原始矩阵（直接影响矩阵）为

$$Ori=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &0 &7.12 &7.52 &8.92 &7.36 &8.24 &8.56 &8.4 &8.92 &1.28 &1.4 &8.08 &8.92 &2.36 &7.12 &1.04\\ \hline C2 &8.32 &0 &5.12 &8.88 &8.92 &2.16 &2.96 &7.16 &8.92 &1.56 &5.16 &4.12 &5 &5.12 &4.12 &2.68\\ \hline C3 &7.44 &2.32 &0 &8.8 &3.08 &7.12 &1.04 &8.92 &8.56 &0.24 &0.26 &5.52 &6.64 &4.48 &6.52 &3.08\\ \hline C4 &7.04 &5.12 &2.04 &0 &8.76 &7.16 &2.48 &7.28 &8.16 &1.2 &2.12 &5.08 &8.6 &3.12 &6.08 &2.2\\ \hline C5 &8.92 &8.56 &7.96 &7.08 &0 &4.84 &4.76 &6.56 &8.72 &1.76 &2.12 &8.48 &8.8 &4.16 &6 &3.04\\ \hline E1 &2.12 &1.5 &0.92 &7.12 &2.48 &0 &6.2 &5.04 &8.76 &3.28 &4.52 &2.24 &8.72 &3.5 &8.6 &5.64\\ \hline E2 &2 &7.04 &1.08 &8.72 &3.24 &8.2 &0 &7.08 &8.4 &8.6 &8 &2.12 &8.4 &6.2 &6.84 &8.24\\ \hline E3 &7.02 &3.2 &2.5 &8.96 &4.24 &4.88 &5.24 &0 &6.12 &2.24 &6.2 &2 &8.6 &3.52 &4.22 &6.2\\ \hline F1 &8.56 &8.92 &6.52 &8.6 &8.6 &7.2 &8.2 &8.4 &0 &2.2 &2 &1.12 &2 &5.08 &2.04 &6\\ \hline F2 &1.04 &1.52 &0.096 &1.08 &1.12 &3.04 &3.52 &2.04 &1.56 &0 &7.8 &1.52 &1.5 &6 &2.52 &6.5\\ \hline F3 &4.5 &6.04 &1 &1.16 &2.52 &8 &8.04 &8.96 &2.5 &8.25 &0 &1.24 &1.4 &7.04 &7.72 &7.5\\ \hline U1 &7.08 &8.72 &7.08 &6.52 &1.5 &6.28 &3.2 &8 &4.56 &2.04 &4 &0 &8.4 &7.2 &5.56 &3\\ \hline U2 &7.04 &4.96 &6.08 &7.52 &5.02 &7.2 &7 &8.12 &6.5 &2.12 &2 &8.88 &0 &1.24 &0.64 &8.2\\ \hline D1 &2 &1.04 &3.56 &6.96 &6.52 &8.6 &6.96 &6.4 &2.28 &1.02 &8.2 &1.02 &6.5 &0 &7.64 &8.92\\ \hline D2 &6.08 &7.02 &1.12 &8.32 &7 &8.24 &4.04 &8.92 &8.68 &4.12 &3.32 &8.12 &5.08 &8.92 &0 &3.04\\ \hline D3 &1.2 &1 &0.92 &8.76 &2.12 &8.96 &8.24 &7.28 &0.8 &6.52 &7.16 &8.52 &3.2 &7.2 &8.92 &0\\ \hline \end{array} $$

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_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &0 &0.066 &0.069 &0.082 &0.068 &0.076 &0.079 &0.077 &0.082 &0.012 &0.013 &0.074 &0.082 &0.022 &0.066 &0.01\\ \hline C2 &0.077 &0 &0.047 &0.082 &0.082 &0.02 &0.027 &0.066 &0.082 &0.014 &0.048 &0.038 &0.046 &0.047 &0.038 &0.025\\ \hline C3 &0.069 &0.021 &0 &0.081 &0.028 &0.066 &0.01 &0.082 &0.079 &0.002 &0.002 &0.051 &0.061 &0.041 &0.06 &0.028\\ \hline C4 &0.065 &0.047 &0.019 &0 &0.081 &0.066 &0.023 &0.067 &0.075 &0.011 &0.02 &0.047 &0.079 &0.029 &0.056 &0.02\\ \hline C5 &0.082 &0.079 &0.073 &0.065 &0 &0.045 &0.044 &0.06 &0.08 &0.016 &0.02 &0.078 &0.081 &0.038 &0.055 &0.028\\ \hline E1 &0.02 &0.014 &0.008 &0.066 &0.023 &0 &0.057 &0.046 &0.081 &0.03 &0.042 &0.021 &0.08 &0.032 &0.079 &0.052\\ \hline E2 &0.018 &0.065 &0.01 &0.08 &0.03 &0.076 &0 &0.065 &0.077 &0.079 &0.074 &0.02 &0.077 &0.057 &0.063 &0.076\\ \hline E3 &0.065 &0.029 &0.023 &0.083 &0.039 &0.045 &0.048 &0 &0.056 &0.021 &0.057 &0.018 &0.079 &0.032 &0.039 &0.057\\ \hline F1 &0.079 &0.082 &0.06 &0.079 &0.079 &0.066 &0.076 &0.077 &0 &0.02 &0.018 &0.01 &0.018 &0.047 &0.019 &0.055\\ \hline F2 &0.01 &0.014 &0.001 &0.01 &0.01 &0.028 &0.032 &0.019 &0.014 &0 &0.072 &0.014 &0.014 &0.055 &0.023 &0.06\\ \hline F3 &0.041 &0.056 &0.009 &0.011 &0.023 &0.074 &0.074 &0.083 &0.023 &0.076 &0 &0.011 &0.013 &0.065 &0.071 &0.069\\ \hline U1 &0.065 &0.08 &0.065 &0.06 &0.014 &0.058 &0.029 &0.074 &0.042 &0.019 &0.037 &0 &0.077 &0.066 &0.051 &0.028\\ \hline U2 &0.065 &0.046 &0.056 &0.069 &0.046 &0.066 &0.064 &0.075 &0.06 &0.02 &0.018 &0.082 &0 &0.011 &0.006 &0.076\\ \hline D1 &0.018 &0.01 &0.033 &0.064 &0.06 &0.079 &0.064 &0.059 &0.021 &0.009 &0.076 &0.009 &0.06 &0 &0.07 &0.082\\ \hline D2 &0.056 &0.065 &0.01 &0.077 &0.064 &0.076 &0.037 &0.082 &0.08 &0.038 &0.031 &0.075 &0.047 &0.082 &0 &0.028\\ \hline D3 &0.011 &0.009 &0.008 &0.081 &0.02 &0.083 &0.076 &0.067 &0.007 &0.06 &0.066 &0.078 &0.029 &0.066 &0.082 &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_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &0.173 &0.219 &0.181 &0.301 &0.221 &0.271 &0.236 &0.292 &0.278 &0.102 &0.137 &0.211 &0.272 &0.166 &0.228 &0.16\\ \hline C2 &0.219 &0.134 &0.145 &0.265 &0.213 &0.191 &0.167 &0.249 &0.245 &0.09 &0.15 &0.157 &0.208 &0.167 &0.179 &0.15\\ \hline C3 &0.2 &0.143 &0.091 &0.253 &0.153 &0.221 &0.14 &0.249 &0.23 &0.072 &0.099 &0.16 &0.211 &0.152 &0.188 &0.144\\ \hline C4 &0.202 &0.173 &0.115 &0.182 &0.205 &0.225 &0.158 &0.241 &0.233 &0.084 &0.12 &0.161 &0.232 &0.145 &0.188 &0.142\\ \hline C5 &0.244 &0.225 &0.183 &0.278 &0.153 &0.236 &0.2 &0.269 &0.267 &0.102 &0.138 &0.211 &0.263 &0.176 &0.213 &0.17\\ \hline E1 &0.142 &0.129 &0.089 &0.225 &0.138 &0.151 &0.179 &0.206 &0.218 &0.102 &0.136 &0.124 &0.214 &0.141 &0.197 &0.165\\ \hline E2 &0.172 &0.202 &0.11 &0.28 &0.174 &0.261 &0.158 &0.265 &0.251 &0.167 &0.195 &0.148 &0.247 &0.194 &0.218 &0.219\\ \hline E3 &0.193 &0.15 &0.111 &0.251 &0.161 &0.204 &0.179 &0.173 &0.207 &0.096 &0.154 &0.131 &0.225 &0.146 &0.171 &0.175\\ \hline F1 &0.224 &0.213 &0.158 &0.275 &0.215 &0.242 &0.218 &0.267 &0.178 &0.102 &0.133 &0.137 &0.194 &0.174 &0.172 &0.186\\ \hline F2 &0.074 &0.075 &0.044 &0.1 &0.073 &0.115 &0.105 &0.11 &0.09 &0.045 &0.128 &0.07 &0.091 &0.118 &0.098 &0.126\\ \hline F3 &0.16 &0.165 &0.088 &0.18 &0.139 &0.224 &0.199 &0.241 &0.168 &0.149 &0.107 &0.115 &0.158 &0.178 &0.199 &0.186\\ \hline U1 &0.208 &0.206 &0.159 &0.249 &0.15 &0.228 &0.171 &0.258 &0.21 &0.096 &0.144 &0.121 &0.239 &0.187 &0.194 &0.158\\ \hline U2 &0.206 &0.176 &0.151 &0.258 &0.176 &0.236 &0.203 &0.258 &0.225 &0.099 &0.128 &0.198 &0.168 &0.137 &0.154 &0.2\\ \hline D1 &0.15 &0.132 &0.116 &0.238 &0.179 &0.24 &0.196 &0.232 &0.177 &0.091 &0.177 &0.125 &0.211 &0.12 &0.207 &0.203\\ \hline D2 &0.213 &0.208 &0.119 &0.282 &0.21 &0.261 &0.193 &0.283 &0.26 &0.123 &0.152 &0.201 &0.228 &0.216 &0.16 &0.172\\ \hline D3 &0.142 &0.133 &0.093 &0.251 &0.141 &0.244 &0.206 &0.239 &0.164 &0.139 &0.173 &0.184 &0.185 &0.187 &0.218 &0.128\\ \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

$$\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{16 \times4}} &Di &Ci &Mi &Ri\\ \hline C1 &3.449 &2.923 &6.372 &0.526\\ \hline C2 &2.93 &2.683 &5.614 &0.247\\ \hline C3 &2.704 &1.953 &4.658 &0.751\\ \hline C4 &2.807 &3.868 &6.675 &-1.06\\ \hline C5 &3.328 &2.701 &6.028 &0.627\\ \hline E1 &2.555 &3.55 &6.106 &-0.995\\ \hline E2 &3.263 &2.907 &6.17 &0.356\\ \hline E3 &2.724 &3.832 &6.556 &-1.108\\ \hline F1 &3.088 &3.403 &6.491 &-0.315\\ \hline F2 &1.462 &1.66 &3.121 &-0.198\\ \hline F3 &2.656 &2.27 &4.926 &0.386\\ \hline U1 &2.978 &2.454 &5.433 &0.524\\ \hline U2 &2.972 &3.347 &6.319 &-0.375\\ \hline D1 &2.796 &2.604 &5.4 &0.191\\ \hline D2 &3.281 &2.982 &6.263 &0.299\\ \hline D3 &2.827 &2.683 &5.511 &0.144\\ \hline \end{array} $$

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

$$D=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times2}} &M &|R|\\ \hline C1 &6.3725 &0.526\\ \hline C2 &5.6136 &0.2468\\ \hline C3 &4.6576 &0.7513\\ \hline C4 &6.6751 &1.0603\\ \hline C5 &6.028 &0.627\\ \hline E1 &6.106 &0.995\\ \hline E2 &6.1695 &0.3563\\ \hline E3 &6.556 &1.1084\\ \hline F1 &6.4907 &0.3153\\ \hline F2 &3.1215 &0.1984\\ \hline F3 &4.9263 &0.3864\\ \hline U1 &5.4328 &0.5241\\ \hline U2 &6.3188 &0.3749\\ \hline D1 &5.4002 &0.1913\\ \hline D2 &6.2634 &0.2995\\ \hline D3 &5.5107 &0.1438\\ \hline \end{array} $$

