云模型双重积分算子的层次多属性决策方法

doi:10.3778/j.issn.1673-9418.2103009

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (8): 1898-1909.DOI: 10.3778/j.issn.1673-9418.2103009

云模型双重积分算子的层次多属性决策方法

王铁旦, 张雨晴, 彭定洪()

昆明理工大学管理与经济学院质量发展研究院，昆明 650093

收稿日期:2021-03-03 修回日期:2021-05-31 出版日期:2022-08-01 发布日期:2021-06-09
通讯作者: +E-mail: pengdinghong2006@163.com。
作者简介:王铁旦（1971—），男，云南昆明人，高级工程师，主要研究方向为模糊决策法。
张雨晴（1996—），女，河北承德人，硕士研究生，主要研究方向为模糊决策法。
彭定洪（1982—），男，云南曲靖人，博士，教授，主要研究方向为系统工程与模糊决策分析。
基金资助:
国家自然科学基金(71861018);国家自然科学基金(61364016);云南省哲学社会科学规划项目(YB2019067);中国博士后科学基金(2015T80990)

Hierarchical Multi-attribute Decision-Making Method with Twofold Integral Operator of Cloud Model

WANG Tiedan, ZHANG Yuqing, PENG Dinghong()

Institute of Quality Development, Faculty of Management and Economics, Kunming University of Science and Technology, Kunming 650093, China

Received:2021-03-03 Revised:2021-05-31 Online:2022-08-01 Published:2021-06-09
About author:WANG Tiedan, born in 1971, senior engineer. His research interest is fuzzy decision methods.
ZHANG Yuqing, born in 1996, M.S. candidate. Her research interest is fuzzy decision methods.
PENG Dinghong, born in 1982, Ph.D., professor. His research interests include system engineering and fuzzy decision analysis.
Supported by:
the National Natural Science Foundation of China(71861018);the National Natural Science Foundation of China(61364016);the Philosophy and Social Science Planning Programs of Yunnan Province(YB2019067);the Postdoctoral Science Foundation of China(2015T80990)

摘要/Abstract

摘要：

由于实际决策问题的复杂性，大多数指标间都具有相互依赖的特征，因此针对不确定环境下指标间具有关联性的多属性决策问题，提出了一种以层次结构为框架、云模型双重积分（C-TI）为聚合算子的决策方法。首先为了体现决策者在确定指标权重时的不确定思维提出了云模型模糊测度;其次在云模型模糊测度的基础上构建了云模型双重积分算子，该算子将云模型作为决策信息的表征，充分体现出决策信息的模糊性和随机性，且应用结合Choquet积分与Sugeno积分优势的双重积分聚合各指标值，其不仅可以有效处理指标间的交互性问题，还能够考虑到指标值与权重的侧重关系;随后讨论了云模型双重积分算子的定理和性质以及特定条件的变化形式，并给出了相应证明;最后构建出云模型双重积分算子的层次多属性决策方法，并将该方法应用到企业社会责任的评估问题上，通过与其他模型的比较，进一步说明所构建的方法适用于处理模糊与随机的不确定问题，并且能够在指标具有关联性的条件下同时考虑到整体的决策效果。

关键词: 云模型, 双重积分, 聚合算子, 模糊测度, 层次多属性决策

Abstract:

Due to the complexity of the actual decision-making problem, most indicators have the characteristics of mutual dependence. Therefore, for the multi-attribute decision-making problem with correlation between indicators under uncertain environment, a decision method with hierarchical structure as the framework and cloud model twofold integral (C-TI) as the aggregation operator is proposed. Firstly, in order to reflect the uncertain thinking of decision makers in determining the index weights, a cloud model fuzzy measure is proposed. Secondly, the cloud model twofold integral operator is constructed on the basis of cloud model fuzzy measure, which takes the cloud model as the representation of decision information, fully reflects the fuzzy and random nature of decision information. And twofold integral combining the advantages of Choquet integral and Sugeno integral is applied to aggregating the index values, which can not only effectively deal with the interaction problem between indexes, but also take into account the focus relationship between the index value and the weight. Subsequently, the theorems and properties of the cloud model twofold integral operator and the variation form of specific conditions are then discussed and the corresponding proofs are given. Finally, a hierarchical multi-attribute decision making method for cloud model twofold integral operator is constructed and applied to the corporate social responsibility assessment problem. Compared with other models, it is further shown that the method constructed in this paper is suitable for dealing with fuzzy and stochastic uncertainty problems and can take into account the overall decision effect while the indicators are correlated.

