原始矩阵（直接影响矩阵）为

$$Ori=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0 &7 &7 &7 &7 &7 &7 &7 &7 &7 &7 &9\\ \hline A2 &8 &0 &0 &3 &3 &3 &4 &3 &3 &3 &4 &0\\ \hline A3 &8 &0 &0 &3 &0 &3 &4 &7 &3 &7 &5 &0\\ \hline K1 &8 &0 &4 &0 &0 &3 &4 &5 &3 &7 &6 &0\\ \hline K2 &8 &0 &0 &1 &0 &3 &4 &4 &5 &7 &7 &1\\ \hline K3 &8 &5 &3 &2 &7 &0 &7 &7 &7 &7 &7 &1\\ \hline X1 &8 &0 &4 &3 &0 &0 &0 &3 &4 &3 &0 &3\\ \hline X2 &8 &6 &3 &4 &0 &0 &1 &0 &0 &8 &3 &1\\ \hline X3 &8 &8 &0 &5 &0 &0 &1 &0 &0 &5 &3 &1\\ \hline R1 &8 &8 &3 &3 &0 &0 &0 &0 &0 &0 &0 &1\\ \hline R2 &8 &8 &4 &9 &4 &4 &4 &4 &4 &4 &0 &1\\ \hline R3 &4 &8 &0 &7 &3 &7 &7 &7 &7 &0 &2 &0\\ \hline \end{array} $$

规范直接关系矩阵求解过程

$$ \require{cancel} \require{AMScd} \begin{CD} O @>>>N \\ \end{CD} $$
$$\mathcal{N}=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0 &0.061 &0.061 &0.061 &0.061 &0.061 &0.061 &0.061 &0.061 &0.061 &0.061 &0.078\\ \hline A2 &0.069 &0 &0 &0.026 &0.026 &0.026 &0.035 &0.026 &0.026 &0.026 &0.035 &0\\ \hline A3 &0.069 &0 &0 &0.026 &0 &0.026 &0.035 &0.061 &0.026 &0.061 &0.043 &0\\ \hline K1 &0.069 &0 &0.035 &0 &0 &0.026 &0.035 &0.043 &0.026 &0.061 &0.052 &0\\ \hline K2 &0.069 &0 &0 &0.009 &0 &0.026 &0.035 &0.035 &0.043 &0.061 &0.061 &0.009\\ \hline K3 &0.069 &0.043 &0.026 &0.017 &0.061 &0 &0.061 &0.061 &0.061 &0.061 &0.061 &0.009\\ \hline X1 &0.069 &0 &0.035 &0.026 &0 &0 &0 &0.026 &0.035 &0.026 &0 &0.026\\ \hline X2 &0.069 &0.052 &0.026 &0.035 &0 &0 &0.009 &0 &0 &0.069 &0.026 &0.009\\ \hline X3 &0.069 &0.069 &0 &0.043 &0 &0 &0.009 &0 &0 &0.043 &0.026 &0.009\\ \hline R1 &0.069 &0.069 &0.026 &0.026 &0 &0 &0 &0 &0 &0 &0 &0.009\\ \hline R2 &0.069 &0.069 &0.035 &0.078 &0.035 &0.035 &0.035 &0.035 &0.035 &0.035 &0 &0.009\\ \hline R3 &0.035 &0.069 &0 &0.061 &0.026 &0.061 &0.061 &0.061 &0.061 &0 &0.017 &0\\ \hline \end{array} $$

综合影响矩阵求解过程

$$\begin{CD} N @>>>T \\ \end{CD} $$

综合影响矩阵如下

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

$$T=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0.067 &0.1 &0.082 &0.097 &0.078 &0.083 &0.092 &0.095 &0.092 &0.104 &0.093 &0.09\\ \hline A2 &0.097 &0.02 &0.014 &0.044 &0.037 &0.037 &0.05 &0.042 &0.042 &0.048 &0.051 &0.011\\ \hline A3 &0.101 &0.026 &0.017 &0.047 &0.011 &0.038 &0.05 &0.077 &0.042 &0.085 &0.06 &0.012\\ \hline K1 &0.101 &0.025 &0.051 &0.022 &0.012 &0.038 &0.051 &0.061 &0.042 &0.085 &0.068 &0.012\\ \hline K2 &0.101 &0.027 &0.016 &0.032 &0.012 &0.038 &0.051 &0.052 &0.06 &0.084 &0.077 &0.021\\ \hline K3 &0.118 &0.076 &0.046 &0.048 &0.075 &0.018 &0.083 &0.085 &0.084 &0.096 &0.085 &0.024\\ \hline X1 &0.09 &0.018 &0.045 &0.042 &0.008 &0.011 &0.013 &0.04 &0.047 &0.044 &0.014 &0.035\\ \hline X2 &0.096 &0.07 &0.039 &0.052 &0.01 &0.013 &0.024 &0.017 &0.015 &0.088 &0.041 &0.018\\ \hline X3 &0.093 &0.086 &0.012 &0.059 &0.011 &0.013 &0.024 &0.015 &0.014 &0.061 &0.041 &0.018\\ \hline R1 &0.087 &0.08 &0.035 &0.038 &0.009 &0.011 &0.013 &0.014 &0.012 &0.015 &0.014 &0.016\\ \hline R2 &0.113 &0.094 &0.053 &0.101 &0.049 &0.052 &0.058 &0.06 &0.058 &0.069 &0.027 &0.022\\ \hline R3 &0.079 &0.093 &0.017 &0.083 &0.039 &0.074 &0.082 &0.083 &0.082 &0.033 &0.042 &0.011\\ \hline \end{array} $$

