核心不同的距离公式，最常见的有曼哈顿距离、欧式距离、切比雪夫等等。
另外一个关键是得到的评价值两列为正向指标
核心不同的距离公式，最常见的有曼哈顿距离、欧式距离、切比雪夫等等。
另外一个关键是得到的评价值两列为正向指标
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：$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
欧几里得距离、欧式距离公式
$$ CEV = \sqrt {\sum_\limits{j=1}^m { \omega_{j}^2 \left({n_{ij}-Min(n_j) } \right)} ^2} $$
切比雪夫 Chebyshev
$$CEV=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &CEV1 &CEV2\\ \hline 2011 &0.1088 &0.0426\\ \hline 2012 &0.1119 &0.0423\\ \hline 2013 &0.1152 &0.0421\\ \hline 2014 &0.1212 &0.0419\\ \hline 2015 &0.2001 &0.1569\\ \hline 2016 &0.2046 &0.1578\\ \hline 2017 &0.2462 &0.1965\\ \hline 牛逼 &0.22 &0.1234\\ \hline 很好 &0.1368 &0.0689\\ \hline 良 &0.0719 &0.0416\\ \hline 很垃圾 &0.0143 &0.0143\\ \hline \end{array} $$$$ Q_i = \left( 1-k \right) \left(\frac{ \sqrt {CEV1_i^2 - Min(CEV1_i)^2}} {\sqrt {Max(CEV1_i)^2 -Min(CEV1_i)^2}} \right) + k\left(\frac{\sqrt{ CEV2_i^2 - Min(CEV2_i)^2}}{\sqrt{Max(CEV2)^2 -Min(CEV2_i)^2}} \right) $$ $$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\ \hline 2011 &0.439 &0.2046\\ \hline 2012 &0.4514 &0.2032\\ \hline 2013 &0.4649 &0.202\\ \hline 2014 &0.4895 &0.2011\\ \hline 2015 &0.8122 &0.7976\\ \hline 2016 &0.8307 &0.8019\\ \hline 2017 &1 &1\\ \hline 牛逼 &0.8934 &0.6256\\ \hline 很好 &0.5535 &0.3437\\ \hline 良 &0.2868 &0.1992\\ \hline 很垃圾 &0 &0\\ \hline \end{array} $$
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。
$$\begin{array}{c|c|c|c|c|c|c}{M_{10 \times1}} &拐点对应的k值\\ \hline 0 &0\\ \hline 1 &0.2627\\ \hline 2 &0.3209\\ \hline 3 &0.8996\\ \hline 4 &0.9088\\ \hline 5 &0.9173\\ \hline 6 &0.9353\\ \hline 7 &0.9475\\ \hline 8 &0.9649\\ \hline 9 &1\\ \hline \end{array} $$上述两列都是正向指标，数值越大越好。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times10}} &k=0 &k=0.263 &k=0.321 &k=0.9 &k=0.909 &k=0.917 &k=0.935 &k=0.947 &k=0.965 &k=1\\ \hline 2011 &9 &9 &9 &9 &8 &7 &7 &6 &6 &6\\ \hline 2012 &8 &8 &8 &9 &9 &9 &8 &8 &7 &7\\ \hline 2013 &7 &7 &7 &7 &8 &9 &9 &9 &9 &8\\ \hline 2014 &6 &6 &6 &6 &6 &6 &7 &8 &9 &9\\ \hline 2015 &4 &4 &4 &3 &3 &3 &3 &3 &3 &3\\ \hline 2016 &3 &3 &2 &2 &2 &2 &2 &2 &2 &2\\ \hline 2017 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\ \hline 牛逼 &2 &3 &4 &4 &4 &4 &4 &4 &4 &4\\ \hline 很好 &5 &5 &5 &5 &5 &5 &5 &5 &5 &5\\ \hline 良 &10 &10 &10 &10 &10 &10 &10 &10 &10 &10\\ \hline 很垃圾 &11 &11 &11 &11 &11 &11 &11 &11 &11 &11\\ \hline \end{array} $$
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 | $2017\succ 牛逼\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 2 | 0<$k$<0.262651 | 0.262651 | $2017\succ 牛逼\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 3 | 0.262651 | 0 | $2017\succ 2016\succ 牛逼 = 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 4 | 0.262651<$k$<0.320895 | 0.058244 | $2017\succ 2016\succ 牛逼\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 5 | 0.320895 | 0 | $2017\succ 2016\succ 牛逼\succ 2015 = 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 6 | 0.320895<$k$<0.899648 | 0.578754 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 7 | 0.899648 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2013\succ 2011\succ 2012 = 良\succ 很垃圾$ |
