优势关系下多粒度粗糙集的三支决策

引用本文

尚蓉, 马建敏, 赵东辉. 优势关系下多粒度粗糙集的三支决策[J]. 郑州大学学报(理学版), 2026, 58(5): 86-94.

SHANG Rong, MA Jianmin, ZHAO Donghui. Three-way Decisions of Multi-granulation Rough Sets Based on Dominance Relation[J]. Journal of Zhengzhou University(Natural Science Edition), 2026, 58(5): 86-94.

基金项目

国家自然科学基金项目(61772019); 科技部国家重点研发计划项目(2022YFC3303100); 陕西省自然科学基金项目(2021JQ-2B)

通信作者

马建敏(1978—), 女, 教授，主要从事粗糙集、概念格和粒计算、统计学习方法和数据分析研究，E-mail: cjm-zm@126.com

作者简介

尚蓉(1999—), 女, 硕士研究生，主要从事粗糙集与三支决策研究，E-mail: shangrong9911@163.com

文章历史

收稿日期：2024-12-06

Contents Abstract Full text Figures/Tables PDF

优势关系下多粒度粗糙集的三支决策

尚蓉¹, 马建敏¹, 赵东辉²

1. 长安大学理学院陕西西安 710064;
2. 杭州数澜科技有限公司浙江杭州 311100

收稿日期：2024-12-06

基金项目：国家自然科学基金项目(61772019); 科技部国家重点研发计划项目(2022YFC3303100); 陕西省自然科学基金项目(2021JQ-2B)

作者简介：尚蓉(1999—), 女, 硕士研究生，主要从事粗糙集与三支决策研究，E-mail: shangrong9911@163.com.

通信作者：马建敏(1978—), 女, 教授，主要从事粗糙集、概念格和粒计算、统计学习方法和数据分析研究，E-mail: cjm-zm@126.com.

摘要：多粒度粗糙集和三支决策都是用来处理不确定性问题的理论方法, 两者的结合可以为处理决策问题提供一种新的视角。针对数值型偏序关系数据集, 引入优势粗糙集的思想, 讨论多个属性粒度结构下的多粒度粗糙集模型及其上的三支决策方法。首先, 基于优势关系建立多个属性下的粒结构, 并在此基础上提出优势关系下的优势乐观多粒度粗糙集、优势悲观多粒度粗糙集和优势平均多粒度粗糙集, 讨论它们的性质及关系。根据不同粒结构下的损失函数, 讨论优势关系下多粒度粗糙集的三支决策规则。

关键词：优势乐观/悲观多粒度粗糙集优势平均多粒度粗糙集多粒度结构平均期望损失

Three-way Decisions of Multi-granulation Rough Sets Based on Dominance Relation

SHANG Rong¹, MA Jianmin¹, ZHAO Donghui²

1. School of Science, Chang'an University, Xi'an 710064, China;
2. Hangzhou DTWave Technology Co., Ltd., Hangzhou 311100, China

Abstract: Both multi-granulation rough set and three-way decisions were theoretical methods used to deal with uncertainty problems. The combination of both provided a new perspective for decision-making problems. For numerical partial relation dataset, by applying the idea of dominant rough set, multi-granulation rough set models with multiple attribute granulation structures and the corresponding three-way decisions were discussed. Firstly, the granulation structures under multiple attribute subsets were established based on the dominance relations. On which, the dominant optimistic multi-granulation rough set, dominant pessimistic multi-granulation rough set and dominant average multi-granulation rough set were then proposed. The properties and relationships of them were also investigated. According to the loss functions based on different granulation structures, three-way decision rules of three kinds of dominant multi-granulation rough sets were depicted.

Key words: dominant optimistic/pessimistic multi-granulation rough set dominant average multi-granulation rough set multi-granulation structure average expected loss

0 引言

Pawlak^[1]提出的粗糙集理论, 其核心思想是通过等价关系引入上、下近似集来表示目标概念, 实现对数据的分类。现已经被广泛应用于特征选择、故障诊断、机器学习^[2-4]等领域。相较于传统的粗糙集理论, Qian等^[5]提出的多粒度粗糙集模型从多个粒结构角度提供更有效的分类信息。Qian等^[6]将多粒度粗糙集与决策理论粗糙集相结合, 提出均值、乐观和悲观多粒度决策理论粗糙集。胡谦等^[7]从多粒度近似空间出发, 计算每个对象在不同粒度下的重要度, 构建对象之间的相似度。于鹏等^[8]在平均型聚合算子的基础上构造了一种新的多粒度粗糙集模型。Sun等^[9]提出模糊邻域多粒度粗糙集模型, 研究了该模型的不确定性测度方法。

由于等价条件过于严格, 使得面对复杂问题时容错能力较弱且不易解决噪声问题。对此, 学者们研究了更宽松条件下的情况。张杰等^[10]基于相似关系, 提出局部粗糙集理论模型, 减少了时间复杂度。Yang等^[11]通过容差关系, 研究多源模糊环境下的乐观、悲观和广义多粒度粗糙集模型。Chen等^[12]为了提高容错性, 提出基于变精度领域的邻域多粒度粗糙集, 研究不同属性子集下权重的获取方法。Hu等^[13]在有序信息系统中, 研究了基于矩阵的动态优势的多粒度粗糙集方法, 提高了知识更新的计算效率。

Yao^[14-15]在粗糙集的基础上提出三支决策的概念, 其主要思想是“三分而治”。随着研究的深入, 多粒度思想和三支决策相结合已成为研究复杂决策问题的重要趋势。王茜等^[16]提出不完备邻域加权限制多粒度决策理论粗糙集, 通过可变三支决策得到决策规则。Pang等^[17]在计算权重时引入多粒化思想, 并利用多粒属性权重确定各方案的评价值, 最终将方案集划分为接受、不确定和拒绝三个区域。徐伟华等^[18]提出两种区间犹豫模糊多粒度粗糙集, 并给出三支决策规则。薛占熬等^[19]提出集对优势关系下的多粒度决策粗糙集, 并构建了可变三支决策模型。

在决策过程中, 优势关系可提供明确的偏好准则, 判断哪些对象在属性集上优于其他对象。通过优势关系研究多粒度粗糙集, 可以帮助决策者在不同粒度下进行更精细的分析, 提高决策的可靠性。此外, 结合三支决策理论研究成本损失, 有利于降低决策风险。因此, 本文研究具有偏好关系的数值型数据集上基于优势关系的多粒度粗糙集及其对应的三支决策规则。首先提出多粒度结构下的乐观、悲观和平均多粒度粗糙集。并通过不同粒结构下的损失函数, 考虑每种多粒度粗糙集下阈值的获取规则, 分别得到三种多粒度粗糙集的三支决策规则。

1 预备知识 1.1 多粒度粗糙集

称四元组IS=(U, AT, V, f)为信息系统^[20], 其中U为非空有限对象集, 称为论域, AT为非空有限属性集。$V=\bigcup\limits_{a \in A T} V_a$, V_a是属性a的值域。∀x∈U, a∈AT, f(x, a)∈V_a。

定义1^[6] 设IS=(U, AT, V, f)为信息系统, A₁, A₂, …, A_m⊆AT。∀X⊆U, X关于A₁, A₂, …, A_m的乐观多粒度下、上近似分别定义为

$ \begin{aligned} & \underline{\sum\limits_{i=1}^m A_i^o}(X)=\left\{x \in U \mid[x]_{A_1} \subseteq X \vee \cdots \vee[x]_{A_m} \subseteq X\right\}, \\ & \overline{\sum\limits_{i=1}^m A_i^o}(X)=\sim\left(\sum\limits_{i=1}^m A_i^o(\sim X)\right) 。\end{aligned} $

其中：[x]_{A_i}是x关于属性子集A_i的等价类；~X是X的补集。称$\left(\underline{\sum\limits_{i=1}^m A_i^o}(X), \overline{\sum\limits_{i=1}^m A_i^o}(X)\right)$为X的乐观多粒度粗糙集。

X关于属性子集A₁, A₂, …, A_m的悲观多粒度下、上近似分别定义为

$ \begin{aligned} & {\underline{\sum\limits_{i=1}^m A_i^P}(X)}=\left\{x \in U \mid[x]_{A_1} \subseteq X \wedge \cdots \wedge[x]_{A_m} \subseteq X\right\}, \\ & \overline{\sum\limits_{i=1}^m A_i^P}(X)=\sim\left(\underline{\sum\limits_{i=1}^m A_i{ }^P}(\sim X)\right) 。\end{aligned} $

称$\left(\underline{\sum\limits_{i=1}^m A_i^P}(X), \overline{\sum\limits_{i=1}^m A_i^P}(X)\right)$为X的悲观多粒度粗糙集。

1.2 三支决策

设U为有限非空论域。∀X⊆U, U被划分为三个互不相交的部分, 即正域POS(X)、边界域BND(X)和负域NEG(X)。记Ω={X, ~X}为状态集, 分别表示对象属于X和不属于X。行动集Λ={a_P, a_B, a_N}, a_P、a_B、a_N分别表示采取接受、延迟和拒绝的三种行动。表 1给出相应的损失函数, 其中λ_PP, λ_BP和λ_NP为对象属于状态X时, 分别采取行动a_P, a_B和a_N对应的损失, λ_PN, λ_BN和λ_NN为对象不属于状态X时, 分别采取a_P, a_B和a_N行动的损失^[14-15]。

表 1 损失函数 Tab. 1 Loss function

给定损失函数, 采取行动a_i(i=P, B, N)的期望损失可表示为^[14]

$ \begin{aligned} & R\left(a_P \mid[x]_R\right)=\lambda_{P P} \operatorname{Pr}\left(X \mid[x]_R\right)+\lambda_{P N} \operatorname{Pr}\left(\sim X \mid[x]_R\right), \\ & R\left(a_B \mid[x]_R\right)=\lambda_{B P} \operatorname{Pr}\left(X \mid[x]_R\right)+\lambda_{B N} \operatorname{Pr}\left(\sim X \mid[x]_R\right), \\ & R\left(a_N \mid[x]_R\right)=\lambda_{N P} \operatorname{Pr}\left(X \mid[x]_R\right)+\lambda_{N N} \operatorname{Pr}\left(\sim X \mid[x]_R\right), \end{aligned} $

其中：[x]_R表示对象x在等价关系R下的等价类；Pr (X [x]_R)表示条件概率。根据贝叶斯决策过程, 可得最小风险决策规则如下^[14]:

(P) 若R (a_P [x]_R)≤R (a_B [x]_R)且R (a_P [x]_R)≤R (a_N [x]_R), 则x∈POS(X),

(B) 若R (a_B [x]_R)≤R (a_P [x]_R)且R (a_B [x]_R)≤R (a_N [x]_R), 则x∈BND(X),

(N) 若R (a_N [x]_R)≤R (a_P [x]_R)且R (a_N [x]_R)≤ R (a_B [x]_R), 则x∈NEG(X)。

当0≤λ_PP≤λ_BP＜λ_NP, 0≤λ_NN≤λ_BN＜λ_PN时, 上述规则简化为^[14]

(P1) 若Pr (X [x]_R)≥α, 则x∈POS(X),

(B1) 若β＜Pr (X [x]_R)＜α, 则x∈BND(X),

(N1) 若Pr (X [x]_R)≤β, 则x∈NEG(X), 其中：

$ \begin{aligned} & \alpha=\frac{\lambda_{P N}-\lambda_{B N}}{\left(\lambda_{P N}-\lambda_{B N}\right)+\left(\lambda_{B P}-\lambda_{P P}\right)}, \\ & \beta=\frac{\lambda_{B N}-\lambda_{N N}}{\left(\lambda_{B N}-\lambda_{N N}\right)+\left(\lambda_{N P}-\lambda_{B P}\right)^{\circ}} \end{aligned} $

2 优势关系下的多粒度粗糙集 2.1 优势乐观多粒度粗糙集

设IS=(U, AT, V, f)为信息系统。∀B⊆AT, 属性子集B的优势关系^[21]可定义为R_B^≥={(x, y)∈ U×U|f(x, a)≥f(y, a), ∀a∈B}。显然, R_B^≥满足自反性和传递性, 但不满足对称性。对任意x∈U, x关于属性子集B的优势类记为[x]_B^≥={y∈U|f(x, a)≥f(y, a), ∀a∈B}。

∀B₁, B₂, …, B_m⊆AT, R_{B_i}^≥(1≤i≤m)为B_i的优势关系。记任意属性子集的单粒度结构为G_{B_i}^≥={[x]_{B_i}^≥|x∈U}, 则信息系统IS上基于B₁, B₂, …, B_m的多粒度结构记为G={G_{B_i}^≥|B_i⊆AT}。

定义2 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 阈值0≤β＜α≤1, X基于多粒度结构G的优势乐观多粒度下、上近似分别定义为

$ \begin{aligned} & \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant O, \alpha}}(X)=\left\{x \in U \mid \bigvee_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha\right\}, \\ & \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant O, \beta}}(X)=U-\left\{x \in U \mid \bigvee_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \beta\right\} 。\end{aligned} $

当$\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant 0, \alpha}}(X) \neq \quad \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant 0, \beta}}(X)$时, 称$\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant 0, \alpha}}(X) , \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant 0, \beta}}(X)$为优势乐观多粒度粗糙集。由此下、上近似定义正域、边界域和负域分别为

$ \begin{aligned} & P O S_{\sum\limits_{i=1}^m R_{B_i}^≥}^O(X)=\underline{{\sum\limits_{i=1}^m R_{B_i}}}{ }^{O, \alpha}(X), \\ & B N D_{\sum\limits_{i=1}^m R_{B_i}^≥}^O(X)=\overline{\sum\limits_{i=1}^m R_{B_i}^{Z^O, \beta}}(X)-\underline{\sum\limits_{i=1}^m R_{B_i}{ }^{O, \alpha}}(X), \\ & N E G_{\sum\limits_{i=1}^m R_{B_i}^≥}^O(X)=U-\overline{\sum\limits_{i=1}^m R_{B_i}^{Z^O}}(X) \text { 。} \end{aligned} $

