原始矩阵（直接影响矩阵）为

$$Ori=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &0 &2 &1 &0 &9 &0 &2 &9 &1 &0 &2 &1\\ \hline H2 &3 &0 &0 &0 &9 &8 &0 &7 &0 &3 &2 &1\\ \hline H3 &3 &9 &0 &3 &0 &0 &3 &0 &0 &0 &2 &6\\ \hline K1 &3 &9 &4 &0 &0 &0 &1 &5 &0 &0 &2 &5\\ \hline K2 &6 &9 &0 &0 &0 &0 &0 &0 &5 &0 &2 &1\\ \hline K3 &8 &5 &2 &0 &1 &0 &2 &0 &9 &0 &2 &1\\ \hline X1 &3 &0 &0 &0 &0 &0 &0 &0 &9 &0 &2 &1\\ \hline X2 &0 &6 &2 &5 &0 &4 &1 &0 &0 &2 &2 &1\\ \hline X3 &9 &0 &0 &0 &0 &8 &7 &8 &8 &0 &2 &1\\ \hline S1 &0 &0 &3 &3 &3 &5 &8 &8 &8 &0 &0 &1\\ \hline S2 &0 &3 &0 &0 &0 &8 &4 &8 &8 &0 &0 &1\\ \hline S3 &5 &0 &0 &4 &5 &8 &8 &8 &8 &0 &2 &0\\ \hline \end{array} $$

规范直接关系矩阵求解过程 $$ \require{cancel} \require{AMScd} \begin{CD} O @>>>N \\ \end{CD} $$

$$\mathcal{N}=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &0 &0.042 &0.021 &0 &0.188 &0 &0.042 &0.188 &0.021 &0 &0.042 &0.021\\ \hline H2 &0.063 &0 &0 &0 &0.188 &0.167 &0 &0.146 &0 &0.063 &0.042 &0.021\\ \hline H3 &0.063 &0.188 &0 &0.063 &0 &0 &0.063 &0 &0 &0 &0.042 &0.125\\ \hline K1 &0.063 &0.188 &0.083 &0 &0 &0 &0.021 &0.104 &0 &0 &0.042 &0.104\\ \hline K2 &0.125 &0.188 &0 &0 &0 &0 &0 &0 &0.104 &0 &0.042 &0.021\\ \hline K3 &0.167 &0.104 &0.042 &0 &0.021 &0 &0.042 &0 &0.188 &0 &0.042 &0.021\\ \hline X1 &0.063 &0 &0 &0 &0 &0 &0 &0 &0.188 &0 &0.042 &0.021\\ \hline X2 &0 &0.125 &0.042 &0.104 &0 &0.083 &0.021 &0 &0 &0.042 &0.042 &0.021\\ \hline X3 &0.188 &0 &0 &0 &0 &0.167 &0.146 &0.167 &0 &0 &0.042 &0.021\\ \hline S1 &0 &0 &0.063 &0.063 &0.063 &0.104 &0.167 &0.167 &0.167 &0 &0 &0.021\\ \hline S2 &0 &0.063 &0 &0 &0 &0.167 &0.083 &0.167 &0.167 &0 &0 &0.021\\ \hline S3 &0.104 &0 &0 &0.083 &0.104 &0.167 &0.167 &0.167 &0.167 &0 &0.042 &0\\ \hline \end{array} $$

