VIKOR-AISM求解过程
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原始矩阵如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{11 \times15}} &-PN1 &-PN2 &-PE1 &-PS1 &-PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
长沙市 &0.661588683 &146.9350744 &0.043311213 &599.9407733 &0.6849 &0.5192 &0.826985854 &0.038881916 &8.86877403 &8.83 &0.966896021 &0.78 &13401 &79530 &16.5\\
\hline
湘潭市 &0.748374512 &626.7880364 &0.09121877 &552.2593791 &0.5208 &0.4614 &0.760728218 &0.05563393 &9.527951886 &8.72 &0.880647764 &0.92 &9501 &40670 &7.5\\
\hline
株洲市 &0.606220096 &342.2009569 &0.085235629 &344.6101936 &0.5748 &0.6153 &0.693779904 &0.0538521 &8.541555024 &10.83 &0.898236186 &1.63 &9328 &40431 &35.5\\
\hline
衡阳市 &0.577109355 &174.1524344 &0.173427086 &468.0600914 &0.4699 &0.3918 &0.652561815 &0.054744627 &8.247884272 &9.48 &0.750494029 &0.76 &8506 &24371 &29\\
\hline
娄底市 &0.481827622 &565.6801661 &0.146611606 &467.3155107 &0.375 &0.482 &0.602284528 &0.050775071 &8.163587747 &9.51 &0.663296322 &1.37 &3951 &22362 &26.3\\
\hline
益阳市 &0.760441029 &203.6084196 &0.225841133 &350.1948052 &0.411 &0.544 &0.745071834 &0.069372335 &7.818242566 &6.98 &0.893090909 &0.82 &6773 &20496 &21.4\\
\hline
邵阳市 &0.736517219 &205.5230669 &0.239861998 &341.2985017 &0.3413 &0.5711 &0.841455491 &0.043308195 &10.09644899 &8.5 &0.643912176 &4.02 &4373 &12797 &40\\
\hline
优 &0.4 &100 &0.05 &400 &0.3 &0.7 &0.8 &0.1 &10 &20 &0.95 &12 &30000 &50000 &150\\
\hline
良 &0.6 &200 &0.1 &600 &0.5 &0.5 &0.7 &0.08 &8 &15 &0.9 &9 &20000 &40000 &100\\
\hline
中 &0.7 &300 &0.2 &800 &0.6 &0.4 &0.6 &0.05 &6 &10 &0.8 &6 &15000 &30000 &50\\
\hline
差 &0.8 &400 &0.3 &1000 &0.7 &0.3 &0.5 &0.02 &4 &5 &0.7 &3 &10000 &20000 &20\\
\hline
\end{array} $$
采用的归一方法如下
极差法
正向指标公式：$$ n_{ij} = \frac{{o_{ij}-min(o_{j})}}{{max(o_{j})-min(o_{j})}} $$
负向指标公式：$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{11 \times15}} &-PN1 &-PN2 &-PE1 &-PS1 &-PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
长沙市 &0.346 &0.911 &1 &0.607 &0.038 &0.548 &0.958 &0.236 &0.799 &0.255 &1 &0.002 &0.363 &1 &0.063\\
\hline
湘潭市 &0.129 &0 &0.813 &0.68 &0.448 &0.404 &0.764 &0.445 &0.907 &0.248 &0.733 &0.014 &0.213 &0.418 &0\\
\hline
株洲市 &0.484 &0.54 &0.837 &0.995 &0.313 &0.788 &0.568 &0.423 &0.745 &0.389 &0.787 &0.077 &0.206 &0.414 &0.196\\
\hline
衡阳市 &0.557 &0.859 &0.493 &0.808 &0.575 &0.23 &0.447 &0.434 &0.697 &0.299 &0.33 &0 &0.175 &0.173 &0.151\\
\hline
娄底市 &0.795 &0.116 &0.598 &0.809 &0.813 &0.455 &0.3 &0.385 &0.683 &0.301 &0.06 &0.054 &0 &0.143 &0.132\\
\hline
益阳市 &0.099 &0.803 &0.289 &0.986 &0.723 &0.61 &0.718 &0.617 &0.626 &0.132 &0.771 &0.005 &0.108 &0.115 &0.098\\
\hline
邵阳市 &0.159 &0.8 &0.234 &1 &0.897 &0.678 &1 &0.291 &1 &0.233 &0 &0.29 &0.016 &0 &0.228\\
\hline
优 &1 &1 &0.974 &0.911 &1 &1 &0.879 &1 &0.984 &1 &0.948 &1 &1 &0.557 &1\\
\hline
良 &0.5 &0.81 &0.779 &0.607 &0.5 &0.5 &0.586 &0.75 &0.656 &0.667 &0.793 &0.733 &0.616 &0.408 &0.649\\
\hline
