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：$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
切比雪夫 Chebyshev
$$CEV=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &CEV1 &CEV2\\ \hline 2011 &0.0426 &0.0745\\ \hline 2012 &0.0423 &0.074\\ \hline 2013 &0.0421 &0.0736\\ \hline 2014 &0.0419 &0.0733\\ \hline 2015 &0.1569 &0.0729\\ \hline 2016 &0.1578 &0.0724\\ \hline 2017 &0.1965 &0.0884\\ \hline 牛逼 &0.1234 &0.0779\\ \hline 很好 &0.0689 &0.0564\\ \hline 良 &0.0416 &0.0282\\ \hline 很垃圾 &0.0143 &0.0078\\ \hline \end{array} $$$$ Q_i = \left( 1-k \right) \left(\frac{ \sqrt {CEV1_i^2 - Min(CEV1_i)^2}} {\sqrt {Max(CEV1_i)^2 -Min(CEV1_i)^2}} \right) + k\left(\frac{\sqrt{ CEV2_i^2 - Min(CEV2_i)^2}}{\sqrt{Max(CEV2)^2 -Min(CEV2_i)^2}} \right) $$ $$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\ \hline 2011 &0.2046 &0.8412\\ \hline 2012 &0.2032 &0.8361\\ \hline 2013 &0.202 &0.8316\\ \hline 2014 &0.2011 &0.8283\\ \hline 2015 &0.7976 &0.8233\\ \hline 2016 &0.8019 &0.8181\\ \hline 2017 &1 &1\\ \hline 牛逼 &0.6256 &0.8807\\ \hline 很好 &0.3437 &0.6349\\ \hline 良 &0.1992 &0.3079\\ \hline 很垃圾 &0 &0\\ \hline \end{array} $$
所谓拐点，就是上述线段中的交点
所谓排序分析，即每个决策系数k对应的Q值的优劣排序，数值越低越优。两个拐点之间要素的排序是稳定一致的
拐点处（交点），存在着至少一次，某两个要素的排序是一致的。
交点坐标位置接近，以至于观测不到交点，下面会变换坐标，使得拐点等距，这样方便观测拐点具体的值。
由上图得到交点加上k=0,k=1即得到所有拐点，结果如下。
$$\begin{array}{c|c|c|c|c|c|c}{M_{17 \times1}} &拐点对应的k值\\ \hline 0 &0\\ \hline 1 &0.4028\\ \hline 2 &0.4113\\ \hline 3 &0.4188\\ \hline 4 &0.4245\\ \hline 5 &0.4506\\ \hline 6 &0.738\\ \hline 7 &0.7499\\ \hline 8 &0.9628\\ \hline 9 &0.9707\\ \hline 10 &0.9709\\ \hline 11 &0.978\\ \hline 12 &0.9789\\ \hline 13 &0.9833\\ \hline 14 &0.9863\\ \hline 15 &0.9917\\ \hline 16 &1\\ \hline \end{array} $$上述两列都是正向指标，数值越大越好。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times17}} &k=0 &k=0.403 &k=0.411 &k=0.419 &k=0.424 &k=0.451 &k=0.738 &k=0.75 &k=0.963 &k=0.971 &k=0.971 &k=0.978 &k=0.979 &k=0.983 &k=0.986 &k=0.992 &k=1\\ \hline 2011 &6 &6 &5 &5 &5 &5 &5 &5 &5 &4 &3 &3 &3 &3 &3 &3 &3\\ \hline 2012 &7 &7 &7 &6 &6 &6 &6 &6 &6 &6 &6 &5 &5 &4 &4 &4 &4\\ \hline 2013 &8 &8 &8 &8 &7 &7 &7 &7 &7 &7 &7 &7 &6 &6 &6 &5 &5\\ \hline 2014 &9 &9 &9 &9 &9 &8 &8 &8 &8 &8 &8 &8 &8 &8 &7 &7 &6\\ \hline 2015 &3 &3 &3 &3 &3 &3 &2 &3 &3 &4 &4 &4 &5 &5 &6 &7 &7\\ \hline 2016 &2 &2 &2 &2 &2 &3 &4 &4 &5 &5 &6 &7 &7 &8 &8 &8 &8\\ \hline 2017 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\ \hline 牛逼 &4 &4 &4 &4 &4 &4 &4 &3 &2 &2 &2 &2 &2 &2 &2 &2 &2\\ \hline 很好 &5 &6 &7 &8 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9 &9\\ \hline 良 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10\\ \hline 很垃圾 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11\\ \hline \end{array} $$
拐点与区段的排序如下：其中拐点中交点的位置有相等的情况出现。
| 序号 | 性质与对应k值 | 区段大小 | Q值排序 |
|---|---|---|---|
| 1 | 0 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 2 | 0<$k$<0.402835 | 0.402835 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 3 | 0.402835 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 很好 = 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 4 | 0.402835<$k$<0.411273 | 0.008439 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 很好\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 5 | 0.411273 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 很好 = 2013\succ 2014\succ 良\succ 很垃圾$ |
