An aggregating method to big group decision-making problem for the public participation problem under the Chinese situation 1

Abstract

A big group decision making problem is investigated, under the intuitionistic fuzzy environment and multiple constrains. Firstly, this paper proposes a distance formula between different intuitionistic fuzzy numbers, to measure their dissimilarity degree. A numerical example is introduced to demonstrate the advantages of the proposed distance measure over the others’ formula. Secondly, an optimizing model is constructed to calculate the criteria’s weight values, making the proposed method suitable for weight unknown problems. Thirdly, clustering idea is introduced to handle big data caused by big decision group. Here, a clustering algorithm is given which could classify the participating people. Meanwhile, computer experiments is utilized to handle the calculation question with respect to big data. Clustering result is based on many times of evolutions, which are obtained by a computer procedure. To derive final ranking results, an extended TOPSIS method is applied depending on the proposed distance measure and clustering results. In summary, a decision making algorithm is clearly shown in form of flow chart. Finally, an experimental analysis for selecting proper library construction is given to illustrate the efficiency and reasonableness of the proposed method.

Keywords

Big group decision making Intuitionistic fuzzy number Clustering analysis TOPSIS method public participation

1. Introduction

Constructing public utilities, a fundamental business has great significance to the country and people. Such as the construction of Public Culture and Sports Facilities, the construction of Compulsory Education Facilities, the construction about daily life business involving water, electricity, gas and others, all of these are closely related to social development and people’s life. In China, most public utilities are supported by the government. And almost all people have a relationship with the public construction project. How the government’s investment in public projects can obtain the max benefit and satisfaction is a meaningful issue. One effective means of informing public demand is that give the public an opportunity to participate in the public project’s decision-making process. As G. P. Whitator [2] said public participation should be considered for public service projects, especially those directly serving the general public. The new public management theory [3] proposed transformation of the western government. This theory compared the public to a client and compared the government to a beneficiary. The education, medical, cultural and other services for public can think of as doctor for the baby delivery. T. Bovaird [4] had clearly pointed out that the management of traditional public services, which did not consider the cooperative relationship between multiple stakeholders, would not meet the need of area. T. Bovaird built a conceptual framework for collaborative production, including users and service society, and listed a number of cases about increasing cooperation and improving local services level. In the process of studying the disposal of abandoned nuclear power plants, A. Bond [5] gave the reasons for influencing public participation, including whether public attitudes are valued and adopted, the public understanding degree of the project’s information, the openness degree of the project information given by government, and whether the government’s power aroused transparently, and so on. M. Doelle and A. John Sinclair [6] pointed out that the public should not only participate in the whole process of environmental assessment, but also should participate in early stages of evaluation. According to research on the repair of rivers, B. Junker [7] said that the public scope should be extended to groups who were not be directly affected by the project, considering the public projects’ long term effect. Result from R.A. Irvnr and J. Stansbury [8] showed it was valuable for the government to introduce the views of the public during decision-making process.

Data showed that from 2012 to 2014, Chinese government’s investment in public construction projects had an increasing trend year by year. China pay attention to the construction of public projects and the urgency of public participation. Many Chinese Scholars have studied the public constructing problem considering the public participation. Allowing the public to participate more directly would improve democratic construction from a micro perspective. That will lay the foundation for improving the level of macro-democratic system. Anning Refining Event in Yunnan province of China is an example. Initially the government adopted a closed-door policy about Anning Refining Event, which meant the public could not get information from the government. After the event being exposed to the public, people pay great attention to the project considering the protection of their own interests. With the interaction between the government and the public becoming deeper, opposition and questioning sound were relatively small. Finally, public changed to a relative understanding attitude to the government. Y. Wang [18] said although the introduction of public participation would bring some pressure on the government’s decision-making capacity, but it would also increase the ability of decision-makers to cope with new challenges. Degree of public participation in public projects is not high enough. The development of the relevant mechanism is not mature yet, and unified operational processes have not been formed. Meanwhile, the public’s desire to join the public project decision-making process has become increasingly strong. It is necessary to consider the introduction of public participation in the decision-making process of public projects, lowering the administrative burden of policy implementation and reducing risk conflicts [31].

After stating the importance of introducing the public, the next part will focus on the main problems about this decision making process. Public-participation faces two of the most important issues: one is how to aggregate the public’s attitudes; the other is how to process the initial decision-making data effectively and retain the public’s preference information as much as possible. Most of the existing studies illustrating the advantages of public-participation are from a political and theoretical point of view. Studies on the specific methods which could deal with public’s decision-making information are relatively few. Public projects are typical multi-criteria decision making (MCDM) problem. Generally speaking, public projects are mostly big and hard to determine immediately. Fuzzy set theory’s development is an advantage to the complex decision making problem. In recent years, multi-criteria decision-making methods based on fuzzy theory have been studied and widely used ([9 , 32]). Peng et al. [45] studied the coal mine safety evaluating problem depending on linguistic intuitionistic MCDM method. Wang et al. [45] pay attention to the risk evaluation of construaction project which is also meaningful to public project. Tian et al. [46] did research on failure modes and their effects, by introducing a fuzzy best-worst method. Ji et al. [48] solved the hotel location selection problem under multi-hesitant fuzzy linguistic environment.

Fuzzy set theory can be more comprehensive and closer to the mentality of decision makers. As the public’s cognitive level is quite different, they may be troubled by some reality problems, such as complex background, pressure from inside and outside, making it difficult for decision makers to give precise decisions. So a more practical description method is needed to collect the public’s attitude under the fuzzy environment. Meanwhile, most current papers have a common problem that they are not suit for the big group decision making problem. Even though pick up some representatives, it is still a big number.

To make up this limitation, this paper introduces the clustering idea. As the public utilities’ constructing being complex, intuitionistic fuzzy number (IFN, [20]) which can describe the decision maker’s preference degree, non-preference degree and hesitant degree. Many researches have studied and applied the intuitionistic fuzzy set to deal with MCDM problem ([21 , 25]). Clustering method is based on dissimilarity. Different distance formulas were proposed based on intuitionistic fuzzy set ([10 , 33]). Owning to the logical ability, TOPSIS has been widely used to handle the MCDM problems. Many have studied TOPSIS method under the IFS’s environment to rank the alternatives. T. Wang et al. [11] extended an OWA-TOPSIS mehtod under intuitionistic fuzzy environment for MCDM problems. D. Jo Shi and S. Kumar [13] also discussed an intuitionistic fuzzy TOPSIS method depending on distance measure and intuitionistic fuzzy entropy. Distance formula between any two IFNs, which are used to measure the similarity degree, is an important tool for TOPSIS mehtod. Great attention has been pay to construct distance measures. The Hamming distance, the Euclidean distance and their normalized formulas, were defined by E. Szmidt and J. Kacprzyk [26] firstly in 2000. Many researchers gave some different distance measures. D. Feng and C. Cheng [14] gave similarity measure. H.B. Mitchell [15] proposed D.Feng’s [14] similarity measure may calculate counter-intuitive results sometimes. P. Grzegorzewski [16] also introduce some distance measures. T. Chen [17] pointed out that P. Grzegorzewski’s [16] inequalities of Euclidean distance and normalized one under IFS environment are in-valid. F. Shen et al. [10] proposed researches from [14 , 22] did not satisfy the properties of distance measures sometimes. J. Ye [22] and Zhang et al. [24] gave distance measure. Chen et al. [23] proposed that Ye’s and Zhang’s distance measure may face the question of divided by zero. Chen et al. [12] pointed out that these distance measures may obtain unreasonable results in a certain situations. F. Shen et al. [10] proposed that conclusions from Wang et al. [11] and Jo Shi Kumar [13] have their own drawbacks during the real decision making process. While, distance formula is core to the TOPSIS method, determining the rationality degree of decision results. F. Shen et al. [28] pointed out that Chen’s [27] distance formula maybe invalid by a number example. Researches on big group decision making problems are from three points. Firstly, some papers focused on improving algorithms efficiency and practicability ([49 , 51–53]). In order to ensure the decision quality, Xu et.al. [51] raised a point that establish conflict large-group emergency decision-making model while protecting minority opinions. Secondly, as the describing tool is important to express decision-makers attitude, some authors pay attention to this part by introducing different fuzzy numbers ([47 , 55]). Three kinds of uncertain preference, thus the interval number, triangular fuzzy number and linguistic value, are transformed into contact numbers for large group decision making problems [53]. Thirdly, some scholars combined different theories to deal with big group decision making problems ([55 –58]). Li et al. rank the alternatives based on the decision makers except deviation, with linguistic assessment information.

