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
$$matrix=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &a(正向指标) &-b(负向指标)\\ \hline 2011 &0.0426 &0.0775\\ \hline 2012 &0.0423 &0.0763\\ \hline 2013 &0.0421 &0.0737\\ \hline 2014 &0.0419 &0.0799\\ \hline 2015 &0.1569 &0.0503\\ \hline 2016 &0.1578 &0.0469\\ \hline 2017 &0.1965 &0.0432\\ \hline 牛逼 &0.1234 &0.0749\\ \hline 很好 &0.0689 &0.0794\\ \hline 良 &0.0416 &0.0839\\ \hline 很垃圾 &0.0143 &0.0884\\ \hline \end{array} $$$$ Q_i = \left( 1-k \right) \left(\frac{ \sqrt {a_i^2 - Min(a_i)^2}} {\sqrt {Max(a_i)^2 -Min(a_i)^2}} \right) + k\left(\frac{\sqrt{ Max(b_i)^2 - b_i^2 }}{\sqrt{Max(b)^2 -Min(b_i)^2}} \right) $$
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。
$$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\ \hline 2011 &0.2046 &0.5515\\ \hline 2012 &0.2032 &0.5788\\ \hline 2013 &0.202 &0.6323\\ \hline 2014 &0.2011 &0.4896\\ \hline 2015 &0.7976 &0.9424\\ \hline 2016 &0.8019 &0.9717\\ \hline 2017 &1 &1\\ \hline 牛逼 &0.6256 &0.6076\\ \hline 很好 &0.3437 &0.5029\\ \hline 良 &0.1992 &0.3603\\ \hline 很垃圾 &0 &0\\ \hline \end{array} $$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。
$$\begin{array}{c|c|c|c|c|c|c}{M_{9 \times1}} &拐点对应的k值\\ \hline 0 &0\\ \hline 1 &0.0223\\ \hline 2 &0.0312\\ \hline 3 &0.0482\\ \hline 4 &0.5228\\ \hline 5 &0.6493\\ \hline 6 &0.7412\\ \hline 7 &0.9449\\ \hline 8 &1\\ \hline \end{array} $$上述两列都是正向指标，数值越大越好。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times9}} &k=0 &k=0.022 &k=0.031 &k=0.048 &k=0.523 &k=0.649 &k=0.741 &k=0.945 &k=1\\ \hline 2011 &6 &6 &7 &8 &8 &8 &8 &7 &7\\ \hline 2012 &7 &8 &8 &8 &7 &7 &6 &6 &6\\ \hline 2013 &8 &8 &7 &6 &6 &5 &5 &5 &4\\ \hline 2014 &9 &9 &9 &9 &9 &9 &9 &9 &9\\ \hline 2015 &3 &3 &3 &3 &3 &3 &3 &3 &3\\ \hline 2016 &2 &2 &2 &2 &2 &2 &2 &2 &2\\ \hline 2017 &1 &1 &1 &1 &1 &1 &1 &1 &1\\ \hline 牛逼 &4 &4 &4 &4 &4 &4 &4 &5 &5\\ \hline 很好 &5 &5 &5 &5 &6 &7 &8 &8 &8\\ \hline 良 &10 &10 &10 &10 &10 &10 &10 &10 &10\\ \hline 很垃圾 &11 &11 &11 &11 &11 &11 &11 &11 &11\\ \hline \end{array} $$
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<$k$<0.022262 | 0.022262 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 3 | 0.022262 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013 = 2014\succ 良\succ 很垃圾$ |
| 4 | 0.022262<$k$<0.031194 | 0.008932 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2013\succ 2012\succ 2014\succ 良\succ 很垃圾$ |
| 5 | 0.031194 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2011 = 2012\succ 2014\succ 良\succ 很垃圾$ |
| 6 | 0.031194<$k$<0.048213 | 0.017019 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2011\succ 2012\succ 2014\succ 良\succ 很垃圾$ |
| 7 | 0.048213 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011 = 2014\succ 良\succ 很垃圾$ |
| 8 | 0.048213<$k$<0.522751 | 0.474538 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 9 | 0.522751 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013 = 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 10 | 0.522751<$k$<0.649271 | 0.12652 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 很好\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 11 | 0.649271 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 很好\succ 2012 = 2011\succ 2014\succ 良\succ 很垃圾$ |
| 12 | 0.649271<$k$<0.741211 | 0.09194 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 很好\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 13 | 0.741211 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 很好\succ 2011 = 2014\succ 良\succ 很垃圾$ |
| 14 | 0.741211<$k$<0.944921 | 0.20371 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013\succ 2012\succ 2011\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
| 15 | 0.944921 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 2013 = 2012\succ 2011\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
| 16 | 0.944921<$k$<1 | 0.055079 | $2017\succ 2016\succ 2015\succ 2013\succ 牛逼\succ 2012\succ 2011\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
| 17 | 1 | 0 | $2017\succ 2016\succ 2015\succ 2013\succ 牛逼\succ 2012\succ 2011\succ 很好\succ 2014\succ 良\succ 很垃圾$ |
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<$k$< 0.022262 | 0.022262 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 2 | 0.022262<$k$< 0.031194 | 0.008932 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2013 \succ 2012 \succ 2014 \succ 良 \succ 很垃圾$ |
| 3 | 0.031194<$k$< 0.048213 | 0.017019 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2011 \succ 2012 \succ 2014 \succ 良 \succ 很垃圾$ |
| 4 | 0.048213<$k$< 0.522751 | 0.474538 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 5 | 0.522751<$k$< 0.649271 | 0.12652 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 很好 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 6 | 0.649271<$k$< 0.741211 | 0.09194 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 2012 \succ 很好 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 7 | 0.741211<$k$< 0.944921 | 0.20371 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 2013 \succ 2012 \succ 2011 \succ 很好 \succ 2014 \succ 良 \succ 很垃圾$ |
| 8 | 0.944921<$k$< 1 | 0.055079 | $2017 \succ 2016 \succ 2015 \succ 2013 \succ 牛逼 \succ 2012 \succ 2011 \succ 很好 \succ 2014 \succ 良 \succ 很垃圾$ |
| 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=0.944921 2013=0.055079 |
| 4 | 很好=0.522751 2013=0.42217 牛逼=0.055079 |
| 5 | 2011=0.031194 2013=0.491557 很好=0.12652 2012=0.350729 |
| 6 | 2012=0.62332 2013=0.008932 2011=0.275808 很好=0.09194 |
| 7 | 2013=0.022262 2012=0.025951 2011=0.692998 很好=0.258789 |
| 8 | 2014=1 |
| 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 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2012 \succ 2011 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |