文章快速检索 高级检索

Partially consistent reduction in interval-valued fuzzy ordered decision information system
SHI Derong, XU Weihua
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
Abstract: In practical problems, some attribute-values of things are within a certain range and this is often used to describe uncertainties in an information system. The attribute-value is often expressed by a fuzzy interval, and the information system in this case is then called an interval-valued fuzzy information system. This paper establishes an interval-valued fuzzy decision ordered information system by introducing dominance relationships. This partially consistent function was built to simplify knowledge expression. A judgment theorem for partially consistent reduction was obtained, and from the recognizable attribute set and recognizable matrix, a partially consistent reduction method for an inconsistent interval-valued fuzzy ordered information system was derived. Furthermore, by combination with a specific case study on venture investment, the significance of partially consistent reduction is explained. This experiment enriches the rough set method for interval-valued fuzzy ordered decision information systems.
Key words: rough set     ordered information system     partially consistent reduction     recognizable matrix     interval-valued

1 基于区间值模糊的决策序信息系统

AT是有限条件属性集，AT={a1, a2, …, ap}；

DT是有限决策属性集，DT={d1, d2, …, dq}；

FU与AT的关系集，其中F={f:U→Va, a∈DT}，Vaa的有限值域；

GU与DT的关系集，其中G={g:U→Vd, d∈DT}，Vdd的有限值域。

I=(U, AT∪DT, F, G)为一个决策信息系统，若对任意f∈Fa∈AT和xi∈U都有

I=(U, AT, F)为区间值模糊信息系统。对任意的a∈AT，在区间值模糊信息系统中可对属性值进行比较，定义

2 区间值模糊决策序信息系统的部分一致约简

1) 对∀x∈U，当BA时，有δB(x)⊆δA(x)；

2) 对∀x, yU，当[y]A⊆[x]A时，有δA(x)⊆δA(y)。

3 区间值模糊决策序信息系统的部分一致约简方法

1) [x]A⊂[y]A

2) I=(U, AT∪d, F, G)；

3) [x]A∩[y]A⊂[x]A且[x]A∩[y]A⊂[y]A

1) 如果[x]A⊂[y]A则至少存在一个z∈[y]A，但z∉[x]A，由z∉[x]A可知，至少存在一个a∈A，使得f(x, a)>f(y, a)。因为z∈[y]A, 所以f(y, a)≤f(z, a)。于是得到f(x, a)>f(y, a), 因此a∈Dis≤ATδ(x, y)，即有A∩Dis≤ATδ(x, y)≠Ø。

2) 如果[x]A∩[y]A=Ø，必然至少存在一个a∈A使得f(x, a)>f(y, a)，即A∩Dis≤ATδ(x, y)≠Ø。否则，若对于所有的a∈A都有f(x, a)≤f(y, a)，则y∈[x]A，与[x]A∩[y]A=Ø矛盾。

3) 如果[x]A∩[y]A⊂[x]A且[x]A∩[y]A⊂[y]A，证明与(1)相同。因为此时也至少存在一个z∈[y]A，但是z∉[x]A。由此必要性即证。

4 区间值模糊决策序信息系统的部分一致约简方法

 U a1 a2 a3 d x1 [0.1, 0.3] [0.2, 0.3] [0.1, 0.4] 3 x2 [0.3, 0.5] [0.2, 0.6] [0.2, 0.8] 2 x3 [0.1, 0.5] [0.1, 0.4] [0.2, 0.7] 1 x4 [0.2, 0.7] [0.1, 0.5] [0.3, 0.7] 2 x5 [0.3, 0.6] [0.3, 0.7] [0.2, 0.9] 3 x6 [0.3, 0.9] [0.2, 0.7] [0.3, 0.8] 1

 Dis≥Aμ x1 x2 x3 x4 x5 x6 x1 Ø Ø Ø Ø Ø Ø x2 Ø Ø Ø Ø Ø Ø x3 Ø Ø Ø Ø Ø Ø x4 Ø Ø Ø Ø Ø Ø x5 A A A A Ø a2, a3 x6 Ø Ø Ø Ø Ø Ø