Key words: cloud model, twofold integral, aggregation operator, fuzzy measure, hierarchical multi-attribute decision making

中图分类号:

C934
TP301.6

王铁旦, 张雨晴, 彭定洪. 云模型双重积分算子的层次多属性决策方法[J]. 计算机科学与探索, 2022, 16(8): 1898-1909.

WANG Tiedan, ZHANG Yuqing, PENG Dinghong. Hierarchical Multi-attribute Decision-Making Method with Twofold Integral Operator of Cloud Model[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(8): 1898-1909.

图/表 8

表1 企业社会责任指标体系

Table 1 Index system of corporate social responsibility

主指标	子指标	指标评价
人权 $C 1$	基本权益保障 $C 11$	员工的基本权益是否得到保障
	员工福利 $C 12$	员工的福利是否具体落实
	员工发展 $C 13$	员工是否得到平等的发展机会
环境和资源 $C 2$	资源利用 $C 21$	企业的资源是否得到合理运用
	环境管理 $C 22$	企业是否对环境进行有效管理
	可持续发展能力 $C 23$	企业保持优势，持续赢利的能力
公益实践 $C 3$	环保倡议 $C 31$	企业是否积极倡议环保活动
公益实践 $C 3$	关爱弱势群体 $C 32$	企业是否主动关注关爱弱势群体
道德行为 $C 4$	政治参与 $C 41$	企业参与政治的情况
	合规运营 $C 42$	企业在运营管理方面是否符合规定
	信息披露 $C 43$	企业披露的信息是否真实完整
公共责任 $C 5$	守法行为 $C 51$	产品、服务及运营方面是否达到法律要求
	客户和员工满意度 $C 52$	顾客和员工的整体满意度如何
	社区建设 $C 53$	企业参与社区建设的程度
慈善行为 $C 6$	捐赠行为 $C 61$	企业对社会的捐赠力度
慈善行为 $C 6$	善因营销 $C 62$	企业对其相关事业的捐赠、资助

表1 企业社会责任指标体系

Table 1 Index system of corporate social responsibility

主指标	子指标	指标评价
人权 $C 1$	基本权益保障 $C 11$	员工的基本权益是否得到保障
	员工福利 $C 12$	员工的福利是否具体落实
	员工发展 $C 13$	员工是否得到平等的发展机会
环境和资源 $C 2$	资源利用 $C 21$	企业的资源是否得到合理运用
	环境管理 $C 22$	企业是否对环境进行有效管理
	可持续发展能力 $C 23$	企业保持优势，持续赢利的能力
公益实践 $C 3$	环保倡议 $C 31$	企业是否积极倡议环保活动
公益实践 $C 3$	关爱弱势群体 $C 32$	企业是否主动关注关爱弱势群体
道德行为 $C 4$	政治参与 $C 41$	企业参与政治的情况
	合规运营 $C 42$	企业在运营管理方面是否符合规定
	信息披露 $C 43$	企业披露的信息是否真实完整
公共责任 $C 5$	守法行为 $C 51$	产品、服务及运营方面是否达到法律要求
	客户和员工满意度 $C 52$	顾客和员工的整体满意度如何
	社区建设 $C 53$	企业参与社区建设的程度
慈善行为 $C 6$	捐赠行为 $C 61$	企业对社会的捐赠力度
慈善行为 $C 6$	善因营销 $C 62$	企业对其相关事业的捐赠、资助