中 &0.25 &0.62 &0.39 &0.304 &0.25 &0.25 &0.293 &0.375 &0.328 &0.333 &0.483 &0.466 &0.424 &0.258 &0.298\\
\hline
差 &0 &0.431 &0 &0 &0 &0 &0 &0 &0 &0 &0.174 &0.199 &0.232 &0.108 &0.088\\
\hline
\end{array} $$
正极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times15}} &-PN1 &-PN2 &-PE1 &-PS1 &-PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
\mathbf{Zone^+} &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\
\hline
\end{array} $$负极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times15}} &-PN1 &-PN2 &-PE1 &-PS1 &-PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
\mathbf{Zone^-} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
\hline
\end{array} $$采用的是熵权法（EWM）求权重
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times15}} &-PN1 &-PN2 &-PE1 &-PS1 &-PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
EWM所得权重 &0.0732 &0.0427 &0.0445 &0.0327 &0.0612 &0.0428 &0.0401 &0.0438 &0.0309 &0.0624 &0.061 &0.1774 &0.0952 &0.0777 &0.1144\\
\hline
权重大小顺序 &5 &12 &9 &14 &7 &11 &13 &10 &15 &6 &8 &1 &3 &4 &2\\
\hline
\end{array} $$
VIKOR的最大化群体效益和最小化反对意见的个别遗憾
最大化群体效益 |
最小化反对意见的个别遗憾 |
| $$ S_i = \sum_\limits{j=1}^m{ \omega_{j} \left(\frac{Zone_j^+ -n_{ij}}{Zone_j^+ -Zone_j^-} \right)} \quad \quad $$ |
$$ R_i = \max_\limits{j=1} { \left( \omega_{j} (\frac{Zone_j^+ -n_{ij}}{Zone_j^+ -Zone_j^-} )\right)} \quad \quad $$ |
代入权重值等即得(S R)两列矩阵，两列都为负向指标
$$ \begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &期望值 &遗憾值\\
\hline
长沙市 &0.5761 &0.1773\\
\hline
湘潭市 &0.6944 &0.175\\
\hline
株洲市 &0.5948 &0.1638\\
\hline
衡阳市 &0.685 &0.1774\\
\hline
娄底市 &0.7069 &0.1679\\
\hline
益阳市 &0.6818 &0.1766\\
\hline
邵阳市 &0.6503 &0.1261\\
\hline
优 &0.047 &0.0344\\
\hline
良 &0.3614 &0.0474\\
\hline
中 &0.6368 &0.0948\\
\hline
差 &0.8956 &0.1422\\
\hline
\end{array} $$
妥协解的公式
公式 |
| $$ Q_i = \left( 1-k \right) \left(\frac{S_i - Min(S_i)}{Max(S_i) -Min(S_i)} \right) + k\left(\frac{R_i - Min(R_i)}{Max(R_i) -Min(R_i)} \right) $$ |
截距方式分析k的值——也是常规方法
一般的论文对于下面的公式
$$ Q_i = \left( 1-k \right) \left(\frac{S_i - Min(S_i)}{Max(S_i) -Min(S_i)} \right) + k\left(\frac{R_i - Min(R_i)}{Max(R_i) -Min(R_i)} \right) $$
其中的$k$随便说一下取0.5就拉倒了。这个好比小学生的四舍五入一样天经地义。事实上这个值很有得商榷的。它是一个敏感性有强有弱的范围。
$$ 对于每一行 令a_i =\frac{S_i - Min(S_i)}{Max(S_i) -Min(S_i)} \quad \quad b_i =\frac{R_i - Min(R_i)}{Max(R_i) -Min(R_i)} $$
$$ Q_i = \left( 1-k \right) a_i + kb_i \quad \quad $$
对于 $x,y$样本
$$
\begin{cases}
\left( 1-k \right) a_x + kb_x \\
\left( 1-k \right) a_y + kb_y
\end{cases}
$$
以上问题就变成了求两条线段是否在$[0,1]$值域内有相交的问题，此题属于初中的知识范畴，不再详细描述。
$$
\left( 1-k \right) a_x + kb_x =\left( 1-k \right) a_y + kb_y
$$
$$
a_x-k a_x + kb_x =a_y-k a_y + kb_y
$$
$$
a_x- a_y=-k a_y + kb_y +k a_x - kb_x
$$
$$
a_x- a_y=(- a_y + b_y + a_x - b_x)k
$$
$$
k =\frac{a_x- a_y}{( a_x- a_y + b_y - b_x)}
$$
基础矩阵如下
$$Base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &a_i &b_i\\
\hline 长沙市 &0.6235 &0.999\\
\hline 湘潭市 &0.7629 &0.9836\\
\hline 株洲市 &0.6455 &0.9051\\
\hline 衡阳市 &0.7518 &1\\
\hline 娄底市 &0.7776 &0.9338\\
\hline 益阳市 &0.7481 &0.9946\\
\hline 邵阳市 &0.7109 &0.6411\\
\hline 优 &0 &0\\
\hline 良 &0.3705 &0.0908\\
\hline 中 &0.695 &0.4223\\
\hline 差 &1 &0.7537\\
\hline \end{array} $$
拐点k值分析
$$Qk_{matrix}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times21}} &k=0 &k=0.093 &k=0.11 &k=0.189 &k=0.196 &k=0.199 &k=0.229 &k=0.281 &k=0.327 &k=0.402 &k=0.502 &k=0.508 &k=0.511 &k=0.553 &k=0.572 &k=0.606 &k=0.701 &k=0.703 &k=0.9 &k=0.966 &k=1\\
\hline 长沙市 &0.624 &0.658 &0.665 &0.695 &0.697 &0.698 &0.71 &0.729 &0.746 &0.774 &0.812 &0.814 &0.815 &0.831 &0.838 &0.851 &0.887 &0.887 &0.962 &0.986 &0.999\\
\hline 湘潭市 &0.763 &0.783 &0.787 &0.805 &0.806 &0.807 &0.813 &0.825 &0.835 &0.851 &0.874 &0.875 &0.876 &0.885 &0.889 &0.896 &0.918 &0.918 &0.962 &0.976 &0.984\\
\hline 株洲市 &0.645 &0.67 &0.674 &0.695 &0.696 &0.697 &0.705 &0.718 &0.73 &0.75 &0.776 &0.777 &0.778 &0.789 &0.794 &0.803 &0.827 &0.828 &0.879 &0.896 &0.905\\
\hline 衡阳市 &0.752 &0.775 &0.779 &0.799 &0.801 &0.801 &0.809 &0.821 &0.833 &0.851 &0.876 &0.878 &0.879 &0.889 &0.894 &0.902 &0.926 &0.926 &0.975 &0.991 &1\\
\hline 娄底市 &0.778 &0.792 &0.795 &0.807 &0.808 &0.809 &0.813 &0.821 &0.829 &0.84 &0.856 &0.857 &0.857 &0.864 &0.867 &0.872 &0.887 &0.887 &0.918 &0.928 &0.934\\
\hline 益阳市 &0.748 &0.771 &0.775 &0.795 &0.796 &0.797 &0.805 &0.817 &0.829 &0.847 &0.872 &0.873 &0.874 &0.884 &0.889 &0.897 &0.921 &0.921 &0.97 &0.986 &0.995\\
\hline 邵阳市 &0.711 &0.704 &0.703 &0.698 &0.697 &0.697 &0.695 &0.691 &0.688 &0.683 &0.676 &0.675 &0.675 &0.672 &0.671 &0.669 &0.662 &0.662 &0.648 &0.643 &0.641\\
\hline 优 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
\hline 良 &0.37 &0.344 &0.34 &0.318 &0.316 &0.315 &0.306 &0.292 &0.279 &0.258 &0.23 &0.228 &0.228 &0.216 &0.21 &0.201 &0.174 &0.174 &0.119 &0.1 &0.091\\
\hline 中 &0.695 &0.67 &0.665 &0.643 &0.641 &0.641 &0.632 &0.618 &0.606 &0.585 &0.558 &0.556 &0.556 &0.544 &0.539 &0.53 &0.504 &0.503 &0.45 &0.432 &0.422\\
\hline 差 &1 &0.977 &0.973 &0.953 &0.952 &0.951 &0.944 &0.931 &0.919 &0.901 &0.876 &0.875 &0.874 &0.864 &0.859 &0.851 &0.827 &0.827 &0.778 &0.762 &0.754\\
\hline \end{array} $$
排序分析
上述是负向指标，数值越小越好，每一列数值最小的排第一。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times21}} &k=0 &k=0.093 &k=0.11 &k=0.189 &k=0.196 &k=0.199 &k=0.229 &k=0.281 &k=0.327 &k=0.402 &k=0.502 &k=0.508 &k=0.511 &k=0.553 &k=0.572 &k=0.606 &k=0.701 &k=0.703 &k=0.9 &k=0.966 &k=1\\
\hline 长沙市 &3 &3 &3 &4 &5 &6 &6 &6 &6 &6 &6 &6 &6 &6 &6 &6 &7 &7 &8 &9 &10\\
\hline 湘潭市 &9 &9 &9 &9 &9 &9 &9 &10 &10 &9 &9 &9 &10 &10 &9 &9 &9 &9 &8 &8 &8\\
\hline 株洲市 &4 &4 &5 &4 &4 &4 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5 &6 &6 &6 &6\\
\hline 衡阳市 &8 &8 &8 &8 &8 &8 &8 &8 &9 &9 &10 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11\\
\hline 娄底市 &10 &10 &10 &10 &10 &10 &9 &8 &7 &7 &7 &7 &7 &7 &8 &8 &8 &7 &7 &7 &7\\
\hline 益阳市 &7 &7 &7 &7 &7 &7 &7 &7 &7 &8 &8 &8 &8 &9 &9 &10 &10 &10 &10 &9 &9\\
\hline 邵阳市 &6 &6 &6 &6 &5 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4\\
\hline 优 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\