加权超矩阵求解：

　求解就是每列归一化，加权超矩阵每列加起来的为1。$加权超矩阵 \mathcal{ \omega} $如下：

注意：当综合影响矩阵中存在某一列的值全部为0的时候需要特殊处理。 $$\omega=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0.0584 &0.1394 &0.1922 &0.1451 &0.2222 &0.1949 &0.1561 &0.1484 &0.1565 &0.1284 &0.1517 &0.3112\\ \hline A2 &0.0845 &0.0284 &0.0334 &0.0663 &0.1047 &0.0879 &0.0844 &0.0661 &0.0712 &0.0595 &0.0833 &0.0381\\ \hline A3 &0.0886 &0.0358 &0.0395 &0.0711 &0.0327 &0.0886 &0.0852 &0.1204 &0.0709 &0.1047 &0.0977 &0.0411\\ \hline K1 &0.0887 &0.0355 &0.1187 &0.0337 &0.0337 &0.0898 &0.086 &0.0955 &0.0719 &0.1045 &0.1115 &0.041\\ \hline K2 &0.0886 &0.0381 &0.038 &0.0478 &0.0353 &0.0903 &0.0862 &0.0808 &0.1018 &0.1032 &0.1248 &0.0713\\ \hline K3 &0.1036 &0.1061 &0.1077 &0.072 &0.2126 &0.0423 &0.1407 &0.1323 &0.1423 &0.1182 &0.1394 &0.0824\\ \hline X1 &0.0789 &0.0249 &0.1057 &0.0623 &0.0228 &0.0259 &0.0226 &0.0629 &0.0796 &0.0543 &0.0232 &0.1199\\ \hline X2 &0.0836 &0.0986 &0.0914 &0.0781 &0.0294 &0.0302 &0.0406 &0.0258 &0.0248 &0.1082 &0.0672 &0.0632\\ \hline X3 &0.0816 &0.1199 &0.0288 &0.0892 &0.03 &0.0295 &0.04 &0.0241 &0.0242 &0.0749 &0.0665 &0.0616\\ \hline R1 &0.0758 &0.1117 &0.0812 &0.0575 &0.0254 &0.0258 &0.0223 &0.0216 &0.0207 &0.0188 &0.0223 &0.0566\\ \hline R2 &0.099 &0.1312 &0.1238 &0.152 &0.1387 &0.1214 &0.0983 &0.0937 &0.0975 &0.0845 &0.0444 &0.0748\\ \hline R3 &0.0687 &0.1305 &0.0396 &0.1249 &0.1124 &0.1735 &0.1377 &0.1286 &0.1385 &0.0407 &0.0681 &0.0389\\ \hline \end{array} $$

极限超矩阵求解：

$$limit W=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632 &0.1632\\ \hline A2 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696 &0.0696\\ \hline A3 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735 &0.0735\\ \hline K1 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766 &0.0766\\ \hline K2 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772 &0.0772\\ \hline K3 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118 &0.1118\\ \hline X1 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584 &0.0584\\ \hline X2 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626 &0.0626\\ \hline X3 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058 &0.058\\ \hline R1 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048 &0.048\\ \hline R2 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035 &0.1035\\ \hline R3 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978 &0.0978\\ \hline \end{array} $$

权重的求解

$$\omega=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times1}} &权重\\ \hline A1 &0.16316\\ \hline A2 &0.0696\\ \hline A3 &0.07349\\ \hline K1 &0.07662\\ \hline K2 &0.0772\\ \hline K3 &0.11175\\ \hline X1 &0.0584\\ \hline X2 &0.06259\\ \hline X3 &0.05796\\ \hline R1 &0.04795\\ \hline R2 &0.1035\\ \hline R3 &0.09777\\ \hline \end{array} $$$$\omega=\begin{array}{c|c|c|c|c|c|c}{M_{1 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline 权重 &0.16316 &0.0696 &0.07349 &0.07662 &0.0772 &0.11175 &0.0584 &0.06259 &0.05796 &0.04795 &0.1035 &0.09777\\ \hline \end{array} $$

归一化求子系统的权重

$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{4 \times1}} &权重\\ \hline A &0.3062\\ \hline K &0.2656\\ \hline X &0.179\\ \hline R &0.2492\\ \hline \end{array} $$$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{1 \times4}} &A &K &X &R\\ \hline 权重 &0.3062 &0.2656 &0.179 &0.2492\\ \hline \end{array} $$