表2 子指标的初始评价结果

Table 2 Initial evaluation results of sub-index

企业	专家	$C 11$	$C 12$	$C 13$	$C 21$	$C 22$	$C 23$	$C 31$	$C 32$	$C 41$	$C 42$	$C 43$	$C 51$	$C 52$	$C 53$	$C 61$	$C 62$
$A 1$	$d 1$	0.61	0.32	0.58	0.36	0.70	0.21	0.44	0.85	0.89	0.74	0.33	0.67	0.44	0.42	0.54	0.70
	$d 2$	0.70	0.40	0.56	0.29	0.78	0.30	0.40	0.70	0.73	0.58	0.38	0.78	0.51	0.35	0.65	0.80
	$d 3$	0.59	0.31	0.70	0.60	0.68	0.25	0.23	0.90	0.60	0.79	0.31	0.66	0.67	0.35	0.50	0.75
$A 2$	$d 1$	0.22	0.57	0.54	0.67	0.64	0.38	0.56	0.70	0.24	0.85	0.40	0.78	0.27	0.48	0.78	0.40
	$d 2$	0.35	0.47	0.40	0.78	0.66	0.40	0.68	0.80	0.31	0.90	0.41	0.76	0.29	0.49	0.76	0.50
	$d 3$	0.31	0.62	0.35	0.72	0.70	0.41	0.70	0.81	0.40	0.71	0.46	0.60	0.31	0.51	0.60	0.52
$A 3$	$d 1$	0.81	0.65	0.79	0.34	0.46	0.35	0.74	0.79	0.79	0.60	0.37	0.70	0.78	0.83	0.68	0.73
	$d 2$	0.82	0.60	0.80	0.37	0.40	0.39	0.68	0.65	0.81	0.50	0.38	0.71	0.80	0.71	0.67	0.65
	$d 3$	0.73	0.54	0.63	0.25	0.37	0.21	0.70	0.54	0.74	0.40	0.50	0.60	0.65	0.64	0.65	0.61
$A 4$	$d 1$	0.27	0.67	0.74	0.74	0.48	0.64	0.52	0.53	0.44	0.66	0.31	0.39	0.38	0.21	0.34	0.61
	$d 2$	0.33	0.78	0.70	0.55	0.37	0.76	0.52	0.52	0.51	0.79	0.32	0.37	0.42	0.20	0.39	0.76
	$d 3$	0.31	0.64	0.82	0.64	0.50	0.51	0.67	0.43	0.67	0.63	0.40	0.50	0.44	0.16	0.35	0.59

表2 子指标的初始评价结果

Table 2 Initial evaluation results of sub-index

企业	专家	$C 11$	$C 12$	$C 13$	$C 21$	$C 22$	$C 23$	$C 31$	$C 32$	$C 41$	$C 42$	$C 43$	$C 51$	$C 52$	$C 53$	$C 61$	$C 62$
$A 1$	$d 1$	0.61	0.32	0.58	0.36	0.70	0.21	0.44	0.85	0.89	0.74	0.33	0.67	0.44	0.42	0.54	0.70
	$d 2$	0.70	0.40	0.56	0.29	0.78	0.30	0.40	0.70	0.73	0.58	0.38	0.78	0.51	0.35	0.65	0.80
	$d 3$	0.59	0.31	0.70	0.60	0.68	0.25	0.23	0.90	0.60	0.79	0.31	0.66	0.67	0.35	0.50	0.75
$A 2$	$d 1$	0.22	0.57	0.54	0.67	0.64	0.38	0.56	0.70	0.24	0.85	0.40	0.78	0.27	0.48	0.78	0.40
	$d 2$	0.35	0.47	0.40	0.78	0.66	0.40	0.68	0.80	0.31	0.90	0.41	0.76	0.29	0.49	0.76	0.50
	$d 3$	0.31	0.62	0.35	0.72	0.70	0.41	0.70	0.81	0.40	0.71	0.46	0.60	0.31	0.51	0.60	0.52
$A 3$	$d 1$	0.81	0.65	0.79	0.34	0.46	0.35	0.74	0.79	0.79	0.60	0.37	0.70	0.78	0.83	0.68	0.73
	$d 2$	0.82	0.60	0.80	0.37	0.40	0.39	0.68	0.65	0.81	0.50	0.38	0.71	0.80	0.71	0.67	0.65
	$d 3$	0.73	0.54	0.63	0.25	0.37	0.21	0.70	0.54	0.74	0.40	0.50	0.60	0.65	0.64	0.65	0.61
$A 4$	$d 1$	0.27	0.67	0.74	0.74	0.48	0.64	0.52	0.53	0.44	0.66	0.31	0.39	0.38	0.21	0.34	0.61
	$d 2$	0.33	0.78	0.70	0.55	0.37	0.76	0.52	0.52	0.51	0.79	0.32	0.37	0.42	0.20	0.39	0.76
	$d 3$	0.31	0.64	0.82	0.64	0.50	0.51	0.67	0.43	0.67	0.63	0.40	0.50	0.44	0.16	0.35	0.59