In this paper, we propose a distance measure for IFNs, which is also used to construct criteria’s weight calculating model and support TOPSIS method. The proposed distance formula is simpler than the one proposed by F. Shen et al. [28], but it could also overcome the drawbacks of reference [26] and [27]. We also prove some properties of it. With the development of computer technology, there exist many softwares to reduce the computational burden and increase the ranking results’ accuracy degree. Multi-agent based modeling method is used to simulate the agents behavior and obtain the revolution results. Considering big group decision making is inevitable to face the big data’s calculation problem. Each participant can be treated as an agent, obtaining evolutions results by computer experiments.

The rest of this paper is organized as follows. In section 2, we present some basic definitions and methods needed. Section 3 provides a new distance formula between IFNs. Analysis of large group clustering method is discussed in Section 4. In Section 5, based on TOPSIS method, a large Group Decision Making Method is given under Intuitionistic Fuzzy environment. Algorithm of the proposed method is summarized in Section 6, along with a flow chart and detailed description. In Section 7, a numerical example is applied to illustrate the effectiveness and reasonableness of the proposed algorithm. Finally, conclusions and future researches are shown in Section 8.

2. Preliminaries

As the proposed method is under the environment of intuitionistic fuzzy sets, some basic definitions about IFN and decision matrix are given.

Atanassov [20] introduced the concept of Intuitionistic Fuzzy Set(IFS), as an extension of Zadeh’s Fuzzy Set.

Definition 1. [20] Let X = {x₁, x₂,. . . , x_n} be a set of n alternatives. The intuitionistic fuzzy set (IFS) is defined as U = {〈x, u_U (x) , v_U (x) 〉|x ∈ X}, where 0 ≤ u_U (x) + v_U (x) ≤1, 0 ≤ u_U (x) , v_U (x) ≤1, and u_U (x) means the membership degree of x to X, v_U (x) means the non-membership degree of x to X. And calculate the hesitant degree of x to X by π = 1 - u_U (x) - v_U (x). The intuitionistic fuzzy number (IFN) is written by (u_U (x) , v_U (x)), without confusion (u, v) for simple.

Wang and Xin [34] gave a distance measure and its properties.

Definition 2. [34] Let d be a mapping d : IFSs (X) × IFSs (X) → [0, 1]. d (A, B) is said to be a distance measure between A ∈ IFSs (X) and B ∈ IFSs (X), if d (A, B) satisfies the next:

(1) 0 ≤ d (A, B) ≤1;

(2) d (A, B) =0 iff A = B;

(3) d (A, B) = d (B, A);

(4) if A ⊆ B ⊆ C, A, B, C ∈ IFSs (X), then d (A, C) ≥ d (A, B) and d (A, C) ≥ d (B, C).

Many different forms of distance have been proposed, such as Sumidt and Kacprzyk [26] introduced the normalized Euclidean distance for any two IFSs.

In the theoretical research and practical application of multi-criteria decision-making, it is often encountered how to calculate the difference between different fuzzy numbers. Referencing to X. Xu and X. Chen’s [29] research, this paper introduces several calculation formulas, as the basement to build a decision-making model.

Suppose α and β be any two IFNs, which are defined in sets A and B. Let A = {α₁, α₂, ⋯ , α_n}, B = {β₁, β₂, ⋯ , β_n}, α = (μ_α (x_i) , ν_α (x_i)), β = (μ_β (x_i) , ν_β (x_i)), α ∈ A, β ∈ B, A, B ⊂ X, X = {x₁, x₂, ⋯ , x_n}:

Intuitionistic fuzzy weighted distance degenerates to weighted Hamming distance $\begin{matrix} I F W D (A, B) = & \sum_{i = 1}^{n} w_{i} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |) \end{matrix}$ (1)

Intuitionistic fuzzy weighted distance degenerates to standard weighted Hamming distance when $W = {\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n}}$ : $\begin{matrix} I F W D (A, B) = & [\sum_{i = 1}^{n} \frac{1}{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |)^{λ}]^{1 / λ} \end{matrix}$ (2)

Intuitionistic fuzzy weighted distance degenerates to weighted Euclidean distance when λ = 2: $\begin{matrix} I F W D (A, B) = & [\sum_{i = 1}^{n} w_{i} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |)^{2}]^{1 / 2} \end{matrix}$ (3)

Then get normal weighted Euclidean distance when $W = {\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n}}$ , λ = 2: $\begin{matrix} I F W D (A, B) = & [\sum_{i = 1}^{n} \frac{1}{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |)^{2}]^{1 / 2} \end{matrix}$ (4)

E. Szmidt and J. Kacprzyk [26] thought it was necessary to consider the hesitant degree π (x) =1 - μ (x) - ν (x) and gave the following equations.

Hamming distance h (A, B): $\begin{matrix} \tilde{h} (A, B) = \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | π_{A} (x_{i}) - π_{B} (x_{i}) |) \end{matrix}$ (5)

Normal Hamming distance Nh (A, B): $\begin{matrix} \tilde{N h} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | π_{A} (x_{i}) - π_{B} (x_{i}) |) \end{matrix}$ (6)

Intuitionistic fuzzy weighted distance Nh (A, B): $\begin{matrix} \tilde{I F W D} (A, B) = [\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | π_{A} (x_{i}) - π_{B} (x_{i}) |)^{λ}]^{1 / λ} \end{matrix}$ (7) where w_i ∈ W, W = {w₁, w₂, ⋯ , w_n} is the weighted vector for criteria during the public partition progress.

Example 1. There are two known patterns represented by the IFNs P₁ and P₂ in the universe of discourse X = {x₁, x₂, x₃}, respectively, where

P₁ = (〈x₁, 0.4, 0.5〉, 〈x₂, 0.6, 0〉, 〈x₃, 0.1, 0.5〉),

P₂ = (〈x₁, 0.5, 0〉, 〈x₂, 0.5, 0.2〉, 〈x₃, 0.4, 0.5〉).

We want to classify an unknown pattern represent by an IFN into one of the patterns, where

Q = (〈x₁, 0.3, 0.3〉, 〈x₂, 0.7, 0.1〉, 〈x₃, 0.2, 0.6〉).

For convenience, let the weight of element x_i, $w_{i} = \frac{1}{3}$ where i = 1, 2, 3. Applying the traditional distance measure (7), obtain $\tilde{IFWD} (P_{1}, Q) = 0.4667$ , $\tilde{IFWD} (P_{2}, Q) = 0.4667$ , then we would not know how to divide the pattern. In the following, we will introduce a novel distance measure to cope this kind of situations.

