DEMATEL-ANP流程图如下:
重点:
第一、直接影响矩阵同ANP/AHP的原始矩阵有着根本的不同。ANP与AHP是基于正互反判断矩阵
第二、必须是基于综合影响矩阵T的求解。这是由于影响的传递性决定的。
有向网络如上图:其中A对C无直接影响。很显然A对C的影响是由A对B施加影响,B对C施加影响。A对C的影响是通过B为中介获得的。
$ T=\mathcal{N}+\mathcal{N}^2+\mathcal{N}^3+\cdots+\mathcal{N}^k $
其中:$ \mathcal{N}\times\mathcal{N} $的意义为增加的间接影响。即矩阵与矩阵相乘得到的值为矩阵要素之间增加的间接影响。
$ \mathcal{N}^\infty $为零阵。意思是影响不停的传递下去,其值为零。
$ \mathcal{N}^\infty $为零阵。前提要求是主对角线的值全部为0
综合影响矩阵$ T= \sum\limits_{k=1}^\infty {\mathcal{N}^k} $的意义为最初的直接影响加上所有的间接影响。
综合影响矩阵$ T=\mathcal{N}(I-\mathcal{N})^{-1} $
原始矩阵(直接影响矩阵)为
$$Ori=\begin{array}{|c|c|c|c|c|c|c|}\hline {M_{16 \times16}} &A1 &A2 &A3 &A4 &A5 &B1 &B2 &B3 &C1 &C2 &C3 &C4 &D1 &D2 &D3 &D4\\ \hline A1 &0 &37 &38 &36 &32 &40 &32 &34 &38 &32 &33 &31 &23 &28 &35 &33\\ \hline A2 &43 &0 &40 &41 &36 &40 &38 &40 &45 &39 &43 &43 &41 &38 &37 &35\\ \hline A3 &37 &41 &0 &43 &41 &38 &36 &41 &45 &42 &41 &43 &32 &34 &41 &39\\ \hline A4 &39 &40 &42 &0 &39 &40 &43 &42 &38 &37 &39 &41 &46 &43 &40 &42\\ \hline A5 &32 &28 &31 &33 &0 &26 &24 &25 &29 &27 &21 &23 &22 &21 &27 &24\\ \hline B1 &41 &39 &40 &42 &39 &0 &44 &46 &38 &38 &42 &39 &43 &41 &40 &42\\ \hline B2 &33 &34 &32 &35 &27 &38 &0 &39 &32 &30 &26 &27 &28 &32 &33 &32\\ \hline B3 &36 &37 &33 &38 &34 &41 &37 &0 &33 &29 &28 &30 &31 &32 &37 &32\\ \hline C1 &46 &46 &39 &42 &35 &40 &38 &35 &0 &41 &37 &38 &31 &36 &41 &36\\ \hline C2 &33 &31 &32 &30 &31 &28 &26 &28 &36 &0 &34 &33 &27 &26 &29 &25\\ \hline C3 &31 &32 &29 &32 &33 &31 &32 &28 &35 &30 &0 &26 &29 &31 &33 &22\\ \hline C4 &33 &32 &30 &33 &29 &33 &34 &33 &31 &32 &31 &0 &28 &30 &32 &24\\ \hline D1 &33 &35 &32 &34 &25 &34 &32 &30 &29 &24 &28 &32 &0 &34 &32 &31\\ \hline D2 &34 &37 &32 &33 &25 &31 &33 &28 &32 &27 &25 &34 &30 &0 &31 &29\\ \hline D3 &35 &36 &31 &42 &31 &33 &32 &28 &34 &33 &27 &34 &29 &33 &0 &31\\ \hline D4 &44 &37 &42 &45 &38 &44 &42 &43 &46 &35 &39 &42 &38 &40 &44 &0\\ \hline \end{array} $$
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_{16 \times16}} &A1 &A2 &A3 &A4 &A5 &B1 &B2 &B3 &C1 &C2 &C3 &C4 &D1 &D2 &D3 &D4\\ \hline A1 &0.068 &0.109 &0.108 &0.11 &0.098 &0.112 &0.102 &0.104 &0.11 &0.099 &0.099 &0.1 &0.086 &0.094 &0.106 &0.097\\ \hline A2 &0.129 &0.079 &0.122 &0.128 &0.113 &0.124 &0.12 &0.121 &0.13 &0.117 &0.121 &0.124 &0.116 &0.116 &0.12 &0.11\\ \hline