﻿ 犹豫模糊集的<i>α</i>-截集及其应用
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 智能系统学报  2017, Vol. 12 Issue (3): 362-370  DOI: 10.11992/tis.201704026 0

### 引用本文

ZHENG Tingting, SANG Xiaoshuang, MA Binbin. α-cut sets of hesitant fuzzy sets and their applications[J]. CAAI Transactions on Intelligent Systems, 2017, 12(3): 362-370. DOI: 10.11992/tis.201704026.

### 文章历史

α-cut sets of hesitant fuzzy sets and their applications
ZHENG Tingting, SANG Xiaoshuang, MA Binbin
School of Mathematical Sciences, Anhui University, Hefei 230601, China
Abstract: The typical cut set is a bridge between fuzzy sets and clarity sets. The hesitant fuzzy set (HFS) theory, as an extension of the classical fuzzy set theory, has not been thoroughly studied till date; furthermore, there is less discussion regarding the relation between the HFS and classical type-Ⅰ fuzzy set theory or other fuzzy set theories. This study analyzed the relations between the HFS and type-1 fuzzy set theory and between HFS and interval type-2 fuzzy set theory, proposed the concept of α-cut sets of HFS, and discussed their properties. Meanwhile, the decomposition (representation) theorems and the more general extension principles of HFS based on α-cut sets were deduced. The corresponding properties were studied. The results of the simulation prove the rationality of the α-cut set concept and provide a novel method for hesitant fuzzy multiple attribute decision-making and clustering analysis. All these conclusions deeply enrich the fundamental theory of HFS.
Key words: hesitant fuzzy set    type-1 fuzzy set    interval type-2 fuzzy set    α-cut set    decomposition theorem    extension principle    multiple attribute decision-making    clustering analysis

1 预备知识

1.1 Ⅰ型模糊集(T1FS)

 $\begin{array}{*{20}{c}} {{\mu _A}:X \to \left[ {0,1} \right]}\\ {x \mapsto {\mu _A}\left( x \right)} \end{array}$

 $\begin{array}{l} {A_\alpha } = \left\{ {x \in X\left| {{\mu _A}\left( x \right) \ge \alpha } \right.} \right\}\\ {A_{\underline \alpha }} = \left\{ {x \in X\left| {{\mu _A}\left( x \right) > \alpha } \right.} \right\} \end{array}$

1.2 区间Ⅱ型模糊集(IT2FS)

 $\tilde A = \left\{ {\left\langle {x,\left( {u,{{\tilde A}_x}\left( u \right)} \right)} \right\rangle \left| {x \in X,\mu \in \left[ {0,1} \right]} \right.} \right\}$

 $\begin{array}{*{20}{c}} {\tilde A:X \to {{\left[ {0,1} \right]}^{\left[ {0,1} \right]}}}\\ {x \mapsto {{\tilde A}_x}} \end{array}$

 $\begin{matrix} {{{\tilde{A}}}_{x}}:\left[ 0,1 \right]\to \left[ 0,1 \right] \\ u\mapsto {{{\tilde{A}}}_{x}}\left( u \right) \\ \end{matrix}$

 $\tilde A = \left\{ {\left\langle {x,{J_{\tilde A}}\left( x \right)} \right\rangle \left| {x \in X} \right.} \right\},$ (1)

 ${J_{\tilde A}}:X \to D\left[ {0,1} \right]x \mapsto {J_{\tilde A}}\left( x \right)$

1) 若设xXJÃ(x)={uÃi(x)|i=1, 2, …, mx}，J$\tilde{B}$(x)={u$\tilde{B}$j(x)|j=1, 2, …, nx}，则

JÃC(x)={1-uÃi(x)|i=1, 2, …, mx}；

JÃ$\tilde{B}$(x)={max{uÃi(x), u$\tilde{B}$j(x)}|i=1, 2, …, mx, j=1, 2, …, nx}；

JÃ$\tilde{B}$(x)={min{uÃi(x), u$\tilde{B}$j(x)} |i=1, 2, …, mx, j=1, 2, …, nx}。

2) 若设xXJÃ(x)=[JÃL(x), JÃR(x)]，J$\tilde{B}$(x)=[J$\tilde{B}$L(x), J$\tilde{B}$R(x)]，则