性质1 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥: B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 下列性质成立:

$ \text { 1) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta}}(X)=\left\{x \in U \mid \bigwedge\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)>\beta\right\} \text {, } $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, 1-\alpha}}(X)=\sim \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \alpha}}(\sim X) \text { 。} $

性质2 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 下面的1)、2)成立，

$ \text { 1) } \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \alpha}}(X)=\bigcup\limits_{i=1}^m {\underline{{R^\geqslant_{B_i}}}}_\alpha(X) \text {, } $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta}}(X)=\bigcap\limits_{i=1}^m \overline{R_{{B_{i_{\beta}}}}^{\geqslant}}(X) \text {, } $

其中：$\underline{R_{B_i}^{\geqslant}}_\alpha(X)=\left\{x \in U \mid \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha\right.$；$\overline{R_{B_{i_\beta}}^{\geqslant}}(X)=\left\{x \in U \mid \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)>\beta\right\}$。

性质3 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X, X_i⊆U, α, α_i∈(0, 1), β, β_i∈(0, 1), i=1, 2, 性质1)~3)成立，

$ \begin{aligned} & \text { 1) } X_1 \subseteq X_2 \Rightarrow \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant O, \alpha}}\left(X_1\right) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \alpha}}\left(X_2\right), \\ & \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta}}\left(X_1\right) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta}}\left(X_2\right) ; \end{aligned} $

$ \text { 2) } \alpha_1 \leqslant \alpha_2 \Rightarrow \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{o, \alpha_2}}}(X) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant 0, \alpha_1}}(X) \text {, } $

$ \text { 3) } \beta_1 \leqslant \beta_2 \Rightarrow \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta_2}}(X) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta_1}}(X) \text { 。} $

2.2 优势悲观多粒度粗糙集

定义3 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 阈值0≤β＜α≤1, X基于多粒度结构G的优势悲观多粒度下、上近似分别定义为

$ \begin{aligned} & \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \alpha}}(X)=\left\{x \in U \mid \bigwedge_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha\right\}, \\ & \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \beta}}(X)=U-\left\{x \in U \mid \bigwedge_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \beta\right\}_{\circ} \end{aligned} $

当$\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \alpha}}(X) \neq \quad \overline{\sum\limits_{i=1}^m R_{B_i}^{\not P^{P, \beta}}}(X)$时, 称$\left(\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \alpha}}}(X), \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \beta}}}(X)\right)$为优势悲观多粒度粗糙集。由此下、上近似定义正域、边界域和负域分别为

$ \begin{gathered} P O S_{\sum\limits_{i=1}^P R_{B_i}^\geqslant}(X)=\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \alpha}}(X), \\ B N D_{\sum\limits_{i=1}^P R_{B_i}^\geqslant}^P(X)=\overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \beta}}}(X)-\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \alpha}}}(X), \\ N E G_{\sum\limits_{i=1}^P R_{B_i}^\geqslant}(X)=U-\overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \beta}}}(X) \text { 。} \end{gathered} $

性质4 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 性质1)、2)成立

$ \text { 1) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geq^{P, \beta}}}(X)=\left\{x \in U \mid \bigvee_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{≥}\right)>\beta\right\} \text {, } $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, 1-\alpha}}}(X)=\sim \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \alpha}}}(\sim X) \text { 。} $

性质5 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 性质1)、2)成立：

$ \text { 1) }\underline{ \sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \alpha}}(X)=\bigcap\limits_{i=1}^m \underline{R_{{B_i}}^{\geqslant}}_\alpha(X) \text {, } $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \beta}}(X)=\bigcup\limits_{i=1}^m \overline{R_{B_{i_\beta}}^{\geqslant}}(X) \text { 。} $

性质6 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X, X_i⊆U, α, α_i∈(0, 1), β, β_i∈(0, 1), i=1, 2, 性质1)~3)成立，

$ \begin{array}{r}\text { 1) } X_1 \subseteq X_2 \Rightarrow \underline{\sum\limits_{i=1}^m R_{B_i}^{≥, \alpha}}\left(X_1\right) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{≥^{P, \alpha}}}\left(X_2\right) \text {, } \\ \underline{\sum\limits_{i=1}^m R_{B_i}^{≥, \beta}}\left(X_1\right) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{≥^{P, \beta}}}\left(X_2\right) \text {, }\end{array} $

