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
$$D1=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &－d^+(负向指标) &d^-(正向指标)\\ \hline 2011 &0.1722 &0.0426\\ \hline 2012 &0.1696 &0.0423\\ \hline 2013 &0.1639 &0.0421\\ \hline 2014 &0.1776 &0.0419\\ \hline 2015 &0.0918 &0.1569\\ \hline 2016 &0.0855 &0.1578\\ \hline 2017 &0.0788 &0.1965\\ \hline 牛逼 &0.1666 &0.1234\\ \hline 很好 &0.1765 &0.0689\\ \hline 良 &0.1865 &0.0416\\ \hline 很垃圾 &0.1965 &0.0143\\ \hline \end{array} $$$$D2=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &－d^+(负向指标) &d^-(正向指标)\\ \hline 2011 &0.0775 &0.0745\\ \hline 2012 &0.0763 &0.074\\ \hline 2013 &0.0737 &0.0736\\ \hline 2014 &0.0799 &0.0733\\ \hline 2015 &0.0503 &0.0729\\ \hline 2016 &0.0469 &0.0724\\ \hline 2017 &0.0432 &0.0884\\ \hline 牛逼 &0.0749 &0.0779\\ \hline 很好 &0.0794 &0.0564\\ \hline 良 &0.0839 &0.0282\\ \hline 很垃圾 &0.0884 &0.0078\\ \hline \end{array} $$$$ S_i^+ =C_i^+ ＝ \frac{ d_i^-} { d_i^- + d_i^+} \quad 正理想点贴近度（相似度），正向指标$$
$$ S_i^- =C_i^- ＝ \frac{ d_i^+} { d_i^- + d_i^+} \quad 负理想点贴近度（相似度），负向指标$$
$$M=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &S_i^+(正向指标) &-S_i^-(负向指标)\\ \hline 2011 &0.1981 &0.5099\\ \hline 2012 &0.1996 &0.5075\\ \hline 2013 &0.2043 &0.5003\\ \hline 2014 &0.1909 &0.5215\\ \hline 2015 &0.6311 &0.4084\\ \hline 2016 &0.6485 &0.3929\\ \hline 2017 &0.7137 &0.3284\\ \hline 牛逼 &0.4256 &0.4903\\ \hline 很好 &0.2806 &0.5846\\ \hline 良 &0.1823 &0.7483\\ \hline 很垃圾 &0.0678 &0.9185\\ \hline \end{array} $$$$ Q_i = \left( 1-k \right) \left(\frac{ \sqrt {a_i^2 - Min(a_i)^2}} {\sqrt {Max(a_i)^2 -Min(a_i)^2}} \right) + k\left(\frac{\sqrt{ Max(b_i)^2 - b_i^2 }}{\sqrt{Max(b)^2 -Min(b_i)^2}} \right) $$
上述妥协解中，需要把负向指标转化为正向指标，原则即两个指标同方向。
$$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\ \hline 2011 &0.262 &0.8906\\ \hline 2012 &0.2643 &0.8925\\ \hline 2013 &0.2712 &0.898\\ \hline 2014 &0.2511 &0.8815\\ \hline 2015 &0.883 &0.9591\\ \hline 2016 &0.9078 &0.9678\\ \hline 2017 &1 &1\\ \hline 牛逼 &0.5913 &0.9055\\ \hline 很好 &0.3832 &0.8259\\ \hline 良 &0.2381 &0.6209\\ \hline 很垃圾 &0 &0\\ \hline \end{array} $$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。
$$\begin{array}{c|c|c|c|c|c|c}{M_{6 \times1}} &拐点对应的k值\\ \hline 0 &0\\ \hline 1 &0.6083\\ \hline 2 &0.6412\\ \hline 3 &0.6519\\ \hline 4 &0.7038\\ \hline 5 &1\\ \hline \end{array} $$上述两列都是正向指标，数值越大越好。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times6}} &k=0 &k=0.608 &k=0.641 &k=0.652 &k=0.704 &k=1\\ \hline 2011 &8 &8 &8 &8 &7 &7\\ \hline 2012 &7 &7 &7 &6 &6 &6\\ \hline 2013 &6 &6 &5 &5 &5 &5\\ \hline 2014 &9 &9 &9 &9 &9 &8\\ \hline 2015 &3 &3 &3 &3 &3 &3\\ \hline 2016 &2 &2 &2 &2 &2 &2\\ \hline 2017 &1 &1 &1 &1 &1 &1\\ \hline 牛逼 &4 &4 &4 &4 &4 &4\\ \hline 很好 &5 &6 &7 &8 &9 &9\\ \hline 良 &10 &10 &10 &10 &10 &10\\ \hline 很垃圾 &11 &11 &11 &11 &11 &11\\ \hline \end{array} $$
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 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 2 | 0<$k$<0.608334 | 0.608334 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 3 | 0.608334 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013 = 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 4 | 0.608334<$k$<0.641178 | 0.032844 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 很好\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 5 | 0.641178 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 很好\succ 2012 = 2011\succ 2014\succ 良\succ 很垃圾$ |
| 6 | 0.641178<$k$<0.651918 | 0.01074 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 很好\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 7 | 0.651918 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 很好 = 2014\succ 良\succ 很垃圾$ |
| 8 | 0.651918<$k$<0.703777 | 0.051859 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
| 9 | 0.703777 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 2014\succ 很好 = 良\succ 很垃圾$ |
| 10 | 0.703777<$k$<1 | 0.296223 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
| 11 | 1 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 2014\succ 很好\succ 良\succ 很垃圾$ |
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<$k$< 0.608334 | 0.608334 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 2 | 0.608334<$k$< 0.641178 | 0.032844 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 很好 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 3 | 0.641178<$k$< 0.651918 | 0.01074 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 2012 \succ 很好 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 4 | 0.651918<$k$< 0.703777 | 0.051859 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 2012 \succ 2011 \succ 很好 \succ 2014 \succ 良 \succ 很垃圾$ |
| 5 | 0.703777<$k$< 1 | 0.296223 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 很好 \succ 良 \succ 很垃圾$ |
| 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=1 |
| 2 | 2015=1 |
| 3 | 牛逼=1 |
| 4 | 很好=0.608334 2013=0.391666 |
| 5 | 2013=0.608334 很好=0.032844 2012=0.358822 |
| 6 | 2012=0.641178 很好=0.01074 2011=0.348082 |
| 7 | 2011=0.651918 很好=0.051859 2014=0.296223 |
| 8 | 2014=0.703777 很好=0.296223 |
| 9 | 良=1 |
| 10 | 很垃圾=1 |
| 情境 | 最优妥协解 |
|---|---|
| 优胜情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 劣汰情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |