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：$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
耦合度公式 $$ C=\left( \frac{\prod \limits_{i=1}^{n} U_i } { \left( \frac{1}{n}\sum \limits_{i=1}^{n}{U_i} \right )^n } \right ) ^\frac{1}{n} $$
子系统数目为n
n=2
$$ C=\frac{2\sqrt{U_1U_2}}{U_1+U_2} $$
n=3
$$ C=\frac{3 \left ({U_1U_2U_3} \right )^ {\frac{1}{3}} }{U_1+U_2+U_3}=\frac{3 \sqrt[3] {U_1U_2U_3} }{U_1+U_2+U_3} $$
n=4
$$ C=\frac{4 \left ({U_1U_2U_3U_4} \right )^ {\frac{1}{4}} }{U_1+U_2+U_3+U_4}=\frac{4 \sqrt[4] {U_1U_2U_3U_4} }{U_1+U_2+U_3+U_4} $$
协调发展度公式 $$ D=\sqrt{CT} $$
$$ T=\sum \limits_{i=1}^{n}{\omega _i U_i} $$
$$a_i= DS_i \qquad b_i=DR_i $$
$$ Q_i = \left( 1-k \right) a_i + kb_i \quad \quad $$
对于 $x,y$样本
$$ \begin{cases} \left( 1-k \right) a_x + kb_x \\ \left( 1-k \right) a_y + kb_y \end{cases} $$
以上问题就变成了求两条线段是否在$[0,1]$值域内有相交的问题，此题属于初中的知识范畴，不再详细描述。
$$ \left( 1-k \right) a_x + kb_x =\left( 1-k \right) a_y + kb_y $$
$$ a_x-k a_x + kb_x =a_y-k a_y + kb_y $$
$$ a_x- a_y=-k a_y + kb_y +k a_x - kb_x $$
$$ a_x- a_y=(- a_y + b_y + a_x - b_x)k $$
$$ k =\frac{a_x- a_y}{( a_x- a_y + b_y - b_x)} $$
| 序号 | 聚类特征-对应k值区段 | Q值排序 |
|---|---|---|
| 1 | 0<$k$< 0.004014 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2007 \prec 2009 \prec 2003 \prec 2012 \prec 2004 \prec 2010 \prec 2013 \prec 2014 \prec 2011 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 2 | 0.004014<$k$< 0.049882 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2007 \prec 2009 \prec 2003 \prec 2012 \prec 2004 \prec 2013 \prec 2010 \prec 2014 \prec 2011 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 3 | 0.049882<$k$< 0.124429 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2007 \prec 2009 \prec 2012 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2014 \prec 2011 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 4 | 0.124429<$k$< 0.126978 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2007 \prec 2009 \prec 2012 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 5 | 0.126978<$k$< 0.190098 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2007 \prec 2012 \prec 2009 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 6 | 0.190098<$k$< 0.257706 | $2002 \prec 2005 \prec 2000 \prec 2001 \prec 2006 \prec 2008 \prec 2012 \prec 2007 \prec 2009 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 7 | 0.257706<$k$< 0.283736 | $2002 \prec 2005 \prec 2000 \prec 2006 \prec 2001 \prec 2008 \prec 2012 \prec 2007 \prec 2009 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 8 | 0.283736<$k$< 0.286144 | $2002 \prec 2005 \prec 2006 \prec 2000 \prec 2001 \prec 2008 \prec 2012 \prec 2007 \prec 2009 \prec 2003 \prec 2004 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 9 | 0.286144<$k$< 0.329085 | $2002 \prec 2005 \prec 2006 \prec 2000 \prec 2001 \prec 2008 \prec 2012 \prec 2007 \prec 2009 \prec 2004 \prec 2003 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 10 | 0.329085<$k$< 0.346663 | $2002 \prec 2005 \prec 2006 \prec 2000 \prec 2008 \prec 2001 \prec 2012 \prec 2007 \prec 2009 \prec 2004 \prec 2003 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 11 | 0.346663<$k$< 0.349996 | $2002 \prec 2005 \prec 2006 \prec 2000 \prec 2008 \prec 2012 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2003 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 12 | 0.349996<$k$< 0.361471 | $2002 \prec 2005 \prec 2006 \prec 2008 \prec 2000 \prec 2012 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2003 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 13 | 0.361471<$k$< 0.380141 | $2002 \prec 2005 \prec 2006 \prec 2008 \prec 2012 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2003 \prec 2013 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 14 | 0.380141<$k$< 0.385831 | $2002 \prec 2005 \prec 2006 \prec 2008 \prec 2012 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2013 \prec 2003 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 15 | 0.385831<$k$< 0.401853 | $2002 \prec 2005 \prec 2006 \prec 2012 \prec 2008 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2013 \prec 2003 \prec 2010 \prec 2011 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 16 | 0.401853<$k$< 0.425998 | $2002 \prec 2005 \prec 2006 \prec 2012 \prec 2008 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2013 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 17 | 0.425998<$k$< 0.447576 | $2002 \prec 2005 \prec 2012 \prec 2006 \prec 2008 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2004 \prec 2013 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 18 | 0.447576<$k$< 0.482857 | $2002 \prec 2005 \prec 2012 \prec 2006 \prec 2008 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 