$ \text { 2) } \alpha_1 \leqslant \alpha_2 \Rightarrow \underline{\sum\limits_{i=1}^{\mathrm{m}} R_{B_i}^{\geqslant P, \alpha_2}}(X) \subseteq \underline{\sum\limits_{i=1}^{\mathrm{m}} R_{B_i}^{\geqslant P, \alpha_1}}(X) \text {, } $

$ \text { 3) } \beta_1 \leqslant \beta_2 \Rightarrow \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \beta_2}}}(X) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant^{P, \beta_1}}}(X) \text { 。} $

2.3 优势平均多粒度粗糙集

定义4 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 阈值0≤β＜α≤1, X基于多粒度结构G的优势平均多粒度下、上近似分别定义为

$ \begin{aligned} & \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}(X)=\left\{x \in U \left\lvert\, \frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha\right.\right\}, \\ & \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \beta}}(X)=U-\left\{x \in U \left\lvert\, \frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \beta\right.\right\}_{\circ} \end{aligned} $

当$\underline{\sum\limits_{i=1}^m R_{B_i}{ }^{M, \alpha}}(X) \neq \overline{\sum\limits_{i=1}^m R_{B_i}{ }^{M, \beta}}(X)$时, 称$\underline{\sum\limits_{i=1}^m R_{B_i}{ }^{M, \alpha}}(X) , \overline{\sum\limits_{i=1}^m R_{B_i}{ }^{M, \beta}}(X)$为优势平均多粒度粗糙集。由此下、上近似定义正域、边界域和负域分别为

$ \begin{gathered} P O S_{\sum\limits_{i=1}^M R_{B_i}^≥}^M(X)=\underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}(X), \\ B N D_{\sum\limits_{i=1}^M R_{B_i}^≥}^M(X)=\overline{\sum\limits_{i=1}^m R_{B_i}^{\nexists M, \beta}}(X)-\underline{\sum\limits_{i=1}^m R_{B_i}^{≥ M, \alpha}(X),} \\ N E G_{\sum\limits_{i=1}^M R_{B_i}^≥}^M(X)=U-\overline{\sum\limits_{i=1}^m R_{B_i}^{≥ M, \beta}}(X) \text { 。} \end{gathered} $

性质7 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 性质1)、2)成立,

$ \begin{aligned} &\text { 1) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \beta}}(X)=\\ &\left\{x \in U \left\lvert\, \frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)>\beta\right.\right\}, \end{aligned} $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{≥^{M, 1-\alpha}}}(X)=\sim \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}(\sim X) \text { 。} $

性质8 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X, X_i⊆U, α, α_i∈(0, 1), β, β_i∈(0, 1), i=1, 2, 性质1)~3)成立，

$ \begin{array}{l} & \text { 1) } X_1 \subseteq X_2 \Rightarrow \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}\left(X_1\right) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}\left(X_2\right), \\ & \overline{\sum\limits_{i=1}^m R_{\beta_i}^{\geqslant M, \beta}}\left(X_1\right) \subseteq \overline{\sum\limits_{i=1}^m R_{\beta_i}^{\geqslant M, \beta}}\left(X_2\right), \end{array} $

$ \text { 2) } \alpha_1 \leqslant \alpha_2 \Rightarrow \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha_2}}(X) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha_1}}(X) \text {, } $

$ \text { 3) } \beta_1 \leqslant \beta_2 \Rightarrow \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \beta_2}}(X) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \beta_1}}(X) \text { 。} $

性质9 设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 0≤β＜α≤1, 性质1)、2)成立，

$ \text { 1) } \underline{\sum\limits_{i=1}^m R_{B_i}^{≥^{P, \alpha}}}(X) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \alpha}}(X) \subseteq \underline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \alpha}}(X) \text {, } $

$ \text { 2) } \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant o, \beta}}(X) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant M, \beta}}(X) \subseteq \overline{\sum\limits_{i=1}^m R_{B_i}^{\geqslant P, \beta}}(X) \text { 。} $

性质9给出了优势关系下平均多粒度粗糙集、乐观多粒度粗糙集和悲观多粒度粗糙集之间的关系。

3 优势多粒度粗糙集的三支决策

设IS=(U, AT, V, f)为信息系统, B₁, B₂, …, B_m⊆AT, G={G_{B_i}^≥|B_i⊆AT}为其多粒度结构。∀X⊆U, 记Ω={X, ~X}为状态集, Λ={a_P, a_B, a_N}为行动集。λ_PPⁱ, λ_BPⁱ和λ_NPⁱ分别表示对象属于状态X时, 在第i个粒结构G_{B_i}下采取行动a_P, a_B和a_N对应的风险损失值, λ_PNⁱ, λ_BNⁱ和λ_NNⁱ分别表示对象不属于状态X时, 在第i个粒结构G_{B_i}下采取a_P, a_B和a_N行动的风险损失值。表 2给出了对象在粒结构G_{B_i}下的损失函数。

表 2 粒结构G_{B_i}下的损失函数 Tab. 2 Loss function based on granular structureG_{B_i}

根据表 2, 在单粒度结构G_{B_i}^≥下分别采取行动a_P, a_B, a_N的期望损失为

$ \left\{\begin{array}{l} R\left(a_P \mid[x]_{B_i}^{\geqslant}\right)=\lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right), \\ R\left(a_B \mid[x]_{B_i}^{\geqslant}\right)=\lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right), \\ R\left(a_N \mid[x]_{B_i}^{\geqslant}\right)=\lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)。\end{array}\right. $

(1)

3.1 优势乐观多粒度粗糙集的三支决策

公式(1)给出了基于B_i的单粒度结构G_{B_i}^≥下分别采取行动a_P, a_B, a_N的期望损失。考虑多粒度结构G={G_{B_i}^≥|B_i⊆AT}下优势乐观多粒度粗糙集的决策规则及阈值的获取。根据贝叶斯决策准则, 最小风险决策规则为

(OP) 若存在B_i, 使R (a_P [x]_{B_i}^≥)≤R (a_B [x]_{B_i}^≥) 且R (a_P [x]_{B_i}^≥)≤R (a_N [x]_{B_i}^≥), 则$x \in {POS}_{\sum\limits_{i=1}^m R_{B_i}^{≥}}^o(X)$。

(ON) 若对任意B_i, 使R (a_N [x]_{B_i}^≥)≤R (a_P [x]_{B_i}^≥)且R (a_N [x]_{B_i}^≥)≤R (a_B [x]_{B_i}^≥), 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^≥}^o(X)$。(OB) 否则, $x \in B N D_{\sum\limits_{i=1}^m R_{B_i}^≥}^o(X)$。

对规则(OP), 存在B_i⊆AT,

$ \begin{aligned} & R\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\lambda_{P N}^i-\lambda_{B N}^i}{\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}, \\ & \operatorname{R}\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \leqslant \operatorname{R}\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\lambda_{P N}^i-\lambda_{N N}^i}{\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)} \circ \end{aligned} $

对规则(ON), 对任意B_i⊆AT,

$ \begin{aligned} & R\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+ \\ & \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+ \\ & \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \frac{\lambda_{P N}^i-\lambda_{N N}^i}{\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)}, \\ & R\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \end{aligned} $

$ \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\lambda_{B N}^i-\lambda_{N N}^i}{\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)}, $

记

$ \left\{\begin{array}{l} \alpha_i=\frac{\lambda_{P N}^i-\lambda_{B N}^i}{\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}, \\ \beta_i=\frac{\lambda_{B N}^i-\lambda_{N N}^i}{\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)}, \\ \gamma_i=\frac{\lambda_{P N}^i-\lambda_{N N}^i}{\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)} 。\end{array}\right. $

(2)

当0≤λ_PPⁱ≤λ_BPⁱ＜λ_NPⁱ, 0≤λ_NNⁱ≤λ_BNⁱ＜λ_PNⁱ, 且(λ_PNⁱ-λ_BNⁱ)(λ_NPⁱ-λ_BPⁱ)>(λ_BPⁱ-λ_PPⁱ)(λ_BNⁱ-λ_NNⁱ)时, 可得0≤β_i＜γ_i＜α_i≤1。取阈值α_O=α_i, β_O=∧β_i, 得0≤β_O≤β_i＜α_O≤1, 故上述规则化简为

(OP1) 若存在B_i, 使Pr (X [x]_{B_i}^≥)≥α_O, 则$x \in P O S_{\sum\limits_{i=1}^m R_{B_i}^≥}^o(X)$。

(ON1) 若对任意B_i, 有Pr (X [x]_{B_i}^≥)≤β_O, 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^{≥}}^o(X)$。

(OB1) 否则, $x \in B N D_{\sum\limits_{i=1}^m R_{B_i}^{≥}}^o(X)$。

3.2 优势悲观多粒度粗糙集的三支决策

根据公式(1)给出的期望损失, 考虑多粒度结构G={G_{B_i}^≥|B_i⊆AT}下优势悲观多粒度粗糙集的决策规则及阈值的获取。根据贝叶斯决策准则可得最小风险决策规则为

(PP) 若对任意B_i, 使R (a_P [x]_{B_i}^≥)≤R (a_B [x]_{B_i}^≥)且R (a_P [x]_{B_i}^≥)≤R (a_N [x]_{B_i}^≥), 则$x \in \operatorname{POS}_{\sum\limits_{i=1}^m R_{B_i}^≥}^P(X)$。

(PN) 若存在B_i, 使R (a_N [x]_{B_i}^≥)≤R (a_P [x]_{B_i}^≥)且R (a_N [x]_{B_i}^≥)≤R (a_B [x]_{B_i}^≥), 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^≥}^P(X)$。

(PB) 否则, $x \in B N D_{\sum\limits_{i=1}^P R_{B_i}^≥}^{P}(X)$。

对规则(PP), 对任意B_i⊆AT,

$ \begin{aligned} & R\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\lambda_{P N}^i-\lambda_{B N}^i}{\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}=\alpha_i, \\ & R\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+ \\ & \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+ \end{aligned} $

$ \begin{aligned} & \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \\ & \frac{\lambda_{P N}^i-\lambda_{N N}^i}{\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)}=\gamma_{i \circ} \end{aligned} $

对规则(PN), 存在B_i⊆AT,

$ \begin{aligned} & R\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\lambda_{P N}^i-\lambda_{N N}^i}{\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)}=\gamma_i, \\ & R\left(a_N \mid[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \\ & \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\lambda_{B N}^i-\lambda_{N N}^i}{\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)}=\beta_{i \circ} \end{aligned} $

当0≤λ_PPⁱ≤λ_BPⁱ＜λ_NPⁱ, 0≤λ_NNⁱ≤λ_BNⁱ＜λ_PNⁱ, 且(λ_PNⁱ-λ_BNⁱ)(λ_NPⁱ-λ_BPⁱ)>(λ_BPⁱ-λ_PPⁱ)(λ_BNⁱ-λ_NNⁱ)时, 有0≤β_i＜γ_i＜α_i≤1。取α_P=∨α_i, β_P=β_i, 则0≤β_P＜α_i≤α_P≤1, 悲观三支决策规则化简为

(PP1) 若对任意B_i, Pr (X [x]_{B_i}^≥)≥α_P, 则$x \in \operatorname{POS}_{\sum\limits_{i=1}^m R_{B_i}^≥}^P(X)$。

(PN1) 若存在B_i, Pr (X [x]_{B_i}^≥)≤β_P, 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^≥}^P(X)$。

(PB1) 否则, $x \in B N D_{\sum\limits_{i=1}^m R_{B_i}^≥}^{{ }_P}(X)$。

3.3 优势平均多粒度粗糙集的三支决策

考虑多粒度结构G={G_{B_i}^≥|B_i⊆AT}下平均优势多粒度粗糙集的决策规则及阈值的获取，则采取行动a_P, a_B, a_N的平均期望损失定义为

$ \left\{\begin{array}{l} R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)=\frac{1}{m} \sum\limits_{i=1}^m R\left(a_P \mid[x]_{B_i}^{\geqslant}\right)= \\ \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right), \\ R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)=\frac{1}{m} \sum\limits_{i=1}^m R\left(a_B \mid[x]_{B_i}^{\geqslant}\right)= \\ \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right), \\ R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)=\frac{1}{m} \sum\limits_{i=1}^m R\left(a_N \mid[x]_{B_i}^{\geqslant}\right)= \\ \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right)。\end{array}\right. $

根据贝叶斯决策准则得最小风险决策规则为

（MP）若$R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，且$R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，则 $x \in P O S_{\sum\limits_{i=1}^≥ R_{B_i}^m}^M(X)$。

（MN）若$R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，且$R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，则 $x \in NEG_{\sum\limits_{i=1}^≥ R_{B_i}^m}^M(X)$。

（MB）若$R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，且$R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right)$，则 $x \in BND_{\sum\limits_{i=1}^≥ R_{B_i}^m}^M(X)$。

应用Pr (X [x]_{B_i}^≥)+Pr (~X [x]_{B_i}^≥)=1化简上述规则。因为对任意a_i≥0, b_i≥0, $\sum\limits_{i=1}^m a_i b_i \leqslant\sum\limits_{i=1}^m a_i \sum\limits_{i=1}^m b_i$, 所以当0≤λ_PPⁱ≤λ_BPⁱ＜λ_NPⁱ和0≤λ_NNⁱ≤λ_BNⁱ＜λ_PNⁱ时, 关系1)~3)成立，

1) $\sum\limits_{i=1}^m\left[\left(\lambda_{B P}^i-\lambda_{P P}^i\right)+\left(\lambda_{P N}^i-\lambda_{B N}^i\right)\right] \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant\sum\limits_{i=1}^m\left[\left(\lambda_{B P}^i-\lambda_{P P}^i\right)+\left(\lambda_{P N}^i-\lambda_{B N}^i\right)\right] \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)$,