5 结论

1) 通过分析部分一致约简的性质得到了对应的判定定理；

2) 在上述基础上建立了辨识矩阵，给出了获取部分一致约简的具体方法，并且用两种情形对实例进行了对比分析。

3) 通过比较可以知道，本文对部分一致约简进行了更精确地刻画，可以简化在时间上的求解过程。

 [1] PAWLAK Z. Rough sets:theoretical aspects of reasoning about data[M]. Boston: Kluwer Academic Publishers, 1991 . [2] PAWLAK Z, GRZYMALA-BUSSE J, SLOWINSKI R, et al. Rough sets[J]. Communications of the ACM , 1995, 38 (11) : 89-95 [3] 王珏, 苗夺谦, 周育健. 关于Rough Set理论与应用的综述[J]. 模式识别与人工智能 , 1996, 9 (4) : 337-344 WANG Jue, MIAO Duoqian, ZHOU Yujian. Rough set theory and its application:a survey[J]. Pattern recognition and artificial intelligence , 1996, 9 (4) : 337-344 [4] 苗夺谦, 王珏. 基于粗糙集的多变量决策树构造方法[J]. 软件学报 , 1997, 8 (6) : 425-431 MIAO Duoqian, WANG Jue. Rough sets based approach for multivariate decision tree construction[J]. Journal of software , 1997, 8 (6) : 425-431 [5] 张小红, 裴道武, 代建华. 模糊数学与Rough集理论[M]. 北京: 清华大学出版社, 2013 . ZHANG Xiaohong, PEI Daowu, DAI Jianhua. Fuzzy mathematics and the rough set theory[M]. Beijing: Tsinghua University Press, 2013 . [6] 徐伟华, 张先韬, 王巧荣. 序信息系统中变精度粗糙集属性约简的MATLAB实现[J]. 重庆理工大学学报:自然科学版 , 2013, 27 (1) : 107-115 XU Weihua, ZHANG Xiantao, WANG Qiaorong. Experimental computing on attribute reduction by Matlab in dominance-based variable precision rough set[J]. Journal of Chongqing university of technology:natural science , 2013, 27 (1) : 107-115 [7] 张文修, 米据生, 吴伟志. 不协调目标信息系统的知识约简[J]. 计算机学报 , 2003, 26 (1) : 12-18 ZHANG Wenxiu, MI Jusheng, WU Weizhi. Knowledge reductions in inconsistent information systems[J]. Chinese journal of computers , 2003, 26 (1) : 12-18 [8] 徐伟华, 张文修. 基于优势关系下不协调目标信息系统的知识约简[J]. 计算机科学 , 2006, 33 (2) : 182-184 XU Weihua, ZHANG Wenxiu. Knowledge reductions in inconsistent information systems based on dominance relations[J]. Computer science , 2006, 33 (2) : 182-184 [9] 韦碧鹏, 吕跃进, 李金海. 优势关系下粗糙集模型的属性约简[J]. 智能系统学报 , 2014, 9 (2) : 251-258 WEI Bipeng, LV Yuejin, LI Jinhai. attribute reduction based on the rough set model under α dominance relation[J]. CAAI transactions on intelligent systems , 2014, 9 (2) : 251-258 [10] 张文修, 梁怡, 吴伟志. 信息系统与知识发现[M]. 北京: 科学出版社, 2003 . ZHANG Wenxiu, LIANG Yi, WU Weizhi. Information system and knowledge discovery[M]. Beijing: Science Press, 2003 . [11] 张楠, 苗夺谦, 岳晓冬. 区间值信息系统的知识约简[J]. 计算机研究与发展 , 2010, 47 (8) : 1362-1371 ZHANG Nan, MIAO Duoqian, YUE Xiaodong. Approaches to knowledge reduction in interval-valued information system[J]. Journal of computer research and development , 2010, 47 (8) : 1362-1371 [12] 于莹莹, 曾雪兰, 孙兴星. 优势关系下的区间值信息系统及其属性约简[J]. 计算机工程与应用 , 2011, 47 (35) : 122-124 YU Yingying, ZENG Xuelan, SUN Xingxing. Interval-valued information system based on dominance relation and its attribute reduction[J]. Computer engineering and applications , 2011, 47 (35) : 122-124 [13] GRECO S, MATARAZZO B, SLOWINSKI R. Rough approximation of a preference relation by dominance relations[J]. European journal of operational research , 1999, 117 (1) : 63-83 DOI:10.1016/S0377-2217(98)00127-1 [14] 徐伟华, 张文修. 基于优势关系下的协调近似空间[J]. 计算机科学 , 2005, 32 (9) : 164-165 XU Weihua, ZHANG Wenxiu. Consistent approximation spaces based on dominance relations[J]. Computer science , 2005, 32 (9) : 164-165 [15] 徐伟华. 序信息系统与粗糙集[M]. 北京: 科学出版社, 2013 . XU Weihua. Ordered information systems and rough sets theory[M]. Beijing: Science Press, 2013 . [16] 徐伟华, 张晓燕, 张文修. 优势关系下不协调目标信息系统的部分一致约简[J]. 模糊系统与数学 , 2009, 23 (6) : 155-161 XU Weihua, ZHANG Xiaoyan, ZHANG Wenxiu. Partially consistent reduction in inconsistent information systems based on dominance relations[J]. Fuzzy systems and mathemATics , 2009, 23 (6) : 155-161
DOI: 10.11992/tis.201606013

0

#### 文章信息

SHI Derong, XU Weihua

Partially consistent reduction in interval-valued fuzzy ordered decision information system

CAAI Transactions on Intelligent Systems, 2016, 11(4): 469-474
http://dx.doi.org/10.11992/tis.201606013