JÃC(x)=[1-JÃR(x), 1-JÃL(x)]；

JÃ$\tilde{B}$(x)=[max{JÃL(x), J$\tilde{B}$L(x)}, max{JÃR(x), J$\tilde{B}$R(x)}]；

JÃ$\tilde{B}$(x)=[min{JÃL(x), J$\tilde{B}$L(x)}, min{JÃR(x), J$\tilde{B}$R(x)}]。

1.3 犹豫模糊集(HFS)

Torra和Narukawa在直觉模糊集和模糊多值集的基础上首次提出犹豫模糊集的概念[1-2]，它可以描述决策中某些犹豫不定的情况，例如某个专家可能会给某个元素定义一组可能的隶属度。

 $E = \left\{ {\left\langle {x,{h_E}\left( x \right)} \right\rangle \left| {x \in X} \right.} \right\},$ (2)

 $\begin{array}{l} {h_E}:X \to P\left( {\left[ {0,1} \right]} \right)\\ x \mapsto {h_E}\left( x \right) \end{array}$

 $\begin{array}{l} h_E^ - \left( x \right) = \min \left\{ {r\left| {r \in {h_E}\left( x \right)} \right.} \right\}\\ h_E^ + \left( x \right) = \max \left\{ {r\left| {r \in {h_E}\left( x \right)} \right.} \right\} \end{array}$

1)hEC(x)={1-r|rhE(x)}；

2)hE1E2(x)={rhE1(x)∪hE2(x)|r≥max{h-E1(x), h-E2(x)}}, 或者等价于{max{r1, r2}|r1hE1(x), r2hE2(x)}；

3)hE1E2(x)={rhE1(x)∪hE2(x)|r≤min{h+E1(x), h+E2(x)}}, 或者等价于{min{r1, r2}r1hE1(x), r2hE2(x)}。

2 犹豫模糊集的α-截集 2.1 犹豫模糊集与离散区间二型模糊集的关系

 $\begin{array}{*{20}{c}} {{J_{{{\tilde A}_1} \cup {{\tilde A}_2}}}\left( x \right) = \left\{ {0.4,0.6,0.8,1} \right\} = {h_{{E_1} \cup {E_2}}}\left( x \right)}\\ {{J_{{{\tilde A}_1} \cup {{\tilde A}_2}}}\left( x \right) = \left\{ {0.2,0.4,0.6} \right\}{h_{{E_1} \cup {E_2}}}\left( x \right)}\\ {{J_{\tilde A_1^C}} = \left\{ {0.4,0.6,0.8} \right\}{h_{E_1^C}}\left( x \right)} \end{array}$
2.2 犹豫模糊集的截集

 ${\mu _{{E_\alpha }}}\left( x \right) = \left\{ \begin{array}{l} \min \left\{ {r \in {h_E}\left( x \right)\left| {r \ge \alpha } \right.} \right\},\\ \;\;\;\;\;\;\;\left\{ {r \in {h_E}\left( x \right)\left| {r \ge \alpha } \right.} \right\} \ne \emptyset \\ 0,\;\;\;\;\;\left\{ {r \in {h_E}\left( x \right)\left| {r \ge \alpha } \right.} \right\} = \emptyset \end{array} \right.$ (3)

 ${\mu _{{E^\alpha }}}\left( x \right) = \left\{ \begin{array}{l} \max \left\{ {r \in {h_E}\left( x \right)\left| {r \le \alpha } \right.} \right\},\\ \;\;\;\;\;\;\;\left\{ {r \in {h_E}\left( x \right)\left| {r \le \alpha } \right.} \right\} \ne \emptyset \\ 0,\;\;\;\;\;\left\{ {r \in {h_E}\left( x \right)\left| {r \le \alpha } \right.} \right\} = \emptyset \end{array} \right.$ (4)

