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  智能系统学报  2017, Vol. 12 Issue (6): 883-888  DOI: 10.11992/tis.201706034
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引用本文  

梁美社, 米据生, 赵天娜. 广义优势多粒度直觉模糊粗糙集及规则获取[J]. 智能系统学报, 2017, 12(6), 883-888. DOI: 10.11992/tis.201706034.
LIANG Meishe, MI Jusheng, ZHAO Tianna. Generalized dominance-based multi-granularity intuitionistic fuzzy rough set and acquisition of decision rules[J]. CAAI Transactions on Intelligent Systems, 2017, 12(6), 883-888. DOI: 10.11992/tis.201706034.

基金项目

国家自然科学基金项目(61573127,61300121,61502144);河北省自然科学基金项目(A2014205157);河北省高校创新团队领军人才培育计划项目(LJRC022);河北师范大学研究生创新项目基金项目(CXZZSS2017046).

通信作者

梁美社. E-mail:liangmeishe@163.com.

作者简介

梁美社,男,1986年生,讲师,博士研究生,主要研究方向为粗糙集理论、粒计算;
米据生,男,1966年生,教授、博士生导师,主要研究方向为粗糙集、概念格,近似推理;
赵天娜,女,1992年生,硕士研究生,主要研究方向为粗糙集、概念格

文章历史

收稿日期:2017-06-09
网络出版日期:2017-11-09
广义优势多粒度直觉模糊粗糙集及规则获取
梁美社1,2, 米据生1, 赵天娜1    
1. 河北师范大学 数学与信息科学学院,河北 石家庄 050024;
2. 石家庄职业技术学院 科技发展与校企合作部,河北 石家庄 050081
摘要:优势关系粗糙集模型是研究序信息系统中数据挖掘的主要方法。为了丰富现有优势关系粗糙集模型,使其更加有效地应用于实际问题,本文首先在直觉模糊决策信息系统中利用三角模和三角余模定义了3种优势关系,得到了3种优势类;其次构造了广义优势关系多粒度直觉模糊粗糙集模型,讨论了该模型的主要性质;随后给出如何从直觉模糊决策信息系统中获取逻辑连接词为“或”的决策规则;最后通过实例说明该模型在处理直觉模糊决策序关系信息系统时是有效的。
关键词多粒度    粗糙集    三角模    优势关系    直觉模糊信息系统    决策规则    
Generalized dominance-based multi-granularity intuitionistic fuzzy rough set and acquisition of decision rules
LIANG Meishe1,2, MI Jusheng1, ZHAO Tianna1    
1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China;
2. Office of Science & Technology Administration, Shijiazhuang University of Applied Technology, Shijiazhuang 050081, China
Abstract: The dominance relation rough set model is the main method of data mining when researching order information systems. In this paper, we attempt to enrich the present model and make it more effective for practical problems using the following methods: Firstly, defining three types of dominance relations using triangular norms and co-norms in an intuitionistic fuzzy decision information system; here, three types of dominance class were obtained; secondly, establishing a generalized dominance-based multi-granularity intuitionistic fuzzy rough set model and discussing its properties; thirdly, establishing the decision rules for obtaining the logic connective " OR” in the intuitionistic fuzzy decision information system; and finally, using an example to illustrate the effectiveness of the model.
Key words: multi-granularity    rough set    triangular norm    dominance relation    intuitionistic fuzzy information system    decision rules    

直觉模糊集是Atanassov[1-3]在系统研究Zadeh[4]模糊集理论的基础上于1986年提出的。与传统Zadeh模糊集相比,由于同时考虑了元素的隶属度、非隶属度和犹豫度3个方面的信息,因此在表达和处理模糊性、不确定性等问题的时候更具灵活性和实用性。近年来将直觉模糊集理论[5-6]与粗糙集理论结合研究受到了广泛关注。

经典的粗糙集理论[7-8]建立在一个等价关系之上,即处理单个粒空间上的目标近似逼近理论。考虑到多个属性之间的关系可能是相互独立的,文献[9-12]从多个角度、多个层次出发,提出了多粒度粗糙集的概念。此后,许多学者开始了多粒度粗糙集的相关研究[13-18]。实际问题中,经常需要考虑描述对象的属性具有顺序性,如距离远近、人口密度等,S. Greco[19-20]等提出了基于优势关系的粗糙集模型,并将该方法引入到模糊信息系统中。以往研究中要么是建立在经典关系或者模糊关系上[21]的多粒度优势关系粗糙集,要么是考虑在单个粒度上的优势直觉模糊粗糙集,并未考虑将二者结合起来研究。

本文主要考虑在直觉模糊语义下,通过引入三角模和三角余模,定义了强、弱、平均3种优势关系,得到了与之对应的3种优势类。在此基础上提出了广义优势关系多粒度直觉模糊粗糙集模型。通过讨论该模型的主要性质,进而获取决策规则。