综合影响矩阵求解过程 $$\begin{CD} N @>>>T \\ \end{CD} $$

　　综合影响矩阵如下

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

$$T=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &0.094 &0.157 &0.043 &0.038 &0.243 &0.095 &0.095 &0.279 &0.112 &0.021 &0.092 &0.055\\ \hline H2 &0.193 &0.141 &0.035 &0.041 &0.267 &0.272 &0.081 &0.272 &0.14 &0.083 &0.104 &0.062\\ \hline H3 &0.166 &0.273 &0.024 &0.094 &0.103 &0.122 &0.136 &0.146 &0.11 &0.023 &0.097 &0.162\\ \hline K1 &0.167 &0.295 &0.108 &0.047 &0.107 &0.132 &0.1 &0.248 &0.105 &0.029 &0.102 &0.149\\ \hline K2 &0.216 &0.256 &0.018 &0.021 &0.097 &0.109 &0.063 &0.136 &0.178 &0.022 &0.089 &0.05\\ \hline K3 &0.288 &0.195 &0.063 &0.027 &0.122 &0.123 &0.13 &0.163 &0.284 &0.019 &0.102 &0.061\\ \hline X1 &0.137 &0.042 &0.011 &0.014 &0.04 &0.074 &0.06 &0.093 &0.24 &0.006 &0.073 &0.04\\ \hline X2 &0.094 &0.218 &0.07 &0.129 &0.073 &0.176 &0.083 &0.114 &0.094 &0.06 &0.088 &0.064\\ \hline X3 &0.301 &0.115 &0.034 &0.04 &0.091 &0.264 &0.221 &0.298 &0.138 &0.02 &0.107 &0.062\\ \hline S1 &0.159 &0.138 &0.099 &0.109 &0.133 &0.232 &0.266 &0.308 &0.306 &0.021 &0.076 &0.079\\ \hline S2 &0.143 &0.166 &0.032 &0.039 &0.072 &0.29 &0.172 &0.295 &0.288 &0.023 &0.065 &0.06\\ \hline S3 &0.293 &0.17 &0.046 &0.13 &0.206 &0.317 &0.281 &0.354 &0.343 &0.025 &0.134 &0.064\\ \hline \end{array} $$

由综合影响矩阵求影响度、被影响度、中心度、原因度 \begin{CD} T @>>>\{D|C|M|R \} \\ \end{CD}

　　影响度、被影响度、中心度与原因度是四种度量要素在系统里影响程度的度量值。都是根据综合影响矩阵计算得出。

求解原理

影响度 $D$	$$ D_i=\sum \limits_{j=1}^{n}{t_{ij}},(i=1,2,3,\cdots,n) $$
被影响度 $C$	$$ C_i=\sum \limits_{j=1}^{n}{t_{ji}},(i=1,2,3,\cdots,n) $$
中心度 $M$	$$ M_i=D_i+C_i $$
原因度 $ R$	$$ R_i=D_i-C_i $$

结果

影响度、被影响度、中心度、原因度

$$\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times4}} &Di &Ci &Mi &Ri\\ \hline H1 &1.323 &2.251 &3.574 &-0.928\\ \hline H2 &1.691 &2.166 &3.857 &-0.475\\ \hline H3 &1.457 &0.584 &2.041 &0.872\\ \hline K1 &1.589 &0.728 &2.317 &0.86\\ \hline K2 &1.255 &1.553 &2.808 &-0.298\\ \hline K3 &1.578 &2.206 &3.784 &-0.629\\ \hline X1 &0.83 &1.689 &2.519 &-0.859\\ \hline X2 &1.264 &2.706 &3.969 &-1.442\\ \hline X3 &1.69 &2.337 &4.027 &-0.648\\ \hline S1 &1.925 &0.352 &2.278 &1.573\\ \hline S2 &1.646 &1.13 &2.776 &0.516\\ \hline S3 &2.363 &0.907 &3.27 &1.456\\ \hline \end{array} $$

绘制中心度与原因度建构的散点图图表

从两列的决策矩阵到关系矩阵

$$ \begin{CD} D=\left[ d_{ij} \right]_{n \times 2}@>取偏序>>A=\left[a_{ij} \right]_{n \times n} \\ \end{CD} $$

中心度，原因度绝对值组成的决策矩阵D

$$D=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times2}} &M &|R|\\ \hline H1 &3.5741 &0.9277\\ \hline H2 &3.8566 &0.4747\\ \hline H3 &2.0408 &0.8723\\ \hline K1 &2.3174 &0.8605\\ \hline K2 &2.8085 &0.2976\\ \hline K3 &3.7841 &0.6287\\ \hline X1 &2.5185 &0.8589\\ \hline X2 &3.9693 &1.4422\\ \hline X3 &4.0269 &0.6476\\ \hline S1 &2.2776 &1.5731\\ \hline S2 &2.7762 &0.5156\\ \hline S3 &3.2701 &1.456\\ \hline \end{array} $$