表3 子指标的初始权重

Table 3 Initial weight of sub-index

专家	$C 11$	$C 12$	$C 13$	$C 21$	$C 22$	$C 23$	$C 31$	$C 32$	$C 41$	$C 42$	$C 43$	$C 51$	$C 52$	$C 53$	$C 61$	$C 62$
$d 1$	0.15	0.10	0.21	0.08	0.25	0.14	0.02	0.04	0.30	0.20	0	0.10	0.28	0.03	0.18	0
$d 2$	0.30	0.12	0.33	0.10	0.28	0.23	0.03	0.05	0.35	0	0.10	0.15	0	0.04	0.24	0.09
$d 3$	0	0.20	0	0.02	0.31	0.01	0	0.06	0.41	0.32	0.17	0.21	0.36	0.07	0	0.13

表3 子指标的初始权重

Table 3 Initial weight of sub-index

专家	$C 11$	$C 12$	$C 13$	$C 21$	$C 22$	$C 23$	$C 31$	$C 32$	$C 41$	$C 42$	$C 43$	$C 51$	$C 52$	$C 53$	$C 61$	$C 62$
$d 1$	0.15	0.10	0.21	0.08	0.25	0.14	0.02	0.04	0.30	0.20	0	0.10	0.28	0.03	0.18	0
$d 2$	0.30	0.12	0.33	0.10	0.28	0.23	0.03	0.05	0.35	0	0.10	0.15	0	0.04	0.24	0.09
$d 3$	0	0.20	0	0.02	0.31	0.01	0	0.06	0.41	0.32	0.17	0.21	0.36	0.07	0	0.13

