﻿ 采用粒计算的属性权重确定方法
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A method for ascertaining the weight of attributes based on granular computing
ZHOU Danchen
Institute of Machinery Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621900, China
Abstract: Aiming at multi-attribute decision making problem without single decision attribute in which the attribute values are in the forms of continuous numbers, a method for ascertaining attributes weight based on fusion of fuzzy quotient spaces theory and rough sets theory is put forward through comparison and analysis of the macrocosmic and microcosmic theoretical models of granular computing. Firstly, the quotient space family with hierarchical structure was established by applying fuzzy quotient spaces theory. The sample clustering resulted in a series of granular quotient spaces were used as classification of single decision attribute in corresponding granular space. Furthermore, the significances of every attribute in all quotient spaces were calculated by applying rough sets theory. Finally, objective attributes weight was ascertained by integrating the significances of every attribute in different granular quotient spaces. The application case showed that the proposed method is reasonable, effective and practical.
Key words: attributes weight     granular computing     fuzzy quotient space     rough set     multi-attribute decision making

1 粒计算理论基础 1.1 模糊商空间理论

1) ∀xX,R(x,x)=1,

2) ∀x,yX,R(x,y)=R(y,x),

3) ∀x,y,zX,R(x,z)≥supy(min(R(x,y),R(y,z)),

1.2 粗糙集理论

2 融合模糊商空间和粗糙集理论的属性权重确定方法 2.1 学术思想

2.2 流程步骤

 图 1 融合模糊商空间和粗糙集理论的属性权重确定流程Fig. 1 Flow chart of ascertaining attributes weight

1) 根据需要处理的样本对象属性(指标)项目,设有待处理的n个样本的组成集合

X={x1,x2,…,xi,…,xn},每个样本用m个指标特征值向量表示:xi={yi1,yi2,…,yik,…,yim},这样就可以构建一个样本初始属性表,如表 1所示。

 样本 条件属性集 a1 a2 … ak … am x1 y11 y12 … y1k … y1m x2 y21 y22 … y2k … y2m    …  …  xi yi1 yi2 … yik … yim    …  …  xn yn1 yn2 … ynk … ynm

2)由于样本初始属性表中各属性的量纲不同、取值范围不同和极性不同,因此必须首先采用线性比例变换法、极差变换法、比重变换法、向量标准化法等方法,消除属性量纲和数量级的影响,并进行极性转换,将属性值统一规范到固定的区间上,使得多个属性能够进行比较。

3)并行的计算步骤,即一方面基于模糊商空间理论,为创建具有分层递阶结构的商空间族,需采用数量积法、相关系数法、夹角余弦法、最大最小法、算术平均最小法等方法根据规范化处理后的属性值计算样本xixj之间的相似关系R(xi,xj)=,构建所有样本的模糊相似矩阵R=[rij]n×n,由于rij=rji,所以R既是自反矩阵也是对称矩阵,如式(3)所示。

4)该步骤同样为并行的计算步骤,即一方面需根据得到的样本集模糊相似矩阵创建具有分层递阶结构的商空间族,另一方面需根据离散化后的样本属性表分析删除各个属性后的等价类U/ind(C-{ak})。

5)以商空间族{X(λ)|0≤λ≤1}所形成的一系列粒度商空间的样本聚类结果作为相应粒度空间下单一决策属性的分类,按照式(2)分别计算各粒度商空间下各个属性的重要度。

6) 根据式(4)计算得出各属性最终重要度:

7)根据各个属性最终重要度的大小,采用归一化方法确定各属性的权重,如式(5)所示。

3 应用算例

 操作工编号 总工时a1 生产工时率a2/% 产品合格率a3/% 按期完成率a4/% 加工难度a5 加班时间a6 x1 3 681.0 99.6 100 98.70 高 294.5 x2 3 135.6 96.5 99.65 100 较高 509.0 x3 1 916.5 97.0 99.39 97.01 一般 277.1 x4 4 098.0 98.7 99.05 96.23 高 601.8 x5 2 076.9 98.1 99.96 97.92 较高 532.2 x6 3 937.6 99.0 95.67 95.58 高 462.6 x7 4 194.3 98.3 98.92 94.67 较高 711.9 x8 3 456.4 98.4 99.78 98.31 较高 346.7 x9 4 515.1 99.2 100 100 较高 543.8 x10 2 365.6 100 97.53 100 一般 132.2 x11 2 654.3 96.9 99.74 87.00 一般 259.7 x12 2 975.2 87.0 100 99.61 高 387.3 x13 3 777.2 99.0 100 99.87 较高 474.2 x14 4 354.7 99.5 99.91 94.02 高 636.5 x15 1 306.9 95.3 99.91 98.44 一般 375.7

1)由于"加工难度"属性为定性指标,直接按照"高=100,较高=50,一般=0"的规则进行处理。对于其余连续型属性值,采用式(6)所示的极差变换法进行属性规范化处理,将每一个属性值统一于同一数值范围[0,100],规范化后的属性数据表如表 3所示。

 样本 a1 a2 a3 a4 a5 a6 x1 74 97 100 90 100 72 x2 57 73 92 100 50 35 x3 19 77 86 77 0 75 x4 87 90 78 71 100 19 x5 24 85 99 84 50 31 x6 82 92 0 66 100 43 x7 90 87 75 59 50 0 x8 67 88 95 87 50 63 x9 100 94 100 100 50 29 x10 33 100 43 100 0 100 x11 42 76 94 0 0 78 x12 52 0 100 97 100 56 x13 77 92 100 99 50 41 x14 95 96 98 54 100 13 x15 0 64 98 88 0 58

2)采用夹角余弦法计算每一个样本之间的相似关系,其计算公式如式(7)所示,从而建立所有样本的模糊相似矩阵,如式(8)所示。

3) 根据文献[24]给出的基于模糊相似矩阵的分层递阶结构聚类算法,得到一个有序的样本商空间族{X(λ)|0≤λ≤1},不同粒度的商空间分别对应不同的样本聚类结果,如表 4所示。图 2所示为与表 4相对应的根据分层递阶结构绘制的样本聚类结构图。

 商空间族 样本聚类结果 聚类数 X(1) {{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8},{x9},{x10},{x11},{x12},{x13},{x14},{x15}} 15 X(0.994) {{x1},{x2,x13},{x3},{x4},{x5},{x6},{x7},{x8},{x9},{x10},{x11},{x12},{x14},{x15}} 14 X(0.993) {{x1},{x2,x9,x13},{x3},{x4},{x5},{x6},{x7},{x8},{x10},{x11},{x12},{x14},{x15}} 13 X(0.991) {{x1},{x2,x9,x13},{x3},{x4,x14},{x5},{x6},{x7},{x8},{x10},{x11},{x12},{x15}} 12 X(0.990) {{x1},{x2,x8,x9,x13},{x3},{x4,x14},{x5},{x6},{x7},{x10},{x11},{x12},{x15}} 11 X(0.979) {{x1,x2,x8,x9,x13},{x3},{x4,x14},{x5},{x6},{x7},{x10},{x11},{x12},{x15}} 10 X(0.978) {{x1,x2,x8,x9,x13},{x3,x15},{x4,x14},{x5},{x6},{x7},{x10},{x11},{x12}} 9 X(0.975) {{x1,x2,x5,x7,x8,x9,x13},{x3,x15},{x4,x14},{x6},{x10},{x11},{x12}} 7 X(0.971) {{x1,x2,x4,x5,x7,x8,x9,x13,x14},{x3,x15},{x6},{x10},{x11},{x12}} 6 X(0.944) {{x1,x2,x4,x5,x7,x8,x9,x13,x14},{x3,x10,x15},{x6},{x11},{x12}} 5 X(0.926) {{x1,x2,x3,x4,x5,x7,x8,x9,x10,x13,x14,x15},{x6},{x11},{x12}} 4 X(0.905) {{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x13,x14,x15},{x11},{x12}} 3 X(0.888) {{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15},{x11}} 2 X(0.865) {{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15}} 1
 图 2 样本聚类结构图Fig. 2The structural clustering graph

4)按照表 5中的离散区间对表 3中的属性值进行离散化处理,得到具有离散化属性值的属性表,如表 6所示。根据表 6,可以分析计算删除各个属性后的等价类。

 y′ik [0,50) [50,80] (80,100] 离散值 1 2 3 含义 低 中等 高

 U a1 a2 a3 a4 a5 a6 x1 2 3 3 3 3 2 x2 2 2 3 3 2 1 x3 1 2 3 2 1 2 x4 3 3 2 2 3 1 x5 1 3 3 3 2 1 x6 3 3 1 2 3 1 x7 3 3 2 2 2 1 x8 2 3 3 3 2 2 x9 3 3 3 3 2 1 x10 1 3 1 3 1 3 x11 1 2 3 1 1 2 x12 2 1 3 3 3 2 x13 2 3 3 3 2 1 x14 3 3 3 2 3 1 x15 1 2 3 3 1 2

5)以表 4中商空间族所形成的14个不同粒度商空间的样本聚类结果作为相应粒度空间下单一决策属性的分类,按照式(2)分别计算各粒度空间下各个属性的重要度,再根据式(4)综合计算得出各属性的最终重要度,计算结果如表 7所示。

 序号 商空间族 属性重要度 a1 a2 a3 a4 a5 a6 1 X(1) 3/15 4/15 3/15 3/15 2/15 2/15 2 X(0.994) 3/15 2/15 3/15 3/15 2/15 2/15 3 X(0.993) 3/15 2/15 3/15 3/15 2/15 2/15 4 X(0.991) 3/15 2/15 3/15 3/15 2/15 2/15 5 X(0.990) 3/15 2/15 3/15 3/15 2/15 0 6 X(0.979) 3/15 2/15 3/15 3/15 2/15 0 7 X(0.978) 3/15 2/15 3/15 3/15 2/15 0 8 X(0.975) 0 2/15 3/15 3/15 2/15 0 9 X(0.971) 0 2/15 3/15 3/15 0 0 10 X(0.944) 0 2/15 3/15 3/15 0 0 11 X(0.926) 0 2/15 3/15 3/15 0 0 12 X(0.905) 0 2/15 0 3/15 0 0 13 X(0.888) 0 0 0 3/15 0 0 14 X(0.865) 0 0 0 0 0 0 综合值 0.1 0.124 0.157 0.186 0.076 0.038

6)按照式(5)确定各属性的权重:wa1=0.147,wa2=0.182,wa3=0.230,wa4=0.273,wa5=0.112,wa6=0.056。由计算结果可以看出,各属性的权重由大到小的排列顺序为:按期完成率>产品合格率>生产工时率>总工时>加工难度>加班时间,这与该企业管理者正从以前单纯强调工作量和加班时间,逐步转变为更加重视任务进度的控制、产品质量的稳定、工作效率的提高这一技能人员工作绩效考核评价新思路是完全吻合的,实现了基于数据驱动的客观权重计算与决策者主观偏好的有机统一,由此初步验证了本文所提出方法的合理性和有效性。

7)为进一步验证该方法的准确性和实用性,采用文献[12, 13, 14, 15]所提出方法进行了属性权重计算(由于篇幅所限,本文不给出详细的计算过程)。其计算结果为:wa1=0.141,wa2=0.172,wa3=0.176,wa4=0.191,wa5=0.190,wa6=0.130,即各属性的权重由大到小的排列顺序为:按期完成率>加工难度>产品合格率>生产工时率>总工时>加班时间。通过对比可知,2种方法属性权重计算结果的总体分布是基本一致的,但本文方法的准确性和实用性主要体现在:①以往方法的属性权重计算结果相对比较平均,差距不大,其标准差仅为0.023,因此对企业管理者新的考核评价思路的体现作用不显著;而本文方法的权重标准差为0.072,为以往方法的3.1倍,相比较而言,更能充分发挥管理者所重视指标的导向作用。②2种方法对于"加工难度"指标权重的计算结果差异较大。考虑到"加工难度"仅仅是个定性指标,目前还难以量化,区分度不强,如果权重过大,在具体实施时将很难操作,因此本文方法相对较低的权重是比较合适的。③本文方法的模糊相似度计算次数仅为以往方法的1/7,且无需进行以往方法求出模糊等价矩阵的7×4次传递闭包计算,因此计算量要小得多。

4 结束语

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DOI: 10.3969/j.issn.1673-4785.201312008

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#### 文章信息

ZHOU Danchen

A method for ascertaining the weight of attributes based on granular computing

CAAI Transactions on Intelligent Systems, 2015, 10(02): 273-280.
DOI: 10.3969/j.issn.1673-4785.201312008