A3 &0.122 &0.125 &0.075 &0.129 &0.119 &0.121 &0.117 &0.122 &0.129 &0.12 &0.118 &0.124 &0.106 &0.111 &0.124 &0.114\\ \hline A4 &0.126 &0.126 &0.125 &0.082 &0.118 &0.125 &0.126 &0.125 &0.123 &0.116 &0.118 &0.123 &0.123 &0.123 &0.124 &0.119\\ \hline A5 &0.09 &0.085 &0.087 &0.092 &0.048 &0.083 &0.079 &0.08 &0.086 &0.08 &0.073 &0.077 &0.073 &0.073 &0.083 &0.075\\ \hline B1 &0.128 &0.125 &0.123 &0.131 &0.118 &0.08 &0.128 &0.13 &0.124 &0.117 &0.121 &0.121 &0.12 &0.121 &0.125 &0.119\\ \hline B2 &0.103 &0.103 &0.099 &0.106 &0.09 &0.107 &0.062 &0.106 &0.101 &0.093 &0.089 &0.092 &0.089 &0.096 &0.101 &0.094\\ \hline B3 &0.11 &0.11 &0.103 &0.113 &0.101 &0.114 &0.108 &0.065 &0.105 &0.096 &0.094 &0.099 &0.096 &0.099 &0.109 &0.097\\ \hline C1 &0.13 &0.129 &0.119 &0.127 &0.111 &0.122 &0.118 &0.114 &0.077 &0.117 &0.113 &0.117 &0.104 &0.112 &0.122 &0.109\\ \hline C2 &0.098 &0.095 &0.094 &0.096 &0.09 &0.091 &0.088 &0.09 &0.101 &0.055 &0.094 &0.095 &0.084 &0.085 &0.092 &0.082\\ \hline C3 &0.097 &0.097 &0.092 &0.099 &0.093 &0.095 &0.095 &0.09 &0.1 &0.09 &0.055 &0.088 &0.087 &0.091 &0.097 &0.079\\ \hline C4 &0.1 &0.099 &0.094 &0.101 &0.09 &0.099 &0.099 &0.097 &0.097 &0.094 &0.092 &0.059 &0.087 &0.092 &0.097 &0.083\\ \hline D1 &0.101 &0.102 &0.097 &0.103 &0.086 &0.101 &0.097 &0.094 &0.095 &0.085 &0.089 &0.096 &0.055 &0.097 &0.098 &0.091\\ \hline D2 &0.101 &0.104 &0.096 &0.101 &0.085 &0.097 &0.097 &0.092 &0.098 &0.088 &0.085 &0.098 &0.089 &0.057 &0.096 &0.088\\ \hline D3 &0.106 &0.106 &0.099 &0.115 &0.095 &0.102 &0.1 &0.095 &0.104 &0.098 &0.091 &0.101 &0.091 &0.098 &0.064 &0.094\\ \hline D4 &0.133 &0.124 &0.127 &0.135 &0.118 &0.131 &0.127 &0.127 &0.133 &0.115 &0.119 &0.126 &0.116 &0.121 &0.13 &0.072\\ \hline \end{array} $$
行向量加权矩阵C如下
$$C=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times16}} &A1 &A2 &A3 &A4 &A5 &B1 &B2 &B3 &C1 &C2 &C3 &C4 &D1 &D2 &D3 &D4\\ \hline A1 &0.0423 &0.0682 &0.0675 &0.0688 &0.0614 &0.0699 &0.0634 &0.0646 &0.0688 &0.0615 &0.062 &0.0622 &0.0538 &0.0588 &0.0661 &0.0608\\ \hline A2 &0.0682 &0.0416 &0.0644 &0.0676 &0.06 &0.0655 &0.0633 &0.0642 &0.0686 &0.0619 &0.0641 &0.0657 &0.0617 &0.0615 &0.0634 &0.0582\\ \hline A3 &0.0649 &0.0666 &0.0402 &0.069 &0.0632 &0.0645 &0.0623 &0.065 &0.0689 &0.0639 &0.0631 &0.0659 &0.0566 &0.0593 &0.0659 &0.0607\\ \hline A4 &0.0654 &0.0654 &0.0651 &0.0428 &0.0613 &0.065 &0.0657 &0.065 &0.0641 &0.0603 &0.0613 &0.064 &0.0641 &0.064 &0.0646 &0.0618\\ \hline A5 &0.0715 &0.0674 &0.0687 &0.073 &0.0383 &0.0652 &0.0624 &0.0632 &0.0682 &0.0632 &0.0577 &0.0611 &0.0573 &0.058 &0.0657 &0.0592\\ \hline B1 &0.0664 &0.0647 &0.0639 &0.0676 &0.0612 &0.0413 &0.0662 &0.0671 &0.064 &0.0607 &0.0629 &0.0628 &0.0623 &0.0627 &0.0645 &0.0617\\ \hline B2 &0.0672 &0.0673 &0.0645 &0.0693 &0.0588 &0.0699 &0.0404 &0.0694 &0.0658 &0.061 &0.058 &0.0603 &0.0583 &0.0627 &0.0659 &0.0612\\ \hline B3 &0.0678 &0.0679 &0.0638 &0.0699 &0.0624 &0.0703 &0.0666 &0.0402 &0.0651 &0.0591 &0.0582 &0.0612 &0.0592 &0.0614 &0.0672 &0.0599\\ \hline C1 &0.0707 &0.0701 &0.0646 &0.0689 &0.0601 &0.0662 &0.0639 &0.062 &0.0416 &0.0639 &0.0613 &0.0635 &0.0564 &0.061 &0.0664 &0.0594\\ \hline C2 &0.0688 &0.0667 &0.066 &0.0671 &0.0632 &0.0639 &0.0613 &0.0627 &0.0704 &0.0383 &0.0655 &0.0663 &0.0587 &0.0595 &0.0643 &0.0572\\ \hline C3 &0.067 &0.0672 &0.0635 &0.0684 &0.0645 &0.066 &0.0657 &0.0624 &0.0693 &0.0623 &0.038 &0.0606 &0.0601 &0.0632 &0.0671 &0.0548\\ \hline C4 &0.0679 &0.0666 &0.0637 &0.0685 &0.0608 &0.0669 &0.0667 &0.0657 &0.0657 &0.0632 &0.0623 &0.0396 &0.0588 &0.0619 &0.0658 &0.0559\\ \hline D1 &0.0678 &0.0688 &0.0651 &0.0692 &0.0578 &0.0677 &0.0652 &0.0635 &0.0642 &0.0572 &0.06 &0.0647 &0.037 &0.0649 &0.0658 &0.0611\\ \hline D2 &0.0688 &0.0705 &0.0653 &0.0687 &0.0579 &0.0657 &0.0661 &0.0622 &0.0666 &0.0596 &0.058 &0.0664 &0.0605 &0.0385 &0.0652 &0.0598\\ \hline D3 &0.0681 &0.0682 &0.0633 &0.0736 &0.0612 &0.0657 &0.064 &0.061 &0.0667 &0.0627 &0.0583 &0.0649 &0.0586 &0.0629 &0.0409 &0.06\\ \hline D4 &0.0679 &0.0634 &0.0648 &0.0691 &0.0605 &0.067 &0.0649 &0.0652 &0.0683 &0.0589 &0.061 &0.0643 &0.0592 &0.0619 &0.0666 &0.037\\ \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_{16 \times16}} &A1 &A2 &A3 &A4 &A5 &B1 &B2 &B3 &C1 &C2 &C3 &C4 &D1 &D2 &D3 &D4\\ \hline A1 &0.0399 &0.0649 &0.0665 &0.0636 &0.0644 &0.0672 &0.0629 &0.0644 &0.0658 &0.0642 &0.0652 &0.0626 &0.0583 &0.0611 &0.0645 &0.0654\\ \hline A2 &0.0643 &0.0396 &0.0635 &0.0625 &0.063 &0.0629 &0.0628 &0.064 &0.0656 &0.0647 &0.0674 &0.0661 &0.0669 &0.0639 &0.0618 &0.0627\\ \hline A3 &0.0612 &0.0634 &0.0397 &0.0638 &0.0664 &0.062 &0.0618 &0.0648 &0.0658 &0.0667 &0.0663 &0.0663 &0.0613 &0.0616 &0.0642 &0.0653\\ \hline A4 &0.0616 &0.0622 &0.0642 &0.0396 &0.0644 &0.0625 &0.0652 &0.0647 &0.0613 &0.0629 &0.0644 &0.0645 &0.0695 &0.0665 &0.063 &0.0666\\ \hline A5 &0.0674 &0.0641 &0.0677 &0.0675 &0.0402 &0.0627 &0.0619 &0.0629 &0.0652 &0.066 &0.0607 &0.0615 &0.0621 &0.0602 &0.0641 &0.0638\\ \hline B1 &0.0626 &0.0616 &0.063 &0.0625 &0.0643 &0.0397 &0.0657 &0.0669 &0.0612 &0.0634 &0.066 &0.0632 &0.0675 &0.0652 &0.0629 &0.0664\\ \hline B2 &0.0633 &0.0641 &0.0636 &0.0641 &0.0617 &0.0672 &0.04 &0.0692 &0.0628 &0.0637 &0.0609 &0.0607 &0.0632 &0.0652 &0.0642 &0.0659\\ \hline B3 &0.0639 &0.0646 &0.0629 &0.0646 &0.0655 &0.0675 &0.066 &0.04 &0.0622 &0.0617 &0.0612 &0.0616 &0.0641 &0.0638 &0.0655 &0.0645\\ \hline C1 &0.0667 &0.0667 &0.0637 &0.0637 &0.0631 &0.0636 &0.0634 &0.0618 &0.0398 &0.0667 &0.0644 &0.0639 &0.0611 &0.0634 &0.0647 &0.064\\ \hline C2 &0.0649 &0.0635 &0.0651 &0.062 &0.0664 &0.0614 &0.0608 &0.0625 &0.0673 &0.04 &0.0688 &0.0667 &0.0636 &0.0619 &0.0627 &0.0616\\ \hline C3 &0.0631 &0.064 &0.0626 &0.0632 &0.0677 &0.0634 &0.0652 &0.0622 &0.0662 &0.065 &0.04 &0.061 &0.0651 &0.0657 &0.0655 &0.059\\ \hline C4 &0.064 &0.0634 &0.0628 &0.0633 &0.0639 &0.0643 &0.0661 &0.0655 &0.0628 &0.066 &0.0654 &0.0399 &0.0638 &0.0643 &0.0642 &0.0602\\ \hline D1 &0.064 &0.0655 &0.0642 &0.064 &0.0606 &0.065 &0.0647 &0.0633 &0.0613 &0.0597 &0.0631 &0.0651 &0.0401 &0.0675 &0.0642 &0.0658\\ \hline D2 &0.0649 &0.0671 &0.0644 &0.0635 &0.0608 &0.0631 &0.0656 &0.062 &0.0637 &0.0623 &0.0609 &0.0668 &0.0656 &0.04 &0.0636 &0.0644\\ \hline D3 &0.0642 &0.0649 &0.0624 &0.0681 &0.0642 &0.0631 &0.0635 &0.0608 &0.0637 &0.0655 &0.0613 &0.0654 &0.0635 &0.0654 &0.0399 &0.0646\\ \hline D4 &0.064 &0.0604 &0.0639 &0.0639 &0.0635 &0.0644 &0.0643 &0.065 &0.0653 &0.0615 &0.0641 &0.0647 &0.0642 &0.0644 &0.065 &0.0398\\ \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
$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{4 \times1}} &权重\\ \hline A &0.3128\\ \hline B &0.1876\\ \hline C &0.2499\\ \hline D &0.2497\\ \hline \end{array} $$$$\omega =\begin{array}{c|c|c|c|c|c|c}{M_{1 \times4}} &A &B &C &D\\ \hline 权重 &0.3128 &0.1876 &0.2499 &0.2497\\ \hline \end{array} $$