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 ，到负理想点的公式
$$ d = Max \left( \omega_{j} \left( n_{ij}- {Min(n_j)} \right) \right) $$
欧几里得距离、欧式距离公式，到正理想点的距离
$$ d = \sqrt {\sum_\limits{j=1}^m { \omega_{j}^2 \left({Max(n_j)-n_{ij} } \right)} ^2} $$
$$matrix=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &a(正向指标) &-b(负向指标)\\ \hline 2011 &0.0426 &0.2344\\ \hline 2012 &0.0423 &0.2269\\ \hline 2013 &0.0421 &0.2164\\ \hline 2014 &0.0419 &0.2203\\ \hline 2015 &0.1569 &0.1318\\ \hline 2016 &0.1578 &0.1268\\ \hline 2017 &0.1965 &0.1118\\ \hline 牛逼 &0.1234 &0.1696\\ \hline 很好 &0.0689 &0.2021\\ \hline 良 &0.0416 &0.2451\\ \hline 很垃圾 &0.0143 &0.2954\\ \hline \end{array} $$$$ Q_i =\left( 1-k \right) \left(\frac{a_i - Min(a_i)}{Max(a_i) -Min(a_i)} \right) + k\left(\frac{ Max(b_i)- b_i}{Max(b_i) -Min(b_i)} \right) $$
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。
$$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\ \hline 2011 &0.1551 &0.3322\\ \hline 2012 &0.1537 &0.373\\ \hline 2013 &0.1525 &0.4303\\ \hline 2014 &0.1516 &0.4088\\ \hline 2015 &0.783 &0.8912\\ \hline 2016 &0.7876 &0.9184\\ \hline 2017 &1 &1\\ \hline 牛逼 &0.599 &0.6852\\ \hline 很好 &0.2995 &0.5078\\ \hline 良 &0.1498 &0.2736\\ \hline 很垃圾 &0 &0\\ \hline \end{array} $$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。
$$\begin{array}{c|c|c|c|c|c|c}{M_{7 \times1}} &拐点对应的k值\\ \hline 0 &0\\ \hline 1 &0.0211\\ \hline 2 &0.0261\\ \hline 3 &0.0332\\ \hline 4 &0.0441\\ \hline 5 &0.0564\\ \hline 6 &1\\ \hline \end{array} $$上述两列都是正向指标，数值越大越好。因此排序情况如下：
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times7}} &k=0 &k=0.021 &k=0.026 &k=0.033 &k=0.044 &k=0.056 &k=1\\ \hline 2011 &6 &6 &7 &8 &9 &9 &9\\ \hline 2012 &7 &8 &8 &8 &7 &8 &8\\ \hline 2013 &8 &8 &7 &6 &6 &6 &6\\ \hline 2014 &9 &9 &9 &9 &9 &8 &7\\ \hline 2015 &3 &3 &3 &3 &3 &3 &3\\ \hline 2016 &2 &2 &2 &2 &2 &2 &2\\ \hline 2017 &1 &1 &1 &1 &1 &1 &1\\ \hline 牛逼 &4 &4 &4 &4 &4 &4 &4\\ \hline 很好 &5 &5 &5 &5 &5 &5 &5\\ \hline 良 &10 &10 &10 &10 &10 &10 &10\\ \hline 很垃圾 &11 &11 &11 &11 &11 &11 &11\\ \hline \end{array} $$
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<$k$<0.021059 | 0.021059 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2012\succ 2013\succ 2014\succ 良\succ 很垃圾$ |
| 3 | 0.021059 | 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 | 0.021059<$k$<0.026142 | 0.005083 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2011\succ 2013\succ 2012\succ 2014\succ 良\succ 很垃圾$ |
| 5 | 0.026142 | 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 | 0.026142<$k$<0.033183 | 0.007041 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2011\succ 2012\succ 2014\succ 良\succ 很垃圾$ |
| 7 | 0.033183 | 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 | 0.033183<$k$<0.044138 | 0.010956 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011\succ 2014\succ 良\succ 很垃圾$ |
| 9 | 0.044138 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2011\succ 2014 = 良\succ 很垃圾$ |
| 10 | 0.044138<$k$<0.056352 | 0.012213 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2014\succ 2011\succ 良\succ 很垃圾$ |
| 11 | 0.056352 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2012\succ 2014 = 2011\succ 良\succ 很垃圾$ |
| 12 | 0.056352<$k$<1 | 0.943648 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2014\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
| 13 | 1 | 0 | $2017\succ 2016\succ 2015\succ 牛逼\succ 很好\succ 2013\succ 2014\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
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<$k$< 0.021059 | 0.021059 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2012 \succ 2013 \succ 2014 \succ 良 \succ 很垃圾$ |
| 2 | 0.021059<$k$< 0.026142 | 0.005083 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2013 \succ 2012 \succ 2014 \succ 良 \succ 很垃圾$ |
| 3 | 0.026142<$k$< 0.033183 | 0.007041 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2011 \succ 2012 \succ 2014 \succ 良 \succ 很垃圾$ |
| 4 | 0.033183<$k$< 0.044138 | 0.010955 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |
| 5 | 0.044138<$k$< 0.056352 | 0.012214 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2014 \succ 2011 \succ 良 \succ 很垃圾$ |
| 6 | 0.056352<$k$< 1 | 0.943648 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2014 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
| 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=1 |
| 5 | 2011=0.026142 2013=0.973858 |
| 6 | 2012=0.044228 2013=0.005083 2011=0.007041 2014=0.943648 |
| 7 | 2013=0.021059 2012=0.955772 2011=0.010955 2014=0.012214 |
| 8 | 2014=0.044138 2011=0.955862 |
| 9 | 良=1 |
| 10 | 很垃圾=1 |
| 情境 | 最优妥协解 |
|---|---|
| 优胜情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2011 \succ 2013 \succ 2012 \succ 2014 \succ 良 \succ 很垃圾$ |
| 劣汰情境 | $2017 \succ 2016 \succ 2015 \succ 牛逼 \succ 很好 \succ 2013 \succ 2012 \succ 2011 \succ 2014 \succ 良 \succ 很垃圾$ |