# 基于犹豫模糊不确定语言信息的群决策方法在城市综合管廊风险评估中的应用Group Decision Making Method Based on Hesitant Fuzzy Uncertain Linguistic Information and Its Application in Risk Assessment of Urban Comprehensive Pipeline

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Multiple attribute decision making is an important research direction in the field of modern deci-sion analysis. As an important part of multiple attribute decision making, multiple attribute deci-sion making with hesitant fuzzy and uncertain linguistic as decision environment has been widely applied to all walks of life. This paper is aimed at the risk of urban underground integrated pipe gallery construction, using hesitant fuzzy uncertain linguistic as the evaluation value of decision making and the maximum deviation method to determine the risk assessment indicators and ex-pert’s weight information involved in the assessment process to conduct the city underground comprehensive pipeline corridor construction pilot cities construction risk assessment, and using the TOPSIS method to conduct a comprehensive assessment of the pilot cities, and then select the most suitable city as a pilot city for the construction of underground comprehensive pipeline cor-ridors.

1. 引言

2. 预备知识

2.1. 语言变量和不确定语言变量

$\stackrel{˜}{S}$ 为不确定语言变量集。考虑任意三个不确定语言变量 $\stackrel{˜}{s}=\left[{s}_{\alpha },{s}_{\beta }\right]$${\stackrel{˜}{s}}_{1}=\left[{s}_{\alpha 1},{s}_{\beta 1}\right]$${\stackrel{˜}{s}}_{2}=\left[{s}_{\alpha 2},{s}_{\beta 2}\right]$ ，则其一般的运算定义如下 [11] ：

1) ${\stackrel{˜}{s}}_{1}\oplus {\stackrel{˜}{s}}_{2}=\left[{s}_{\alpha 1},{s}_{\beta 1}\right]\oplus \left[{s}_{\alpha 2},{s}_{\beta 2}\right]=\left[{s}_{\alpha 1}\oplus {s}_{\alpha 2},{s}_{\beta 1}\oplus {s}_{\beta 2}\right]=\left[{s}_{\alpha 1+\alpha 2},{s}_{\beta 1+\beta 2}\right]$

2) $\lambda \stackrel{˜}{s}=\lambda \left[{s}_{\alpha },{s}_{\beta }\right]=\left[\lambda {s}_{\alpha },\lambda {s}_{\beta }\right]=\left[{s}_{\lambda \alpha },{s}_{\lambda \beta }\right]$ ，其中 $\lambda \in \left[0,1\right]$

3) ${\stackrel{˜}{s}}_{1}\oplus {\stackrel{˜}{s}}_{2}={\stackrel{˜}{s}}_{2}\oplus {\stackrel{˜}{s}}_{1}$

4) $\lambda \left({\stackrel{˜}{s}}_{1}\oplus {\stackrel{˜}{s}}_{2}\right)=\lambda {\stackrel{˜}{s}}_{1}\oplus \lambda {\stackrel{˜}{s}}_{2}$ ，其中 $\lambda \in \left[0,1\right]$

5) $\left({\lambda }_{1}+{\lambda }_{2}\right)\stackrel{˜}{s}={\lambda }_{1}\stackrel{˜}{s}\oplus {\lambda }_{2}\stackrel{˜}{s}$ ，其中 ${\lambda }_{1},{\lambda }_{2}\in \left[0,1\right]$

2.2. 犹豫模糊不确定语言集

$A=\left\{〈x,\left[{s}_{\theta \left(x\right)},{s}_{\tau \left(x\right)}\right],{h}_{A}\left(x\right)〉|x\in X\right\}$ (2.1)

1) ${\stackrel{˜}{a}}_{1}\oplus {\stackrel{˜}{a}}_{2}=〈\left[{s}_{\theta \left({a}_{1}\right)+\theta \left({a}_{2}\right)},{s}_{\tau \left({a}_{1}\right)+\tau \left({a}_{2}\right)}\right],{\cup }_{\gamma \left({a}_{1}\right)\in h\left({a}_{1}\right),\gamma \left({a}_{2}\right)\in h\left({a}_{2}\right)}\left(\gamma \left({a}_{1}\right)+\gamma \left({a}_{2}\right)-\gamma \left({a}_{1}\right)\gamma \left({a}_{2}\right)\right)〉$

2) ${\stackrel{˜}{a}}_{1}\otimes {\stackrel{˜}{a}}_{2}=〈\left[{s}_{\theta \left({a}_{1}\right)\theta \left({a}_{2}\right)},{s}_{\tau \left({a}_{1}\right)\tau \left({a}_{2}\right)}\right],{\cup }_{\gamma \left({a}_{1}\right)\in h\left({a}_{1}\right),\gamma \left({a}_{2}\right)\in h\left({a}_{2}\right)}\left(\gamma \left({a}_{1}\right)\gamma \left({a}_{2}\right)\right)〉$

3) $\lambda {\stackrel{˜}{a}}_{1}=〈\left[{s}_{\lambda \theta \left({a}_{1}\right)},{s}_{\lambda \tau \left({a}_{1}\right)}\right],{\cup }_{\gamma \left({a}_{1}\right)\in h\left({a}_{1}\right)}\left(1-{\left(1-\gamma \left({a}_{1}\right)\right)}^{\lambda }\right)〉$

4) ${\stackrel{˜}{a}}_{1}^{\lambda }=〈\left[{s}_{\theta {\left({a}_{1}\right)}^{\lambda }},{s}_{\tau {\left({a}_{1}\right)}^{\lambda }}\right],{\cup }_{\gamma \left({a}_{1}\right)\in h\left({a}_{1}\right)}\left(\gamma {\left({a}_{1}\right)}^{\lambda }\right)〉$

$S\left(\stackrel{˜}{a}\right)=\left(\frac{1}{lh\left(a\right)}{\sum }_{\gamma \left(a\right)\in h\left(a\right)}\gamma \left(a\right)\right)×{s}_{\left(\theta \left(a\right)+\tau \left(a\right)\right)/2}={s}_{\left(\theta \left(a\right)+\tau \left(a\right)\right)×{\sum }_{\gamma \left(a\right)\in h\left(a\right)}\gamma \left(a\right)/2lh\left(a\right)}$ (2.2)

$H\left(\stackrel{˜}{a}\right)=\frac{2}{lh\left(a\right)\left(lh\left(a\right)+1\right)}{\sum }_{i=1}^{lh\left(a\right)}{\sum }_{j=i+1}^{lh\left(a\right)}\left({\gamma }_{\sigma \left(i\right)}\left(a\right)-{\gamma }_{\sigma \left(j\right)}\left(a\right)\right)×{s}_{\left(\theta \left(a\right)+\tau \left(a\right)\right)}$ (2.3)

1) 如果 $S\left({\stackrel{˜}{a}}_{1}\right)>S\left({\stackrel{˜}{a}}_{2}\right)$ ，则称 ${\stackrel{˜}{a}}_{1}$ 优于 ${\stackrel{˜}{a}}_{2}$ ，也即是 ${\stackrel{˜}{a}}_{1}>{\stackrel{˜}{a}}_{2}$

2) 如果 $S\left({\stackrel{˜}{a}}_{1}\right)=S\left({\stackrel{˜}{a}}_{2}\right)$ ，则

i) 如果 $H\left({\stackrel{˜}{a}}_{1}\right)>H\left({\stackrel{˜}{a}}_{2}\right)$ ，则 ${\stackrel{˜}{a}}_{1}>{\stackrel{˜}{a}}_{2}$

ii) 如果 $H\left({\stackrel{˜}{a}}_{1}\right)=H\left({\stackrel{˜}{a}}_{2}\right)$ ，则意味着 ${\stackrel{˜}{a}}_{1}$${\stackrel{˜}{a}}_{2}$ 描述了同样的信息，也即是， ${\stackrel{˜}{a}}_{1}={\stackrel{˜}{a}}_{2}$

2.3. 模糊测度和Choquet积分

1) $\mu \left(\varnothing \right)\text{=}0$$\mu \left(X\right)\text{=}1$

2) 如果 $A,B\in P\left(X\right)$$A\subseteq B$ ，则 $\mu \left(A\right)\le \mu \left(B\right)$ ，其中， $P\left(X\right)$ 为论域 $X$ 的幂集。

${C}_{\mu }\left(f\left({x}_{\left(i\right)}\right)\right)=\underset{i=1}{\overset{n}{\sum }}f\left({x}_{\left(i\right)}\right)\left[\mu \left({A}_{\left(i\right)}\right)-\mu \left({A}_{\left(i+1\right)}\right)\right]$ (2.4)

${\rho }_{S}^{sh}\left(\mu ,N\right)=\underset{T\subseteq N\S}{\sum }\frac{\left(n-t-s\right)!t!}{\left(n-s+1\right)!}\left(\mu \left(S\cup T\right)-\mu \left(T\right)\right)$$\forall S\subseteq N$(2.5)

${\rho }_{S}^{sh}\left({g}_{\lambda },N\right)=\underset{T\subseteq N\S}{\sum }\frac{\left(n-s-t\right)!t!}{\left(n-s+1\right)!}\left({g}_{\lambda }\left(S\cup T\right)-{g}_{\lambda }\left(T\right)\right)$$\forall S\subseteq N$(2.6)

${\rho }_{i}^{sh}\left({g}_{\lambda },N\right)=\underset{S\subseteq N\i}{\sum }\frac{\left(n-s-1\right)!s!}{n!}{g}_{\lambda }\left(i\right)\underset{j\in S}{\prod }\left[1+\lambda {g}_{\lambda }\left(j\right)\right]$$\forall i\subseteq N$(2.7)

${C}_{{\rho }^{sh}\left({g}_{\lambda },N\right)}\left(f\left({x}_{\left(i\right)}\right)\right)=\underset{i=1}{\overset{n}{\sum }}f\left({x}_{\left(i\right)}\right)\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)$ (2.8)