关系矩阵$A$

$$A=\begin{array} {c|c|c|c|c|c|c|c}{M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &1 & & & & & & &1 & & & & \\ \hline H2 & &1 & & & & & &1 &1 & & & \\ \hline H3 &1 & &1 & & & & &1 & &1 & &1\\ \hline K1 &1 & & &1 & & & &1 & & & &1\\ \hline K2 &1 &1 & & &1 &1 & &1 &1 & & &1\\ \hline K3 & & & & & &1 & &1 &1 & & & \\ \hline X1 &1 & & & & & &1 &1 & & & &1\\ \hline X2 & & & & & & & &1 & & & & \\ \hline X3 & & & & & & & & &1 & & & \\ \hline S1 & & & & & & & & & &1 & & \\ \hline S2 &1 & & & & &1 & &1 &1 & &1 &1\\ \hline S3 & & & & & & & & & & & &1\\ \hline \end{array} $$

AISM部分求解

关系矩阵到相乘矩阵

\begin{CD} A@>A+I>>B \\ \end{CD}

$$B=\begin{array} {c|c|c|c|c|c|c|c}{M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &1 & & & & & & &1 & & & & \\ \hline H2 & &1 & & & & & &1 &1 & & & \\ \hline H3 &1 & &1 & & & & &1 & &1 & &1\\ \hline K1 &1 & & &1 & & & &1 & & & &1\\ \hline K2 &1 &1 & & &1 &1 & &1 &1 & & &1\\ \hline K3 & & & & & &1 & &1 &1 & & & \\ \hline X1 &1 & & & & & &1 &1 & & & &1\\ \hline X2 & & & & & & & &1 & & & & \\ \hline X3 & & & & & & & & &1 & & & \\ \hline S1 & & & & & & & & & &1 & & \\ \hline S2 &1 & & & & &1 & &1 &1 & &1 &1\\ \hline S3 & & & & & & & & & & & &1\\ \hline \end{array} $$

相乘矩阵B到可达矩阵R

\begin{CD} B@>连乘或者幂乘>>R \\ \end{CD}

可达矩阵R

$$R=\begin{array} {c|c|c|c|c|c|c|c}{M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &1 & & & & & & &1 & & & & \\ \hline H2 & &1 & & & & & &1 &1 & & & \\ \hline H3 &1 & &1 & & & & &1 & &1 & &1\\ \hline K1 &1 & & &1 & & & &1 & & & &1\\ \hline K2 &1 &1 & & &1 &1 & &1 &1 & & &1\\ \hline K3 & & & & & &1 & &1 &1 & & & \\ \hline X1 &1 & & & & & &1 &1 & & & &1\\ \hline X2 & & & & & & & &1 & & & & \\ \hline X3 & & & & & & & & &1 & & & \\ \hline S1 & & & & & & & & & &1 & & \\ \hline S2 &1 & & & & &1 & &1 &1 & &1 &1\\ \hline S3 & & & & & & & & & & & &1\\ \hline \end{array} $$

由可达矩阵通过对抗的抽取方式得到一对要素的层级分布 \begin{CD} R @>结果优先抽取>原因优先抽取> \frac {up型要素层级分布 }{down 型要素层级分布} \\ \end{CD}

可达矩阵R层级抽取的过程如下

结果优先——UP型抽取过程	原因优先——DOWN型抽取过程
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline H1&H1,X2&H1 \\\hline H2&H2,X2,X3&H2 \\\hline H3&H1,H3,X2,S1,S3&H3 \\\hline K1&H1,K1,X2,S3&K1 \\\hline K2&H1,H2,K2,K3,X2,X3,S3&K2 \\\hline K3&K3,X2,X3&K3 \\\hline X1&H1,X1,X2,S3&X1 \\\hline X2&\color{red}{\fbox{X2}}&\color{red}{\fbox{X2}} \\\hline X3&\color{red}{\fbox{X3}}&\color{red}{\fbox{X3}} \\\hline S1&\color{red}{\fbox{S1}}&\color{red}{\fbox{S1}} \\\hline S2&H1,K3,X2,X3,S2,S3&S2 \\\hline S3&\color{red}{\fbox{S3}}&\color{red}{\fbox{S3}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&H1,H3,K1,K2,X1,S2&H1 \\\hline H2&H2,K2&H2 \\\hline H3&\color{blue}{\fbox{H3}}&\color{blue}{\fbox{H3}} \\\hline K1&\color{blue}{\fbox{K1}}&\color{blue}{\fbox{K1}} \\\hline K2&\color{blue}{\fbox{K2}}&\color{blue}{\fbox{K2}} \\\hline K3&K2,K3,S2&K3 \\\hline X1&\color{blue}{\fbox{X1}}&\color{blue}{\fbox{X1}} \\\hline X2&H1,H2,H3,K1,K2,K3,X1,X2,S2&X2 \\\hline X3&H2,K2,K3,X3,S2&X3 \\\hline S1&H3,S1&S1 \\\hline S2&\color{blue}{\fbox{S2}}&\color{blue}{\fbox{S2}} \\\hline S3&H3,K1,K2,X1,S2,S3&S3 \\\hline \end{array} $$
抽取出X2、X3、S1、S3放置上层，删除后剩余的情况如下	抽取出H3,K1,K2,X1,S2放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline H1&\color{red}{\fbox{H1}}&\color{red}{\fbox{H1}} \\\hline H2&\color{red}{\fbox{H2}}&\color{red}{\fbox{H2}} \\\hline H3&H1,H3&H3 \\\hline K1&H1,K1&K1 \\\hline K2&H1,H2,K2,K3&K2 \\\hline K3&\color{red}{\fbox{K3}}&\color{red}{\fbox{K3}} \\\hline X1&H1,X1&X1 \\\hline S2&H1,K3,S2&S2 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&\color{blue}{\fbox{H1}}&\color{blue}{\fbox{H1}} \\\hline H2&\color{blue}{\fbox{H2}}&\color{blue}{\fbox{H2}} \\\hline K3&\color{blue}{\fbox{K3}}&\color{blue}{\fbox{K3}} \\\hline X2&H1,H2,K3,X2&X2 \\\hline X3&H2,K3,X3&X3 \\\hline S1&\color{blue}{\fbox{S1}}&\color{blue}{\fbox{S1}} \\\hline S3&\color{blue}{\fbox{S3}}&\color{blue}{\fbox{S3}} \\\hline \end{array} $$
抽取出H1、H2、K3放置上层，删除后剩余的情况如下	抽取出H1,H2,K3,S1,S3放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline H3&\color{red}{\fbox{H3}}&\color{red}{\fbox{H3}} \\\hline K1&\color{red}{\fbox{K1}}&\color{red}{\fbox{K1}} \\\hline K2&\color{red}{\fbox{K2}}&\color{red}{\fbox{K2}} \\\hline X1&\color{red}{\fbox{X1}}&\color{red}{\fbox{X1}} \\\hline S2&\color{red}{\fbox{S2}}&\color{red}{\fbox{S2}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline X2&\color{blue}{\fbox{X2}}&\color{blue}{\fbox{X2}} \\\hline X3&\color{blue}{\fbox{X3}}&\color{blue}{\fbox{X3}} \\\hline \end{array} $$
抽取出H3、K1、K2、X1、S2放置上层，删除后剩余的情况如下	抽取出X2,X3放置下层，删除后剩余的情况如下

抽取方式的结果如下

层级	结果优先——UP型	原因优先——DOWN型
第0层	X2,X3,S1,S3	X2,X3
第1层	H1,H2,K3	H1,H2,K3,S1,S3
第2层	H3,K1,K2,X1,S2	H3,K1,K2,X1,S2

计算一般性骨架矩阵 \begin{CD} R @>缩点运算>>R' @>缩边运算>>S' @>以最简菊花链表示回路>>S \\ \end{CD}

一般性骨架矩阵即为不缩点的情况下的最简结构，即边数最少。$S$如下

$$S=\begin{array} {c|c|c|c|c|c|c|c}{M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 & & & & & & & &1 & & & & \\ \hline H2 & & & & & & & &1 &1 & & & \\ \hline H3 &1 & & & & & & & & &1 & &1\\ \hline K1 &1 & & & & & & & & & & &1\\ \hline K2 &1 &1 & & & &1 & & & & & &1\\ \hline K3 & & & & & & & &1 &1 & & & \\ \hline X1 &1 & & & & & & & & & & &1\\ \hline X2 & & & & & & & & & & & & \\ \hline X3 & & & & & & & & & & & & \\ \hline S1 & & & & & & & & & & & & \\ \hline S2 &1 & & & & &1 & & & & & &1\\ \hline S3 & & & & & & & & & & & & \\ \hline \end{array} $$

代入一般性骨架矩阵$S$，绘制对抗层级拓扑图

UP型

　　第0层

　　第1层

　　第2层

DOWN型

　　第0层

　　第1层

　　第2层

权重系列求解

$$ \begin{CD} \{M|R \}@>>> d @>>>\omega \\ \end{CD} $$

　　$ \{M|R \}$ 为要素的中心度与原因度

　　$d$　两者合成距离向量

　　$ \omega $ 权重

求解原理

$$d = \sqrt {M^2 + R^2} $$

$$MR=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times2}} &中心度 &原因度\\ \hline H1 &3.5741 &0.9277\\ \hline H2 &3.8566 &0.4747\\ \hline H3 &2.0408 &0.8723\\ \hline K1 &2.3174 &0.8605\\ \hline K2 &2.8085 &0.2976\\ \hline K3 &3.7841 &0.6287\\ \hline X1 &2.5185 &0.8589\\ \hline X2 &3.9693 &1.4422\\ \hline X3 &4.0269 &0.6476\\ \hline S1 &2.2776 &1.5731\\ \hline S2 &2.7762 &0.5156\\ \hline S3 &3.2701 &1.456\\ \hline \end{array} $$

向量的计算采用欧式距离公式

$$d=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times1}} &向量\\ \hline H1 &3.6925\\ \hline H2 &3.8857\\ \hline H3 &2.2194\\ \hline K1 &2.472\\ \hline K2 &2.8242\\ \hline K3 &3.836\\ \hline X1 &2.661\\ \hline X2 &4.2232\\ \hline X3 &4.0787\\ \hline S1 &2.7681\\ \hline S2 &2.8237\\ \hline S3 &3.5796\\ \hline \end{array} $$

归一化求权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{12 \times1}} &权重\\ \hline H1 &0.0945\\ \hline H2 &0.0995\\ \hline H3 &0.0568\\ \hline K1 &0.0633\\ \hline K2 &0.0723\\ \hline K3 &0.0982\\ \hline X1 &0.0681\\ \hline X2 &0.1081\\ \hline X3 &0.1044\\ \hline S1 &0.0709\\ \hline S2 &0.0723\\ \hline S3 &0.0916\\ \hline \end{array} $$

归一化求子系统的权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{4 \times1}} &权重\\ \hline H &0.2508\\ \hline K &0.2338\\ \hline X &0.2806\\ \hline S &0.2348\\ \hline \end{array} $$

DEMATEL-AISM-Weight

原始矩阵（直接影响矩阵）为

规范直接关系矩阵求解过程 $$ \require{cancel} \require{AMScd} \begin{CD} O @>>>N \\ \end{CD} $$

综合影响矩阵求解过程 $$\begin{CD} N @>>>T \\ \end{CD} $$

由综合影响矩阵求影响度、被影响度、中心度、原因度 \begin{CD} T @>>>\{D|C|M|R \} \\ \end{CD}

求解原理

结果

绘制中心度与原因度建构的散点图图表

从两列的决策矩阵到关系矩阵

AISM部分求解

关系矩阵到相乘矩阵

相乘矩阵B到可达矩阵R

由可达矩阵通过对抗的抽取方式得到一对要素的层级分布 \begin{CD} R @>结果优先抽取>原因优先抽取> \frac {up型要素层级分布 }{down 型要素层级分布} \\ \end{CD}

抽取方式的结果如下

计算一般性骨架矩阵 \begin{CD} R @>缩点运算>>R' @>缩边运算>>S' @>以最简菊花链表示回路>>S \\ \end{CD}

代入一般性骨架矩阵$S$，绘制对抗层级拓扑图

UP型

DOWN型

权重系列求解

求解原理

DEMATEL-AHDT流程图如下：