原始矩阵（直接影响矩阵）为

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

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

$$\mathcal{N}=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &0 &0.047 &0.023 &0 &0.209 &0 &0.047 &0.209 &0.023 &0 &0.047 &0.023\\ \hline H2 &0.07 &0 &0 &0 &0.209 &0.186 &0 &0.163 &0 &0.07 &0.047 &0.023\\ \hline H3 &0.07 &0.209 &0 &0.07 &0 &0 &0.07 &0 &0 &0 &0.047 &0.14\\ \hline K1 &0.07 &0.209 &0.093 &0 &0 &0 &0.023 &0.116 &0 &0 &0.047 &0.116\\ \hline K2 &0.14 &0.209 &0 &0 &0 &0 &0 &0 &0.116 &0 &0.047 &0.023\\ \hline K3 &0.186 &0.116 &0.047 &0 &0.023 &0 &0.047 &0 &0.209 &0 &0.047 &0.023\\ \hline X1 &0.07 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.047 &0.023\\ \hline X2 &0 &0.14 &0.047 &0 &0 &0 &0.023 &0 &0 &0.047 &0.047 &0.023\\ \hline X3 &0.209 &0 &0 &0 &0 &0 &0.023 &0 &0 &0 &0.047 &0.023\\ \hline S1 &0 &0 &0.07 &0.07 &0.07 &0.07 &0 &0 &0 &0 &0 &0.023\\ \hline S2 &0 &0.07 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.023\\ \hline S3 &0.116 &0 &0 &0.093 &0.116 &0.186 &0 &0 &0 &0 &0.047 &0\\ \hline \end{array} $$

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

　　综合影响矩阵如下

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

$$T=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &H1 &H2 &H3 &K1 &K2 &K3 &X1 &X2 &X3 &S1 &S2 &S3\\ \hline H1 &0.085 &0.165 &0.042 &0.01 &0.271 &0.043 &0.063 &0.255 &0.066 &0.023 &0.096 &0.055\\ \hline H2 &0.196 &0.149 &0.034 &0.015 &0.301 &0.232 &0.03 &0.23 &0.088 &0.091 &0.109 &0.063\\ \hline H3 &0.166 &0.296 &0.022 &0.09 &0.121 &0.09 &0.088 &0.093 &0.037 &0.025 &0.101 &0.177\\ \hline K1 &0.166 &0.32 &0.115 &0.025 &0.125 &0.092 &0.05 &0.206 &0.038 &0.032 &0.107 &0.162\\ \hline K2 &0.226 &0.275 &0.015 &0.007 &0.114 &0.062 &0.02 &0.093 &0.148 &0.024 &0.094 &0.051\\ \hline K3 &0.297 &0.201 &0.063 &0.011 &0.137 &0.05 &0.075 &0.096 &0.243 &0.018 &0.104 &0.062\\ \hline X1 &0.082 &0.018 &0.004 &0.003 &0.025 &0.009 &0.005 &0.02 &0.007 &0.002 &0.056 &0.029\\ \hline X2 &0.046 &0.185 &0.057 &0.013 &0.059 &0.047 &0.033 &0.041 &0.018 &0.061 &0.072 &0.046\\ \hline X3 &0.235 &0.042 &0.01 &0.005 &0.063 &0.015 &0.037 &0.057 &0.016 &0.006 &0.071 &0.038\\ \hline S1 &0.065 &0.079 &0.085 &0.081 &0.109 &0.095 &0.017 &0.036 &0.034 &0.007 &0.031 &0.056\\ \hline S2 &0.019 &0.083 &0.003 &0.003 &0.026 &0.021 &0.003 &0.018 &0.008 &0.007 &0.01 &0.029\\ \hline S3 &0.224 &0.122 &0.029 &0.1 &0.199 &0.217 &0.028 &0.078 &0.074 &0.012 &0.098 &0.04\\ \hline \end{array} $$

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

$$\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times4}} &Di &Ci &Mi &Ri\\ \hline H1 &1.173 &1.808 &2.981 &-0.636\\ \hline H2 &1.536 &1.934 &3.47 &-0.399\\ \hline H3 &1.308 &0.479 &1.786 &0.829\\ \hline K1 &1.437 &0.362 &1.799 &1.075\\ \hline K2 &1.128 &1.549 &2.677 &-0.421\\ \hline K3 &1.357 &0.973 &2.33 &0.384\\ \hline X1 &0.26 &0.45 &0.711 &-0.19\\ \hline X2 &0.678 &1.224 &1.902 &-0.546\\ \hline X3 &0.593 &0.775 &1.368 &-0.182\\ \hline S1 &0.696 &0.308 &1.004 &0.387\\ \hline S2 &0.229 &0.947 &1.176 &-0.718\\ \hline S3 &1.222 &0.806 &2.028 &0.417\\ \hline \end{array} $$