3. A novel type of distance formula between IFNs

Chen and Tan [42] proposed a score function S (a) = μ_a - ν_a to calculate the score value of intuitionistic fuzzy number a = (μ_a, ν_a), where -1 ≤ S (a) ≤1. Hong et al. [43] pointed out that the scoring function alone cannot achieve the purpose of comparing different intuitive fuzzy numbers. That is to say, the score function will fail. For example, a₁ = (0.75, 0.15) and a₂ = (0.68, 0.08), their score values are the same S (a₁) = S (a₂) =0.60. However, it is clear that the evaluation results of these two evaluations are different. Hong et al. [43] gave the definition of the accuracy function of intuitionistic fuzzy numbers to differ them. For an IFN a = (μ_a, ν_a), H (a) = μ_a + ν_a is its accuracy function. Then obtain the hesitant degree by π (a) =1 - (μ_a + ν_a) =1 - H (a). Calculate the accuracy values H (a₁) =0.90 and H (a₂) =0.76, obtaining a₁ > a₂, that means a₁ is better. In summary, Z. Xu [35] gave a comparison method for any two different intuitionistic fuzzy numbers a₁ and a₂.

If S (a₁) < S (a₂), then a₁ < a₂ that means a₁ is small than a₂;

If S (a₁) > S (a₂), then a₁ > a₂ that means a₁ is bigger than a₂;

If S (a₁) = S (a₂): and if H (a₁) < H (a₂), then a₁ < a₂ that means a₁ is small than a₂; and if H (a₁) > H (a₂), then a₁ > a₂ that means a₁ is bigger than a₂; and if H (a₁) = H (a₂), then a₁ = a₂ that means a₁ is equal to a₂.

According to the meaning of accuracy function H (a_i) = μ_{a
_i} + ν_{a
_i} and score function S (a_i) = μ_{a
_i} - ν_{a
_i}, their effects on the distance between intuitionistic ambiguities should be discussed.

$\begin{matrix} | H_{A} (x_{i}) - H_{B} (x_{i}) | \\ = & | (1 - μ_{A} (x_{i}) - ν_{A} (x_{i})) - (1 - μ_{B} (x_{i}) - ν_{B} (x_{i})) | \\ = & | (1 - H_{A} (x_{i})) - (1 - H_{B} (x_{i})) | \\ = & | π_{A} (x_{i}) - π_{B} (x_{i}) | \end{matrix}$ (8)

$\begin{matrix} | S_{A} (x_{i}) - S_{B} (x_{i}) | \\ = & | (μ_{A} (x_{i}) - ν_{A} (x_{i})) - (μ_{B} (x_{i}) - ν_{B} (x_{i})) | \\ = & | μ_{A} (x_{i}) - ν_{A} (x_{i}) - μ_{B} (x_{i}) + ν_{B} (x_{i}) | \\ = & | μ_{A} (x_{i}) - μ_{B} (x_{i}) + ν_{B} (x_{i}) - ν_{A} (x_{i}) | \\ \leq & | μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{B} (x_{i}) - ν_{A} (x_{i}) | \end{matrix}$ (9)

A novel distance measure of any two INFs from sets A and B:

$\begin{matrix} \bar{h} (A (x_{i}), B (x_{i})) \\ = | μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) | \end{matrix}$ (10)

Hamming distance $\bar{h} (A, B)$ :

$\begin{array}{l} \bar{h} (A, B) \\ = & \sum_{i = 1}^{i} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) |) \end{array}$ (11)

Normal Hamming distance $\bar{Nh} (A, B)$ : $\begin{array}{l} \bar{N h} (A, B) \\ = & \frac{1}{n} \sum_{n}^{i = 1} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) |) \end{array}$ (12)

Table 1

Comparison results for different distance measures

Method	D₁ (P₁, Q)	D₂ (P₂, Q)	Comparison	Classing result
Reference [26]	0.1278	0.1278	D₁ (P₁, Q) = D₂ (P₂, Q)	Q could not be determined
Reference [27]	0.2796	0.2354	D₁ (P₁, Q) ≥ D₂ (P₂, Q)	Q should be classified into team P₂
This paper’s	1.3750	1.3000	D₁ (P₁, Q) ≥ D₂ (P₂, Q)	Q should be classified into team P₂

Intuitionistic fuzzy weighted distance Nh (A, B): $\begin{array}{l} N h (A, B) \\ = & [\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) |)^{λ}]^{1 / λ} \end{array}$ (13) Some properties about this distance measure will be given below.

Theorem 3. Let A, B and C be any three intuitionistic fuzzy sets. α, β and γ are three IFNs from A, B and C respectively. $\bar{h} (α, β)$ is the proposed distance measure, which satisfies the following properties:

(1) $\bar{h} (α, β)$ is boundary;

(2) $\bar{h} (α, β) = 0$ iff α = β;

(3) $\bar{h} (α, β) = \bar{h} (β, α)$ ;

Proof. (1) $\bar{h} (α, β) = | μ_{A} (α) - μ_{B} (β) | + | ν_{A} (α) - ν_{B} (β) | + | π_{A} (α) - π_{B} (β) | + | S_{A} (α) - S_{B} (β) | .$ Considering the boundaries of intuitionistic fuzzy numbers and their elements, get 0 ≤ μ_A (α) , ν_A (α) , π_A (α) ≤1, S_A (α) = μ_A (α) - ν_A (α), S_B (α) = μ_B (α) - ν_B (α), then -1 ≤ S_A (α) ≤1, -1 ≤ S_B (β) ≤1, 0 ≤ |S_A (α) - S_B (β) |≤2; along with 0 ≤ |μ_A (α) - μ_B (β) |≤1, 0 ≤ |ν_A (α) - ν_B (β) |≤1, 0 ≤ |π_A (α) - π_B (β) |≤1, obtain $0 \leq \bar{h} (α, β) \leq 5$ .

Especially, let $\tilde{h} (α, β) = \frac{1}{4} (| μ_{A} (α) - μ_{B} (β) | + | ν_{A} (α) - ν_{B} (β) | + | π_{A} (α) - π_{B} (β) | + \frac{1}{2} | S_{A} (α) - S_{B} (β) |)$ then $0 \leq | \tilde{h} (α, β) | \leq 1$ .

(2) If $\bar{h} (α, β) = 0$ , then consider the definition of $\bar{h} (α, β)$ and ${\begin{matrix} | μ_{A} (α) - μ_{B} (β) | \geq 0, \\ | ν_{A} (α) - ν_{B} (β) | \geq 0, \\ | π_{A} (α) - π_{B} (β) | \geq 0, \\ | S_{A} (α) - S_{B} (β) | \geq 0. \end{matrix}$ (14)

and get ${\begin{array}{l} | μ_{A} (α) - μ_{B} (β) | = 0, \\ | ν_{A} (α) - ν_{B} (β) | = 0, \\ | π_{A} (α) - π_{B} (β) | = 0, \\ | S_{A} (α) - S_{B} (β) | = 0. \end{array}$ (15) that means μ_A (α) = μ_B (β), ν_A (α) = ν_B (β), π_A (α) = π_B (β), i.e., α = β.

(3) $\bar{h} (α, β) = \bar{h} (β, α)$ , that is obvious.

Equations (5)(6)(7) is proposed by reference [25], which may exist powerless conditions. Such as shown in Example 1, these functions cannot class P₁ and P₂. Equation (16) is an extension of Equation (7), and not able to deal with Example 2. The following Table 1 could show this paper’s class results are the same with different methods.