关系矩阵$A$

$$A=\begin{array} {c|c|c|c|c|c|c|c}{M_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &1 & & &1 & & & &1 & & & & & & & & \\ \hline C2 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 & \\ \hline C3 & & &1 &1 & &1 & &1 & & & & & & & & \\ \hline C4 & & & &1 & & & & & & & & & & & & \\ \hline C5 & & & &1 &1 &1 & &1 & & & & & & & & \\ \hline E1 & & & &1 & &1 & &1 & & & & & & & & \\ \hline E2 &1 & & &1 & & &1 &1 & & & & &1 & & & \\ \hline E3 & & & & & & & &1 & & & & & & & & \\ \hline F1 & & & &1 & & & &1 &1 & & & & & & & \\ \hline F2 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 & &1 & \\ \hline F3 &1 & & &1 &1 &1 & &1 & & &1 &1 & & & & \\ \hline U1 &1 & & &1 &1 &1 & &1 & & & &1 & & & & \\ \hline U2 &1 & & &1 & & & &1 & & & & &1 & & & \\ \hline D1 &1 &1 & &1 &1 &1 &1 &1 &1 & & &1 &1 &1 &1 & \\ \hline D2 &1 & & &1 & & & &1 &1 & & & &1 & &1 & \\ \hline D3 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 &1\\ \hline \end{array} $$

AISM部分求解

关系矩阵到相乘矩阵

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

$$B=\begin{array} {c|c|c|c|c|c|c|c}{M_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &1 & & &1 & & & &1 & & & & & & & & \\ \hline C2 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 & \\ \hline C3 & & &1 &1 & &1 & &1 & & & & & & & & \\ \hline C4 & & & &1 & & & & & & & & & & & & \\ \hline C5 & & & &1 &1 &1 & &1 & & & & & & & & \\ \hline E1 & & & &1 & &1 & &1 & & & & & & & & \\ \hline E2 &1 & & &1 & & &1 &1 & & & & &1 & & & \\ \hline E3 & & & & & & & &1 & & & & & & & & \\ \hline F1 & & & &1 & & & &1 &1 & & & & & & & \\ \hline F2 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 & &1 & \\ \hline F3 &1 & & &1 &1 &1 & &1 & & &1 &1 & & & & \\ \hline U1 &1 & & &1 &1 &1 & &1 & & & &1 & & & & \\ \hline U2 &1 & & &1 & & & &1 & & & & &1 & & & \\ \hline D1 &1 &1 & &1 &1 &1 &1 &1 &1 & & &1 &1 &1 &1 & \\ \hline D2 &1 & & &1 & & & &1 &1 & & & &1 & &1 & \\ \hline D3 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 &1\\ \hline \end{array} $$

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

$$R=\begin{array} {c|c|c|c|c|c|c|c}{M_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 &1 & & &1 & & & &1 & & & & & & & & \\ \hline C2 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 & \\ \hline C3 & & &1 &1 & &1 & &1 & & & & & & & & \\ \hline C4 & & & &1 & & & & & & & & & & & & \\ \hline C5 & & & &1 &1 &1 & &1 & & & & & & & & \\ \hline E1 & & & &1 & &1 & &1 & & & & & & & & \\ \hline E2 &1 & & &1 & & &1 &1 & & & & &1 & & & \\ \hline E3 & & & & & & & &1 & & & & & & & & \\ \hline F1 & & & &1 & & & &1 &1 & & & & & & & \\ \hline F2 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 & &1 & \\ \hline F3 &1 & & &1 &1 &1 & &1 & & &1 &1 & & & & \\ \hline U1 &1 & & &1 &1 &1 & &1 & & & &1 & & & & \\ \hline U2 &1 & & &1 & & & &1 & & & & &1 & & & \\ \hline D1 &1 &1 & &1 &1 &1 &1 &1 &1 & & &1 &1 &1 &1 & \\ \hline D2 &1 & & &1 & & & &1 &1 & & & &1 & &1 & \\ \hline D3 &1 &1 & &1 &1 &1 &1 &1 &1 & & & &1 & &1 &1\\ \hline \end{array} $$

