综合-共振-对抗解释结构模型

前置流程图说明

　　对于综合影响矩阵T它是一个典型的模糊矩阵,任意模糊矩阵得到布尔型可达矩阵R的方式有两种。

　　取截距，得到关系矩阵A，由A得到相乘矩阵B，B经过运算得到可达矩阵R

　　由T得到模糊可达矩阵FR，FR再取截距，得到可达矩阵R

　　即两者是等价的。

　　上述论述请参看论文：

　　黄炜黑客与反黑客思维研究的方法论启示——解释结构模型新探

　　兰芳基于FISM-ANP-灰色聚类的软件项目开发风险评价研究

　　目前的论文基本都是错的，一个重要的原因就是认为T+I取截距后得到的是可达矩阵，这是错得离谱。

T-RAISM流程图与说明

　　方法英文全称如下：

　　Base ON Total Matrix Resonant Adversarial Interpretive Structure Modeling Method

　　Total Resonant Adversarial Interpretive Structure Modeling Method

　　中文名如下：

　　综合-共振-对抗解释结构模型（推荐）

　　共振体，共振结构：请参看论文黑客与反黑客思维研究的方法论启示——解释结构模型新探该概念来自化学学科

　　TRAISM中核心概念：

　　第一、阈值集合

　　对于模糊可达矩阵，矩阵中的数值去重后得到的集合即成为阈值集合，有$\ddot \Delta = \{ \lambda_1 , \lambda_2 , … , \lambda_n \}$

　　第二、取截距的方式

　　模糊可达矩阵$ FR=[f_{ij}]_{n \times n} $ 且有 $ f_{ij}∈ \ddot \Delta　$

　　截矩阵$　U=[u_{ij}]_{n \times n} $

　　模糊可达矩阵与截矩阵对应关系如下：

　　$$ u_{ij}= \begin{cases} 1 , \text{ $e_i$}\rightarrow \text{$e_j$ 当： $\tilde f_{ij} ＞ \lambda $} \\ Un , \text{ $e_i$}\rightarrow \text{$e_j$ 当： $ \tilde f_{ij} ＝ \lambda $} \\ 0, \text{ $e_i$}\rightarrow \text{$e_j$ 当：　$\tilde f_{ij} ＜ \lambda $} \end{cases} $$

　　第三、截距阵为母阵，其特点如下

　　由于母阵是由模糊可达矩阵取截距得到，当母阵中的不确定值全部为0的时候，对应的矩阵为可达矩阵。

　　当母阵中的不确定值全部为1的时候，对应的矩阵为可达矩阵。

　　第四、两次运用五项原则

　　第一次使用请参看 T-AISM方法

　　第二次使用请参看特征共振体计算

规范化方法选择（参数选择）：

原始矩阵（直接影响矩阵）为

$$Ori=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &F1 &F2 &F3 &F4 &F5 &F6 &F7 &F8 &F9 &F10 &F11 &F12\\ \hline F1 &0 &2 &1 &2 &1 &3 &3 &1 &1 &3 &17 &11\\ \hline F2 &2 &0 &3 &3 &1 &5 &1 &4 &1 &13 &4 &1\\ \hline F3 &1 &1 &0 &10 &1 &13 &2 &2 &2 &2 &2 &0\\ \hline F4 &13 &10 &14 &0 &5 &9 &19 &9 &8 &9 &9 &6\\ \hline F5 &1 &1 &3 &7 &0 &1 &1 &1 &15 &10 &10 &10\\ \hline F6 &1 &15 &12 &10 &11 &0 &2 &1 &1 &1 &13 &3\\ \hline F7 &2 &4 &4 &5 &5 &6 &0 &8 &2 &3 &2 &14\\ \hline F8 &1 &16 &12 &10 &2 &0 &1 &0 &3 &12 &10 &13\\ \hline F9 &1 &2 &3 &4 &5 &6 &7 &2 &0 &10 &10 &12\\ \hline F10 &5 &0 &13 &13 &0 &0 &0 &0 &1 &0 &3 &11\\ \hline F11 &3 &13 &0 &0 &0 &0 &0 &0 &0 &1 &0 &11\\ \hline F12 &0 &3 &1 &0 &0 &0 &0 &9 &0 &1 &12 &0\\ \hline \end{array} $$