1 预备知识 1.1 直觉模糊信息系统

定义1[1-3]  设U是非空集合,称 ${{A}} \!=\! \left\{\! {\left\langle {{\mu _{{A}}}\left( x \right),{\nu _{{A}}}\left( x \right)} \right\rangle \! |} \right.$ $\left. {x \in U} \right\}$ 为直觉模糊集,其中 ${\mu _{{A}}}\left( x \right)$ ${\nu _{{A}}}\left( x \right) \in \left[ {0,1} \right]$ 分别为U中元素x属于A的隶属度和非隶属度,且对于 $\forall x \in U$ 满足关系式 $0 \leqslant {\mu _{{A}}}\left( x \right) + {\nu _{{A}}}\left( x \right) \leqslant 1$ 。称 $1 - {\mu _{{A}}}\left( x \right) - $ $ {\nu _{{A}}}\left( x \right)$ x属于A的犹豫度或不确定度。用 ${\rm{IFS}}\left( U \right)$ 表示U上全体直觉模糊子集, $P\left( U \right)$ 表示U上全体经典子集。

定义2[1]  对于任意 ${{A}},{{B}} \in {\rm{IFS}}\left( U \right)$ ,即 ${{A}} \!=\! \left\{ {\left\langle {{\mu _{{A}}}\left( x \right),} \right.} \right.$ $\left. {\left. {{\nu _{{A}}}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\},{{B}} = \left\{ {\left\langle {{\mu _{{B}}}\left( x \right),{\nu _{{B}}}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\}$ ,有:

1) ${{A}} \subseteq {{B}} \Leftrightarrow \forall x \in U$ ${\mu _{{A}}}\left( x \right) \leqslant {\mu _{{B}}}\left( x \right)$ ${\nu _{{A}}}\left( x \right) \geqslant {\nu _{{B}}}\left( x \right)$

2) ${{A}} \!=\! {{B}} \Leftrightarrow \forall x \in U$ ${\mu _{{A}}}\left( x \right) = {\mu _{{B}}}\left( x \right)$ ,且 ${\nu _{{A}}}\left( x \right) \! \geqslant \! {\nu _{{B}}}\left( x \right)$

3) $\sim {{A}} = \left\{ {\left\langle {{\nu _{{A}}}\left( x \right),{\mu _{{A}}}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\}$

4) ${{A}} \cap {{B}} \Leftrightarrow \left\{ {\left\langle {{\mu _{{A}}}\left( x \right) \wedge {\mu _{{B}}}\left( x \right),{\nu _{{A}}}\left( x \right) \vee {\nu _{{B}}}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\}$

5) ${{A}} \cup { B} \Leftrightarrow \left\{ {\left\langle {{\mu _{{A}}}\left( x \right) \vee {\mu _{{{B}}}}\left( x \right),{\nu _{{A}}}\left( x \right) \wedge {\nu _{{{B}}}}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\}$

$\alpha = \left( {{u_\alpha },{v_\alpha }} \right)$ ,其中 ${\mu _\alpha }$ ${\nu _\beta } \in \left[ {0,1} \right]$ ,且 $0 \leqslant {\mu _\alpha } + {\nu _\alpha } \leqslant 1$ ,则称α为一个直觉模糊数。全体直觉模糊数集合记为IFN。它的得分函数 $s\left( \alpha \right) = {u_\alpha } - {v_\alpha }$ ,精确函数为 $h\left( \alpha \right) = {u_\alpha } + {v_\alpha }$ ,利用得分函数和精确函数就可以给出比较两个直觉模糊数大小的方法。

定义3[22]  对于 $\forall {\alpha _1},\,{\alpha _2} \in {\rm{INF}}$ ,如果 $\forall a,\,b,\,c \in $ [0,1], $s\left( {{\alpha _1}} \right) < s\left( {{\alpha _2}} \right)$ ,则 ${\alpha _1} < {\alpha _2}$ ;如果 $s\left( {{\alpha _1}} \right) = s\left( {{\alpha _2}} \right)$ ,且若 $h\left( {{\alpha _1}} \right) < h\left( {{\alpha _2}} \right)$ ,则 ${\alpha _1} < {\alpha _2}$ ;若 $h\left( {{\alpha _1}} \right) = h\left( {{\alpha _2}} \right)$ ,则 ${\alpha _1} = $ $ {\alpha _2}$

定义4[1-3]  称 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $U = \left\{ {{x_1},{x_2}, \cdots ,{x_n}} \right\}$ 为对象集, $A = \left\{ {{a_1},{a_2}, \cdots ,{a_m}} \right\}$ 为条件属性集,RUA的直觉模糊二元关系,即 $R = \left\{ {\left\langle {\left( {x,a} \right),{\mu _a}\left( x \right),{\nu _a}\left( x \right)} \right\rangle \left| {\left( {x,a} \right) \in U \times A \cup \left\{ d \right\}} \right.} \right\}$ 。其中 ${\mu _a}:$ $U \times A \to \left[ {0,1} \right],{\nu _a}:U \times A \to \left[ {0,1} \right]$ ,且满足 $\forall \left( {x,a} \right) \in U \times A$ $0 \leqslant {\mu _a}\left( x \right) + {\nu _a}\left( x \right) \leqslant 1$ 。本文中记 $U \times A$ 上的直觉模糊关系全体为 ${\rm{IFR}}\left( {U \times A} \right)$

1.2 三角模算子

定义5[23]  若映射 $N:\left[ {0,1} \right] \to \left[ {0,1} \right]$ $\forall a,b \in \left[ {0,1} \right]$ 若满足以下条件:

1) $N\left( 0 \right) = 1$ $N\left( 1 \right) = 0$ (边界性);

2) $a \leqslant b$ ,则 $N\left( a \right) \geqslant N(b)$ (单调性);