Example 2. There are two patterns P₁ and P₂ represented by IFNs $\tilde{P_{1}}$ and $\tilde{P_{2}}$ in the universe of X = {x₁, x₂, x₃},

$\tilde{P_{1}} = {〈 x_{1}, 0.6, 0.25 〉, 〈 x_{2}, 0, 0.25 〉, 〈 x_{3}, 0.3, 0.25 〉}$ ;

$\tilde{P_{2}} = {〈 x_{1}, 0.1, 0.75 〉, 〈 x_{2}, 0.15, 0.1 〉, 〈 x_{3}, 0.2, 0.35 〉}$ .

Here is an unknown pattern represented by an IFS $\tilde{Q}$ to one of $\tilde{P_{1}}$ and $\tilde{P_{2}}$ , where

$\tilde{Q} = {〈 x_{1}, 0, 0.15 〉, 〈 x_{2}, 0.325, 0.425 〉, 〈 x_{3}, 0.2, 0.25 〉}$ .

4. Analysis of large group clustering method

4.1. Large population clustering under vector space

X. Xu and X. Chen [36] proposed the definition of preference vector in European space, and gave the definition of group decision matrix for multi-objective and multi-criteria based on intuitionistic fuzzy numbers. In n-dimensional Euclidean space Eⁿ, there are T decision-making members form a group Ω. Suppose there are s alternatives, m criteria, then the decision vector $V^{k} = (v_{ij}^{k})_{1 \times m}$ represents the k-th decision maker’s evaluation result for i-th alternative with all criteria. When the decision maker needs to give the evaluation results of all the alternatives, a decision matrix for the decision maker can be obtained. Let the result of the decision maker’s decision use the intuitionistic fuzzy number, making the initial decision matrix.

X. Xu and X. Chen [36] pointed out the three preconditions for decision-making members in the group: Firstly, all decision-makers were evaluated in the same way of thinking; Secondly, the thinking between the decision-making members of the differences, can not be classified as a group of the same class; Thirdly, there are subgroups that gather within the group. On the one hand, the first two cases of research value are low; On the other hand, when the public participation in the decision-making or evaluation of public projects, the public is a typical large group, so the third model is very easy to happen. For this model, suppose there are t subgroups within the group Ω, where the k - th subgroup φ_k has n_k decision makers satisfying $\sum_{k = 1}^{t} n_{k} = T$ and 1 ≤ t ≤ T. Reference to L. Hu and G. Luo’s [37] description method, X. Xu and other articles insteaded relevance by similarity. Here, we introduce the threshold value γ(0 ≤ γ ≤ 1) for clustering the decision members of group Ω. Considering all of the same evaluation criteria, when similarity degree between decision makers is greater than the threshold, they can be classified to the same class. This paper gives the definitions in the following way.

Definition 4. For any two matrix u and v from m × n dimensional linear space, whose similarity degree is defined by $ρ (u, v) = \frac{| u - \bar{u} | \cdot | v - \bar{v} |^{T}}{∥ u - \bar{u} ∥_{2} \cdot ∥ v - \bar{v} ∥_{2}}$ (16) where $\bar{v} = (\bar{v_{1}}, \bar{v_{2}}, \dots, \bar{v_{m}})^{T}$ , $\bar{v_{i}} = \frac{1}{n} \sum_{j = 1}^{n} v_{ij}$ is the vector v_ij consisting of vector set’s average vector, so is $\bar{u}$ ; ∥v_i ∥ ₂ is the 2-norm of n-dimensional Euclidean space v_i = (v_i1, v_i2, ⋯ , v_in) and $∥ v ∥_{2} = (\sum_{i = 1}^{m} \sum_{j = 1}^{n} | v_{ij} |^{2})^{\frac{1}{2}}$ , $u \cdot v = \sum_{i = 1}^{m} \sum_{j = 1}^{n} u_{ij} v_{ij}$ .

Definition 5. For matrix v from sets V = {v₁, v₂, ⋯ , v_T}, the similarity degree of v with V is defined by $ρ (v, V) = \frac{1}{T} \sum_{i = 1}^{T} ρ (v, v^{i})$ (17) where v_i ∈ V, 1 < t ≤ T.

The average of the similarities among all the vectors in the vector set $ρ (v, V) = \frac{1}{C_{T}^{2}} \sum_{1 \leq i < j \leq T} ρ (v_{i}, v_{j})$ , where $C_{T}^{2} = \frac{T (T - 1)}{2}$ . Let ρ = 0 when T = 1.

In order to measure the cluster results, we introduce the concept of Deciation Index, which could be obtained according to Equation (18).

Definition 6. Deviation index of clustering result is calculated by the following equation: $DI = \frac{\sum_{i = 1}^{N} (n_{i} - T / N)}{N}$ (18) where N is the cluster number, T is the number of members, n_i is the number of each cluster, 1 < t ≤ T.

4.2. Large population clustering based on computational experiments

Computational experiments are important for simulating people’s social behavior [38]. According to Definition 4, this section will present a clustering method based on similarity. It is necessary to compute the corresponding calculation program according to the clustering step, and finally output the concrete result. X. Xu et al. [39] only measure the difference according to the conceptual formula of consistency. This model also outputs the average consistency and standard consistency of each evolutionary result. This paper will analyses more parameters to reach more accuracy decision result. Average uniformity refers to the average of the internal consistency of each cluster, which is similar to the mean value in statistics. The average level of consistency for each cluster is examined. Its calculation method is the same as the calculation of the mean. Standard conformance refers to the deviation of the internal consistency and average consistency of each cluster. The specific calculation is calculated as the standard deviation. Simulation process includes that the number of clusters, average consistency, and standard consistency of each cluster. Stop the process, when the evolution process is relatively stable. Select the relatively stable time period for the clustering results. The data of the obtained results are analyzed to determine what kind of threshold, and the optimal number of clusters under the threshold has the clustering results in the optimal number of clusters. Members of subgroups do not exceed the total number of members included in the population can be formed in the population. In the specific clustering process, there always exists a suitable threshold, which decides whether a member can be assigned to this subgroup.

In the process of evaluation of public projects, the similarity between different participating members can be calculated by Method 1. So members with high similarity are divided into the same subgroup. The Algorithm of the clustering method are also summarised in flow chart as shown in Fig. 1

Fig. 1

Algorithm of the proposed Method 1.

Method 1

Step 1. Determine the group Ω and mark the members of it in order 1 ∼ T, whose evaluation results make a new set Ψ. Let Q be a temporary set whose initial setting is empty.

Step 2. According to the reality, determine the threshold value γ, 1 ≤ γ ≤ 1.

Step 3. Let the initial number of cluster k = 1. Chose the vector with order i = 1 from set Ψ.

Step 4. Choose the vector Vⁱ and put it to the cluster Ω_k, then the number of its member n_k = 1.

Step 5. If the set Ψ≠ ∅, then select the next vector Vⁱ in order, i = i + 1; If the set Ψ =∅, turn to Step 7.

Step 6. Calculate the similarity degree ρ (Vⁱ, Ω_k) of vector Vⁱ and vectors from cluster Ω_k. If ρ (Vⁱ, Ω_k) > γ, then assign this alternative to the cluster Ω_k; If ρ (Vⁱ, Ω_k) ≤ γ, then assign it to the temporary set Q and move the set Vⁱ out of Ψ. Turn to Step 5.

Step 7. If Q≠ ∅, let Ψ = Q; If Q =∅, the count of the cluster k = k + 1, and turn to Step 4.

