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
$$ \require{cancel} \require{AMScd} \begin{CD} O @>>>N \\ \end{CD} $$
综合影响矩阵求解过程
$$\begin{CD} N @>>>T \\ \end{CD} $$
综合影响矩阵如下
$T=\mathcal{N}(I-\mathcal{N})^{-1}$
$$T=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0.192 &0.205 &0.156 &0.196 &0.142 &0.155 &0.183 &0.191 &0.182 &0.218 &0.185 &0.156\\ \hline A2 &0.185 &0.06 &0.041 &0.092 &0.069 &0.072 &0.097 &0.087 &0.085 &0.104 &0.1 &0.031\\ \hline A3 &0.197 &0.073 &0.048 &0.101 &0.034 &0.074 &0.099 &0.14 &0.086 &0.161 &0.114 &0.034\\ \hline K1 &0.198 &0.073 &0.096 &0.065 &0.035 &0.075 &0.1 &0.118 &0.088 &0.161 &0.127 &0.034\\ \hline K2 &0.198 &0.077 &0.046 &0.079 &0.036 &0.076 &0.101 &0.104 &0.114 &0.159 &0.138 &0.047\\ \hline K3 &0.245 &0.159 &0.097 &0.115 &0.132 &0.054 &0.158 &0.163 &0.159 &0.193 &0.163 &0.058\\ \hline X1 &0.167 &0.052 &0.081 &0.084 &0.024 &0.032 &0.04 &0.08 &0.088 &0.092 &0.042 &0.063\\ \hline X2 &0.181 &0.131 &0.076 &0.102 &0.03 &0.037 &0.058 &0.049 &0.044 &0.158 &0.084 &0.041\\ \hline X3 &0.175 &0.151 &0.036 &0.112 &0.03 &0.036 &0.057 &0.046 &0.042 &0.117 &0.082 &0.039\\ \hline R1 &0.157 &0.136 &0.064 &0.076 &0.025 &0.03 &0.038 &0.04 &0.036 &0.046 &0.039 &0.035\\ \hline R2 &0.231 &0.178 &0.105 &0.186 &0.093 &0.101 &0.12 &0.126 &0.118 &0.15 &0.078 &0.052\\ \hline R3 &0.178 &0.175 &0.051 &0.158 &0.078 &0.13 &0.151 &0.155 &0.151 &0.095 &0.096 &0.035\\ \hline \end{array} $$
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。 $$\omega=\begin{array}{c|c|c|c|c|c|c}{M_{12 \times12}} &A1 &A2 &A3 &K1 &K2 &K3 &X1 &X2 &X3 &R1 &R2 &R3\\ \hline A1 &0.0832 &0.1396 &0.1735 &0.1436 &0.1957 &0.1782 &0.1521 &0.1468 &0.1526 &0.132 &0.1484 &0.2503\\ \hline A2 &0.0802 &0.0408 &0.0457 &0.0676 &0.0951 &0.0827 &0.0805 &0.0673 &0.0714 &0.0629 &0.0797 &0.0501\\ \hline A3 &0.0856 &0.0496 &0.0531 &0.0737 &0.0465 &0.0848 &0.0826 &0.108 &0.0724 &0.0973 &0.0914 &0.0544\\ \hline K1 &0.086 &0.0495 &0.1075 &0.0477 &0.0476 &0.0862 &0.0836 &0.0908 &0.0736 &0.0973 &0.1015 &0.0545\\ \hline K2 &0.0859 &0.0521 &0.0518 &0.0581 &0.0494 &0.0869 &0.0839 &0.0799 &0.0954 &0.096 &0.1108 &0.0752\\ \hline K3 &0.1065 &0.1082 &0.1086 &0.0842 &0.1809 &0.0622 &0.1316 &0.1256 &0.1333 &0.1165 &0.131 &0.0922\\ \hline X1 &0.0724 &0.0352 &0.0909 &0.0613 &0.033 &0.0363 &0.033 &0.0619 &0.0735 &0.0554 &0.0335 &0.1009\\ \hline X2 &0.0787 &0.0893 &0.0844 &0.0749 &0.0413 &0.0422 &0.0484 &0.0378 &0.0367 &0.0957 &0.0671 &0.065\\ \hline X3 &0.076 &0.1028 &0.0398 &0.0816 &0.0413 &0.0408 &0.0472 &0.0353 &0.0355 &0.0708 &0.0657 &0.0628\\ \hline R1 &0.068 &0.0925 &0.0713 &0.0552 &0.0344 &0.0348 &0.0315 &0.0307 &0.0298 &0.0279 &0.0314 &0.0555\\ \hline R2 &0.1003 &0.1212 &0.1169 &0.1364 &0.1274 &0.1159 &0.1 &0.0968 &0.0992 &0.0905 &0.0625 &0.0832\\ \hline R3 &0.0773 &0.1193 &0.0565 &0.1156 &0.1073 &0.149 &0.1258 &0.1191 &0.1266 &0.0576 &0.077 &0.056\\ \hline \end{array} $$
极限超矩阵求解:
权重的求解
归一化求子系统的权重
$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{4 \times1}} &权重\\ \hline A &0.301\\ \hline K &0.2676\\ \hline X &0.1818\\ \hline R &0.2496\\ \hline \end{array} $$$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{1 \times4}} &A &K &X &R\\ \hline 权重 &0.301 &0.2676 &0.1818 &0.2496\\ \hline \end{array} $$