规范直接关系矩阵求解过程

$$\mathcal{N}=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &F1 &F2 &F3 &F4 &F5 &F6 &F7 &F8 &F9 &F10 &F11 &F12\\ \hline F1 &0 &0.018 &0.009 &0.018 &0.009 &0.027 &0.027 &0.009 &0.009 &0.027 &0.153 &0.099\\ \hline F2 &0.018 &0 &0.027 &0.027 &0.009 &0.045 &0.009 &0.036 &0.009 &0.117 &0.036 &0.009\\ \hline F3 &0.009 &0.009 &0 &0.09 &0.009 &0.117 &0.018 &0.018 &0.018 &0.018 &0.018 &0\\ \hline F4 &0.117 &0.09 &0.126 &0 &0.045 &0.081 &0.171 &0.081 &0.072 &0.081 &0.081 &0.054\\ \hline F5 &0.009 &0.009 &0.027 &0.063 &0 &0.009 &0.009 &0.009 &0.135 &0.09 &0.09 &0.09\\ \hline F6 &0.009 &0.135 &0.108 &0.09 &0.099 &0 &0.018 &0.009 &0.009 &0.009 &0.117 &0.027\\ \hline F7 &0.018 &0.036 &0.036 &0.045 &0.045 &0.054 &0 &0.072 &0.018 &0.027 &0.018 &0.126\\ \hline F8 &0.009 &0.144 &0.108 &0.09 &0.018 &0 &0.009 &0 &0.027 &0.108 &0.09 &0.117\\ \hline F9 &0.009 &0.018 &0.027 &0.036 &0.045 &0.054 &0.063 &0.018 &0 &0.09 &0.09 &0.108\\ \hline F10 &0.045 &0 &0.117 &0.117 &0 &0 &0 &0 &0.009 &0 &0.027 &0.099\\ \hline F11 &0.027 &0.117 &0 &0 &0 &0 &0 &0 &0 &0.009 &0 &0.099\\ \hline F12 &0 &0.027 &0.009 &0 &0 &0 &0 &0.081 &0 &0.009 &0.108 &0\\ \hline \end{array} $$

综合影响矩阵求解过程

　　综合影响矩阵如下

$T=\mathcal{N}(I-\mathcal{N})^{-1}$

$$T=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &F1 &F2 &F3 &F4 &F5 &F6 &F7 &F8 &F9 &F10 &F11 &F12\\ \hline F1 &0.015 &0.06 &0.032 &0.038 &0.019 &0.04 &0.037 &0.03 &0.018 &0.049 &0.189 &0.14\\ \hline F2 &0.037 &0.035 &0.068 &0.065 &0.023 &0.064 &0.026 &0.052 &0.022 &0.141 &0.076 &0.053\\ \hline F3 &0.032 &0.058 &0.047 &0.124 &0.034 &0.141 &0.047 &0.041 &0.037 &0.051 &0.07 &0.044\\ \hline F4 &0.153 &0.179 &0.206 &0.088 &0.084 &0.142 &0.207 &0.133 &0.107 &0.159 &0.195 &0.175\\ \hline F5 &0.036 &0.057 &0.073 &0.103 &0.019 &0.04 &0.04 &0.04 &0.151 &0.131 &0.152 &0.157\\ \hline F6 &0.041 &0.19 &0.156 &0.138 &0.118 &0.045 &0.052 &0.044 &0.043 &0.07 &0.181 &0.093\\ \hline F7 &0.039 &0.088 &0.083 &0.087 &0.063 &0.08 &0.024 &0.102 &0.04 &0.071 &0.085 &0.178\\ \hline F8 &0.045 &0.199 &0.167 &0.145 &0.038 &0.047 &0.045 &0.042 &0.052 &0.165 &0.162 &0.184\\ \hline F9 &0.034 &0.071 &0.075 &0.08 &0.064 &0.08 &0.084 &0.05 &0.021 &0.126 &0.151 &0.17\\ \hline F10 &0.07 &0.04 &0.152 &0.146 &0.016 &0.037 &0.033 &0.031 &0.027 &0.032 &0.081 &0.139\\ \hline F11 &0.033 &0.129 &0.013 &0.012 &0.004 &0.01 &0.005 &0.016 &0.004 &0.03 &0.027 &0.113\\ \hline F12 &0.009 &0.059 &0.028 &0.017 &0.005 &0.008 &0.006 &0.088 &0.006 &0.03 &0.128 &0.03\\ \hline \end{array} $$

模糊可达矩阵FR的求解

FR：模糊可达矩阵的求解
$FB= T +I$
$FB^{(k-1)}≠FB^{k}=FB^{(k+1)}= FR$
$FR 为模糊可达矩阵$
$FB 为模糊相乘矩阵$
$FB 主对角线为1$
$FB= \begin{array} {c|c|c|c|c|c|c|c}{ FB_{n \times n}} &1 &2 &{\cdots} &n \\ \hline 1 & \color{blue}{1}&{b_{12}}&{\cdots}&{b_{1n}}\\ \hline 2 & {b_{21}}&\color{blue}{1}&{\cdots}&{b_{2n}}\\ \hline {\vdots} &{\vdots}&{\vdots}&\color{blue}{1}&{\vdots}\\ \hline n & {b_{n1}}&{b_{n2}}&{\cdots}&\color{blue}{1}\\ \hline \end{array}$
$FB 主对角线为1$
$ 设FC =FB \times FB \quad FC=\left[ c_{ij} \right]_{n \times n} \quad FB=\left[ b_{ij} \right]_{n \times n}$
$ \begin{equation}\begin{split} c_{ij}&=\sum_{k=1}^n b_{ik}\odot b_{kj} \\ &=(b_{i1} \odot b_{1j}) \oplus (b_{i2} \odot b_{2j}) \oplus (b_{i3} \odot b_{3j}) \cdots \oplus \cdots(b_{in} \odot b_{nj})\\ \end{split}\end{equation}$
模糊算子采用查德算子，即最大最小算子，格式如下。
$ \begin{equation}\begin{split} c_{ij}&=\sum_{k=1}^n b_{ik} \land b_{kj} \\ &=(b_{i1} \land b_{1j}) \vee (b_{i2} \land b_{2j}) \vee (b_{i3} \land b_{3j}) \cdots \vee \cdots(b_{in} \land b_{nj})\\ \end{split}\end{equation}$

模糊相乘矩阵求解 FB=T+I即综合影响矩阵中主对角线全部变成1

$$FB=\begin{array} {c|c|c}{M_{12 \times12}} &F1 &F2 &F3 &F4 &F5 &F6 &F7 &F8 &F9 &F10 &F11 &F12\\ \hline F1 &1 &0.0595 &0.0325 &0.0384 &0.0187 &0.0402 &0.0374 &0.0299 &0.0178 &0.0489 &0.189 &0.1397\\ \hline F2 &0.0372 &1 &0.0681 &0.065 &0.0226 &0.0637 &0.0259 &0.0515 &0.0223 &0.1408 &0.0757 &0.0535\\ \hline F3 &0.0324 &0.0582 &1 &0.1245 &0.0344 &0.1411 &0.0471 &0.0406 &0.037 &0.051 &0.0696 &0.044\\ \hline F4 &0.1528 &0.1787 &0.206 &1 &0.0845 &0.1422 &0.2069 &0.1328 &0.1066 &0.1588 &0.1946 &0.1752\\ \hline F5 &0.0357 &0.0571 &0.0733 &0.1029 &1 &0.04 &0.0402 &0.0399 &0.1507 &0.1313 &0.1521 &0.1566\\ \hline F6 &0.0412 &0.1895 &0.1558 &0.1376 &0.1183 &1 &0.0522 &0.0437 &0.043 &0.0698 &0.1809 &0.0933\\ \hline F7 &0.039 &0.0878 &0.083 &0.0868 &0.0635 &0.0798 &1 &0.1023 &0.04 &0.071 &0.0852 &0.1781\\ \hline F8 &0.0453 &0.1987 &0.1668 &0.1449 &0.038 &0.047 &0.0447 &1 &0.0517 &0.1647 &0.1615 &0.1844\\ \hline F9 &0.0336 &0.0708 &0.075 &0.0804 &0.0638 &0.0797 &0.0835 &0.0503 &1 &0.1261 &0.1505 &0.1705\\ \hline F10 &0.0695 &0.0404 &0.152 &0.1465 &0.0159 &0.0368 &0.0329 &0.0313 &0.0275 &1 &0.0812 &0.1387\\ \hline F11 &0.0333 &0.129 &0.013 &0.0117 &0.0037 &0.0097 &0.0049 &0.0159 &0.0039 &0.0301 &1 &0.1134\\ \hline F12 &0.0092 &0.0589 &0.0276 &0.0172 &0.0046 &0.0082 &0.0056 &0.0883 &0.0058 &0.0302 &0.1276 &1\\ \hline \end{array} $$