Step 8. Output clustering results, the calculation is completed.

5. Large Group Decision Making Method Based on TOPSIS under Intuitionistic Fuzzy environment

5.1. An extended method to describe the evaluating results of public participating problem

The IFN contains three kinds of information from decision maker, the satisfactory degree, unsatisfactory degree and hesitant degree, which has a great applicability when dealing with problems with strong ambiguity and uncertainty. In general, decision makers evaluate the alternatives depending on several different criteria. For example, the government invested in the construction of a public project. In public project decision making process, the public should give their results through different evaluation criteria. Suppose there are m alternatives making a set X = {x₁, x₂,. . . , x_m} for the public to choose. The government choose some experts from the expert library arbitrarily. After a comprehensive consideration, they gave the evaluation criteria, summarized to the criteria set C = {C₁, C₂,. . . , C_n}. Considering all the evaluation criteria and alternatives, the initial decision matrix are shown in Equation (19). $\bar{A} = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{mn} \end{matrix}]$ (19) where a_ij = (μ, ν) is an IFN standing for the evaluation results given by decision makers with the j-th evaluation criteria for the i-th alternative, μ standing for the satisfactory degree and ν standing for the unsatisfactory degree.

For a certain public project, there are m programs after the initial screening. In order to take the public’s satisfaction degree into consideration, the government selected T publics in anonymous form to participate in the decision-making process randomly. The public will consider n evaluation criteria and give evaluation values to express their own preferences. Under criteria C_j, the t-th public give his decision results by $b_{ij}^{t} = (u_{b_{ij}^{t}}, v_{b_{ij}^{t}})$ for alternative X_t, $u_{b_{ij}^{t}}$ standing for satisfactory degree of the public project, $v_{b_{ij}^{t}}$ standing for the unsatisfactory degree of the public project, where $u_{b_{ij}^{t}}$ and $v_{b_{ij}^{t}}$ belong to point 0 to 100. 100 point is the highest score which means full satisfaction of the maximum, and 0 point is the lowest score which means the minimum satisfaction degree. If there only collect one value $u_{b_{ij}^{t}}$ or $v_{b_{ij}^{t}}$ , get the other one by calculation. If the sum of $u_{b_{ij}^{t}}$ and $v_{b_{ij}^{t}}$ is bigger than 100, then this decision result should be deleted. For the obtained data $b_{ij}^{t} = (u_{b_{ij}^{t}}, v_{b_{ij}^{t}})$ , let $μ_{b_{ij}^{t}} = u_{b_{ij}^{t}} / 100$ or $ν_{b_{ij}^{t}} = v_{b_{ij}^{t}} / 100$ , get the intuitionistic fuzzy number $a_{b_{ij}^{t}} = (μ_{b_{ij}^{t}}, ν_{b_{ij}^{t}})$ , which satisfies the definition’s condition. Then obtain the initial decision matrix in form of matrix (19).

Public are selected according to the specific public project within the scope of the random selection of the relevant public. There need to pay attention to the choice of public representatives in the scope to cover different ages, education, work, etc.. Because the public’s levels are different with each other, the initial decision-making results will inevitably have a variety of problems about information aggregating. Therefore, it is needed to pre-process the initial decision-making results of the public, removing the unqualified evaluation information. In the public decision-making process, adhering to the principle of people-oriented, participates in the public will obtain equal weight.

5.2. Determination of the weight of public project evaluation criteria under public participation

Before building the model to calculate criteria’s weight, the symbols used in the model are described as follows:

X = {X₁, X₂, ⋯ , X_m}: Alternatives set;

C = {C₁, C₂, ⋯ , C_n}: Criteria set;

W = {w₁, w₂, ⋯ , w_n}: Weights vector set for criteria;

Ω = {D₁, D₂, ⋯ , D_T}: A collection of all the public groups involved in making decisions. T is the number of all public.

Considering the impact of evaluation criteria on the accuracy of evaluation results, it is weighted according to the public decision information. A higher accuracy rating for an alternative should have a higher weight, and a lower accuracy evaluation criteria for the alternatives should have a lower weight. Under criteria C_j, j = 1, 2, ⋯ , n, decision maker D_t, t = 1, 2, ⋯ , T gives his/her decision result $A^{t} = (a_{ij})_{m \times n}^{t}$ for alternative X_i, i = 1, 2, ⋯ , m, where a_ij is an IFN. According to the definition of score function, the higher value S (a_ij) of an alternative X_i under the evaluation criterion C_j, the better alternative X_i. According to the definition of accuracy function, the higher value H (a_ij) of a certain alternative X_i under the evaluation criterion C_j is higher, the lower hesitant degree and then the better alternative X_i. From the perspective of the accuracy information contained in the evaluation results given by the public, the model can be constructed.

M₁: ${\begin{matrix} \max f_{1} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} w_{j} | \frac{S (a_{ij})}{S (a_{ij}) - H (a_{ij})} |; \\ \sum_{j = 1}^{n} w_{j = 1} = 1; \\ 0 \leq w_{j} \leq 1 . \end{matrix}$ (20)

Next weight the criteria according to their distinguish ability. For the n criteria, if the difference degrees are smaller under a criterion, that means this criterion’s distinguish ability is worse, then should give this criteria a smaller weight value; if the difference degree is bigger under a criteria, that means this criterion’s distinguish ability is better, then should give this criteria a bigger weight. Based on the Equation (10), give the following weighted Euclidean distance, i.e. λ = 2. $\begin{array}{l} d_{N h} (A, B) \\ = [\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | + | S_{A} (x_{i}) - S_{B} (x_{i}) |)^{2}]^{1 / 2} \end{array}$ (21)

Calculate the difference degree between any two IFNs by d_Nh (a_ij, a_kj) and construct the next model:

M₂: ${\begin{array}{l} \max f_{2} (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{1 \leq j < k \leq n} w_{j} d_{N h} (a_{i j}, a_{k j}); \\ \sum_{j = 1}^{n} w_{j} = 1; \\ 0 \leq w_{j} \leq 1. \end{array}$ (22)

Considering the impact on the weight distribution from two aspects: the accuracy degree of the evaluation results and the distribution degree of the criteria, introduce the adjustment factor α and construct the following model:

M: ${\begin{matrix} maxf (w) = α f_{1} (w) + (1 - α) f_{2} (w); \\ \sum_{j = 1}^{n} w_{j} = 1; \\ 0 \leq w_{j} \leq 1; \\ 0 \leq α \leq 1 . \end{matrix}$ (23)

α is determined by the decision makers depending on the reality. Next calculate the model M by constructing Lagrange function L (w, λ), where λ is the Lagrange multiplier:

$L (w, λ) = f (w) + λ (\sum_{j = 1}^{n} w_{j} - 1)$ (24)

Differentiating L (w, λ) with respect to w_j and λ. And let their results be 0, then the following equations is obtained:

${\begin{matrix} \frac{\partial L}{\partial w} = 0; \\ \frac{\partial L}{\partial λ} = 0. \end{matrix}$ (25)

Obtaining the normalized value of w_j,

$w_{j} = \frac{\sum_{i = 1}^{m} [α | \frac{S (a_{ij})}{S (a_{ij}) - H (a_{ij})} | + Sum (d_{IFWD})]}{\sum_{j = 1}^{n} \sum_{j = 1}^{m} [α | \frac{S (a_{ij})}{S (a_{ij}) - H (a_{ij})} | + Sum (d_{IFWD})]}$ (26) where Sum (d_IFWD) = ∑_1≤j<k≤n (1 - α) d_IFWD (a_ij, a_kj).