表4 子指标的云模型评价结果

Table 4 Cloud model evaluation results of sub-index

企业	$C 11$	$C 12$		$C 13$	$C 21$
$A 1$	（0.63,0.044,0.003 4）	（0.34,0.038,0.002 4）		（0.61,0.058,0.005 7）	（0.42,0.122,0.026 4）
$A 2$	（0.29,0.049,0.004 4）	（0.55,0.056,0.005 8）		（0.43,0.073,0.009 7）	（0.72,0.038,0.003 0）
$A 3$	（0.79,0.038,0.002 4）	（0.60,0.038,0.003 0）		（0.74,0.073,0.009 1）	（0.32,0.047,0.003 9）
$A 4$	（0.30,0.022,0.009 0）	（0.70,0.056,0.005 4）		（0.75,0.044,0.003 7）	（0.64,0.064,0.009 0）
企业	$C 22$	$C 23$		$C 31$	$C 32$
$A 1$	（0.72,0.040,0.002 8）	（0.25,0.031,0.002 0）		（0.36,0.084,0.012 4）	（0.82,0.078,0.010 8）
$A 2$	（0.67,0.022,0.000 9）	（0.40,0.011,0.000 2）		（0.65,0.058,0.005 7）	（0.77,0.047,0.003 7）
$A 3$	（0.41,0.033,0.002 1）	（0.32,0.071,0.008 9）		（0.71,0.022,0.000 9）	（0.66,0.087,0.015 7）
$A 4$	（0.45,0.053,0.004 9）	（0.64,0.084,0.015 6）		（0.57,0.067,0.007 5）	（0.49,0.042,0.003 0）
企业	$C 41$	$C 42$		$C 43$	$C 51$
$A 1$	（0.74,0.100,0.021 1）	（0.70,0.082,0.012 0）		（0.34,0.027,0.001 3）	（0.70,0.051,0.004 4）
$A 2$	（0.32,0.056,0.006 4）	（0.82,0.073,0.009 7）		（0.42,0.024,0.001 0）	（0.71,0.076,0.009 7）
$A 3$	（0.78,0.027,0.001 3）	（0.50,0.067,0.010 0）	（0.42,0.056,0.005 2）		（0.67,0.047,0.003 7）
$A 4$	（0.54,0.087,0.013 9）	（0.69,0.064,0.007 2）	（0.34,0.038,0.002 4）		（0.42,0.053,0.004 9）
企业	$C 52$	$C 53$	$C 61$		$C 62$
$A 1$	（0.54,0.087,0.013 9）	（0.37,0.031,0.001 6）	（0.56,0.058,0.006 0）		（0.75,0.033,0.002 5）
$A 2$	（0.29,0.013,0.000 4）	（0.49,0.011,0.000 2）	（0.71,0.076,0.002 0）		（0.47,0.049,0.004 1）
$A 3$	（0.74,0.062,0.006 6）	（0.73,0.069,0.009 2）	（0.67,0.011,0.000 2）		（0.66,0.044,0.003 7）
$A 4$	（0.41,0.022,0.000 9）	（0.19,0.020,0.000 7）	（0.36,0.020,0.000 7）		（0.65,0.071,0.008 6）

表4 子指标的云模型评价结果

Table 4 Cloud model evaluation results of sub-index

企业	$C 11$	$C 12$		$C 13$	$C 21$
$A 1$	（0.63,0.044,0.003 4）	（0.34,0.038,0.002 4）		（0.61,0.058,0.005 7）	（0.42,0.122,0.026 4）
$A 2$	（0.29,0.049,0.004 4）	（0.55,0.056,0.005 8）		（0.43,0.073,0.009 7）	（0.72,0.038,0.003 0）
$A 3$	（0.79,0.038,0.002 4）	（0.60,0.038,0.003 0）		（0.74,0.073,0.009 1）	（0.32,0.047,0.003 9）
$A 4$	（0.30,0.022,0.009 0）	（0.70,0.056,0.005 4）		（0.75,0.044,0.003 7）	（0.64,0.064,0.009 0）
企业	$C 22$	$C 23$		$C 31$	$C 32$
$A 1$	（0.72,0.040,0.002 8）	（0.25,0.031,0.002 0）		（0.36,0.084,0.012 4）	（0.82,0.078,0.010 8）
$A 2$	（0.67,0.022,0.000 9）	（0.40,0.011,0.000 2）		（0.65,0.058,0.005 7）	（0.77,0.047,0.003 7）
$A 3$	（0.41,0.033,0.002 1）	（0.32,0.071,0.008 9）		（0.71,0.022,0.000 9）	（0.66,0.087,0.015 7）
$A 4$	（0.45,0.053,0.004 9）	（0.64,0.084,0.015 6）		（0.57,0.067,0.007 5）	（0.49,0.042,0.003 0）
企业	$C 41$	$C 42$		$C 43$	$C 51$
$A 1$	（0.74,0.100,0.021 1）	（0.70,0.082,0.012 0）		（0.34,0.027,0.001 3）	（0.70,0.051,0.004 4）
$A 2$	（0.32,0.056,0.006 4）	（0.82,0.073,0.009 7）		（0.42,0.024,0.001 0）	（0.71,0.076,0.009 7）
$A 3$	（0.78,0.027,0.001 3）	（0.50,0.067,0.010 0）	（0.42,0.056,0.005 2）		（0.67,0.047,0.003 7）
$A 4$	（0.54,0.087,0.013 9）	（0.69,0.064,0.007 2）	（0.34,0.038,0.002 4）		（0.42,0.053,0.004 9）
企业	$C 52$	$C 53$	$C 61$		$C 62$
$A 1$	（0.54,0.087,0.013 9）	（0.37,0.031,0.001 6）	（0.56,0.058,0.006 0）		（0.75,0.033,0.002 5）
$A 2$	（0.29,0.013,0.000 4）	（0.49,0.011,0.000 2）	（0.71,0.076,0.002 0）		（0.47,0.049,0.004 1）
$A 3$	（0.74,0.062,0.006 6）	（0.73,0.069,0.009 2）	（0.67,0.011,0.000 2）		（0.66,0.044,0.003 7）
$A 4$	（0.41,0.022,0.000 9）	（0.19,0.020,0.000 7）	（0.36,0.020,0.000 7）		（0.65,0.071,0.008 6）