称映射N为模糊补映射(或模糊负算子)。

$\forall a \in \left[ {0,1} \right]$ 均有 $N\left( a \right) = 1 - a$ 成立,称N为标准模糊补算子,记为Ns

定义6[24]  若映射 $T : \left[ {0,1} \right] \, \times \, \left[ {0,1} \right] \, \to \, \left[ {0,1} \right]$ ,若 $\forall a,b,c \in \left[ {0,1} \right]$ ,满足以下条件:

1) $T\left( {a,1} \right) = a$ (边界性);

2) 若 $b \leqslant c$ ,则 $T\left( {a,b} \right) \leqslant T\left( {a,c} \right)$ (单调性);

3) $T\left( {a,b} \right){\rm{ = }}T\left( {b,a} \right)$ (交换性);

4) $T\left( {a,T\left( {b,c} \right)} \right){\rm{ = }}T\left( {T\left( {a,b} \right),c} \right)$ (结合性);

则称T为三角模(t-模)。

定义7[24]  若映射 $S:\left[ {0,1} \right] \, \times \, \left[ {0,1} \right] \, \to \, \left[ {0,1} \right]$ ,若 $\forall a,b,c \in \left[ {0,1} \right]$ ,满足以下条件:

1) $S\left( {a,0} \right) = a$ (边界性);

2) 若 $b \leqslant c$ ,则 $S\left( {a,b} \right) \leqslant S\left( {a,c} \right)$ (单调性);

3) $S\left( {a,b} \right){\rm{ = }}S\left( {b,a} \right)$ (交换性);

4) $S\left( {a,S\left( {b,c} \right)} \right){\rm{ = }}S\left( {S\left( {a,b} \right),c} \right)$ (结合性);

S为三角模余模(t-余模)。

TST关于模糊补算子N满足是对偶的当且仅当 $\forall a,b \in \left[ {0,1} \right]$ $N\left( {T\left( {a,b} \right)} \right) = {S_T}\left( {N\left( a \right),N\left( b \right)} \right)$ $N\left( {{S_T}\left( {a,b} \right)} \right) = T\left( {N\left( a \right),N\left( b \right)} \right)$ ,称 $\left( {T,{S_T},N} \right)$ 为对偶三元组。常见的对偶三元组有:

$\min \text{-} \max :\left( {\min \left( {a,b} \right),\max \left( {a,b} \right),{N_S}} \right)$

${\rm{product \text{-} sum}}:\left( {ab,a + b - 1,{N_S}} \right)$

${\rm{Lukasiewicz}}:\left( {\max (0,a + b - 1),\min (1,a + b),{N_S}} \right)$

2 多粒度直觉模糊粗糙集 2.1 直觉模糊信息系统中的优势关系

定义8 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \subseteq A$ $\forall x,y \in U$ ,称 $R_{f,B}^{\leqslant} = \left\{ {\left( {x,y} \right) \in U \times U:{f_a}\left( x \right) \leqslant {f_a}\left( y \right),} \right.$ $\left. {\forall a \in B} \right\}$ 为直觉模糊信息系统中属性子集B的普通优势关系,其中 ${f_a}\left( x \right) = \left\langle {{u_a}\left( x \right),{v_a}\left( x \right)} \right\rangle $

定义9 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \subseteq A$ $\forall x,y \in U$ ,称 $R_{T,B}^{\leqslant } = \left\{ {\left( {x,y} \right) \in U \times U:{T_a}\left( x \right) \leqslant {T_a}\left( y \right),} \right.$ $\left. {\forall a \in B} \right\}$ 为直觉模糊信息系统中属性子集B的强优势关系, 其中 ${T_a}\left( x \right) = T\left( {{u_a}\left( x \right),N\left( {{v_a}\left( x \right)} \right)} \right)$

定义10 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \subseteq A$ $\forall x,y \in U$ ,称 $R_{S,B}^ \leqslant = \left\{ {\left( {x,y} \right) \in U \times U:{S_a}\left( x \right) \leqslant {S_a}\left( y \right),} \right.$ $\left. {\forall a \in B} \right\}$ 为直觉模糊信息系统中属性子集B的弱优势关系,其中 ${S_a}\left( x \right) = S\left( {{u_a}\left( x \right),N\left( {{v_a}\left( x \right)} \right)} \right)$

定义11 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \! \subseteq \! A$ $\forall x,y \! \in \! U$ ,称 $R_{AV,B}^{\leqslant} \!=\! \left\{ {\left( {x,y} \right) \! \in \! U \! \times \! U:A{V_a} \! \left( x \right) \! \leqslant \! A{V_a}\left( y \right),} \right.$ $\left. {\forall a \in B} \right\}$ 为直觉模糊信息系统中属性子集B的平均优势关系,其中 $A{V_a}\left( x \right) = \displaystyle\frac{1}{2}\left( {T\left( {{u_a}\left( x \right),N\left( {{v_a}\left( x \right)} \right)} \right) + {S_T}\left( {{u_a}\left( x \right),} \right.} \right.$ $\left. {\left. {N\left( {{v_a}\left( x \right)} \right)} \right)} \right)$

根据以上定义的3种优势关系,我们可以得到相应的3种优势类。

定义12 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \subseteq A$ $\forall x,y \in U$ ,称 $\left[ x \right]_{T,B}^ \leqslant $ 为对象x的强优势类,其中 $\left[ x \right]_{T,B}^ \leqslant = \left\{ {y \in U:\left( {x,y} \right) \in R_{T,B}^ \leqslant } \right\}$

