.

G.W.

Wei ,

X.F.

Zhao ,

H.J.

Wang and

Lin , GRA Model for Selecting an ERP System in Trapezoidal Intuitionistic Fuzzy Setting, Information: An International Interdisciplinary Journal 13(4) (2010), 1143–1148.

57.

Li and

Wei , GRA method for multiple criteria group decision making with incomplete weight information under hesitant fuzzy setting, Journal ofIntelligent and Fuzzy Systems 27 (2014), 1095–1105.

58.

G.D.

Li ,

Yamaguchi and

Nagai , A grey-based decision-making approach to the supplier selection problem, Mathematical and Computer Modelling 46((3-4)) (2007), 573–581.

59.

Wei ,

H-J.

Wang ,

Lin and

Zhao , Grey Relational Analysis Method for Intuitionistic Fuzzy Multiple Attribute Decision Making with Preference Information on Alternatives, Int J Computational Intelligence Systems 4(2) (2011), 164–173.

60.

P.S.

Park and

S.H.

Kim , Tools for interactive multi-attribute decision making with incompletely identified information, European Journal of Operational Research 98 (1997), 111–123.

61.

S.H.

Kim ,

S.H.

Choi and

J.K.

Kim , An interactive procedure for multiple attribute group decision making with incomplete information: Rangebased approach, European Journal of Operational Research 118 (1999), 139–152.

62.

S.H.

Kim and

B.S.

Ahn , Interactive group decision making procedure under incomplete information, European Journal of Operational Research 116 (1999), 498–507.

63.

P.S.

Park ,

S.H.

Kim and

W.C.

Yoon , Establishing strict dominance between alternatives with special type of incomplete information, European Journal of Operational Research 96 (1996), 398–406.

64.

G.W.

Wei and

Gao , The generalized Dice similarity measures for picture fuzzy sets and their applications, Informatica 29(1) (2018), 107–124.

65.

Wang ,

Gao and

Wei , Methods for MADM with Picture Fuzzy Muirhead Mean Operators and Their Application for Evaluating the Financial Investment Risk, Symmetry 11(1) (2019), 6.

66.

Torra , Aggregation of linguistic labels when semantics is based on antonyms, International Journal of Intelligent Systems 16 (2001), 513–524.

67.

Wang ,

Gao ,

G.W.

Wei and

Wei , Methods for Multiple-Attribute Group Decision Making with q-Rung Interval-Valued Orthopair Fuzzy Information and Their Applications to the Selection of Green Suppliers, Symmetry 11(1) (2019), 56.

68.

Degani and

Bortolan , The problem of linguistic approximation in clinical decision making, International Journal ofApproximate Reasoning 2 (1988), 143–162.

69.

Wu ,

Wang ,

Wei and

Wei , Research on Construction Engineering Project Risk Assessment with Some 2-Tuple Linguistic Neutrosophic Hamy Mean Operators, Sustainability 10(5) (2018), 1536.

70.

G.W.

Wei ,

F.E.

Alsaadi ,

Hayat and

Alsaedi , Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics 9(4) (2018), 713–719.

71.

G.W.

Wei , TODIM method for picture fuzzy multiple attribute decision making, Informatica 29(3) (2018), 555–566.

72.

X.M.

Deng ,

G.W.

Wei ,

Gao and

Wang , Models for safety assessment of construction project with some 2-tuple linguistic Pythagorean fuzzy Bonferroni mean operators, IEEE Access 6 (2018), 52105–52137.

73.

B.S.

Ahn , The uncertain OWA aggregation with weighting functions having a constant level of orness, International Journal of Intelligent Systems 21 (2006), 469–483.