 ${\mu _{{E_\alpha }}}\left( {{x_1}} \right) = \left\{ \begin{array}{l} 0.2,\;\;\;0 \le \alpha \le 0.2\\ 0.4,\;\;\;0.2 < \alpha \le 0.4\\ 0.6,\;\;\;\;0.4 < \alpha \le 0.6\\ 0,\;\;\;\;\;\;\;0.6 < \alpha \le 1 \end{array} \right.$
 ${\mu _{{E_{\underline \alpha }}}}\left( {{x_1}} \right) = \;\left\{ \begin{array}{l} 0.2,\;\;\;\;0 \le \alpha \le 0.2\\ 0.4,\;\;\;\;0.2 \le \alpha < 0.4\\ 0.6,\;\;\;\;0.4 \le \alpha < 0.6\\ 0,\;\;\;\;\;\;\;0.6 < \alpha \le 1 \end{array} \right.$
 ${\mu _{{E^\alpha }}}\left( {{x_1}} \right) = \left\{ \begin{array}{l} 0,\;\;\;\;\;\;\;0 \le \alpha < 0.2\\ 0.2,\;\;\;\;0.2 \le \alpha < 0.4\\ 0.4,\;\;\;\;0.4 \le \alpha < 0.6\\ 0.6,\;\;\;\;0.6 \le \alpha \le 1 \end{array} \right.$
 ${\mu _{{E^{\underline \alpha }}}}\left( {{x_1}} \right) = \left\{ \begin{array}{l} 0,\;\;\;\;\;\;\;0 \le \alpha \le 0.2\\ 0.2,\;\;\;\;0.2 < \alpha \le 0.4\\ 0.4,\;\;\;\;0.4 < \alpha \le 0.6\\ 0.6,\;\;\;\;0.6 < \alpha \le 1 \end{array} \right.$
 ${\mu _{{E_\alpha }}}\left( {{x_2}} \right) = \left\{ \begin{array}{l} 0.3,\;\;\;0 \le \alpha \le 0.3\\ 0.7,\;\;\;0.3 < \alpha \le 0.7\\ 0,\;\;\;\;\;\;\;0.7 < \alpha \le 1 \end{array} \right.$
 ${\mu _{{E_{\underline \alpha }}}}\left( {{x_2}} \right) = \;\left\{ \begin{array}{l} 0.3,\;\;\;\;0 \le \alpha \le 0.3\\ 0.7,\;\;\;\;0.3 \le \alpha < 0.7\\ 0,\;\;\;\;\;\;\;0.7 \le \alpha \le 1 \end{array} \right.$
 ${\mu _{{E^\alpha }}}\left( {{x_2}} \right) = \left\{ \begin{array}{l} 0,\;\;\;\;\;\;\;0 \le \alpha < 0.3\\ 0.3,\;\;\;\;0.3 \le \alpha < 0.7\\ 0.7,\;\;\;\;0.7 \le \alpha \le 1 \end{array} \right.$
 ${\mu _{{E^{\underline \alpha }}}}\left( {{x_2}} \right) = \left\{ \begin{array}{l} 0,\;\;\;\;\;\;\;0 \le \alpha \le 0.3\\ 0.3,\;\;\;\;0.3 < \alpha \le 0.7\\ 0.7,\;\;\;\;0.7 < \alpha \le 1 \end{array} \right.$

 ${E_\alpha } = \left\{ \begin{array}{l} \left\{ {\left( {{x_1},0.2} \right),\left( {{x_2},0.3} \right)} \right\},\;\;0 \le \alpha \le 0.2\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;0.2 < \alpha \le 0.3\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;0.3 < \alpha \le 0.4\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;0.4 < \alpha \le 0.6\\ \left\{ {\left( {{x_2},0.7} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.6 < \alpha \le 0.7\\ \emptyset ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.7 < \alpha \le 1 \end{array} \right.$
 ${E_{\underline \alpha }} = \left\{ \begin{array}{l} \left\{ {\left( {{x_1},0.2} \right),\left( {{x_2},0.3} \right)} \right\},\;\;0 \le \alpha < 0.2\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;0.2 \le \alpha < 0.3\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;0.3 \le \alpha < 0.4\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;0.4 \le \alpha < 0.6\\ \left\{ {\left( {{x_2},0.7} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.6 \le \alpha < 0.7\\ \emptyset ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.7 \le \alpha \le 1 \end{array} \right.$
 ${E^\alpha } = \left\{ \begin{array}{l} \emptyset ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 \le \alpha < 0.2\\ \left\{ {\left( {{x_1},0.2} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.2 \le \alpha < 0.3\\ \left\{ {\left( {{x_1},0.2} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.3 \le \alpha < 0.4\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.4 \le \alpha < 0.6\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.6 \le \alpha < 0.7\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.7 \le \alpha \le 1 \end{array} \right.$
 ${E^{\underline \alpha }} = \left\{ \begin{array}{l} \emptyset ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 \le \alpha \le 0.2\\ \left\{ {\left( {{x_1},0.2} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0.2 < \alpha \le 0.3\\ \left\{ {\left( {{x_1},0.2} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.3 < \alpha \le 0.4\\ \left\{ {\left( {{x_1},0.4} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.4 < \alpha \le 0.6\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.3} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.6 < \alpha \le 0.7\\ \left\{ {\left( {{x_1},0.6} \right),\left( {{x_2},0.7} \right)} \right\},\;\;\;\;\;\;\;\;\;\;\;0.7 < \alpha \le 1 \end{array} \right.$