类似地,将 $R_{T,B}^ \leqslant $ 替换成 $R_{f,B}^ \leqslant ,R_{S,B}^ \leqslant ,R_{AV,B}^ \leqslant $ 可以得到对象x的普通优势类 $\left[ x \right]_{f,B}^ \leqslant $ 、弱优势类 $\left[ x \right]_{S,B}^ \leqslant $ 、平均优势类 $\left[ x \right]_{AV,B}^ \leqslant $

定理1 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $B \subseteq A$ $\forall x,y \in U$ ,有 $\left[ x \right]_{AV,B}^{\leqslant} \subseteq \left[ x \right]_{T,B}^{\leqslant} \cup \left[ x \right]_{S,B}^{\leqslant} $

证明 假设 $y \notin \left[ x \right]_{T,B}^{\leqslant} \cup \left[ x \right]_{S,B}^{\leqslant} $ ,有 $y \notin \left[ x \right]_{T,B}^{\leqslant} $ $y \notin \left[ x \right]_{S,B}^{\leqslant} $ ${T_a}\left( x \right) > {T_a}\left( y \right)$ ${S_a}\left( x \right) > {S_a}\left( y \right)$ 。根据定义11可得 $A{V_a}\left( x \right) > A{V_a}\left( y \right)$ ,由定义12可知 $y \notin \left[ x \right]_{AV,B}^{\leqslant} $ ,从而证得 $\left[ x \right]_{AV,B}^{\leqslant} \subseteq \left[ x \right]_{T,B}^{\leqslant} \cup \left[ x \right]_{S,B}^{\leqslant} $

定理2 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $A{V^1},A{V^2},A{V^3}$ 分别由常见的对偶三元组生成,对于 $\left( {U,A,R} \right)$ $U \!=\! \left\{ {{x_1},{x_2}, \cdots ,{x_m}} \right\}$ $\left[ x \right]_{A{V^1},B}^{\leqslant} = \left[ x \right]_{A{V^2},B}^{\leqslant} = \left[ x \right]_{A{V^3},B}^{\leqslant} $

证明 只需要证明对于 $\forall \left( {x,a} \right) \in U \times A$ 均有 $A{V_a}^1\left( x \right) = A{V_a}^3\left( x \right) = A{V_a}^3\left( x \right)$ 即可。

$\begin{array}{c}AV_a^1\left( x \right) = \displaystyle\frac{1}{2}\left( {\min \left( {{u_a}\left( x \right),N\left( {{v_a}\left( x \right)} \right)} \right) + } \right.\\\left. {\max \left( {{u_a}\left( x \right),N\left( {{v_a}\left( x \right)} \right)} \right)} \right) = \\{\left( {{u_a}\left( x \right) + N\left( {{v_a}\left( x \right)} \right)} \right)/2}\end{array}$
$\begin{array}{c}AV_a^2\left( x \right) = \displaystyle\frac{1}{2}\left( {{u_a}\left( x \right) \cdot N\left( {{v_a}\left( x \right)} \right) + {u_a}\left( x \right) + } \right.\\\left. {N\left( {{v_a}\left( x \right)} \right) - {u_a}\left( x \right) \cdot N\left( {{v_a}\left( x \right)} \right)} \right) = \\\left( {{u_a}\left( x \right) + N\left( {{v_a}\left( x \right)} \right)} \right)/2\end{array}$
$\begin{array}{c}AV_a^3\left( x \right) = \displaystyle\frac{1}{2}\left( {\max \left( {0,{u_a}\left( x \right) + N\left( {{v_a}\left( x \right)} \right) - 1} \right) + } \right.\\\left. {\min \left( {1,{u_a}\left( x \right) + N\left( {{v_a}\left( x \right)} \right)} \right)} \right) = \\\displaystyle\frac{1}{2}\left( {\max \left( {0,{u_a}\left( x \right) - {v_a}\left( x \right)} \right) + } \right.\\\left. {\min \left( {1,{u_a}\left( x \right) + 1 - {v_a}\left( x \right)} \right)} \right)\end{array}$

分两种情况讨论 $AV_a^3\left( x \right)$

1) 若 ${u_a}\left( x \right) - {v_a}\left( x \right) \geqslant 0$ $\min \left( {1,{u_a}\left( x \right) + 1 - {v_a}\left( x \right)} \right) = 1$ $\max \left( {0,{u_a}\left( x \right) - {v_a}\left( x \right)} \right) = {u_a}\left( x \right) - {v_a}\left( x \right)$ ,则有 $AV_a^3 = {u_a}\left( x \right) + $ $ N\left( {{v_a}\left( x \right)} \right)$

2) 若 ${u_a}\left( x \right) - {v_a}\left( x \right) < 0$ ,有 $\min \left( {1,{u_a}\left( x \right) + 1 - {v_a}\left( x \right)} \right) = $ ${u_a}\left( x \right) \!+\! 1 \!-\! {v_a}\left( x \right)$ $\max \left( {0,{u_a}\left( x \right) \!-\! {v_a}\left( x \right)} \right) \!=\! 0$ ,则 $AV_a^3 \!=\! {u_a}\left( x \right) \!+ $ $ N\left( {{v_a}\left( x \right)} \right)$ ,证毕。