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

$$D=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times2}} &M &|R|\\ \hline H1 &2.9811 &0.6356\\ \hline H2 &3.4701 &0.3986\\ \hline H3 &1.786 &0.829\\ \hline K1 &1.7994 &1.0746\\ \hline K2 &2.6771 &0.4211\\ \hline K3 &2.3299 &0.3836\\ \hline X1 &0.7106 &0.1898\\ \hline X2 &1.9021 &0.5457\\ \hline X3 &1.3676 &0.1818\\ \hline S1 &1.0037 &0.3874\\ \hline S2 &1.1756 &0.7184\\ \hline S3 &2.0283 &0.4165\\ \hline \end{array} $$

关系矩阵$A$

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

AISM部分求解

关系矩阵到相乘矩阵

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

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

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

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

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
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline H1&\color{red}{\fbox{H1}}&\color{red}{\fbox{H1}} \\\hline H2&\color{red}{\fbox{H2}}&\color{red}{\fbox{H2}} \\\hline H3&H3,K1&H3 \\\hline K1&\color{red}{\fbox{K1}}&\color{red}{\fbox{K1}} \\\hline K2&H1,K2&K2 \\\hline K3&H1,H2,K2,K3&K3 \\\hline X1&H1,H2,H3,K1,K2,K3,X1,X2,S1,S2,S3&X1 \\\hline X2&H1,X2&X2 \\\hline X3&H1,H2,H3,K1,K2,K3,X2,X3,S3&X3 \\\hline S1&H1,H2,H3,K1,K2,X2,S1,S2,S3&S1 \\\hline S2&H3,K1,S2&S2 \\\hline S3&H1,K2,S3&S3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&H1,K2,K3,X1,X2,X3,S1,S3&H1 \\\hline H2&H2,K3,X1,X3,S1&H2 \\\hline H3&H3,X1,X3,S1,S2&H3 \\\hline K1&H3,K1,X1,X3,S1,S2&K1 \\\hline K2&K2,K3,X1,X3,S1,S3&K2 \\\hline K3&K3,X1,X3&K3 \\\hline X1&\color{blue}{\fbox{X1}}&\color{blue}{\fbox{X1}} \\\hline X2&X1,X2,X3,S1&X2 \\\hline X3&\color{blue}{\fbox{X3}}&\color{blue}{\fbox{X3}} \\\hline S1&X1,S1&S1 \\\hline S2&X1,S1,S2&S2 \\\hline S3&X1,X3,S1,S3&S3 \\\hline \end{array} $$
抽取出H1、H2、K1放置上层，删除后剩余的情况如下	抽取出X1,X3放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline H3&\color{red}{\fbox{H3}}&\color{red}{\fbox{H3}} \\\hline K2&\color{red}{\fbox{K2}}&\color{red}{\fbox{K2}} \\\hline K3&K2,K3&K3 \\\hline X1&H3,K2,K3,X1,X2,S1,S2,S3&X1 \\\hline X2&\color{red}{\fbox{X2}}&\color{red}{\fbox{X2}} \\\hline X3&H3,K2,K3,X2,X3,S3&X3 \\\hline S1&H3,K2,X2,S1,S2,S3&S1 \\\hline S2&H3,S2&S2 \\\hline S3&K2,S3&S3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&H1,K2,K3,X2,S1,S3&H1 \\\hline H2&H2,K3,S1&H2 \\\hline H3&H3,S1,S2&H3 \\\hline K1&H3,K1,S1,S2&K1 \\\hline K2&K2,K3,S1,S3&K2 \\\hline K3&\color{blue}{\fbox{K3}}&\color{blue}{\fbox{K3}} \\\hline X2&X2,S1&X2 \\\hline S1&\color{blue}{\fbox{S1}}&\color{blue}{\fbox{S1}} \\\hline S2&S1,S2&S2 \\\hline S3&S1,S3&S3 \\\hline \end{array} $$
抽取出H3、K2、X2放置上层，删除后剩余的情况如下	抽取出K3,S1放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline K3&\color{red}{\fbox{K3}}&\color{red}{\fbox{K3}} \\\hline X1&K3,X1,S1,S2,S3&X1 \\\hline X3&K3,X3,S3&X3 \\\hline S1&S1,S2,S3&S1 \\\hline S2&\color{red}{\fbox{S2}}&\color{red}{\fbox{S2}} \\\hline S3&\color{red}{\fbox{S3}}&\color{red}{\fbox{S3}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&H1,K2,X2,S3&H1 \\\hline H2&\color{blue}{\fbox{H2}}&\color{blue}{\fbox{H2}} \\\hline H3&H3,S2&H3 \\\hline K1&H3,K1,S2&K1 \\\hline K2&K2,S3&K2 \\\hline X2&\color{blue}{\fbox{X2}}&\color{blue}{\fbox{X2}} \\\hline S2&\color{blue}{\fbox{S2}}&\color{blue}{\fbox{S2}} \\\hline S3&\color{blue}{\fbox{S3}}&\color{blue}{\fbox{S3}} \\\hline \end{array} $$
抽取出K3、S2、S3放置上层，删除后剩余的情况如下	抽取出H2,X2,S2,S3放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline X1&X1,S1&X1 \\\hline X3&\color{red}{\fbox{X3}}&\color{red}{\fbox{X3}} \\\hline S1&\color{red}{\fbox{S1}}&\color{red}{\fbox{S1}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&H1,K2&H1 \\\hline H3&\color{blue}{\fbox{H3}}&\color{blue}{\fbox{H3}} \\\hline K1&H3,K1&K1 \\\hline K2&\color{blue}{\fbox{K2}}&\color{blue}{\fbox{K2}} \\\hline \end{array} $$
抽取出X3、S1放置上层，删除后剩余的情况如下	抽取出H3,K2放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline X1&\color{red}{\fbox{X1}}&\color{red}{\fbox{X1}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline H1&\color{blue}{\fbox{H1}}&\color{blue}{\fbox{H1}} \\\hline K1&\color{blue}{\fbox{K1}}&\color{blue}{\fbox{K1}} \\\hline \end{array} $$
抽取出X1放置上层，删除后剩余的情况如下	抽取出H1,K1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	H1,H2,K1	H1,K1
第1层	H3,K2,X2	H3,K2
第2层	K3,S2,S3	H2,X2,S2,S3
第3层	X3,S1	K3,S1
第4层	X1	X1,X3