5.3. Mrthodology to intuitionistic fuzzy TOPSIS for Public Participation

The TOPSIS method was first proposed by C.L. Hwang and K. Yoon, originally solving the multi-criteria decision-making problem with only one decision maker [40]. The core idea of TOPSIS is to choose the solution which is the closest to the positive ideal and the farthest from negative ideal. J. Nan et al. [41] extended TOPSIS proposed by C.L. Hwang and K. Yoon [40], applying it to multi-criteria decision making based on intuitionistic fuzzy sets and giving a corresponding multi-criteria decision-making method [41]. However, article such as J. Nan et al. [41] is distinguishable from the types of evaluation criteria. Obviously, there is a difference between the positive and negative ideals of benefit-based standards and cost-based standards. Due to the ease of understanding and operability of the TOPSIS method, it has been favored by many scholars since it was put forward. This study will also use the idea of TOPSIS to rank alternatives of public projects. In order to improve the efficiency of scientific decision-making, this paper will calculate the distance between the decision result and the positive and negative ideal of each alternative depending on the weighted Euclidean distance formula which is proposed in the previous paper. Public participation in public project decision-making is a multi-criteria group decision-making problem. The decision matrix B = (b_ij) _m×n involved in the following steps is obtained after preprocessing the decision data information, where b_ij = (μ_{b
_ij}, ν_{b
_ij}) is IFN, representing the evaluation of alternative X_i under criterion C_j. In this paper, the intuitionistic fuzzy TOPSIS method depending on the proposed distance measure for public participating problem, is detailed as follows.

Method 2

Step 1. Under criterion C_j, determine the positive solution $b_{j}^{+} = (μ_{b_{j}^{+}}, ν_{b_{j}^{+}})$ and negative solution $b_{j}^{-} = (μ_{b_{j}^{-}}, ν_{b_{j}^{-}})$ :

$\begin{matrix} b_{j}^{+} = {\begin{matrix} (max μ_{b_{ij}}, min ν_{b_{ij}}), j \in J_{1}; \\ (max μ_{b_{ij}}, min ν_{b_{ij}}), j \in J_{2} . \end{matrix} \end{matrix}$ (27)

$\begin{matrix} b_{j}^{-} = {\begin{matrix} (min μ_{b_{ij}}, max ν_{b_{ij}}), j \in J_{1}; \\ (max μ_{b_{ij}}, min ν_{b_{ij}}), j \in J_{2} . \end{matrix} \end{matrix}$ (28) where, J₁ represents an benefit evaluation index set, J₂ represents the cost evaluation index set.

Step 2. Calculate the criteria’s weight value w_j based on the Equation (10).

Step 3. Calculate the distance from each solution to the positive ideal solution: $\begin{matrix} d_{i}^{+} = & [\sum_{j = 1}^{n} w_{j} (| μ_{i} (x_{j}) - μ_{b_{j}^{+}} | + | ν_{i} (x_{j}) - ν_{b_{j}^{+}} | \\ + | π_{i} (x_{j}) - π_{d_{j}^{+}} | + | S_{i} (x_{j}) - S_{d_{j}^{+}} |)^{2}]^{\frac{1}{2}} \end{matrix}$ (29)

Calculate the distance from each solution to the negative ideal solution: $\begin{matrix} d_{i}^{-} = & [\sum_{j = 1}^{n} w_{j} (| μ_{i} (x_{j}) - μ_{b_{j}^{-}} | + | ν_{i} (x_{j}) - ν_{b_{j}^{-}} | \\ + | π_{i} (x_{j}) - π_{d_{j}^{-}} | + | S_{i} (x_{j}) - S_{d_{j}^{-}} |)^{2}]^{\frac{1}{2}} \end{matrix}$ (30)

Step 4. Calculate the ratio of distance from the negative idea to the sum: $R_{i} = \frac{d_{j}^{-}}{d_{j}^{+} + d_{j}^{-}}$ (31) where i = 1, 2, ⋯ , T, T standing for the number of alternatives.

Depending on the meanings of distances $d_{j}^{+}$ and $d_{j}^{-}$ , the bigger value of R_i, the better alternative X_i; the smaller value of R_i, the worse alternative X_i. Therefore, we can rank public project programs and select the best one.

6. Algorithm of the proposed method for public participating problem

Summary the above analysis, develop a method for solving the big group decision making problem, especially for the public projects with public participating. The algorithm is detailed in the following stages: firstly, select the participating public members and determine the criteria set and alternative; secondly, obtain the decision making result and aggregative the information; lastly, obtain the ranking result and make the judgements. Combing Fig. 2 gives the algorithm of the proposed method.

Fig. 2

Algorithm of the proposed method.

Step 1. Relevant government departments and the expert group determine the initial set of alternative X; evaluation criteria system C; to meet the statistical laws, under the premise of selecting T to participate in public decision-making process of the public.

Step 2. The public groups that participate the public projects are given the initial decision-making results according to their true attitude. After the rational screening, the initial decision-making matrix are A_t, t = 1, 2, ⋯ , T.

Step 3. Based on the similarity of large group clustering steps, apply Method 1 the initial decision data are clustered.

Step 4. Based on the evolutions’ result, choose the best threshold value. Calculate the satisfactory degree and unsatisfactory degree of each cluster under each criterion, where each participant’s weight is equal.

Step 5. According to the decision data after clustering in Step 4, the weighted criteria are used to calculate the evaluation standard weight, and the normal weight of the criteria is obtained by using Equation (26).

Step 6. Using the method given above, the positive and negative ideal solutions of the evaluation results of the alternatives are found by Equation (27) and Equation (28), and the evaluation results of the alternatives are calculated by using Equation (29) and Equation (30). To calculate the positive and negative ideal solution distance, optimally use Equation (31).

Step 7. Output the final sorting result and make a summary analysis of the decision-making process.

Step 8. Concluded.

7. A Numerical example

Public utilities are important to the whole society, which concern almost all public’s everyday life. In 2017, October 18th, the 19th National Congress of the Communist Party of China was taken place. It is a meeting of great importance during the decisive stage in building a moderately prosperous society in all respects and at a critical moment as socialism with Chinese characteristics has entered a new era. This congress pointed out that, public cultural services have been improved, art and literature are thriving, and cultural programs and industries are going strong. Library is a main utility for public’s daily life about public culture. This section considers an illustrative example of library constructing project choosing problem. After a preliminary investigation, 5 library-constructing plans {X₁, X₂, …, X₅} are determined as alternatives. Based on the experts discussion, offer 5 criteria for public to evaluate the alternatives. Applying the proposed method, obtain the raking results in the following steps.

Step 1. Let X = {X₁, X₂, …, X₅} be an alternative set with 5 alternatives; T = {T₁, T₂, …, T₁₀₀} be a decision maker set with 100 members; C = {C₁, C₂, …, C₅} be a criteria set with 5 criteria.

Step 2. Each decision maker will evaluate every alternative under each criterion by the extended IFN description method described in Section 5.1. Then get 100 decision matrixes and show one in Table 2 as example.