1)EαEαEαEα

2)α1 < α2Eα1Eα2, ${{E}^{\underline{{{a}_{1}}}}}\subseteq {{E}^{\underline{{{a}_{2}}}}}$

α1 < α2$\underset{x\in X}{\mathop{\wedge }}\, \left\{ h_{E}^{+}\left( x \right) \right\}$，则Eα1Eα2

α1 < α2 < $\underset{x\in X}{\mathop{\wedge }}\, \left\{ h_{E}^{+}\left( x \right) \right\}$，则${{E}_{\underline{{{a}_{1}}}}}\subseteq {{E}_{\underline{{{a}_{2}}}}}$

3)(EF)αEαFα；(EF)αEαFα；(EF)αEαFα；(EF)αEαFα

4)(EF)αEαFα；(EF)αEαFα；(EF)αEαFα；(EF) αEαFα

5) 设{αttT}满足a=$\underset{t\in T}{\mathop{\wedge }}\, \left\{ {{\alpha }_{t}} \right\}$b=$\underset{t\in T}{\mathop{\wedge }}\, \left\{ {{\alpha }_{t}} \right\}$ < $\underset{x\in X}{\mathop{\wedge }}\, \left\{ h_{E}^{+}\left( x \right) \right\}$，则

 $\begin{array}{l} \bigcap\limits_{t \in T} {{E_{{\alpha _t}}} = {E_a}} ;\bigcap\limits_{t \in T} {{E_{\underline {{\alpha _t}} }} = {E_{\underline a }}} ;\bigcap\limits_{t \in T} {{E_{{\alpha _t}}} = {E_b}} ;\bigcap\limits_{t \in T} {{E_{\underline {{\alpha _t}} }} = {E_{\underline b }}} ;\\ \bigcap\limits_{t \in T} {{E^{{\alpha _t}}} = {E^a}} ;\bigcap\limits_{t \in T} {{E^{\underline {{\alpha _t}} }} = {E^{\underline a }}} ;\bigcap\limits_{t \in T} {{E^{{\alpha _t}}} = {E^b}} ;\bigcap\limits_{t \in T} {{E^{\underline {{\alpha _t}} }} = {E^{\underline b }}} ; \end{array}$

6) 若α < $\underset{x\in X}{\mathop{\wedge }}\, \left\{ h_{E}^{+}\left( x \right) \right\}$，则

 $\begin{array}{l} {\left( {{E_\alpha }} \right)^C} = {\left( {{E^C}} \right)^{1 - \alpha }};{\left( {{E^C}} \right)_\alpha } = {\left( {{E^{1 - \alpha }}} \right)^C};\\ {\left( {{E_{\underline \alpha }}} \right)^C} = {\left( {{E^C}} \right)^{\underline {1 - \alpha } }};{\left( {{E^C}} \right)_{\underline \alpha }} = {\left( {{E^{\underline {1 - \alpha } }}} \right)^C}; \end{array}$

7)E0=E0=E1=EE 1=E0=E0=Ø。

① 当α>max{hE+(x), hF+(x)}时，{rhE(x)|rα}={rhF(x)|rα} ={rhEF(x)|rα}=Ø，故μEα(x)=μFα(x)=μ(EF)α(x)=0，从而μ(EF)α(x)=max{μEα(x), μFα(x)}=μEαFα(x)=0。

② 当max{hE+(x), hF+(x)}≥α >max{hE-(x), hF-(x)}时，有{rhE(x)∪hF(x) |rα} ⊇{rhE(x) |rα}且{rhE(x)∪hF(x)|rα} ⊇{rhF(x) rα}，故μ(EF)α(x)=min{rhEF(x)rα}=min{rhE(x) ∪hF(x)|r≥max{hE-(x), hF-(x), α}}=min{rhE(x)∪hF(x) |rα}≤max{min{rhE(x) |rα}, min{rhF(x)|rα}}=max{μEα(x), μFα(x)}=μEαFα(x)。