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

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

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

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

　　$ \{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

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

$$MR=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times2}} &中心度 &原因度\\ \hline H1 &2.9811 &0.6356\\ \hline H2 &3.4701 &0.3986\\ \hline H3 &1.786 &0.829\\ \hline K1 &1.7994 &1.0746\\ \hline K2 &2.6771 &0.4211\\ \hline K3 &2.3299 &0.3836\\ \hline X1 &0.7106 &0.1898\\ \hline X2 &1.9021 &0.5457\\ \hline X3 &1.3676 &0.1818\\ \hline S1 &1.0037 &0.3874\\ \hline S2 &1.1756 &0.7184\\ \hline S3 &2.0283 &0.4165\\ \hline \end{array} $$

向量的计算采用欧式距离公式

$$d=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times1}} &向量\\ \hline H1 &3.0481\\ \hline H2 &3.4929\\ \hline H3 &1.9691\\ \hline K1 &2.0958\\ \hline K2 &2.71\\ \hline K3 &2.3613\\ \hline X1 &0.7356\\ \hline X2 &1.9788\\ \hline X3 &1.3796\\ \hline S1 &1.0759\\ \hline S2 &1.3777\\ \hline S3 &2.0706\\ \hline \end{array} $$

归一化求权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{12 \times1}} &权重\\ \hline H1 &0.1255\\ \hline H2 &0.1438\\ \hline H3 &0.081\\ \hline K1 &0.0863\\ \hline K2 &0.1115\\ \hline K3 &0.0972\\ \hline X1 &0.0303\\ \hline X2 &0.0814\\ \hline X3 &0.0568\\ \hline S1 &0.0443\\ \hline S2 &0.0567\\ \hline S3 &0.0852\\ \hline \end{array} $$

归一化求子系统的权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{4 \times1}} &权重\\ \hline H &0.3503\\ \hline K &0.295\\ \hline X &0.1685\\ \hline S &0.1862\\ \hline \end{array} $$

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：