${C}_{{\rho }^{sh}\left({g}_{\lambda },N\right)}\left(f\left({x}_{\left(i\right)}\right)\right)=\underset{i=1}{\overset{n}{\otimes }}f{\left({x}_{\left(i\right)}\right)}^{{\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)}$ (2.9)

3. 犹豫模糊不确定语言多属性群决策方法

3.1. Shapley Choquet积分算子

$HFULSC{A}_{{g}_{\lambda }}\left({\stackrel{˜}{a}}_{1},{\stackrel{˜}{a}}_{2},\cdots ,{\stackrel{˜}{a}}_{n}\right)=\underset{i=1}{\overset{n}{\oplus }}\left({\stackrel{˜}{a}}_{\left(i\right)}\right)\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{A\left(i+1\right)}^{sh}\left({g}_{\lambda },N\right)\right)$ (3.1)

$\begin{array}{l}HFULSC{A}_{{g}_{\lambda }}\left({\stackrel{˜}{a}}_{1},{\stackrel{˜}{a}}_{2},\cdots ,{\stackrel{˜}{a}}_{n}\right)\\ =〈\left[{s}_{{\sum }_{i=1}^{n}\theta \left({a}_{i}\right)\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)},{s}_{{\sum }_{i=1}^{n}\tau \left({a}_{i}\right)\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)}\right]\end{array}$ $,\left({\cup }_{\gamma \left({a}_{\left(i\right)}\right)\in h\left({a}_{\left(i\right)}\right)}\left(1-\underset{i=1}{\overset{n}{\prod }}{\left(1-\gamma \left({a}_{\left(i\right)}\right)\right)}^{\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)}\right)〉$ (3.2)

$HFULSC{G}_{{g}_{\lambda }}\left({\stackrel{˜}{a}}_{1},{\stackrel{˜}{a}}_{2},\cdots ,{\stackrel{˜}{a}}_{n}\right)=\underset{i=1}{\overset{n}{\otimes }}{\left({\stackrel{˜}{a}}_{\left({a}_{i}\right)}\right)}^{{\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)}$ (3.3)

$\begin{array}{l}HFULSC{G}_{{g}_{\lambda }}\left({\stackrel{˜}{a}}_{1},{\stackrel{˜}{a}}_{2},\cdots ,{\stackrel{˜}{a}}_{n}\right)\\ =〈\left[{s}_{{\prod }_{i=1}^{n}\theta {\left({a}_{i}\right)}^{\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)}},{s}_{{\prod }_{i=1}^{n}\tau {\left({a}_{i}\right)}^{\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)}}\right]\end{array}$ $,\left({\cup }_{\gamma \left({a}_{\left(i\right)}\right)\in h\left({a}_{\left(i\right)}\right)}\underset{i=1}{\overset{n}{\prod }}\gamma {\left({a}_{\left(i\right)}\right)}^{\left({\rho }_{{A}_{\left(i\right)}}^{sh}\left({g}_{\lambda },N\right)-{\rho }_{{A}_{\left(i+1\right)}}^{sh}\left({g}_{\lambda },N\right)\right)}\right)〉$ (3.4)

3.2. 犹豫模糊不确定语言集的距离测度

(3.5)

(3.6)

3.3. 基于Shapley Choquet积分的群决策TOPSIS方法

3.3.1. 问题描述

3.3.2. Shapley模糊测度的确定

1998年Wang [22] 率先提出了基于离差最大化的客观赋权方法，即在一个多属性决策问题，就决策属性而言，如果在某一个决策属性下，对于不同的方案，决策专家给出的决策属性值差异很小，则说明该属性无法很好地区分不同方案，从而该属性应该被赋予一个较小的权重。反之，属性应该被赋予较大的权重。

(3.7)

。 (3.8)

(3.9)

(3.10)

。 (3.11)

(3.12)

3.3.3. 基于Shapley Choquet积分的模糊多属性群决策TOPSIS方法

(3.13)

(3.14)

。 (3.15)

4. 犹豫模糊不确定语言TOPSIS法在城市地下综合管廊建设风险评估中的应用

4.1. 问题描述

Table 1. Hesitant fuzzy uncertain linguistic decision matrix given by expert

Table 2. Hesitant fuzzy uncertain linguistic decision matrix given by expert

Table 3. Hesitant Fuzzy Uncertain Linguistic decision matrix given by expert

4.2. 城市地下综合管廊建设风险评估

5. 结论与展望

1) 定义了犹豫模糊不确定语言元的得分函数和精确函数；基于Shapley Chouqet积分，全面考虑各个相关元素之间每一个联盟的相互作用的特性，提出用于集结信息的犹豫模糊不确定语言Shapley Chouqet算术平均算子和犹豫模糊不确定语言Shapley Chouqet几何平均算子；之后定义了度量犹豫模糊不确定语言集之间距离的汉明距离公式。

2) 基于离差最大化方法，通过构建数学模型，分别求解决策专家集和属性集的Shapley模糊测度。

3) 将经典的TOPSIS方法拓展到犹豫模糊不确定语言环境中，提出了基于Shapley Chouqet积分的多属性群决策TOPSIS方法。

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