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
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline C1&C1,C4,E3&C1 \\\hline C2&C1,C2,C4,C5,E1,E2,E3,F1,U2,D2&C2 \\\hline C3&C3,C4,E1,E3&C3 \\\hline C4&\color{red}{\fbox{C4}}&\color{red}{\fbox{C4}} \\\hline C5&C4,C5,E1,E3&C5 \\\hline E1&C4,E1,E3&E1 \\\hline E2&C1,C4,E2,E3,U2&E2 \\\hline E3&\color{red}{\fbox{E3}}&\color{red}{\fbox{E3}} \\\hline F1&C4,E3,F1&F1 \\\hline F2&C1,C2,C3,C4,C5,E1,E2,E3,F1,F2,F3,U1,U2,D2&F2 \\\hline F3&C1,C4,C5,E1,E3,F3,U1&F3 \\\hline U1&C1,C4,C5,E1,E3,U1&U1 \\\hline U2&C1,C4,E3,U2&U2 \\\hline D1&C1,C2,C4,C5,E1,E2,E3,F1,U1,U2,D1,D2&D1 \\\hline D2&C1,C4,E3,F1,U2,D2&D2 \\\hline D3&C1,C2,C4,C5,E1,E2,E3,F1,U2,D2,D3&D3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C1&C1,C2,E2,F2,F3,U1,U2,D1,D2,D3&C1 \\\hline C2&C2,F2,D1,D3&C2 \\\hline C3&C3,F2&C3 \\\hline C4&C1,C2,C3,C4,C5,E1,E2,F1,F2,F3,U1,U2,D1,D2,D3&C4 \\\hline C5&C2,C5,F2,F3,U1,D1,D3&C5 \\\hline E1&C2,C3,C5,E1,F2,F3,U1,D1,D3&E1 \\\hline E2&C2,E2,F2,D1,D3&E2 \\\hline E3&C1,C2,C3,C5,E1,E2,E3,F1,F2,F3,U1,U2,D1,D2,D3&E3 \\\hline F1&C2,F1,F2,D1,D2,D3&F1 \\\hline F2&\color{blue}{\fbox{F2}}&\color{blue}{\fbox{F2}} \\\hline F3&F2,F3&F3 \\\hline U1&F2,F3,U1,D1&U1 \\\hline U2&C2,E2,F2,U2,D1,D2,D3&U2 \\\hline D1&\color{blue}{\fbox{D1}}&\color{blue}{\fbox{D1}} \\\hline D2&C2,F2,D1,D2,D3&D2 \\\hline D3&\color{blue}{\fbox{D3}}&\color{blue}{\fbox{D3}} \\\hline \end{array} $$
抽取出C4、E3放置上层，删除后剩余的情况如下	抽取出F2,D1,D3放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline C1&\color{red}{\fbox{C1}}&\color{red}{\fbox{C1}} \\\hline C2&C1,C2,C5,E1,E2,F1,U2,D2&C2 \\\hline C3&C3,E1&C3 \\\hline C5&C5,E1&C5 \\\hline E1&\color{red}{\fbox{E1}}&\color{red}{\fbox{E1}} \\\hline E2&C1,E2,U2&E2 \\\hline F1&\color{red}{\fbox{F1}}&\color{red}{\fbox{F1}} \\\hline F2&C1,C2,C3,C5,E1,E2,F1,F2,F3,U1,U2,D2&F2 \\\hline F3&C1,C5,E1,F3,U1&F3 \\\hline U1&C1,C5,E1,U1&U1 \\\hline U2&C1,U2&U2 \\\hline D1&C1,C2,C5,E1,E2,F1,U1,U2,D1,D2&D1 \\\hline D2&C1,F1,U2,D2&D2 \\\hline D3&C1,C2,C5,E1,E2,F1,U2,D2,D3&D3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C1&C1,C2,E2,F3,U1,U2,D2&C1 \\\hline C2&\color{blue}{\fbox{C2}}&\color{blue}{\fbox{C2}} \\\hline C3&\color{blue}{\fbox{C3}}&\color{blue}{\fbox{C3}} \\\hline C4&C1,C2,C3,C4,C5,E1,E2,F1,F3,U1,U2,D2&C4 \\\hline C5&C2,C5,F3,U1&C5 \\\hline E1&C2,C3,C5,E1,F3,U1&E1 \\\hline E2&C2,E2&E2 \\\hline E3&C1,C2,C3,C5,E1,E2,E3,F1,F3,U1,U2,D2&E3 \\\hline F1&C2,F1,D2&F1 \\\hline F3&\color{blue}{\fbox{F3}}&\color{blue}{\fbox{F3}} \\\hline U1&F3,U1&U1 \\\hline U2&C2,E2,U2,D2&U2 \\\hline D2&C2,D2&D2 \\\hline \end{array} $$
抽取出C1、E1、F1放置上层，删除后剩余的情况如下	抽取出C2,C3,F3放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline C2&C2,C5,E2,U2,D2&C2 \\\hline C3&\color{red}{\fbox{C3}}&\color{red}{\fbox{C3}} \\\hline C5&\color{red}{\fbox{C5}}&\color{red}{\fbox{C5}} \\\hline E2&E2,U2&E2 \\\hline F2&C2,C3,C5,E2,F2,F3,U1,U2,D2&F2 \\\hline F3&C5,F3,U1&F3 \\\hline U1&C5,U1&U1 \\\hline U2&\color{red}{\fbox{U2}}&\color{red}{\fbox{U2}} \\\hline D1&C2,C5,E2,U1,U2,D1,D2&D1 \\\hline D2&U2,D2&D2 \\\hline D3&C2,C5,E2,U2,D2,D3&D3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C1&C1,E2,U1,U2,D2&C1 \\\hline C4&C1,C4,C5,E1,E2,F1,U1,U2,D2&C4 \\\hline C5&C5,U1&C5 \\\hline E1&C5,E1,U1&E1 \\\hline E2&\color{blue}{\fbox{E2}}&\color{blue}{\fbox{E2}} \\\hline E3&C1,C5,E1,E2,E3,F1,U1,U2,D2&E3 \\\hline F1&F1,D2&F1 \\\hline U1&\color{blue}{\fbox{U1}}&\color{blue}{\fbox{U1}} \\\hline U2&E2,U2,D2&U2 \\\hline D2&\color{blue}{\fbox{D2}}&\color{blue}{\fbox{D2}} \\\hline \end{array} $$
抽取出C3、C5、U2放置上层，删除后剩余的情况如下	抽取出E2,U1,D2放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline C2&C2,E2,D2&C2 \\\hline E2&\color{red}{\fbox{E2}}&\color{red}{\fbox{E2}} \\\hline F2&C2,E2,F2,F3,U1,D2&F2 \\\hline F3&F3,U1&F3 \\\hline U1&\color{red}{\fbox{U1}}&\color{red}{\fbox{U1}} \\\hline D1&C2,E2,U1,D1,D2&D1 \\\hline D2&\color{red}{\fbox{D2}}&\color{red}{\fbox{D2}} \\\hline D3&C2,E2,D2,D3&D3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C1&C1,U2&C1 \\\hline C4&C1,C4,C5,E1,F1,U2&C4 \\\hline C5&\color{blue}{\fbox{C5}}&\color{blue}{\fbox{C5}} \\\hline E1&C5,E1&E1 \\\hline E3&C1,C5,E1,E3,F1,U2&E3 \\\hline F1&\color{blue}{\fbox{F1}}&\color{blue}{\fbox{F1}} \\\hline U2&\color{blue}{\fbox{U2}}&\color{blue}{\fbox{U2}} \\\hline \end{array} $$
抽取出E2、U1、D2放置上层，删除后剩余的情况如下	抽取出C5,F1,U2放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline C2&\color{red}{\fbox{C2}}&\color{red}{\fbox{C2}} \\\hline F2&C2,F2,F3&F2 \\\hline F3&\color{red}{\fbox{F3}}&\color{red}{\fbox{F3}} \\\hline D1&C2,D1&D1 \\\hline D3&C2,D3&D3 \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C1&\color{blue}{\fbox{C1}}&\color{blue}{\fbox{C1}} \\\hline C4&C1,C4,E1&C4 \\\hline E1&\color{blue}{\fbox{E1}}&\color{blue}{\fbox{E1}} \\\hline E3&C1,E1,E3&E3 \\\hline \end{array} $$
抽取出C2、F3放置上层，删除后剩余的情况如下	抽取出C1,E1放置下层，删除后剩余的情况如下
$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} & R_{e} & T_{e} \\\hline F2&\color{red}{\fbox{F2}}&\color{red}{\fbox{F2}} \\\hline D1&\color{red}{\fbox{D1}}&\color{red}{\fbox{D1}} \\\hline D3&\color{red}{\fbox{D3}}&\color{red}{\fbox{D3}} \\\hline \end{array} $$	$$\begin{array} {c\|c\|c\|c\|c\|c\|c\|c}{} &Q_{e} & T_{e} \\\hline C4&\color{blue}{\fbox{C4}}&\color{blue}{\fbox{C4}} \\\hline E3&\color{blue}{\fbox{E3}}&\color{blue}{\fbox{E3}} \\\hline \end{array} $$
抽取出F2、D1、D3放置上层，删除后剩余的情况如下	抽取出C4,E3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	C4,E3	C4,E3
第1层	C1,E1,F1	C1,E1
第2层	C3,C5,U2	C5,F1,U2
第3层	E2,U1,D2	E2,U1,D2
第4层	C2,F3	C2,C3,F3
第5层	F2,D1,D3	F2,D1,D3