\hline 良 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2\\
\hline 中 &5 &4 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3 &3\\
\hline 差 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &10 &9 &8 &7 &7 &6 &5 &5 &5 &5 &5\\
\hline \end{array} $$
聚类特征
| 序号 |
聚类特征-对应k值区段 |
Q值排序 |
| 1 | 0<$k$< 0.09298 | $优 \succ 良 \succ 长沙市 \succ 株洲市 \succ 中 \succ 邵阳市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 2 | 0.093<$k$< 0.11022 | $优 \succ 良 \succ 长沙市 \succ 中 \succ 株洲市 \succ 邵阳市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 3 | 0.1102<$k$< 0.18941 | $优 \succ 良 \succ 中 \succ 长沙市 \succ 株洲市 \succ 邵阳市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 4 | 0.1894<$k$< 0.1962 | $优 \succ 良 \succ 中 \succ 株洲市 \succ 长沙市 \succ 邵阳市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 5 | 0.1962<$k$< 0.19858 | $优 \succ 良 \succ 中 \succ 株洲市 \succ 邵阳市 \succ 长沙市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 6 | 0.1986<$k$< 0.22919 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 娄底市 \succ 差$ |
|---|
| 7 | 0.2292<$k$< 0.28066 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 益阳市 \succ 衡阳市 \succ 娄底市 \succ 湘潭市 \succ 差$ |
|---|
| 8 | 0.2807<$k$< 0.32731 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 益阳市 \succ 娄底市 \succ 衡阳市 \succ 湘潭市 \succ 差$ |
|---|
| 9 | 0.3273<$k$< 0.40155 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 益阳市 \succ 衡阳市 \succ 湘潭市 \succ 差$ |
|---|
| 10 | 0.4015<$k$< 0.50194 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 益阳市 \succ 湘潭市 \succ 衡阳市 \succ 差$ |
|---|
| 11 | 0.5019<$k$< 0.50785 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 益阳市 \succ 湘潭市 \succ 差 \succ 衡阳市$ |
|---|
| 12 | 0.5078<$k$< 0.51123 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 益阳市 \succ 差 \succ 湘潭市 \succ 衡阳市$ |
|---|
| 13 | 0.5112<$k$< 0.55251 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 差 \succ 益阳市 \succ 湘潭市 \succ 衡阳市$ |
|---|
| 14 | 0.5525<$k$< 0.5723 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 差 \succ 娄底市 \succ 益阳市 \succ 湘潭市 \succ 衡阳市$ |
|---|
| 15 | 0.5723<$k$< 0.6055 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 长沙市 \succ 差 \succ 娄底市 \succ 湘潭市 \succ 益阳市 \succ 衡阳市$ |
|---|
| 16 | 0.6055<$k$< 0.70079 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 株洲市 \succ 差 \succ 长沙市 \succ 娄底市 \succ 湘潭市 \succ 益阳市 \succ 衡阳市$ |
|---|
| 17 | 0.7008<$k$< 0.70275 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 差 \succ 株洲市 \succ 长沙市 \succ 娄底市 \succ 湘潭市 \succ 益阳市 \succ 衡阳市$ |
|---|
| 18 | 0.7027<$k$< 0.90007 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 差 \succ 株洲市 \succ 娄底市 \succ 长沙市 \succ 湘潭市 \succ 益阳市 \succ 衡阳市$ |
|---|
| 19 | 0.9001<$k$< 0.96573 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 差 \succ 株洲市 \succ 娄底市 \succ 湘潭市 \succ 长沙市 \succ 益阳市 \succ 衡阳市$ |
|---|
| 20 | 0.9657<$k$< 1 | $优 \succ 良 \succ 中 \succ 邵阳市 \succ 差 \succ 株洲市 \succ 娄底市 \succ 湘潭市 \succ 益阳市 \succ 长沙市 \succ 衡阳市$ |
|---|
区段截取方式Q排序——k值区段截取分析
区段截取方式Q排序——夹逼方式
扯蛋模型