Table 2
The decision result by T₁

T ₁ C ₁ C ₂ C ₃ C ₄ C ₅

X ₁ (85,10) (65,35) (75,20) (80,10) (45,40)

X ₂ (80,16) (75,15) (80,15) (85,14) (70,20)

X ₃ (70,20) (70,20) (85,10) (76,16) (84,10)

X ₄ (95,0) (80,10) (80,13) (84,13) (74,15)

X ₅ (60,25) (65,25) (65,23) (62,36) (68,20)

T ₁	C ₁	C ₂	C ₃	C ₄	C ₅
X ₁	(85,10)	(65,35)	(75,20)	(80,10)	(45,40)
X ₂	(80,16)	(75,15)	(80,15)	(85,14)	(70,20)
X ₃	(70,20)	(70,20)	(85,10)	(76,16)	(84,10)
X ₄	(95,0)	(80,10)	(80,13)	(84,13)	(74,15)
X ₅	(60,25)	(65,25)	(65,23)	(62,36)	(68,20)

Fig. 3

Deviation index of different thresholds.

Fig. 4

Numbers of cluster under different threshold values.

Step 3. Applying Method 1 can obtain the cluster results, which show that under different threshold values, the numbers of clusters, average-consensus degree and standard-consensus degree are different. After analysing all results, find the cluster number and consensus degree are the best when γ ∈ [0.71, 0.85].

Consider all of the cluster numbers when γ ∈ [0.70, 0.85], which are shown in Fig. 4. From Fig. 4, know that when γ is too small, the cluster number is also too small to separate the decision makers. On the other hand, when γ is too big, the cluster number is also too big to concentrate the decision makers opinions. Based on Definition 6, deviation index could measure the average distance for different cluster results, which is similar to the concept of variance in statistics.

Calculating all of the difference indexes of different thresholds could determine the proper value, results summarized into Fig. 3. It is obvious to see that the smallest number of DI comes to appearance when γ = 0.81.

Fig. 5

Numbers of cluster and consensus degree when threshold value γ = 0.70.

Fig. 6

Numbers of cluster and consensus degree when threshold value is 0.81.

Table 3

The cluster result of γ = 0.81

Evolutionary times	Number of clusters	Average consistency	Standard consistency
485	7	0.8462	0.0699
427	7	0.8255	0.0912
457	7	0.8299	0.0154
460	7	0.8216	0.0230
423	7	0.8172	0.0173

Table 4

The 485th evolutionary result

Cluster number	Number of members	Member number	Consistency degree
1	54	1,3,4,5,6,7,12,13,14,15,18,19,20,21,24,25,27,28,29,31,
		32,36,39,40,42,43,44,49,50,51,57,59,61,63,64,66,67,
		68,70,76,78,80,82,86,87,89,91,92,93,94,95,96,98,99	0.8349
2	12	2,22,26,45,54,56,60,71,73,84,88,97	0.8549
3	11	23,34,47,48,52,53,65,77,79,83,85	0.8251
4	7	30,38,72,74,75,81,90	0.8194
5	5	9,10,55,62,69	0.7710
6	5	8,16,17,33,41	0.8241
7	2	11,37	0.9938

Table 5

Satisfaction and dissatisfaction with different criteria after clustering

		Clus-1	Clus-2	Clus-3	Clus-4	Clus-5	Clus-6
Standard 1	Satisfactory degree	0.798	0.733	0.767	0.751	0.695	0.759
	Unsatisfactory degree	0.173	0.169	0.170	0.191	0.226	0.162
Standard 2	Satisfactory degree	0.736	0.738	0.744	0.726	0.679	0.728
	Unsatisfactory degree	0.194	0.191	0.180	0.180	0.198	0.202
Standard 3	Satisfactory degree	0.788	0.801	0.797	0.794	0.733	0.793
	Unsatisfactory degree	0.157	0.140	0.159	0.157	0.190	0.160
Standard 4	Satisfactory degree	0.759	0.771	0.763	0.753	0.709	0.756
	Unsatisfactory degree	0.191	0.174	0.194	0.204	0.230	0.190
Standard 5	Satisfactory degree	0.757	0.766	0.767	0.752	0.692	0.761
	Unsatisfactory degree	0.180	0.176	0.171	0.182	0.250	0.180

Figure 5 tells that the cluster result are too small to separate the decision makers. The distribution is more uniform when γ = 0.81, meaning the progress more stable from Fig. 6. After 500 evolutions, the results become stable. Choosing the least 100 times, first consider the number of members contained in each class, considering both the average consistency and the standard consistency. First of all, as shown in Table 3, according to the average consistency from top to bottom, considering the standard consistency, can conclude that the result of the 485th evolution is the best.

Step 4. According to the result of the 485th evolution, the public’s initial decision-making matrix is classified. There are only two members in category 7 of category 2 and their consistency is as high as 99.38%, only 6 categories remain, and the evaluation results of each category member are averaged to obtain the initial values of Table 5. In addition, an initial decision matrix (19) $\bar{A_{1}}$ is obtained, containing six clusters and five evaluation criteria, where the first three evaluation criteria are benefit-based and the latter two evaluation criteria are cost-based.

Table 6

Evaluation criteria weight under different adjustment factors

	C ₁	C ₂	C ₃	C ₄	C ₅
α = 0.1	0.201	0.181	0.248	0.179	0.191
α = 0.3	0.205	0.180	0.244	0.179	0.191
α = 0.5	0.212	0.179	0.237	0.179	0.194
α = 0.7	0.222	0.176	0.226	0.178	0.197
α = 0.9	0.245	0.171	0.202	0.177	0.204

Step 5. According to the decision data after clustering progress in Step 4, the weighted standard is used to calculate the evaluation standard weights, and the criteria’s weights under different adjustment coefficients are shown in Table 6.

Step 6. Find the positive and negative ideal solutions of the alternatives’ evaluation results according to the Equations (27) and (28). Calculate the distance from the evaluation result alternative solution to the positive ideal and negative ideal solution, and take the optimality into Equations (31) to get the public item decision model to sort the alternatives. Finally, the result of comprehensive distance calculation is obtained, as shown in Tables 7 to 13.

Table 7

The overall distance of cluster 1

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.512	0.514	0.516	0.519	0.527
X ₂	0.561	0.560	0.557	0.553	0.544
X ₃	0.228	0.227	0.226	0.224	0.219
X ₄	0.320	0.321	0.323	0.326	0.333
X ₅	0.721	0.719	0.718	0.714	0.708

Table 8

The overall distance of cluster 2

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.463	0.465	0.468	0.472	0.482
X ₂	0.568	0.566	0.564	0.560	0.550
X ₃	0.140	0.140	0.140	0.141	0.142
X ₄	0.341	0.343	0.345	0.348	0.356
X ₅	0.662	0.661	0.661	0.661	0.660

Table 9

The overall distance of cluster 3

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.521	0.522	0.525	0.530	0.539
X ₂	0.589	0.587	0.585	0.581	0.573
X ₃	0.245	0.244	0.244	0.243	0.242
X ₄	0.363	0.365	0.367	0.372	0.381
X ₅	0.686	0.685	0.685	0.684	0.683

Table 10

The overall distance of cluster 4

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.543	0.544	0.546	0.549	0.556
X ₂	0.579	0.577	0.575	0.570	0.559
X ₃	0.305	0.304	0.301	0.297	0.287
X ₄	0.356	0.357	0.360	0.366	0.376
X ₅	0.714	0.713	0.711	0.708	0.702

$& \bar{A_{1}} = & [\begin{matrix} (0.798, 0.173) & (0.736, 0.194) & (0.788, 0.157) & (0.759, 0.191) & (0.757, 0.180) \\ (0.773, 0.169) & (0.738, 0.191) & (0.801, 0.140) & (0.771, 0.174) & (0.766, 0.176) \\ (0.767, 0.170) & (0.744, 0.180) & (0.797, 0.159) & (0.763, 0.194) & (0.767, 0.171) \\ (0.751, 0.191) & (0.726, 0.180) & (0.794, 0.157) & (0.753, 0.204) & (0.752, 0.182) \\ (0.695, 0.226) & (0.679, 0.198) & (0.733, 0.790) & (0.709, 0.230) & (0.692, 0.250) \\ (0.759, 0.162) & (0.728, 0.202) & (0.793, 0.160) & (0.756, 0.190) & (0.761, 0.180) \end{matrix}]$ (32)

Step 7. In this case, in order to balance the influence of different elements in the standard weight calculation model on the evaluation results, and also to make the results more representative, take the adjustment coefficient α = 0.5. The distance results of α = 0.5 from Tables 7 to 13 are screened out and normalized by weights. Then, the distribution of evaluation standard weight is shown in the form of a line graph as shown in Fig. 7.

