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Application of the set-pair analysis connection number in decision-making of black-start vague set
ZHAO Yuling, ZHANG Lian
Zhejiang University of Water Resources and Electric Power, Hangzhou 310018,China
Abstract: A vague set is a kind of fuzzy set that contains uncertainty. So uncertainty analysis is required in the application of it. To solve this problem,the proposed decision-making method of a black-start vague set is based on connection number analysis of set-pair. First,rewrite the expert weights,index weights and targeted values represented by vague sets into the connection number form. Next,obtain the decision model of a vague set for black-start based on connection number,analyze the uncertainty of the model calculations,study the changes in the ranking of each project in conditions of uncertainty and select the optimal project. Practical application shows that the method can deal with the problem of optimizing black-start schemes of the correlation and uncertainty in the selected evaluation indexes effectively and clearly. It is simple in algorithm and convenient to make decision on spot.
Key words: black-start decision     vague set     correlation index     uncertainty     connection number     set-pair analysis

1 集对分析与联系数简介 1.1 集对分析

1.2 联系数

 (1)

2 联系数的运算

2.1 加法运算

2.2 乘法运算

 (2)

3 vague集向联系数的转换 3.1 vague的概念

 图 1 Vague集 Fig. 1 Vague set

3.2 vague集的不确定度

 (1)

πV(x)为vague集V的不确定度。

3.3 vague集的不确定性及其特征联系数

 (4)

 (5)

 (6)
 (7)
4 联系数表示的黑启动决策模型 4.1 黑启动群决策的vague数据

4.1.1 黑启动方案的评价指标数据

4.1.2 指标权重W(ck)

4.1.3 专家人数与专家权重

4.2 群决策模型

4.2.1 基本模型

 (8)

 (9)

Sv代表第v个方案(v=1,2,…,m)，M(Sv)表示第v个方案的综合评价值，pv(ck)表示第v个方案的指标k(k=1,2,…,n)的值，W(ck)表示指标ck的权重，W(Ej)表示专家权重。

m个方案的优劣评价准则为：M(Sv)值大的优于M(Sv)值小的。

4.2.2 关联模型

4.3 专家关联权重与指标关联权重的计算

1)利用式(5)~(7)把各vague集转化为a+bi形式的联系数。

2)把Q个专家之间的关联权重W(E1,E2,…,EQ)一一折算给各个关联专家，即得平均关联权重W(E1,E2,…,EQ)的计算公式：

3)根据上述分析得出每个专家共2p/2个平均关联权重的平均值:

4)用类似于前3步的方法计算各指标的(独立)权重。

4.4 不确定性分析

5 实例

 方案 机组状态c1 爬坡速率c2 机组容量c3 变电站个数c4 1 (0.30,0.50) (0.96,0.04) (1.00,0.00) (0.20,0.80) 2 (0.60,0.17) (0.53,0.47) (0.67,0.33) (0.25,0.75) 3 (0.45,0.33) (1.00,0.00) (0.42,0.58) (0.33,0.67) 4 (0.30,0.50) (0.50,0.50) (0.42,0.58) (0.33,0.67) 5 (0.60,0.17) (0.27,0.73) (0.42,0.58) (1.00,0.00) 6 (0.90,0.00) (0.91,0.09) (0.67,0.33) (0.25,0.75)

e3)，权重、关联权重分别为

W({e1})=(0.30,0.60)

W({e2})=(0.40,0.50)

W({e3})=(0.40,0.30)

W({e1,e2})=(0.60,0.20)

W({e1,e3})=(0.70,0.10)

W({e2,e3})=(0.70,0.20)

W({e1,e2,e3})=(1.00,0.00)

W({c1})=(0.10,0.65)

W({c2})=(0.25,0.55)

W({c3})=(0.20,0.50)

W({c4})=(0.20,0.70)

W({c1,c2})=(0.30,0.40)

W({c2,c3})=(0.50,0.25)

W({c2,c4})=(0.45,0.35)

W({c3,c4})=(0.40,0.30)

W({c1,c3})=(0.30,0.30)

W({c1,c4})=(0.30,0.40)

W({c2,c3,c4})=(0.85,0.10)

W({c1,c3,c4})=(0.70,0.20)

W({c1,c2,c4})=(0.65,0.30)

W({c1,c1,c3})=(0.75,0.20)

W({c1,c2,c3,c4})=(1.00,0.00)

1)根据式(5)～(7)计算3位专家权重如表 2

 W1) W(e2) W(e3) 平均 0.32075+0.075i 0.34575+0.0625i 0.35825+0.1125i

2)根据式(5)～(7)计算专家e1给出的黑启动指标权重及计及专家权重后的各指标权重，见表 3

 专家 指标c1 指标c2 指标c3 指标c4 平均权重联系数 (0.1875+0.1021i)× (0.32075+0.075i)= 0.0601+0.0546i (0.2344+0.0783i)× (0.32075+0.075i)= 0.0752+0.0486i (0.227+0.1053i)× (0.32075+0.075i)= 0.0728+0.0587i (0.2291+0.0709i)× (0.32075+0.075i)= 0.0735+0.0452i