例1 假设一场选举中有6个候选人 $\left\{ {{x_1},{x_2}, \cdots ,} \right.$ $\left. {{x_6}} \right\}$ ,一名投票者A对6位候选人的支持意向表示为 $ \left\langle \right. $ 0.8,0, 1 $ \left. \right\rangle $ $ \left\langle \right. $ 0.4,0.2 $ \left. \right\rangle $ $ \left\langle \right. $ 0.5,0.3 $ \left. \right\rangle $ $ \left\langle \right. $ 0.5,0.4 $ \left. \right\rangle $ $ \left\langle \right. $ 0.6,0.4 $ \left. \right\rangle $ $ \left\langle \right. $ 0.4,0.1 $ \left. \right\rangle $ ,利用定义9~定义12的优势关系,计算6位候选人优势关系如下:

1) ${x_4} \leqslant {x_2} \leqslant {x_3} \leqslant {x_5} \leqslant {x_6} \leqslant {x_1}$

2) ${x_2} = {x_6} \leqslant {x_4} = {x_3} \leqslant {x_5} \leqslant {x_1}$

3) ${x_4} = {x_5} \leqslant {x_3} \leqslant {x_2} \leqslant {x_6} = {x_1}$

4) ${x_4} \leqslant {x_2} = {x_3} = {x_5} \leqslant {x_6} \leqslant {x_1}$

结果显示, $\left[ x \right]_{f,B}^{\leqslant} $ 过多关注支持与反对的绝对差, $\left[ x \right]_{T,B}^{\leqslant} $ 侧重于表达相对于属性子集B支持程度绝对高于x的对象集合, $\left[ x \right]_{S,B}^{\leqslant} $ 侧重于表达相对于属性子集B支持程度可能高于x的对象集合, $\left[ x \right]_{AV,B}^{\leqslant} $ 则侧重于表达相对于属性子集B支持程度平均高于x的对象集合。若 $\forall \left( {x,a} \right) \in U \times A$ ,都有 ${\mu _a}\left( x \right) + {\nu _a}\left( x \right) = 1$ ,则 $\left[ x \right]_{f,B}^{\leqslant} = \left[ x \right]_{T,B}^{\leqslant} = \left[ x \right]_{S,B}^{\leqslant} = \left[ x \right]_{AV,B}^{\leqslant} $ 均为普通模糊信息系统中的优势类。

定理3 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $U = \left\{ {{x_1},{x_2}, \cdots ,{x_m}} \right\}$ 。对于 $\forall B \subseteq A$ $\left[ x \right]_{ \cdot ,B}^{\leqslant} $ 是由 $R_{ \cdot ,B}^{\leqslant} $ 生成的优势类, ·表示TSAV 3种算子,则

1) $R_{ \cdot ,B}^{\leqslant} $ 满足自反性和传递性;

2) ${x_j} \in \left[ {{x_k}} \right]_{ \cdot ,B}^{\leqslant} \Leftrightarrow \left[ {{x_j}} \right]_{ \cdot ,B}^{\leqslant} \subseteq \left[ {{x_k}} \right]_{ \cdot ,B}^{\leqslant} $

3) $\left[ {{x_i}} \right]_{ \cdot ,B}^{\leqslant} = \cup \left\{ {\left[ {{x_j}} \right]_{ \cdot ,B}^{\leqslant} :{x_j} \in \left[ {{x_i}} \right]_{ \cdot ,B}^{\leqslant} } \right\}$

4) $U = \bigcup\nolimits_{i = 1}^m {\left[ {{x_i}} \right]_{ \cdot ,B}^{\leqslant} } $

5) $\left[ {{x_i}} \right]_{ \cdot ,B}^{\leqslant} = \left[ {{x_j}} \right]_{ \cdot ,B}^{\leqslant} \Leftrightarrow {T_{{a_i}}}\left( {{x_i}} \right) = {T_{{a_i}}}\left( {{x_j}} \right)$ $\forall {a_i} \in B$

证明 我们以 $\left[ x \right]_{T,B}^{\leqslant} $ 为例,只证明2),其余均可直接由定义14直接证明。

充分性:若 ${x_l} \in \left[ {{x_j}} \right]_{T,B}^{\leqslant} $ ,则对于 $\forall {a_i} \in B$ ${T_{{a_i}}}\left( {{x_j}} \right){\leqslant} $ $ {T_{{a_i}}}\left( {{x_l}} \right)$ ;由于 ${x_j} \in \left[ {{x_k}} \right]_{T,B}^{\leqslant} $ ,则对于 $\forall {a_i} \in B$ ${T_{{a_i}}}\left( {{x_k}} \right) {\leqslant}$ $ {T_{{a_i}}}\left( {{x_j}} \right)$ ,成立,从而 ${x_l} \in \left[ {{x_k}} \right]_{T,B}^{\leqslant} $ ,即 $\left[ {{x_j}} \right]_{T,B}^{\leqslant} \subseteq \left[ {{x_k}} \right]_{T,B}^{\leqslant} $

必要性:根据 $R_{T,B}^{\leqslant} $ 满足自反性有 ${x_j} \in \left[ {x{}_j} \right]_{T,B}^{\leqslant} $ ,由于 $\left[ {{x_j}} \right]_{T,B}^{\leqslant} \subseteq \left[ {{x_k}} \right]_{T,B}^{\leqslant} $ 成立,故有 ${x_j} \in \left[ {x{}_k} \right]_{T,B}^{\leqslant} $