Table 11

The overall distance of cluster 5

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.495	0.497	0.499	0.503	0.513
X ₂	0.642	0.641	0.638	0.634	0.626
X ₃	0.209	0.208	0.206	0.203	0.196
X ₄	0.206	0.208	0.210	0.214	0.222
X ₅	0.564	0.563	0.562	0.560	0.556

Table 12

The overall distance of cluster 6

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.500	0.501	0.502	0.505	0.512
X ₂	0.622	0.620	0.617	0.611	0.599
X ₃	0.307	0.307	0.306	0.305	0.302
X ₄	0.323	0.325	0.329	0.335	0.348
X ₅	0.742	0.740	0.738	0.734	0.726

Table 13

The overall distance 7

	α = 0.1	α = 0.3	α = 0.5	α = 0.7	α = 0.9
X ₁	0.477	0.479	0.481	0.484	0.492
X ₂	0.540	0.538	0.536	0.532	0.523
X ₃	0.214	0.213	0.212	0.211	0.207
X ₄	0.305	0.306	0.308	0.312	0.319
X ₅	0.659	0.658	0.657	0.655	0.650

It can be clearly seen in Fig. 7, the different clusters are relatively uniform. Further, based on the drastic results of the members of each cluster contains the basic situation analysis, it can be seen that members generally have the same characteristics of their own more likely to be gathered into the same category. When planning

Fig. 7

Ranking results of different alternatives.

Fig. 8

Ranking result between different distance measures.

Table 14

Comparison results between different distance measures

Distance measure	Criteria weights	Weighted Distances	Best One
Reference [25]	(0.1891,0.2018,0.2265,0.1371,0.2456)	(0.5765,0.6303,0.6900,0.5345,0.5855)	A ₃
Reference [27]	(0.0994,0.2630,0.2475,0.2259,0.1642)	(0.0333,0.2815,0.2510,-0.2925,0.0351)	A ₂
α = 0.1	(0.1206,0.2452, 0.2343,0.1775,0.2224)	(0.4882,0.5950,0.4624,0.3195,0.3025)	A ₂
α = 0.2	(0.1363,0.2352,0.2325,0.1682,0.2278)	(0.4819,0.5787,0.4612,0.3237,0.3012)	A ₂
α = 0.3	(0.1530,0.2246,0.2306,0.1584,0.2334)	(0.4752,0.5614,0.4599,0.3281,0.2999)	A ₂
α = 0.4	(0.1705,0.2135,0.2286,0.1481,0.2393)	(0.4681,0.5431,0.4585,0.3329,0.2984)	A ₂
α = 0.5	(0.1891,0.2018,0.2265,0.1371,0.2456)	(0.4606,0.5236,0.4569,0.3382,0.2969)	A ₂
α = 0.6	(0.2087,0.1893,0.2242,0.1255,0.2522)	(0.4526,0.5027,0.4551,0.3439,0.2953)	A ₂
α = 0.7	(0.2295,0.1761,0.2218,0.1133,0.2592)	(0.4441,0.4805,0.4531,0.3501,0.2935)	A ₂
α = 0.8	(0.2517,0.1621,0.2193,0.1002,0.2667)	(0.4350,0.4567,0.4507,0.3570,0.2916)	A ₂
α = 0.9	(0.2753,0.1472,0.2166,0.0863,0.2747)	(0.4252,0.4311,0.4480,0.3646,0.2896)	A ₃

and constructing a public project, they need to consider the needs of different types of audiences a fully nd achieve a true menu-based approach.

This numerical example shows the efficiency of proposed method. There are two points which should be noticed. Firstly, Fig. 4 shows the cluster numbers and member numbers under different thresholds. Determining the threshold value is a key point for the whole process. Except for analyzing the cluster results qualitatively, we introduce a novel definition Deviation Index (DI), to calculate the average deviation values. In this example, DI reaches the smallest value when γ = 0.81. However, it is not a constant value. That is to say, the best threshold value may change in other cases. Depending on many numerical experiments, we find that the best value always in the vicinity of 0.8. This conclusion maybe useful for some situations with low precision requirements. Secondly, after determining the threshold, we should confirm the cluster number. As shown in Table 3, this example carried out 500 times evolutions. And then choose the evolution result with the required cluster results, considering the average consistency and standard consistency. The evolution result may have certain error under different evolution times. According to our experiments, this kind of error has little effect on the ranking result.

In order to illustrate the superiority of the proposed distance measure between the IFDs, a comparison between it and some existing distance measures is shown in Table 14. Either the criteria weights or the weighted distances are different under different situations. Especially, the weighted values are changing based on accommodation coefficient α. Taking the situation α = 0.5 for example, the best alternative and the most important criterion are the same with reference [27]. Ranking results are shown in Fig. 8, where numbers in the figure stands for the ranking orders. We could see the overall ranking results are similar.Distance measure from reference [27] could overcome drawbacks of the previous methods. Obtaining the same best alternative could show the reasonableness of this paper’s method. What is more, the proposed distance measure is simpler and easy to understand.

8. Conclusions

This article considers a reality problem which introduces the public into the decision-making process of public projects. Based on intuitionistic fuzzy sets theory, this paper proposes an extended-IFN description method to describe public’s satisfactory attitude and unsatisfactory attitude. The proposed method is easy to understand and effective to utilize, making it suitable for the public’s diversity. A novel distance measure is given to compare similarity degree between different IFNs. A numerical example used by F. Shen et al. [28] is introduced, showing the effectiveness and reasonableness of the proposed distance formula. Furthermore, three methods are proposed to handle the big group decision making problem. Firstly, a clustering method is given depending on the similarity degree. Meanwhile, computational experiments is utilized to deal with the big data problem. Secondly, a model to calculate the criteria weight under the IFN’S environment is constructed, which could to deal with criteria unknown cases. Thirdly, a TOPSIS method based on the proposed distance formula is used to rank the alternatives. In summary, an algorithm of all the proposed methods is built. An illustrative example is applied to show its effectiveness. In the further, the big group decision making problem under other environment should been considered, such as kinds of preference relations. In addition, the distance measure also has some space to improve.

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An aggregating method to big group decision-making problem for the public participation problem under the Chinese situation 1

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1. Large population clustering under vector space

5.1. An extended method to describe the evaluating results of public participating problem

Table 2 The decision result by T1 T 1 C 1 C 2 C 3 C 4 C 5 X 1 (85,10) (65,35) (75,20) (80,10) (45,40) X 2 (80,16) (75,15) (80,15) (85,14) (70,20) X 3 (70,20) (70,20) (85,10) (76,16) (84,10) X 4 (95,0) (80,10) (80,13) (84,13) (74,15) X 5 (60,25) (65,25) (65,23) (62,36) (68,20)

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