2) $\sum\limits_{i=1}^m\left[\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)\right] \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant\sum\limits_{i=1}^m\left[\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)\right] \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)$,

3) $\sum\limits_{i=1}^m\left[\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)\right] \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant\sum\limits_{i=1}^m\left[\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)\right] \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)$。

对规则(MP), 则

$ \begin{aligned} & R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}, \end{aligned} $

$ \begin{aligned} & R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)} 。\end{aligned} $

对规则(MN), 则

$ \begin{aligned} & R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)} \\ & R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)} 。\end{aligned} $

对规则(MB), 则

$ \begin{aligned} & R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_P \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{P P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{P N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}, \\ & R\left(a_B \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \leqslant R\left(a_N \mid \sum\limits_{i=1}^m[x]_{B_i}^{\geqslant}\right) \Leftrightarrow \\ & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{B P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{B N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \leqslant \end{aligned} $

$ \begin{aligned} & \frac{1}{m}\left(\sum\limits_{i=1}^m \lambda_{N P}^i \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)+\sum\limits_{i=1}^m \lambda_{N N}^i \operatorname{Pr}\left(\sim X \mid[x]_{B_i}^{\geqslant}\right)\right) \Rightarrow \\ & \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \frac{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)} 。\end{aligned} $

故决策规则化简为

(MP1) 若$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha_M$, 且$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \gamma_M$, 则$x \in \operatorname{POS}_{\sum\limits_{i=1}^m R_{B_i}^{\geqslant}}^M(X)$。

(MN1) 若$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \gamma_M$，且$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \beta_M$, 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^{≥}}^M(X)$。

(MB1) 若$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \alpha_M$，且$\sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \beta_M$, 则$x \in B N D_{\sum\limits_{i=1}^m R_{B_i}^≥}^P(X)$。

其中：阈值α_M, β_M, γ_M分别为

$ \left\{\begin{array}{l} \alpha_M=\frac{1}{m} \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)+\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}, \\ \beta_M=\frac{1}{m} \frac{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{B P}^i\right)}, \\ \gamma_M=\frac{1}{m} \frac{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)+\left(\lambda_{N P}^i-\lambda_{P P}^i\right)} 。\end{array}\right. $

(3)

当$\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right) \sum\limits_{i=1}^m\left(\lambda_{N P}^i-\lambda_{B P}^i\right)>\sum\limits_{i=1}^m\left(\lambda_{B P}^i-\right.\left.\lambda_{P P}^i\right) \sum\limits_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)$时, 有β_M≤α_M。由$\frac{a}{c}>$\frac{b}{d} \Rightarrow \frac{a}{c}>\frac{a+b}{c+d}>\frac{b}{d}$, 可得$\frac{\sum\limits_{i=1}^m\left(\lambda_{N P}^i-\lambda_{B P}^i\right)}{\sum_{i=1}^m\left(\lambda_{B N}^i-\lambda_{N N}^i\right)}>\frac{\sum\limits_{i=1}^m\left(\lambda_{N P}^i-\lambda_{P P}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{N N}^i\right)}>\frac{\sum\limits_{i=1}^m\left(\lambda_{B P}^i-\lambda_{P P}^i\right)}{\sum\limits_{i=1}^m\left(\lambda_{P N}^i-\lambda_{B N}^i\right)}$, 故有0≤β_M＜γ_M＜α_M≤1。由此平均三支决策规则化简为

(MP2) 若$\frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \geqslant \alpha_M$, 则$x \in \operatorname{POS}_{\sum\limits_{i=1}^m R_{B_i}^{\geqslant}}^M(X)$。

(MN2) 若$\frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right) \leqslant \beta_M$, 则$x \in N E G_{\sum\limits_{i=1}^m R_{B_i}^{≥}}^M(X)$。

(MB2) 若$\beta_M＜\frac{1}{m} \sum\limits_{i=1}^m \operatorname{Pr}\left(X \mid[x]_{B_i}^{\geqslant}\right)＜\alpha_M$, 则$x \in B N D_{\sum\limits_{i=1}^m R_{B_i}^≥}^M(X)$。

4 结语

在分析不确定性和复杂数据时, 多粒度粗糙集通过多个角度观察和分析数据。传统的多粒度粗糙集以等价关系为基础构建。本文针对数值型偏序关系数据集, 建立优势关系下的多粒度结构, 提出了优势乐观多粒度粗糙集、优势悲观多粒度粗糙集和优势平均多粒度粗糙集。将这三种多粒度粗糙集和三支决策相结合, 通过损失函数降低了决策风险, 给出了三种多粒度粗糙集下的三支决策规则。下一步将研究不完备环境下基于优势关系的多粒度粗糙集及三支决策模型。

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