③ 当max{hE-(x), hF-(x)}≥α>min{hE-(x), hF-(x)}时，不妨设hE-(x) < αhF-(x)，则μ(EF)α(x)=min{rhE(x)∪hF(x) |rhF-(x)}=hF-(x), μEα(x)=min{rhE(x) |rα}≥α, μFα(x)=min{rhF(x) |rhF-(x)}=hF-(x)≥α；故μ(EF)α(x)=hF-(x) ≤max{μEα(x), μFα(x)}=μEαFα(x)。

④ 当min{hE-(x), hF-(x)}≥α时，也可类似证明μ(EF)α(x)=max{μEα(x), μFα(x)}=μEαFα(x)。

⑤ 这里仅证明结论5) 中$\underset{t\in T}{\mathop{\cup }}\,$Eαt=Eb成立，其余情况类似证明。

xX, tT，因为$\underset{t\in T}{\mathop{\vee }}\,${αt} < $\underset{x\in X}{\mathop{\wedge }}\, \left\{ h_{E}^{+}\left( x \right) \right\}$，所以μEαt(x)=min{rhE(x)|rαt}, μEb(x)=min{rhE(x)|r$\underset{t\in T}{\mathop{\vee }}\,$αt}。

c=μEb(x)，则c=min{rhE(x)| r$\underset{t\in T}{\mathop{\vee }}\,$αt}, 故∀tTc≥min{rhE(x)| rαt}，从而c$\underset{t\in T}{\mathop{\vee }}\,$min{rhE(x)|rαt}=μ$\underset{t\in T}{\mathop{\cup }}\,$Eαt(x), 故Eb$\underset{t\in T}{\mathop{\cup }}\,$Eαt

d=μ$\underset{t\in T}{\mathop{\cup }}\,$Eαt(x)=$\underset{t\in T}{\mathop{\vee }}\,$min{rhE(x)|rαt}，则∀tTd≥min{rhE(x)|rαt}。

⑥ 这里仅证明结论6) 中(Eα)C=(EC)1-α成立，其余类似。

3 犹豫模糊集的分解(表示)定理

 ${E_\alpha } = \bigcup\limits_{\lambda \in \left[ {0,1} \right]} {\lambda {{\left( {{E_\alpha }} \right)}_\lambda }} = \bigcup\limits_{\lambda \in \left[ {0,1} \right)} {\lambda {{\left( {{E_\alpha }} \right)}_{\underline \lambda }}}$

 $\lambda {\left( {{E_\alpha }} \right)_\lambda } = \left\{ \begin{array}{l} \left\{ {\left( {{x_1},\lambda } \right),\left( {{x_2},\lambda } \right)} \right\},\;\;\;\;\;\;0 \le \lambda \le 0.2\\ \left\{ {\left( {{x_1},0} \right),\left( {{x_2},\lambda } \right)} \right\},\;\;\;\;\;\;0.2 < \lambda \le 0.3\\ \left\{ {\left( {{x_1},0} \right),\left( {{x_2},0} \right)} \right\},\;\;\;\;\;\;0.3 < \lambda \le 1 \end{array} \right.$
 $\lambda {\left( {{E_\alpha }} \right)_{\underline \lambda }} = \left\{ \begin{array}{l} \left\{ {\left( {{x_1},\lambda } \right),\left( {{x_2},\lambda } \right)} \right\},\;\;\;\;\;\;0 \le \lambda < 0.2\\ \left\{ {\left( {{x_1},0} \right),\left( {{x_2},\lambda } \right)} \right\},\;\;\;\;\;\;0.2 \le \lambda < 0.3\\ \left\{ {\left( {{x_1},0} \right),\left( {{x_2},0} \right)} \right\},\;\;\;\;\;\;0.3 \le \lambda < 1 \end{array} \right.$

$\underset{\lambda \in \left[0, 1 \right]}{\mathop{\cup }}\,$λ(Eα)λ=$\underset{\lambda \in \left[0, 1 \right]}{\mathop{\cup }}\,$λ(Eα) λ={(x1, 0.2), (x2, 0.3)}=Eα