表5 子指标云模型模糊测度

Table 5 Cloud model fuzzy measure of sub-index

指标	模糊测度	指标	模糊测度
$C 11$	（0.23,0.06,0.011 3）	$C 41$	（0.35,0.04,0.003 0）
$C 12$	（0.14,0.04,0.002 8）	$C 42$	（0.26,0.06,0.007 2）
$C 13$	（0.27,0.06,0.007 2）	$C 43$	（0.14,0.04,0.002 5）
$C 21$	（0.09,0.01,0.000 2）	$C 51$	（0.15,0.04,0.003 0）
$C 22$	（0.28,0.02,0.000 9）	$C 52$	（0.32,0.04,0.003 2）
$C 23$	（0.19,0.05,0.004 1）	$C 53$	（0.05,0.02,0.000 4）
$C 31$	（0.03,0.01,0.000 1）	$C 61$	（0.21,0.03,0.001 8）
$C 32$	（0.05,0.01,0.000 1）	$C 62$	（0.11,0.02,0.000 8）

表5 子指标云模型模糊测度

Table 5 Cloud model fuzzy measure of sub-index

指标	模糊测度	指标	模糊测度
$C 11$	（0.23,0.06,0.011 3）	$C 41$	（0.35,0.04,0.003 0）
$C 12$	（0.14,0.04,0.002 8）	$C 42$	（0.26,0.06,0.007 2）
$C 13$	（0.27,0.06,0.007 2）	$C 43$	（0.14,0.04,0.002 5）
$C 21$	（0.09,0.01,0.000 2）	$C 51$	（0.15,0.04,0.003 0）
$C 22$	（0.28,0.02,0.000 9）	$C 52$	（0.32,0.04,0.003 2）
$C 23$	（0.19,0.05,0.004 1）	$C 53$	（0.05,0.02,0.000 4）
$C 31$	（0.03,0.01,0.000 1）	$C 61$	（0.21,0.03,0.001 8）
$C 32$	（0.05,0.01,0.000 1）	$C 62$	（0.11,0.02,0.000 8）

表6 主指标云模型模糊测度

Table 6 Cloud model fuzzy measure of main indicator

指标	模糊测度
$C 1$	（0.221,0.175,0.007 9）
$C 2$	（0.198,0.075,0.002 7）
$C 3$	（0.040,0.012,0.000 1）
$C 4$	（0.253,0.133,0.004 8）
$C 5$	（0.206,0.094,0.002 7）
$C 6$	（0.170,0.050,0.001 4）

