﻿ 相似度三支决策模糊粗糙集模型的决策代价研究
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 智能系统学报  2020, Vol. 15 Issue (6): 1068-1078  DOI: 10.11992/tis.201909015 0

### 引用本文

ZENG Ting, TANG Xiao, TAN Yang, et al. Decision costs of the similarity three-way decision-theoretic fuzzy rough set model[J]. CAAI Transactions on Intelligent Systems, 2020, 15(6): 1068-1078. DOI: 10.11992/tis.201909015.

### 文章历史

1. 四川师范大学 数学科学学院，四川 成都 610066;
2. 四川师范大学 智能信息与量子信息研究所，四川 成都 610066

Decision costs of the similarity three-way decision-theoretic fuzzy rough set model
ZENG Ting 1,2, TANG Xiao 1,2, TAN Yang 1,2, DING Benxiang 1,2
1. School of Mathematical Science, Sichuan Normal University, Chengdu 610066, China;
2. Institute of Intelligent Information and Quantum Information, Sichuan Normal University, Chengdu 610066, China
Abstract: In the three-way decision-theoretic fuzzy rough set model, several scholars established the objective function based on the similarity three-way decision-theoretic fuzzy rough set model to derive the method for calculating the optimal threshold pair (α, β). However, during this research, the authors did not discuss the description of the decision costs in the similarity three-way decision-theoretic fuzzy rough set model. The new function describing the decision costs is used in the method for calculating the threshold pair (α, β) based on the fuzzy information system. First, in the fuzzy information system, the fuzzy number is associated with the decision costs of three-way decisions by establishing a function describing the decision costs. Then, the numerical description of the decision costs of three-way decisions is obtained by fitting the membership frequency. Finally, two examples are given to illustrate the feasibility and applicability of the method.
Key words: three-way decisions    fuzzy rough set    decision costs    fuzzy number    threshold pair    attribute value    interval value    multiset

1 决策粗糙集模型

1.1 多重集

1.2 基于模糊数的决策粗糙集模型

Yao等[4]基于贝叶斯决策过程，提出决策理论粗糙集模型，该模型利用状态集 $O = \left\{ {A,{A^C}} \right\}$ 和动作集 $a = \left\{ {{a_e},{a_b},{a_r}} \right\}$ 来描述决策过程，其中状态集 $O = \left\{ {A,{A^C}} \right\}$ 中的元素分别表示某事件属于 $A$ 和不属于 $A$ ，动作集 $a = \left\{ {{a_e},{a_b},{a_r}} \right\}$ 中的元素分别表示将对象 $x$ 划分到正域、边界域和负域。

 $\begin{array}{l} \left. 1 \right)R\left( {{a_e}\left| {\left[ x \right]} \right.} \right) = {\lambda _{ep}} \cdot P\left( {A\left| {\left[ x \right]} \right.} \right) + {\lambda _{en}} \cdot P\left( {{A^C}\left| {\left[ x \right]} \right.} \right) \\ \left. 2 \right)R\left( {{a_b}\left| {\left[ x \right]} \right.} \right) = {\lambda _{bp}} \cdot P\left( {A\left| {\left[ x \right]} \right.} \right) + {\lambda _{bn}} \cdot P\left( {{A^C}\left| {\left[ x \right]} \right.} \right) \\ \left. 3 \right)R\left( {{a_r}\left| {\left[ x \right]} \right.} \right) = {\lambda _{rp}} \cdot P\left( {A\left| {\left[ x \right]} \right.} \right) + {\lambda _{rn}} \cdot P\left( {{A^C}\left| {\left[ x \right]} \right.} \right) \\ \end{array}$

1)如果 $R\left( {{a_e}\left| {\left[ x \right]} \right.} \right) \leqslant R\left( {{a_b}\left| {\left[ x \right]} \right.} \right)$ $R\left( {{a_e}\left| {\left[ x \right]} \right.} \right) \leqslant R( {a_r}|$ ${\left[ x \right]} )$ ，则采取接受决策；

2)如果 $R\left( {{a_b}\left| {\left[ x \right]} \right.} \right) \leqslant R\left( {{a_e}\left| {\left[ x \right]} \right.} \right)$ $R\left( {{a_b}\left| {\left[ x \right]} \right.} \right) \leqslant R( {a_r}|$ ${\left[ x \right]} )$ ，则采取延迟决策；

3)如果 $R\left( {{a_r}\left| {\left[ x \right]} \right.} \right) \leqslant R\left( {{a_e}\left| {\left[ x \right]} \right.} \right)$ $R\left( {{a_r}\left| {\left[ x \right]} \right.} \right) \leqslant R( {a_b}|$ ${\left[ x \right]} )$ ，则采取拒绝决策。

 ${\mu _F}\left\{ \begin{array}{l} 0,\quad x < {r_1} \\ \dfrac{{x - {r_1}}}{{{r_2} - {r_1}}},\quad {r_1} \leqslant x < {r_2} \\ 1,\quad {r_2} \leqslant x < {r_3} \\ \dfrac{{{r_4} - x}}{{{r_4} - {r_3}}},\quad {r_3} \leqslant x < {r_4} \\ 0,\quad x \geqslant {r_4} \end{array} \right.$

1)设在一次统计中， $n$ 位专家给出了 $n$ 个区间值 ${\lambda _{ * * }} = \left[ {\lambda _{ * * }^L,\lambda _{ * * }^U} \right]$

2)找出 $n$ 个区间值中的最小值 ${\lambda _{\min }}$ 和最大值 ${\lambda _{\max }}$ ，以 ${\lambda _{\min }}$ 为起点， ${\lambda _{\max }}$ 为终点， $\dfrac{{{\lambda _{\max }} - {\lambda _{\min }}}}{k}$ 为长度( $k \in {N^ * }$ )，作 $k$ 个区间的划分；

3)计算每个区间的隶属频率 $f = \dfrac{m}{n}$ ，其中 $n$ 为随机选择的样本总数， $m$ 为区间样本覆盖 ${\lambda _{ * * }}$ 的频数；

4)以实数 $x$ 为横坐标，隶属频率为纵坐标，绘出 ${\lambda _{ * * }}$ 的模糊分布曲线。

5)对该模糊分布左右两边的曲线进行直线拟合，得到一个梯形分布函数。

 ${M_{ep}} = \left( {{r_{11}},{r_{12}},{r_{13}},{r_{14}}} \right)$
 ${M_{bp}} = \left( {{r_{21}},{r_{22}},{r_{23}},{r_{24}}} \right)$
 ${M_{rp}} = \left( {{r_{31}},{r_{32}},{r_{33}},{r_{34}}} \right)$
 ${M_{en}} = \left( {{s_{11}},{s_{12}},{s_{13}},{s_{14}}} \right)$
 ${M_{bn}} = \left( {{s_{21}},{s_{22}},{s_{23}},{s_{24}}} \right)$
 ${M_{rn}} = \left( {{s_{31}},{s_{32}},{s_{33}},{s_{34}}} \right)$

 $\begin{split}& \;\alpha \!=\! \left( {\dfrac{1}{{1 \!+\! \dfrac{{{r_{24}} - {r_{11}}}}{{{s_{31}} - {s_{24}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{23}} - {r_{12}}}}{{{s_{32}} - {s_{23}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{22}} - {r_{13}}}}{{{s_{33}} - {s_{22}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{21}} - {r_{14}}}}{{{s_{34}} - {s_{21}}}}}}} \right)\\ &\;\beta \!=\! \left( {\dfrac{1}{{1 \!+\! \dfrac{{{r_{34}} - {r_{21}}}}{{{s_{21}} - {s_{14}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{33}} - {r_{22}}}}{{{s_{22}} - {s_{13}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{32}} - {r_{23}}}}{{{s_{23}} - {s_{12}}}}}},\dfrac{1}{{1 \!+\! \dfrac{{{r_{31}} - {r_{24}}}}{{{s_{24}} - {s_{11}}}}}}} \right) \end{split}$

$M = \left( {{r_1},{r_2},{r_3},{r_4}} \right)$ 是梯形模糊数， $\alpha = ( {a_1},{a_2},$ ${a_3},{a_4} ),\beta = \left( {{b_1},{b_2},{b_3},{b_4}} \right)$ ，用模糊满意度[16]来比较两个模糊数的大小， $\eta$ -水平截集的左右断点分别记为 $\alpha _\eta ^L{\text{、}}\alpha _\eta ^U{\text{、}}\beta _\eta ^L{\text{、}}\beta _\eta ^U$ ，则 $\alpha _\eta ^L = {a_2}\eta + {a_1}\left( {1 - \eta } \right),$ $\alpha _\eta ^U = {a_3}\eta +$ ${a_4}\left( {1 - \eta } \right),\beta _\eta ^L = {b_2}\eta + {b_1}\left( {1 - \eta } \right), \beta _\eta ^U = {b_3}\eta +$ ${b_4}\left( {1 - \eta } \right)$

2 基于三支决策模糊粗糙集模型的决策代价

1)接受决策： ${\lambda _e}\left( D \right) = 1 - D$

2)延迟决策： ${\lambda _b}\left( D \right) = \left| {0.5 - D} \right|$

3)拒绝决策： ${\lambda _r}\left( D \right) =D$

1)在一个模糊信息系统 $\varOmega = \left( {U,A,V,f} \right)$ 中，计算每个对象在每个属性下相对于其他对象的距离(此处用曼哈顿距离 ${\rm{di}}{{\rm{s}}_{ij}^{1}} = \dfrac{{\displaystyle\sum\limits_{k = 1}^n {\left| {{v_{ij}} - {v_{kj}}} \right|} }}{n}$ )，得到关于决策代价的曼哈顿距离 $D = {\left( {{\rm{di}}{{\rm{s}}_{ij}^{1}}} \right)_{n \times m}}$ ，其中 $i =$ $1,2, \cdot \cdot \cdot ,n{\text{；}}j = 1,2, \cdot \cdot \cdot ,m$

2)计算决策代价函数 ${\lambda }_{*}\left(D\right)$

3)找出 $n$ 个值中的最小值 ${\lambda _{\min }}$ 和最大值 ${\lambda _{\max }}$ ，以 ${\lambda _{\min }} - \delta - \dfrac{{{\lambda _{\max }} - {\lambda _{\min }}}}{{2k}}$ 为起点， ${\lambda _{\max }} + \delta + \dfrac{{{\lambda _{\max }} - {\lambda _{\min }}}}{{2k}}$ 为终点， $\dfrac{{{\lambda _{\max }} - {\lambda _{\min }}}}{k}$ 为区间长度( $k \in {N^ * }$ )，划分为 $k + 1$ 个区间，其中 $\delta$ 为邻域值；

4)计算每个动作在不同属性下的隶属频率 $f = \dfrac{m}{n}$ ，其中 $n$ 为随机选择的样本总数， $m$ 为区间样本覆盖某个值的 $\delta$ 邻域的频数；

5)以实数 $x$ 为横坐标，隶属频率为纵坐标，对点进行曲线拟合，取决策动作在每个属性下曲线最高点的横坐标，得到一个不同属性下的决策代价组；

6)某个决策动作的决策代价即为

 ${\lambda _ * } = \sqrt {\dfrac{{\displaystyle\sum\limits_{l = 1}^m {\lambda {{_ * ^l}^2}} }}{m}}$
3 实例分析

 Download: 图 1 接受决策属性 ${a_1}$ 频率分布拟合曲线 Fig. 1 Accept the decision attribute ${a_1}$ frequency distribution fitting curve
 Download: 图 2 接受决策属性 ${a_2}$ 频率分布拟合曲线 Fig. 2 Accept the decision attribute ${a_2}$ frequency distribution fitting curve
 Download: 图 3 接受决策属性 ${a_3}$ 频率分布拟合曲线 Fig. 3 Accept the decision attribute ${a_3}$ frequency distribution fitting curve
 Download: 图 4 接受决策属性 ${a_4}$ 频率分布拟合曲线 Fig. 4 Accept the decision attribute ${a_4}$ frequency distribution fitting curve

 Download: 图 5 接受决策属性 ${a_1}$ 频率分布拟合曲线 Fig. 5 Accept the decision attribute ${a_1}$ frequency distribution fitting curve
 Download: 图 6 接受决策属性 ${a_2}$ 频率分布拟合曲线 Fig. 6 Accept the decision attribute ${a_2}$ frequency distribution fitting curve
 Download: 图 7 接受决策属性 ${a_3}$ 频率分布拟合曲线 Fig. 7 Accept the decision attribute ${a_3}$ frequency distribution fitting curve
 Download: 图 8 接受决策属性 ${a_4}$ 频率分布拟合曲线 Fig. 8 Accept the decision attribute ${a_4}$ frequency distribution fitting curve

 Download: 图 9 接受决策属性 ${a_1}$ 频率分布拟合曲线 Fig. 9 Accept the decision attribute ${a_1}$ frequency distribution fitting curve
 Download: 图 10 接受决策属性 ${a_2}$ 频率分布拟合曲线 Fig. 10 Accept the decision attribute ${a_2}$ frequency distribution fitting curve
 Download: 图 11 接受决策属性 ${a_3}$ 频率分布拟合曲线 Fig. 11 Accept the decision attribute ${a_3}$ frequency distribution fitting curve

 Download: 图 12 接受决策属性 ${a_4}$ 频率分布拟合曲线 Fig. 12 Accept the decision attribute ${a_4}$ frequency distribution fitting curve
 Download: 图 13 接受决策属性 ${a_5}$ 频率分布拟合曲线 Fig. 13 Accept the decision attribute ${a_5}$ frequency distribution fitting curve

4 结束语

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