4 犹豫模糊集的扩展原则

Torra等人[1]曾介绍了HFS的扩展原理。

 $\begin{array}{*{20}{c}} {\forall x \in X}\\ {{\Theta _H}\left( x \right) = \bigcup\limits_{r \in {h_1}\left( x \right) \times {h_2}\left( x \right) \times \cdots \times {h_n}\left( x \right)} {\left\{ {\Theta \left( r \right)} \right\}} } \end{array}$

 $\begin{array}{*{20}{c}} {{h_{f\left( E \right)}}:Y \to P\left( {\left[ {0,1} \right]} \right)}\\ {y \mapsto {h_{f\left( E \right)}}\left( y \right)} \end{array}$

f -1(y)={xX|f(x)=y}，则

 ${h_{f\left( E \right)}}\left( y \right) = \left\{ \begin{array}{l} \mathop \vee \limits_{x \in {f^{ - 1}}\left( y \right)} \left\{ {r\left( x \right)\left| {r\left( x \right) \in {h_E}\left( x \right)} \right.} \right\},\;\;\;\;\;{f^{ - 1}}\left( y \right) \ne \emptyset \\ 0,\;\;\;\;{f^{ - 1}}\left( y \right) = \emptyset \end{array} \right.$

f也可诱导一个从HFS(Y)到HFS(X)的犹豫模糊逆函数，满足

 $\begin{array}{*{20}{c}} {{h_{{f^{ - 1}}\left( F \right)}}:X \to P\left( {\left[ {0,1} \right]} \right)}\\ {x \mapsto {h_{{f^{ - 1}}\left( F \right)}}\left( x \right) = \left\{ {r \in {h_F}\left( y \right)\left| {y = f\left( x \right) \in Y} \right.} \right\}。} \end{array}$

1)f(E)αf(Eα)，f(E)αf(Eα)；

2) f-1(F)α=f-1(Fα)，f-1(F)α=f-1(Fα)；

3)f(EG)αf(E)αf(G)αf-1(FH)αf-1(F)αf-1(H)α

4)f(Eα)Cf(EαC)，f-1(FαC)=f-1(F)αC

① 当{rhf(E)(y)|rα}=Ø时，μf(E)α(y)=0。也就是说, ∀xf-1(y)，若r(x)∈hE(x)，则r(x) < α，即μEα(x)=0。故μf(Eα)(y)=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,$μEα(x)=0=μf(E)α(y)。

② 当{rhf(E)(y)|rα}≠Ø时，μf(E)α(y)=min{r′∈hf(E)(y) |r′≥α}=min$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,${r(x)|r(x)∈hE(x) ∧r(x)≥a}。且μf(Eα)(y)=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,${μEα(x)}=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,$min{r(x)|r(x)∈hE(x)∧r(x)≥a}。

2) 对于∀xX，分以下分两种情况讨论：

① 当μf-1(F)α(x)=0时，f-1(F)α=Ø。说明∀rhf-1(F)(x), r < α。考虑到rhF(y)，这里y=f(x), 因此μFα(y)=0⇒μf-1(Fα)(x)=0。故μf-1(F)α(x)=μf-1(Fα)(x)。

② 当μf-1(F)α(x)≠0时，μf-1(F)α(x)=min{rhf-1(F)(x)rα}=min{rhF(y)y= f(x), rα}=μFα(y), 这里y=f(x), 且μf-1(Fα)(x)=μFα(y)。故μf -1(F)α(x)=μf -1(Fα)(x)，即f -1(F)α=f -1(Fα)。

f(EG)αf(EG)α)⊆ f(EαGα) =f(Eα)∪f(Gα)⊆ f(E)αf(G)α

f -1(FH)α=f -1((FH)α)f -1(FαHα) =f -1(Fα)∩f -1(Hα)=f -1(F)αf -1(H)α

④ 因为μf(Eα)C(y)=1-μf(Eα)(y)=1-$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,${μEα(x)}=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\wedge }}\,${1-μEα(x)}, μf(EαC)(y)=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,${μEαC(x)}=$\underset{x\in {{f}^{-1}}\left( y \right)}{\mathop{\vee }}\,${1-μEα(x)}, 所以μf(Eα)C(y)≤μf(EαC)(y)⇒f(Eα)Cf(EαC)。