W(c1)=0.2034+0.1877i

W(c2)=0.2116+0.1732i

W(c3)=0.2144+0.1797i

W(c4)=0.2255+0.1702i

3)把表 1中的各vague集数据改写成联系数，得表 4

 方案 c1 c2 c3 c4 1 0.30+0.20i 0.96+0i 1+0i 0.2+0i 2 0.60+0.23i 0.53+0i 0.67+0i 0.25+0i 3 0.45+0.22i 1+0i 0.42+0i 0.33+0i 4 0.30+0.20i 0.50+0i 0.42+0i 0.33+0i 5 0.60+0.23i 0.27+0i 0.42+0i 1+0i 6 0.90+0.10i 0.91+0i 0.67+0i 0.25+0i

4)利用表 4，并结合第2步得到的各指标权重，采用式(8)算得各方案的综合评价值联系数为

M(s1)=0.523 62+0.514 542i

M(s2)=0.434 211+0.457 318i

M(s3)=0.467 593+0.475 347i

M(s4)=0.331 283+0.352 77i

M(s5)=0.494 72+0.495 011i

M(s6)=0.575 639+0.528 601i

5)对各方案的综合评价值联系数作不确定性计算分析，得表 5

 i=－1 i=－0.5 i=0 i=0.5 i=1 M(s1) 0.0091② 0.2663② 0.5237② 0.7809② 1.0382② M(s2) -0.0231⑥ 0.2056⑤ 0.5237② 0.6629⑤ 0.8915⑤ M(s3) -0.0078④ 0.2299④ 0.4676④ 0.7053④ 0.9429④ M(s4) -0.0215⑤ 0.1549⑥ 0.3313⑥ 0.5077⑥ 0.6840⑥ M(s5) -0.0002③ 0.2472③ 0.4947③ 0.7422③ 0.9897③ M(s6) 0.0470① 0.3113① 0.5756① 0.8299① 1.1042①

 方案 第1次 第2次 第3次 第4次 第5次 第6次 总分 1 16 18 8 12 8 2 272 2 2 8 12 8 3 31 161 3 4 12 12 12 16 8 208 4 1 5 10 10 20 18 159 5 8 16 12 12 12 4 240 6 33 5 10 10 5 1 304 Σ 64 64 64 64 64 64 1344

6 结束语

W({c1})=(0.15,0.7)

W({c2})=(0.2,0.7)

W({c3})=(0.15,0.7)

W({c4})=(0.25,0.55)

W({c1,c2})=(0.3,0.5)

W({c2,c3})=(0.37,0.43)

W({c2,c4})=(0.5,0.3)

W({c3,c4})=(0.45,0.3)

W({c1,c3})=(0.35,0.45)

W({c1,c4})=(0.45,0.30)

W({c2,c3,c4})=(0.75,0.10)

W({c1,c3,c4})=(0.75,0.10)

W({c1,c2,c4})=(0.68,0.15)

W({c1,c2,c3})=(0.55,0.25)

W({c1,c2,c3,c4})=(1.00,0.00)

W({c1})=(0.2,0.5)

W({c2})=(0.1,0.65)

W({c3})=(0.15,0.7)

W({c4})=(0.15,0.7)

W({c1,c2})=(0.35,0.3)

W({c2,c3})=(0.3,0.4)

W({c2,c4})=(0.3,0.4)

W({c3,c4})=(0.35,0.4)

W({c1,c3})=(0.42,0.28)

W({c1,c4})=(0.42,0.28)

W(c2,c3,c4)=(0.6,0.05)

W(c1,c3,c4)=(0.65,0.05)

W(c1,c2,c4)=(0.55,0.15)

W(c1,c2,c3)=(0.6,0.05)

W(c1,c2,c3,c4)=(1.00,0.00)

 专家 指标c1 指标c2 指标c3 指标c4 平均权重联系数 (0.2013+0.0810i)×(0.34575+0.0625i)=0.0696+0.0457i (0.2119+0.0716i)×(0.34575+0.0625i)=0.0733+0.0425i (0.2085+0.0802i)×(0.34575+0.0625i)=0.0721+0.0457i (0.2409+0.0883i)×(0.34575+0.0625i)=0.0833+0.0510i

 专家 指标c1 指标c2 指标c3 指标c4 平均权重联系数 (0.2056+0.1365i)×(0.35825+0.1125i)=0.0737+0.0874i (0.1764+0.1324i)×(0.35825+0.1125i)=0.0631+0.0821i (0.194+0.113i)×(0.35825+0.1125i)=0.0695+0.0753i (0.1910+0.1115i)×(0.35825+0.1125i)=0.0687+0.0740i

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DOI: 10.3969/j.issn.1673-4785.201305002

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#### 文章信息

ZHAO Yuling, ZHANG Lian

Application of the set-pair analysis connection number in decision-making of black-start vague set

CAAI Transactions on Intelligent Systems, 2014, 9(5): 632-640
http://dx.doi.org/10.3969/j.issn.1673-4785.201305002