2.2 多粒度优势直觉模糊粗糙集

根据文献[6-9]所提出的多粒度粗糙集的思想,以下给出优势关系下直觉模糊多粒度粗糙集定义。

定义13 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $U = \left\{ {{x_1},{x_2}, \cdots ,{x_m}} \right\}$ ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A$ $\forall X \in U$ $\left[ x \right]_{T,{A_i}}^{\leqslant} $ 是由 $R_{T,{A_i}}^{\leqslant} $ 诱导产生的强优势关系类,则X在强优势关系下乐观多粒度下上近似集合分别为

$\begin{array}{c}\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( X \right) = \left\{ {x \in U:\left[ x \right]_{T,{A_1}}^{\leqslant} \subseteq X \vee \left[ x \right]_{T,{A_2}}^{\leqslant} } \subseteq \right.\\\left. { X \vee \cdots \vee \left[ x \right]_{T,{A_n}}^{\leqslant} \subseteq X} \right\}\\[5pt]\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( X \right) = \sim \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( { \sim X} \right)\end{array}$

式中 $ \sim X$ 表示集合X的补集。

序对 $\left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( X \right),\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( X \right)} \right)$ 称为强优势关系下X的乐观直觉模糊粗糙集。

定义14 设 $\left( {U,A,R} \right)$ 为一个直觉模糊信息系统, $U = \left\{ {{x_1},{x_2}, \cdots ,{x_m}} \right\}$ ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A$ $\forall X \in U$ $\left[ x \right]_{T,{A_i}}^{\leqslant} $ 是由 $R_{T,{A_i}}^{\leqslant} $ 诱导产生的强优势关系类,则X在强优势关系下悲观多粒度下上近似集合分别为

$\begin{array}{c}\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( X \right) = \left\{ {x \in U:\left[ x \right]_{T,{A_1}}^{\leqslant} \subseteq X \wedge \left[ x \right]_{T,{A_2}}^{\leqslant} } \subseteq \right.\\\left. { X \wedge \cdots \wedge \left[ x \right]_{T,{A_n}}^{\leqslant} \subseteq X} \right\},\\\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( X \right) = \sim \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( { \sim X} \right),\end{array}$

式中 $ \sim X$ 表示集合X的补集。

序对 $\left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( X \right),\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( X \right)} \right)$ 称为强优势关系下的悲观直觉模糊粗糙集。

上述定义的乐观多粒度下近似要求至少有一个粒度满足优势关系,而悲观多粒度下近似则要求在所有粒度空间中满足一致的优势关系。多粒度上近似均由下近似的补集定义得到。

3 多粒度优势粗糙直觉模糊集及决策规则获取

在直觉模糊决策信息系统中,由于被近似的决策属性集合是直觉模糊集合,而不是由决策属性确定的等价类集合,因此需要将上述结论进行推广。

3.1 多粒度优势粗糙直觉模糊集

定义15 设 $\left( {U,A \cup \left\{ d \right\},R} \right)$ 为一个直觉模糊决策信息系统,其中 $U = \left\{ {{x_1}, {x_2}, \cdots ,{x_m}} \right\}$ ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A$ ,决策属性d基于强优势关系 $R_{T,{A_i}}^{\leqslant} $ 的多粒度乐观下上近似集分别为

$\begin{array}{l}\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( d \right)\left( x \right) = \vee _{i = 1}^n\left\{ { \wedge \left\{ {d\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\leqslant}} \right\}} \right\}\\[5pt]\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( d \right)\left( x \right) = \wedge _{i = 1}^n\left\{ { \vee \left\{ {d\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\geqslant} } \right\}} \right\}\end{array}$

序对 $\left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( {{f_d}} \right),\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} O}} } \left( {{f_d}} \right)} \right)$ 称为强优势关系下fd的乐观直觉模糊粗糙集。

定义16 设 $\left( {U,A \cup \left\{ d \right\},R} \right)$ 为一个直觉模糊决策信息系统,其中 $U = \left\{ {{x_1},{x_2}, \cdots ,{x_m}} \right\}$ ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A$ ,决策属性d基于强优势关系 $R_{T,{A_i}}^{\leqslant} $ 的多粒度悲观下上近似集分别为

$\begin{array}{c}\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right)\left( x \right) = \wedge _{i = 1}^n\left\{ { \wedge \left\{ {{f_d}\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\leqslant}} \right\}} \right\}\\[5pt]\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {{f_d}} \right)\left( x \right) = \vee _{i = 1}^n\left\{ { \vee \left\{ {{f_d}\left( y \right):y \in \left[ x \right]_{T,{A_i}}^ {\geqslant} } \right\}} \right\}\end{array}$

序对 $\left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( X \right),\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( X \right)} \right)$ 称为强优势关系下fd的悲观直觉模糊粗糙集。

例2  表1为一个关于风险投资的直觉模糊决策信息系统实例。其中 $U = \left\{ {{x_1},{x_2}, \cdots ,{x_8}} \right\}$ 表示风险投资项目; $A = \left\{ {{a_1},{a_2},{a_3},{a_4}} \right\}$ 为条件属性,表示不同领域专家对投资项目所在位置、人口密度、交通状况和投资额度给出的评价,d为决策属性。