表6 主指标云模型模糊测度

Table 6 Cloud model fuzzy measure of main indicator

指标	模糊测度
$C 1$	（0.221,0.175,0.007 9）
$C 2$	（0.198,0.075,0.002 7）
$C 3$	（0.040,0.012,0.000 1）
$C 4$	（0.253,0.133,0.004 8）
$C 5$	（0.206,0.094,0.002 7）
$C 6$	（0.170,0.050,0.001 4）

表7 指标值排序及其相应指标集的云模型模糊测度

Table 7 Index value sorting and cloud model fuzzy measures of index set

指标	指标值	模糊测度	模糊测度值
$x 1$	（0.34,0.038,0.002 4）	$τ (A 1)$	（1.00,0.138,0.020 1）
$x 2$	（0.61,0.058,0.005 7）	$τ (A 2)$	（0.50,0.064,0.008 3）
$x 3$	（0.04,0.012,0.000 1）	$τ (A 3)$	（0.14,0.004,0.002 8）

表7 指标值排序及其相应指标集的云模型模糊测度

Table 7 Index value sorting and cloud model fuzzy measures of index set

指标	指标值	模糊测度	模糊测度值
$x 1$	（0.34,0.038,0.002 4）	$τ (A 1)$	（1.00,0.138,0.020 1）
$x 2$	（0.61,0.058,0.005 7）	$τ (A 2)$	（0.50,0.064,0.008 3）
$x 3$	（0.04,0.012,0.000 1）	$τ (A 3)$	（0.14,0.004,0.002 8）

表8 主指标云模型评价值

Table 8 Cloud model evaluation results of main indicator

指标	$A 1$	$A 2$	$A 3$	$A 4$
$C 1$	（0.4,0.08,0.009 2）	（0.4,0.11,0.012 9）	（0.7,0.08,0.006 6）	（0.4,0.20,0.015 6）
$C 2$	（0.3,0.04,0.003 6）	（0.6,0.10,0.010 2）	（0.3,0.06,0.006 9）	（0.7,0.08,0.009 8）
$C 3$	（0.4,0.11,0.012 1）	（0.7,0.16,0.005 8）	（0.7,0.17,0.015 4）	（0.5,0.12,0.003 2）
$C 4$	（0.4,0.06,0.006 2）	（0.4,0.06,0.005 8）	（0.5,0.08,0.007 9）	（0.5,0.07,0.007 8）
$C 5$	（0.4,0.07,0.004 4）	（0.4,0.06,0.003 6）	（0.9,0.10,0.007 6）	（0.3,0.04,0.002 5）
$C 6$	（0.6,0.10,0.006 7）	（0.5,0.09,0.005 2）	（0.7,0.11,0.006 2）	（0.4,0.06,0.003 0）

表8 主指标云模型评价值

Table 8 Cloud model evaluation results of main indicator

指标	$A 1$	$A 2$	$A 3$	$A 4$
$C 1$	（0.4,0.08,0.009 2）	（0.4,0.11,0.012 9）	（0.7,0.08,0.006 6）	（0.4,0.20,0.015 6）
$C 2$	（0.3,0.04,0.003 6）	（0.6,0.10,0.010 2）	（0.3,0.06,0.006 9）	（0.7,0.08,0.009 8）
$C 3$	（0.4,0.11,0.012 1）	（0.7,0.16,0.005 8）	（0.7,0.17,0.015 4）	（0.5,0.12,0.003 2）
$C 4$	（0.4,0.06,0.006 2）	（0.4,0.06,0.005 8）	（0.5,0.08,0.007 9）	（0.5,0.07,0.007 8）
$C 5$	（0.4,0.07,0.004 4）	（0.4,0.06,0.003 6）	（0.9,0.10,0.007 6）	（0.3,0.04,0.002 5）
$C 6$	（0.6,0.10,0.006 7）	（0.5,0.09,0.005 2）	（0.7,0.11,0.006 2）	（0.4,0.06,0.003 0）

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云模型双重积分算子的层次多属性决策方法

Hierarchical Multi-attribute Decision-Making Method with Twofold Integral Operator of Cloud Model

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