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

$$S=\begin{array} {c|c|c|c|c|c|c|c}{M_{16 \times16}} &C1 &C2 &C3 &C4 &C5 &E1 &E2 &E3 &F1 &F2 &F3 &U1 &U2 &D1 &D2 &D3\\ \hline C1 & & & &1 & & & &1 & & & & & & & & \\ \hline C2 & & & & &1 & &1 & & & & & & & &1 & \\ \hline C3 & & & & & &1 & & & & & & & & & & \\ \hline C4 & & & & & & & & & & & & & & & & \\ \hline C5 & & & & & &1 & & & & & & & & & & \\ \hline E1 & & & &1 & & & &1 & & & & & & & & \\ \hline E2 & & & & & & & & & & & & &1 & & & \\ \hline E3 & & & & & & & & & & & & & & & & \\ \hline F1 & & & &1 & & & &1 & & & & & & & & \\ \hline F2 & &1 &1 & & & & & & & &1 & & & & & \\ \hline F3 & & & & & & & & & & & &1 & & & & \\ \hline U1 &1 & & & &1 & & & & & & & & & & & \\ \hline U2 &1 & & & & & & & & & & & & & & & \\ \hline D1 & &1 & & & & & & & & & &1 & & & & \\ \hline D2 & & & & & & & & &1 & & & &1 & & & \\ \hline D3 & &1 & & & & & & & & & & & & & & \\ \hline \end{array} $$

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

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

$$MR=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times2}} &中心度 &原因度\\ \hline C1 &6.3725 &0.526\\ \hline C2 &5.6136 &0.2468\\ \hline C3 &4.6576 &0.7513\\ \hline C4 &6.6751 &1.0603\\ \hline C5 &6.028 &0.627\\ \hline E1 &6.106 &0.995\\ \hline E2 &6.1695 &0.3563\\ \hline E3 &6.556 &1.1084\\ \hline F1 &6.4907 &0.3153\\ \hline F2 &3.1215 &0.1984\\ \hline F3 &4.9263 &0.3864\\ \hline U1 &5.4328 &0.5241\\ \hline U2 &6.3188 &0.3749\\ \hline D1 &5.4002 &0.1913\\ \hline D2 &6.2634 &0.2995\\ \hline D3 &5.5107 &0.1438\\ \hline \end{array} $$

向量的计算采用欧式距离公式

$$d=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times1}} &向量\\ \hline C1 &6.3942\\ \hline C2 &5.6191\\ \hline C3 &4.7178\\ \hline C4 &6.7588\\ \hline C5 &6.0606\\ \hline E1 &6.1865\\ \hline E2 &6.1798\\ \hline E3 &6.649\\ \hline F1 &6.4983\\ \hline F2 &3.1278\\ \hline F3 &4.9414\\ \hline U1 &5.458\\ \hline U2 &6.3299\\ \hline D1 &5.4035\\ \hline D2 &6.2706\\ \hline D3 &5.5126\\ \hline \end{array} $$

归一化求权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{16 \times1}} &权重\\ \hline C1 &0.0694\\ \hline C2 &0.061\\ \hline C3 &0.0512\\ \hline C4 &0.0734\\ \hline C5 &0.0658\\ \hline E1 &0.0672\\ \hline E2 &0.0671\\ \hline E3 &0.0722\\ \hline F1 &0.0706\\ \hline F2 &0.034\\ \hline F3 &0.0536\\ \hline U1 &0.0593\\ \hline U2 &0.0687\\ \hline D1 &0.0587\\ \hline D2 &0.0681\\ \hline D3 &0.0598\\ \hline \end{array} $$

归一化求子系统的权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{5 \times1}} &权重\\ \hline C &0.3208\\ \hline E &0.2064\\ \hline F &0.1582\\ \hline U &0.128\\ \hline D &0.1866\\ \hline \end{array} $$

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：