表 1 风险决策直觉模糊信息系统 Tab.1 Intuitionistic fuzzy information system with risk decision

若将例2中每个条件属性都看作一个独立的粒度空间,则根据定义15、定义16,决策属性d关于属性集合的多粒度乐观和悲观下、上近似集如下所示。

$\begin{array}{c}\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} O}} } \left( {{f_d}} \right) = \left\{ {\left\langle {0.6,0.3} \right\rangle ,\left\langle {0.6,0.3} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,} \right.\\[4.5pt]\left\langle {0.5,0.4} \right\rangle ,\left\langle {0.4,0.6} \right\rangle \left. {,\left\langle {0.3,0.6} \right\rangle ,\left\langle {0.0,0.9} \right\rangle ,\left\langle {0.6,0.3} \right\rangle } \right\}\\[4.5pt]\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( {{f_d}} \right) = \left\{ {\left\langle {0.7,0.2} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.4,0.6} \right\rangle ,} \right.\\[4.5pt]\left\langle {0.5,0.4} \right\rangle ,\left. {\left\langle {0.4,0.6} \right\rangle ,\left\langle {0.4,0.6} \right\rangle ,\left\langle {0.4,0.6} \right\rangle ,\left\langle {0.6,0.3} \right\rangle } \right\}\\[4.5pt]\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right) = \left\{ {\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,} \right.\\[4.5pt]\left. {\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle ,\left\langle {0.0,1.0} \right\rangle } \right\}\\[4.5pt]\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}P}} } \left( {{f_d}} \right) = \left\{ {\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle } \right.,\\[4.5pt]\left. {\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,} \right\}\end{array}$

定理4 设 $\left( {U,A \cup \left\{ d \right\},R} \right)$ 为直觉模糊决策信息系统,其中 ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A$ ,则:

1) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) \subseteq {f_d} \subseteq \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)$ $\underline {{\mkern 1mu} \sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}P}} } \left( {{f_d}} \right) \subseteq $ ${f_d} \subseteq \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}P}} } \left( {{f_d}} \right)$

2) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) = \cup _{i = 1}^n\underline {R_{T,{A_i}}^{{\leqslant} O}} \left( {{f_d}} \right)$ $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} O}} } \left( {{f_d}} \right) =\cap _{i = 1}^n $ $ \overline {R_{T,{A_i}}^{ {\leqslant} O}} \left( {{f_d}} \right)$

3) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {{f_d}} \right) = \cap _{i = 1}^n\underline {R_{T,{A_i}}^{ {\leqslant} P}} \left( {{f_d}} \right)$ $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right) =\cup _{i = 1}^n $ $ \overline {R_{T,{A_i}}^{ {\leqslant} P}} \left( {{f_d}} \right)$

4) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( {{f_d}} \right)} \right) = \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant}O}} } \left( {{f_d}} \right)$ $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } $ $ \left( {\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( {{f_d}} \right)} \right) = \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)$

5) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right)} \right) = \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right)$ $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } $ $\left( {\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {{f_d}} \right)} \right) = \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} P}} } \left( {{f_d}} \right)$

证明 这里只给出乐观多粒度情况下的证明,悲观多粒度的可类似得到。

1) $\forall x \in U$ ,根据定义15有 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( d \right)\left( x \right) = $ $ \vee _{i = 1}^n\left\{ { \wedge \left\{ {d\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\leqslant} } \right\}} \right\}$ 。由近似关系的自反性知 $ \wedge \left\{ {d\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\leqslant} } \right\}{\leqslant} {f_d}\left( x \right)$ ,从而 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( d \right)\left( x \right){\leqslant} \vee _{i = 1}^n $ $ \left( {{f_d}\left( x \right)} \right) = {f_d}\left( x \right)$ ,即 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) \subseteq {f_d}$ 成立。同理 ${f_d} \subseteq $ $ \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} O}} } \left( {{f_d}} \right)$

2) $\forall x \in U$ ,由定义15有 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)\left( x \right) = \vee _{i = 1}^n $ $ \left\{ { \wedge \left\{ {d\left( y \right):y \in \left[ x \right]_{T,{A_i}}^{\leqslant}} \right\}} \right\} = \vee _{i = 1}^n\underline {R_{T,{A_i}}^{{\leqslant} O}} \left( {{f_d}} \right)\left( x \right)$ ,即 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) = $ $ \cup _{i = 1}^n\underline {R_{T,{A_i}}^{{\leqslant} O}} \left( {{f_d}} \right)$

3) 同理 $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{ {\leqslant} O}} } \left( {{f_d}} \right) = \cap _{i = 1}^n\overline {R_{T,{A_i}}^{ {\leqslant} O}} \left( {{f_d}} \right)$