 $\begin{array}{*{20}{c}} {\forall \left( {{E_1},{E_2}, \cdots ,{E_n}} \right) \in {\rm{HFS}}\left( {{X_1}} \right) \times {\rm{HFS}}\left( {{X_2}} \right) \times \cdots \times {\rm{HFS}}\left( {{X_n}} \right)}\\ {{h_{f\left( {{E_1},{E_2}, \cdots ,{E_n}} \right)}}:Y \to P\left( {\left[ {0,1} \right]} \right)}\\ {y \mapsto {h_{f\left( {{E_1},{E_2}, \cdots ,{E_n}} \right)}}\left( y \right)} \end{array}$

f -1(y)={(x1, x2, …, xn)|f(x1, x2, …, xn) =y, xiXi, i=1, 2, …, n}，则当f -1(y)≠Ø时

 $\begin{array}{*{20}{c}} {{h_{f\left( {{E_1},{E_2}, \cdots ,{E_n}} \right)}}\left( y \right) = \left\{ {\mathop \vee \limits_{\left( {{x_1},{x_2}, \cdots ,{x_n}} \right) \in {f^{ - 1}}\left( y \right)} \left\{ {\mathop \wedge \limits_{i = 1}^n {r_i}\left( {{x_i}} \right)\left| {{r_i}\left( {{x_i}} \right) \in {h_{{E_i}}}\left( {{x_i}} \right)} \right.,} \right.} \right.}\\ {\left. {\left. {i = 1,2, \cdots ,n} \right\}} \right\}} \end{array}$

f-1(y)=Ø时，hf(E)(y)=0。

5 应用实例 5.1 HFS的α-截集在多属性决策中的应用

1) 当μEα(x)>μEα(y)且hE+(x)≥hE+(y)，或μEα(x)=μEα(y)且hE+(x)>hE+(y)时，则P(Eα(x)>Eα(y))=1；

2) 当μEα(x) < μEα(y)且hE+(x)≤hE+(y)，或μEα(x)=μEα(y)且hE+(x) < hE+(y)时，则P(Eα(x)>Eα(y))=0；

3) 除以上两种情况外，P(Eα(x)>Eα(y))=12。

 $P\left( {{E_\alpha }\left( x \right) > {E_\alpha }\left( y \right)} \right) + P\left( {{E_\alpha }\left( y \right) > {E_\alpha }\left( x \right)} \right) = 1。$

1) 给定α∈[0, 1]，任意xX，计算

 ${E_\alpha }\left( x \right) = \sum\limits_{k = 1}^4 {{\omega _k}{\mu _{{E_{k\alpha }}}}\left( x \right)} ,h_E^ + \left( x \right) = \sum\limits_{k = 1}^4 {{\omega _k}h_{{E_k}}^ + \left( x \right)} 。$

2) 根据定义13，令pij=P(Eα(xi)>Eα(xj))，得到可能度矩阵P=(pij)5×5

3) 令${{p}_{i}}=\sum\limits_{j=1}^{5}{{{p}_{ij}}}$得每个项目的可能值，则可按照pi从大到小顺序来决定方案。

5.2 HFS的α-截集在聚类分析中的应用

Chen[18]和Liao[37]都曾讨论过犹豫模糊环境下的聚类算法，他们的算法都是需要先通过某种犹豫模糊数的相似测度度量构建相关矩阵，从而进行聚类。本文采取的原理是通过截集将HFS转换成T1FS，将犹豫模糊数聚类问题转换成经典模糊数聚类问题，从而使问题解决变得简单。

1) 给定α∈[0, 1]，则可得到第i个公司关于第k个指标在阈值α上的隶属度，即μEkα(xi)。

2) 采用最大最小法建立不同公司间的模糊相似矩阵R=(rij)10×10，其中

 ${r_{ij}} = \frac{{\sum\limits_{k = 1}^5 {\min \left\{ {{\mu _{{E_{{k_\alpha }}}}}\left( {{x_i}} \right),{\mu _{{E_{{k_\alpha }}}}}\left( {{x_j}} \right)} \right\}} }}{{\sum\limits_{k = 1}^5 {\max \left\{ {{\mu _{{E_{{k_\alpha }}}}}\left( {{x_i}} \right),{\mu _{{E_{{k_\alpha }}}}}\left( {{x_j}} \right)} \right\}} }},\;\;i,j = 1,2, \cdots ,10。$

3) 求出R的闭包，即模糊等价矩阵Ȓ

4) 从大到小取不同阈值λ∈[0, 1]，利用模糊等价矩阵的截矩阵Ȓλ进行聚类。

6 结束语