模糊可达矩阵FR

$$FR=\begin{array} {c|c|c}{M_{12 \times12}} &F1 &F2 &F3 &F4 &F5 &F6 &F7 &F8 &F9 &F10 &F11 &F12\\ \hline F1 &1 &0.129 &0.129 &0.129 &0.1183 &0.129 &0.129 &0.129 &0.1183 &0.129 &0.189 &0.1397\\ \hline F2 &0.1408 &1 &0.1408 &0.1408 &0.1183 &0.1408 &0.1408 &0.1328 &0.1183 &0.1408 &0.1408 &0.1408\\ \hline F3 &0.1408 &0.1411 &1 &0.1408 &0.1183 &0.1411 &0.1408 &0.1328 &0.1183 &0.1408 &0.1411 &0.1408\\ \hline F4 &0.1528 &0.1787 &0.206 &1 &0.1183 &0.1422 &0.2069 &0.1328 &0.1183 &0.1588 &0.1946 &0.1781\\ \hline F5 &0.1313 &0.1313 &0.1313 &0.1313 &1 &0.1313 &0.1313 &0.1313 &0.1507 &0.1313 &0.1521 &0.1566\\ \hline F6 &0.1408 &0.1895 &0.1558 &0.1408 &0.1183 &1 &0.1408 &0.1328 &0.1183 &0.1408 &0.1809 &0.1408\\ \hline F7 &0.1276 &0.1276 &0.1276 &0.1276 &0.1183 &0.1276 &1 &0.1276 &0.1183 &0.1276 &0.1276 &0.1781\\ \hline F8 &0.1465 &0.1987 &0.1668 &0.1465 &0.1183 &0.1422 &0.1465 &1 &0.1183 &0.1647 &0.1615 &0.1844\\ \hline F9 &0.129 &0.129 &0.129 &0.129 &0.1183 &0.129 &0.129 &0.129 &1 &0.129 &0.1505 &0.1705\\ \hline F10 &0.1465 &0.1465 &0.152 &0.1465 &0.1183 &0.1422 &0.1465 &0.1328 &0.1183 &1 &0.1465 &0.1465\\ \hline F11 &0.129 &0.129 &0.129 &0.129 &0.1183 &0.129 &0.129 &0.129 &0.1183 &0.129 &1 &0.129\\ \hline F12 &0.1276 &0.1276 &0.1276 &0.1276 &0.1183 &0.1276 &0.1276 &0.1276 &0.1183 &0.1276 &0.1276 &1\\ \hline \end{array} $$

模糊可达矩阵FR中阈值集合分析

模糊可达矩阵FR中除数字0外不重复的值作为元素构成的集合称为阈值集合。

33个结构数量分布图（瀑布图）

特征结构分析

由模糊可达矩阵的阈值集合$\ddot \Delta $ 元素的数目得出在(0,1]的截距值范围内得到 33个结构。

其中模糊可达矩阵(0,1]的截距值范围内取截距，得到的截矩阵为可达矩阵

设截距值为$\lambda $ ，$T$的截距阵为 $A$ ,$R$为 $A$的可达矩阵。

$R$即为模糊可达矩阵 $FR$ 的　$\lambda $ 　截距阵

序号	特征阈值	聚类对应截距$\lambda $区段	层级数	互不连通区域数目	回路数目	最大回路要素数目	该结构的数目
1	0.11834	0<$\lambda$<0.11834	1	1	1	12	21
2	0.12756	0.11834<$\lambda$<0.12756	3	1	1	10	17
3	0.12903	0.12756<$\lambda$<0.12903	5	1	1	8	24
4	0.13127	0.12903<$\lambda$<0.13127	4	1	1	6	8
5	0.13282	0.13127<$\lambda$<0.13282	3	1	1	6	5
6	0.13968	0.13282<$\lambda$<0.13968	4	1	1	5	1
7	0.14078	0.13968<$\lambda$<0.14078	4	1	1	5	18
8	0.14109	0.14078<$\lambda$<0.14109	4	1	2	2	3
9	0.14222	0.14109<$\lambda$<0.14222	4	1	1	2	3
10	0.14647	0.14222<$\lambda$<0.14647	4	1	1	2	9
11	0.15053	0.14647<$\lambda$<0.15053	3	1	0	1	1
12	0.15068	0.15053<$\lambda$<0.15068	3	1	0	1	1
13	0.15199	0.15068<$\lambda$<0.15199	3	1	0	1	1
14	0.15209	0.15199<$\lambda$<0.15209	3	1	0	1	1