4) 由1)可知 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)} \right) \subseteq $ $ \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } $ (fd),因此只要证明 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) \! \subseteq \! \underline {\sum\nolimits_{i = 1}^n \! {R_{T,{A_i}}^{{\leqslant} O}} } \! \left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)} \right)$ 即可。由2)可知 $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant}O}} } \left( {\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)} \right) = \cup _{i = 1}^n\underline {R_{T,{A_i}}^{{\leqslant} O}}$ $ \left( {\underline {\sum\nolimits_{i = 1}^n \, {R_{T,{A_i}}^{{\leqslant} O}} } \, \left( {{f_d}} \right)} \right) = \cup _{i = 1}^n \, \underline {R_{T,{A_i}}^{{\leqslant} O}} \, \left( { \cup _{i = 1}^n \, \underline {R_{T,{A_i}}^{{\leqslant} O}} \, \left( {{f_d}} \right)} \right) \supseteq \cup _{i = 1}^n \cup _{i = 1}^n\underline {R_{T,{A_i}}^{{\leqslant} O}} $ $\left( {\underline {R_{T,{A_i}}^{{\leqslant}O}} \left( {{f_d}} \right)} \right) = \cup _{i = 1}^n\underline {R_{T,{A_i}}^{ {\leqslant} O}} \left( {{f_d}} \right) = \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)$ 。类似的,易证 $\left( {\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)} \right) = \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)$

定理5 设 $\left( {U,A \cup \left\{ d \right\},R} \right)$ 为直觉模糊决策信息系统,其中 ${A_1}, {A_2}, \cdots ,{A_n} \subseteq A,$ 则:

1) $\underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {{f_d}} \right) \subseteq \underline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right)$

2) $\overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} O}} } \left( {{f_d}} \right) \subseteq \overline {\sum\nolimits_{i = 1}^n {R_{T,{A_i}}^{{\leqslant} P}} } \left( {{f_d}} \right)$

证明:根据定义15、定义16,此定义易证。

3.2 决策规则获取

经典粗糙集理论中,下近似中的元素对决策规则的支持是确定的,而边界域中的元素对决策规则的支持是不确定的。在此基础上Greco利用优势关系定义了两种由逻辑连词“且”构成的决策规则。根据多粒度思想,由上一小节构造的优势关系多粒度粗糙直觉模糊集可得到逻辑连接词为“或”的两种决策规则,具体形式如下:

“at least”决策规则:

$\begin{array}{c}{T_{{a_1}}}\left( y \right) \geqslant {T_{{a_1}}}\left( x \right) \vee {T_{{a_2}}}\left( x \right) \vee \cdots \vee {T_{{a_n}}}\left( y \right) \geqslant {T_{{a_n}}}\left( x \right) \to \\[5pt]{f_d}\left( y \right) \geqslant \underline {\sum\nolimits_{i = 1}^n {R_{T,{a_i}}^{ \leqslant O}} } \left( {{f_d}} \right)\left( x \right);\end{array}$

“at most”决策规则:

$\begin{array}{c}{T_{{a_1}}}\left( y \right) \leqslant {T_{{a_1}}}\left( x \right) \vee {T_{{a_2}}}\left( x \right) \vee \cdots \vee {T_{{a_n}}}\left( y \right) \leqslant {T_{{a_n}}}\left( x \right) \to \\[5pt]{f_d}\left( y \right) \leqslant \underline {\sum\nolimits_{i = 1}^n {R_{T,{a_i}}^{ \leqslant P}} } \left( {{f_d}} \right)\left( x \right).\end{array}$

例3 根据例2计算结果,可以生成两种“或”决策规则,这里只给出“at least”规则:

1) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.86 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.56 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.04 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.56 \to {f_d}\left( y \right) \geqslant \left( {0.6,0.3} \right) $

2) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.86 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.08 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.10 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.02 \to {f_d}\left( y \right) \geqslant \left( {0.6,0.3} \right) $

3) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.03 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.02 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.02 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.86 \to {f_d}\left( y \right) \geqslant \left( {0.0,1.0} \right) $

4) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.00 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.81 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.81 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.72 \to {f_d}\left( y \right) \geqslant \left( {0.5,0.4} \right) $

5) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.02 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.01 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 1.0 \,\, \vee $ ${T_{{a_4}}}\left( y \right) \geqslant 0.76 \to {f_d}\left( y \right) \geqslant \left( {0.4,0.6} \right) $

6) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.12 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.06 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.90 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.02 \to {f_d}\left( y \right) \geqslant \left( {0.3,0.6} \right) $

7) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.00 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.02 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.81 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 0.08 \to {f_d}\left( y \right) \geqslant \left( {0.0,0.9} \right) $

8) $ {T_{{a_1}}}\left( y \right) \,\geqslant\, 0.86 \vee {T_{{a_2}}}\left( y \right) \,\geqslant\, 0.81 \vee {T_{{a_3}}}\left( y \right) \,\geqslant\, 0.02 \,\, \vee $ $ {T_{{a_4}}}\left( y \right) \geqslant 1.0 \to {f_d}\left( y \right) \geqslant \left( {0.6,0.3} \right) $

4 结束语

本文将多粒度的基本思想引入到直觉模糊决策信息系统中,利用t-模及t-余模定义了3种新的优势关系。分析这3种优势关系所表达的不同语义,在此基础上提出了广义优势关系下多粒度直觉模糊粗糙集模型。通过该模型的主要性质进行讨论,这种模型使得多粒度方法能够有效处理直觉模糊决策信息系统中直觉模糊概念近似和规则提取等问题。最后结合实例,具体给出了在直觉模糊决策信息系统中,逻辑连接词为“或”的决策规则。

在本文基础上,将深入研究3种优势关系之间的内在联系,以及如何利用辨识矩阵和启发算法获得广义优势关系多粒度粗糙直觉模糊集模型的属性约简。

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