| 6 | 0.411273<$k$<0.418848 | 0.007574 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 很好\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 7 | 0.418848 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 很好 = 2014\succ 良\succ 很垃圾$ |
| 8 | 0.418848<$k$<0.424492 | 0.005644 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
| 9 | 0.424492 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 很好\succ 2014 = 良\succ 很垃圾$ |
| 10 | 0.424492<$k$<0.450553 | 0.026061 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 11 | 0.450553 | 0 | $2017\succ 2015\succ 2016 = 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 12 | 0.450553<$k$<0.738039 | 0.287487 | $2017\succ 2015\succ 2016\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 13 | 0.738039 | 0 | $2017\succ 2015\succ 牛逼\succ 2016 = 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 14 | 0.738039<$k$<0.749867 | 0.011827 | $2017\succ 2015\succ 牛逼\succ 2016\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 15 | 0.749867 | 0 | $2017\succ 牛逼\succ 2015 = 2016\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 16 | 0.749867<$k$<0.962806 | 0.21294 | $2017\succ 牛逼\succ 2015\succ 2016\succ 2011\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 17 | 0.962806 | 0 | $2017\succ 牛逼\succ 2015\succ 2011\succ 2016 = 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 18 | 0.962806<$k$<0.970719 | 0.007913 | $2017\succ 牛逼\succ 2015\succ 2011\succ 2016\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 19 | 0.970719 | 0 | $2017\succ 牛逼\succ 2011\succ 2015 = 2016\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 20 | 0.970719<$k$<0.970863 | 0.000144 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2016\succ 2012\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 21 | 0.970863 | 0 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2012\succ 2016 = 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 22 | 0.970863<$k$<0.978028 | 0.007165 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2012\succ 2016\succ 2013\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 23 | 0.978028 | 0 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2012\succ 2016\succ 2013 = 2014\succ 很好\succ 良\succ 很垃圾$ |
| 24 | 0.978028<$k$<0.978949 | 0.000921 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2012\succ 2013\succ 2016\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 25 | 0.978949 | 0 | $2017\succ 牛逼\succ 2011\succ 2015\succ 2012 = 2013\succ 2016\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 26 | 0.978949<$k$<0.983326 | 0.004377 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2015\succ 2013\succ 2016\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 27 | 0.983326 | 0 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2015\succ 2013\succ 2016\succ 2014 = 很好\succ 良\succ 很垃圾$ |
| 28 | 0.983326<$k$<0.98627 | 0.002944 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2015\succ 2013\succ 2014\succ 2016\succ 很好\succ 良\succ 很垃圾$ |
| 29 | 0.98627 | 0 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2015 = 2014\succ 2016\succ 很好\succ 良\succ 很垃圾$ |
| 30 | 0.98627<$k$<0.991684 | 0.005414 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2015\succ 2014\succ 2016\succ 很好\succ 良\succ 很垃圾$ |
| 31 | 0.991684 | 0 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 2015 = 2016\succ 很好\succ 良\succ 很垃圾$ |
| 32 | 0.991684<$k$<1 | 0.008316 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 2015\succ 2016\succ 很好\succ 良\succ 很垃圾$ |
| 33 | 1 | 0 | $2017\succ 牛逼\succ 2011\succ 2012\succ 2013\succ 2014\succ 2015\succ 2016\succ 很好\succ 良\succ 很垃圾$ |
提取区段的位置
| 序号 | 聚类特征-对应k值区段 | 区段大小 | Q值排序 |
|---|---|---|---|
| 1 | 0<$k$< 0.402835 | 0.402835 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 2 | 0.402835<$k$< 0.411273 | 0.008438 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 很好 \succ 2012 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 3 | 0.411273<$k$< 0.418848 | 0.007575 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 2012 \succ 很好 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 4 | 0.418848<$k$< 0.424492 | 0.005644 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 很好 \succ 2014 \succ 良 \succ 很垃圾$ |
| 5 | 0.424492<$k$< 0.450553 | 0.026061 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 6 | 0.450553<$k$< 0.738039 | 0.287486 | $2017 \succ 2015 \succ 2016 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 7 | 0.738039<$k$< 0.749867 | 0.011828 | $2017 \succ 2015 \succ 牛逼 \succ 2016 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 8 | 0.749867<$k$< 0.962806 | 0.212939 | $2017 \succ 牛逼 \succ 2015 \succ 2016 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 9 | 0.962806<$k$< 0.970719 | 0.007913 | $2017 \succ 牛逼 \succ 2015 \succ 2011 \succ 2016 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 10 | 0.970719<$k$< 0.970863 | 0.000144 | $2017 \succ 牛逼 \succ 2011 \succ 2015 \succ 2016 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 11 | 0.970863<$k$< 0.978028 | 0.007165 | $2017 \succ 牛逼 \succ 2011 \succ 2015 \succ 2012 \succ 2016 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 12 | 0.978028<$k$< 0.978949 | 0.000921 | $2017 \succ 牛逼 \succ 2011 \succ 2015 \succ 2012 \succ 2013 \succ 2016 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 13 | 0.978949<$k$< 0.983326 | 0.004377 | $2017 \succ 牛逼 \succ 2011 \succ 2012 \succ 2015 \succ 2013 \succ 2016 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 14 | 0.983326<$k$< 0.98627 | 0.002944 | $2017 \succ 牛逼 \succ 2011 \succ 2012 \succ 2015 \succ 2013 \succ 2014 \succ 2016 \succ 很好 \succ 良 \succ 很垃圾$ |
| 15 | 0.98627<$k$< 0.991684 | 0.005414 | $2017 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 2015 \succ 2014 \succ 2016 \succ 很好 \succ 良 \succ 很垃圾$ |
| 16 | 0.991684<$k$< 1 | 0.008316 | $2017 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 2015 \succ 2016 \succ 很好 \succ 良 \succ 很垃圾$ |
| 层级，序号越小越优 | 要素所占区段，该层级要素的的占比 |
|---|---|
| 0 | 2017=1 |
| 1 | 2016=0.450553 2015=0.299314 牛逼=0.250133 |
| 2 | 2015=0.671405 2016=0.287486 牛逼=0.011828 2011=0.029281 |
| 3 | 牛逼=0.738039 2016=0.224767 2011=0.007913 2015=0.00823 2012=0.021051 |
| 4 | 很好=0.402835 2011=0.559971 2016=0.008057 2012=0.008086 2015=0.007321 2013=0.01373 |
| 5 | 2011=0.402835 很好=0.008438 2012=0.55959 2016=0.007165 2013=0.008242 2015=0.005414 2014=0.008316 |
| 6 | 2012=0.411273 很好=0.007575 2013=0.55918 2016=0.005298 2014=0.008358 2015=0.008316 |
| 7 | 2013=0.418848 很好=0.005644 2014=0.558834 2016=0.016674 |
| 8 | 2014=0.424492 很好=0.575508 |
| 9 | 良=1 |
| 10 | 很垃圾=1 |
| 情境 | 最优妥协解 |
|---|---|
| 优胜情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 劣汰情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2011 \succ 2012 \succ 2013 \succ 很好 \succ 2014 \succ 良 \succ 很垃圾$ |