| 8 | 0.899648<$k$<0.90879 | 0.009142 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2013\succ 2011\succ 2012\succ 良\succ 很垃圾$ |
| 9 | 0.90879 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2011\succ 2013 = 2012\succ 良\succ 很垃圾$ |
| 10 | 0.90879<$k$<0.917348 | 0.008558 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2011\succ 2013\succ 2012\succ 良\succ 很垃圾$ |
| 11 | 0.917348 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2011\succ 2012\succ 2013 = 良\succ 很垃圾$ |
| 12 | 0.917348<$k$<0.935274 | 0.017925 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2014\succ 2011\succ 2012\succ 2013\succ 良\succ 很垃圾$ |
| 13 | 0.935274 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2014 = 2012\succ 2013\succ 良\succ 很垃圾$ |
| 14 | 0.935274<$k$<0.947497 | 0.012223 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2014\succ 2012\succ 2013\succ 良\succ 很垃圾$ |
| 15 | 0.947497 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2014 = 2013\succ 良\succ 很垃圾$ |
| 16 | 0.947497<$k$<0.964942 | 0.017445 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2014\succ 2013\succ 良\succ 很垃圾$ |
| 17 | 0.964942 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014 = 良\succ 很垃圾$ |
| 18 | 0.964942<$k$<1 | 0.035058 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 19 | 1 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
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<$k$< 0.262651 | 0.262651 | $2017 \succ 牛逼 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
| 2 | 0.262651<$k$< 0.320895 | 0.058244 | $2017 \succ 2016 \succ 牛逼 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
| 3 | 0.320895<$k$< 0.899648 | 0.578753 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
| 4 | 0.899648<$k$< 0.90879 | 0.009142 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2014 \succ 2013 \succ 2011 \succ 2012 \succ 良 \succ 很垃圾$ |
| 5 | 0.90879<$k$< 0.917348 | 0.008558 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2014 \succ 2011 \succ 2013 \succ 2012 \succ 良 \succ 很垃圾$ |
| 6 | 0.917348<$k$< 0.935274 | 0.017926 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2014 \succ 2011 \succ 2012 \succ 2013 \succ 良 \succ 很垃圾$ |
| 7 | 0.935274<$k$< 0.947497 | 0.012223 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2014 \succ 2012 \succ 2013 \succ 良 \succ 很垃圾$ |
| 8 | 0.947497<$k$< 0.964942 | 0.017445 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2012 \succ 2014 \succ 2013 \succ 良 \succ 很垃圾$ |
| 9 | 0.964942<$k$< 1 | 0.035058 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 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 | 牛逼=0.262651 2016=0.737349 |
| 2 | 2016=0.262651 牛逼=0.058244 2015=0.679105 |
| 3 | 2015=0.320895 牛逼=0.679105 |
| 4 | 很好=1 |
| 5 | 2014=0.935274 2011=0.064726 |
| 6 | 2013=0.90879 2011=0.026484 2014=0.012223 2012=0.052503 |
| 7 | 2012=0.929797 2011=0.009142 2013=0.043616 2014=0.017445 |
| 8 | 2011=0.899648 2012=0.0177 2013=0.047594 2014=0.035058 |
| 9 | 良=1 |
| 10 | 很垃圾=1 |
| 情境 | 最优妥协解 |
|---|---|
| 优胜情境 | $2017 \succ 牛逼 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
| 劣汰情境 | $2017 \succ 2016 \succ 牛逼 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |