文章快速检索 高级检索

1. 烟台大学 计算机与控制工程学院, 山东 烟台 264005;
2. 上海大学 计算机工程与科学学院, 上海 200444;
3. 中联重科股份有限公司 麓谷工业园研发中心, 湖南 长沙 410205

Utility-based three-way decisions model
ZHANG Nan1, JIANG Lili1, YUE Xiaodong2, ZHOU Jie3
1. School of Computer and Control Engineering, Yantai University, Yantai 264005, China ;
2. School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China ;
3. Research and Development Center of Lugu Industrial Park, Zoomlion Heavy Industry Science & Technology Development Co., Ltd., Changsha 410205, China
Abstract: Risk decisions are usually influenced by subjective and objective factors. During the process of decision-making, decisions are based not only on the objective risk but also on the subjective activity of decision-makers. In order to introduce decision-makers' subjective attitudes to risk, a model of utility-based three-way decisions (UTWD) was produced by extending the risk function to a utility function. The monotonic relationships between the utility and probabilities of objects in three regions were investigated systematically. The computational methods for positive region utility, boundary region utility, and negative region utility are also given. Finally, an example is given to substantiate the conceptual arguments. The model is an extension of three-way decisions and provides a beneficial investigation into subjective risk measures in three-way decision research.
Key words: utility theory     three-way decisions     rough sets     artificial intelligence     risk decisions

1 经典三支决策模型

2 效用三支决策模型 2.1 效用理论

 图 1 效用函数曲线 Fig. 1 The utility function curves
2.2 基于效用理论的三支决策模型

 函数 aP aB aN X u(λPP) u(λBP) u(λNP) Xc u(λPN) u(λBN) u(λNN)

Ψ(aP|[x])=u(λPP)P(X|[x])+u(λPN)P(Xc|[x])

Ψ(aB|[x])=u(λBP)P(X|[x])+u(λBN)P(Xc|[x])

Ψ(aN|[x])=u(λNP)P(X|[x])+u(λNN)P(Xc|[x])

P) 若Ψ(aP|[x])≥Ψ(aB|[x])并且Ψ(aP|[x])≥Ψ(aN|[x])，则判定[x]⊆POSπ(X);

B) 若Ψ(aB|[x])≥Ψ(aN|[x])并且Ψ(aB|[x])≥Ψ(aP|[x])，则判定[x]⊆BNDπ(X);

N) 若Ψ(aN|[x])≥Ψ(aP|[x])并且Ψ(aN|[x])≥Ψ(aB|[x])，则判定[x]⊆NEGπ(X)。

P1)若P(X|[x])≥αu并且P(X|[x])≥γu，则判定[x]⊆POSπ(X)

B1)若P(X|[x])≥βu并且P(X|[x])≤αu，则判定[x]⊆BNDπ(X)

N1)若P(X|[x])≤γu并且P(X|[x])≤βu，则判定[x]⊆NEGπ(X)其中αuβuγu分别为

u(λPP)≥u(λBP)>u(λNP)和u(λNN)≥u(λBN)> u(λPN)，则αu∈(0, 1]，βu∈[0, 1)，γu∈(0, 1)。进一步通过变换可得：

βu≠0，则有

Δ(αu)度量了在状态[x]⊆X下由边界域移动到正域效用增加值与在状态[x]⊆Xc下由边界域移动到正域效用减少值的比例；Δ(βu)度量了在状态[x]⊆X下由边界域移动到负域效用减少值，与在状态[x]⊆Xc下由边界域移动到负域效用增加值的比例；Δ(γu)度量了在状态[x]⊆X下由正域移动到负域效用减少值与在状态[x]⊆Xc下由正域移动到负域效用增加值的比例。Δ(αu)、Δ(βu)和Δ(γu)分别涉及两种状态下正域与边界域、边界域与负域和正域与负域间的转换。不同决策者对待风险的态度不同将导致获得的效用函数不同，从而在相同客观因素下得到的参数差异较大，影响最终的决策判定结果。

P2)若P(X|[x])≥αu，则[x]⊆POSπ(X)

B2)若βu < P(X|[x]) < αu，则[x]⊆BNDπ(X)

N2)若P(X|[x])≤βu，则[x]⊆NEGπ(X)

1) 单个对象划分到X正域的效用为

2) 单个对象划分到X边界域的效用为

3) 单个对象划分到X负域的效用为

2.3 效用与对象的概率之间的关系讨论

 图 2 效用函数曲线 Fig. 2 The utility function curves

p∈[0, βu]时，

p∈[αu, 1]时，

p∈(αu, βu)时，效用可能随概率增加呈单调上升、不变或下降。在p=αup=βu两点的效用分别采用划分到正域和负域的效用公式计算。在图 2(a)中，满足以下条件

 图 3 效用拟合函数曲线 Fig. 3 The utility fitting function curves

3 实例分析

 函数 c1 c2 d x1 1 1 1 x2 2 1 2 x3 2 2 2 x4 1 1 3 x5 2 2 3 x6 1 3 3 x7 2 1 4 x8 2 2 4 x9 1 3 4

 函数 aP aB aN X 0 1 000 3 500 Xc 2 500 600 0

 函数 风险厌恶型 风险中立型 风险喜好型 aP aB aN aP aB aN aP aB aN X 1 0.830 1 0 1 0.714 3 0 1 0.5 0 Xc 0.5 0.901 2 1 0.285 7 0.828 6 1 0.075 7 0.678 7 1

 类型 αu γu βu 风险厌恶型 0.702 5 0.333 3 0.106 4 风险中立型 0.655 2 0.416 7 0.193 5 风险喜好型 0.546 7 0.480 3 0.391 2

 图 4 效用和概率之间的关系 Fig. 4 The relationships between the utility and probability
4 结论

1) 效用函数可以较合理的为决策主观能动性的量化提供度量标准；

2) 决策的总效用为正域效用，负域效用与边界域效用之和；

3) 决策的效用与概率之间存在区间性单调关系。接下来，如何结合效用函数特性，构建效用三支决策模型的合理属性约简目标函数将是下一步主要研究工作。

 [1] LIU Dun, LIANG Decui, WANG Changchun. A novel three-way decision model based on incomplete information system[J]. Knowledge-based systems , 2016, 91 : 32-45 DOI:10.1016/j.knosys.2015.07.036 [2] LIANG Decui, PEDRYCZ W, LIU Dun, et al. Three-way decisions based on decision-theoretic rough sets under linguistic assessment with the aid of group decision making[J]. Applied soft computing , 2015, 29 : 256-269 DOI:10.1016/j.asoc.2015.01.008 [3] LIU Dun, LI Tianrui, ZHANG Junbo. Incremental updating approximations in probabilistic rough sets under the variation of attributes[J]. Knowledge-based systems , 2015, 73 : 81-96 DOI:10.1016/j.knosys.2014.09.008 [4] MIN Fan, HE Huaping, QIAN Yuhua, et al. Test-cost-sensitive attribute reduction[J]. Information sciences , 2011, 181 (22) : 4928-4942 DOI:10.1016/j.ins.2011.07.010 [5] ZHANG Hengru, MIN Fan. Three-way recommender systems based on random forests[J]. Knowledge-based systems , 2016, 91 : 275-286 DOI:10.1016/j.knosys.2015.06.019 [6] DENG Xiaofei, YAO Yiyu. Decision-theoretic three-way approximations of fuzzy sets[J]. Information sciences , 2014, 279 : 702-715 DOI:10.1016/j.ins.2014.04.022 [7] DENG Xiaofei, YAO Yiyu. A multifaceted analysis of probabilistic three-way decisions[J]. Fundamenta informaticae , 2014, 132 (3) : 291-313 [8] YAO Yiyu. Three-way decisions with probabilistic rough sets[J]. Information sciences , 2010, 180 (3) : 341-353 DOI:10.1016/j.ins.2009.09.021 [9] ZHANG Xianyong, MIAO Duoqian. Reduction target structure-based hierarchical attribute reduction for two-category decision-theoretic rough sets[J]. Information sciences , 2014, 277 : 755-776 DOI:10.1016/j.ins.2014.02.160 [10] HERBERT J P, YAO Jingtao. Game-theoretic rough sets[J]. Fundamenta informaticae , 2011, 108 (3/4) : 267-286 [11] ZHOU Bing, YAO Yiyu, LUO Jigang. Cost-sensitive three-way email spam filtering[J]. Journal of intelligent information systems , 2014, 42 (1) : 19-45 DOI:10.1007/s10844-013-0254-7 [12] HU Baoqing. Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets[J]. Knowledge-based systems , 2016, 91 : 16-31 DOI:10.1016/j.knosys.2015.09.026 [13] 李华雄, 周献中, 李天瑞, 等. 决策粗糙集理论及其研究进展[M]. 北京: 科学出版社, 2011 . [14] 贾修一, 商琳, 周献中, 等. 三支决策理论与应用[M]. 南京: 南京大学出版社, 2012 . [15] YANG Xiaoping, YAO Jingtao. Modelling multi-agent three-way decisions with decision-theoretic rough sets[J]. Fundamenta informaticae , 2012, 115 (2/3) : 157-171 [16] MA Xi'ao, WANG Guoyin, YU Hong, et al. Decision region distribution preservation reduction in decision-theoretic rough set model[J]. Information sciences , 2014, 278 : 614-640 DOI:10.1016/j.ins.2014.03.078 [17] YU Hong, LIU Zhanguo, WANG Guoyin. An automatic method to determine the number of clusters using decision-theoretic rough set[J]. International journal of approximate reasoning , 2014, 55 (1) : 101-115 DOI:10.1016/j.ijar.2013.03.018 [18] 贾修一, 李伟湋, 商琳, 等. 一种自适应求三枝决策中决策阈值的算法[J]. 电子学报 , 2011, 39 (11) : 2520-2525 JIA Xiuyi, LI Weiwei, SHANG Lin, et al. An adaptive learning parameters algorithm in three-way decision-theoretic rough set model[J]. Acta electronica sinica , 2011, 39 (11) : 2520-2525 [19] LIU Dun, LI Tianrui, LIANG Decui. Incorporating logistic regression to decision-theoretic rough sets for classifications[J]. International journal of approximate reasoning , 2014, 55 (1) : 197-210 DOI:10.1016/j.ijar.2013.02.013 [20] 刘盾, 李天瑞, 苗夺谦, 等. 三支决策与粒计算[M]. 北京: 科学出版社, 2013 . [21] YAO Yiyu, WONG S K M, LINGRAS P. A decision-theoretic rough set model[C]//RAS Z W, ZEMANKOVA M, EMRICH M L. Methodologies for Intelligent Systems, Vol. 5. New York:North-Holland, 1990:17-24. [22] YU Hong, CHU Shuangshuang, YANG Dachun. Autonomous knowledge-oriented clustering using decision-theoretic rough set theory[J]. Fundamenta informaticae , 2012, 115 (2/3) : 141-156 [23] 贾修一, 商琳, 周献中, 等. 三支决策理论与应用[M]. 南京: 南京大学出版社, 2012 : 17 -33. [24] 刘盾, 李天瑞, 李华雄. 粗糙集理论:基于三支决策视角[J]. 南京大学学报:自然科学版 , 2013, 49 (5) : 574-581 LIU Dun, LI Tianrui, LI Huaxiong. Rough set theory:a three-way decisions perspective[J]. Journal of Nanjing university:natural sciences , 2013, 49 (5) : 574-581 [25] JIA Xiuyi, LIAO Wenhe, TANG Zhenmin, et al. Minimum cost attribute reduction in decision-theoretic rough set models[J]. Information sciences , 2013, 219 : 151-167 DOI:10.1016/j.ins.2012.07.010 [26] LI Huaxiong, ZHANG Libo, HUANG Bing, et al. Sequential three-way decision and granulation for cost-sensitive face recognition[J]. Knowledge-based systems , 2016, 91 : 241-251 DOI:10.1016/j.knosys.2015.07.040 [27] QIAN Yuhua, ZHANG Hu, SANG Yanli, et al. Multigranulation decision-theoretic rough sets[J]. International journal of approximate reasoning , 2014, 55 (1) : 225-237 DOI:10.1016/j.ijar.2013.03.004 [28] DOU Huili, YANG Xibei, SONG Xiaoning, et al. Decision-theoretic rough set:a multicost strategy[J]. Knowledge-based systems , 2016, 91 : 71-83 DOI:10.1016/j.knosys.2015.09.011 [29] 于洪, 王国胤, 姚一豫. 决策粗糙集理论研究现状与展望[J]. 计算机学报 , 2015, 38 (8) : 1628-1639 YU Hong, WANG Guoyin, YAO Yiyu. Current research and future perspectives on decision-theoretic rough sets[J]. Chinese journal of computers , 2015, 38 (8) : 1628-1639 [30] 于洪, 王国胤, 李天瑞, 等. 三支决策:复杂问题求解方法与实践[M]. 北京: 科学出版社, 2015 : 300 -313. [31] 李华, 胡奇英. 预测与决策教程[M]. 北京: 机械工业出版社, 2012 .
DOI: 10.11992/tis.201606010

0

#### 文章信息

ZHANG Nan, JIANG Lili, YUE Xiaodong, ZHOU Jie

Utility-based three-way decisions model

CAAI Transactions on Intelligent Systems, 2016, 11(4): 459-468
http://dx.doi.org/10.11992/tis.201606010