19 | 0.482857<$k$< 0.511714 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2000 \prec 2001 \prec 2007 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 20 | 0.511714<$k$< 0.549372 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2000 \prec 2007 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 21 | 0.549372<$k$< 0.555117 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2000 \prec 2007 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 22 | 0.555117<$k$< 0.571131 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2000 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 23 | 0.571131<$k$< 0.586998 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2000 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2011 \prec 2003 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 24 | 0.586998<$k$< 0.596125 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2000 \prec 2009 \prec 2001 \prec 2013 \prec 2004 \prec 2011 \prec 2003 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 25 | 0.596125<$k$< 0.636827 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2000 \prec 2009 \prec 2001 \prec 2013 \prec 2004 \prec 2011 \prec 2003 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 26 | 0.636827<$k$< 0.690115 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2000 \prec 2001 \prec 2013 \prec 2004 \prec 2011 \prec 2003 \prec 2014 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 27 | 0.690115<$k$< 0.691086 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2000 \prec 2001 \prec 2013 \prec 2004 \prec 2011 \prec 2014 \prec 2003 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 28 | 0.691086<$k$< 0.724035 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2000 \prec 2013 \prec 2001 \prec 2004 \prec 2011 \prec 2014 \prec 2003 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 29 | 0.724035<$k$< 0.73293 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2013 \prec 2000 \prec 2001 \prec 2004 \prec 2011 \prec 2014 \prec 2003 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 30 | 0.73293<$k$< 0.818162 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2013 \prec 2000 \prec 2001 \prec 2011 \prec 2004 \prec 2014 \prec 2003 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 31 | 0.818162<$k$< 0.831374 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2013 \prec 2000 \prec 2011 \prec 2001 \prec 2004 \prec 2014 \prec 2003 \prec 2010 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 32 | 0.831374<$k$< 0.841236 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2009 \prec 2013 \prec 2000 \prec 2011 \prec 2001 \prec 2004 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 33 | 0.841236<$k$< 0.851966 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2013 \prec 2009 \prec 2000 \prec 2011 \prec 2001 \prec 2004 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 34 | 0.851966<$k$< 0.889288 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2013 \prec 2009 \prec 2011 \prec 2000 \prec 2001 \prec 2004 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2017 \prec 2019 \prec 2015 \prec 2018$ |
| 35 | 0.889288<$k$< 0.918941 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2013 \prec 2009 \prec 2011 \prec 2000 \prec 2001 \prec 2004 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2017 \prec 2015 \prec 2019 \prec 2018$ |
| 36 | 0.918941<$k$< 0.929383 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2013 \prec 2009 \prec 2011 \prec 2000 \prec 2004 \prec 2001 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2017 \prec 2015 \prec 2019 \prec 2018$ |
| 37 | 0.929383<$k$< 0.962432 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2007 \prec 2013 \prec 2009 \prec 2011 \prec 2000 \prec 2004 \prec 2001 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2015 \prec 2017 \prec 2019 \prec 2018$ |
| 38 | 0.962432<$k$< 0.96906 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2013 \prec 2007 \prec 2009 \prec 2011 \prec 2000 \prec 2004 \prec 2001 \prec 2014 \prec 2010 \prec 2003 \prec 2016 \prec 2015 \prec 2017 \prec 2019 \prec 2018$ |
| 39 | 0.96906<$k$< 1 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2013 \prec 2007 \prec 2009 \prec 2011 \prec 2000 \prec 2004 \prec 2014 \prec 2001 \prec 2010 \prec 2003 \prec 2016 \prec 2015 \prec 2017 \prec 2019 \prec 2018$ |
| 排名 | 要素所占区段 |
|---|---|
| 0 | 2002=1 |
| 1 | 2005=1 |
| 2 | 2012=0.574002 2000=0.283736 2006=0.142262 |
| 3 | 2008=0.552978 2001=0.257706 2006=0.082889 2000=0.06626 2012=0.040167 |
| 4 | 2006=0.774849 2008=0.117937 2001=0.071379 2012=0.02436 2000=0.011475 |
| 5 | 2007=0.407315 2008=0.329085 2000=0.193646 2013=0.037568 2001=0.017578 2012=0.014808 |
| 6 | 2007=0.271069 2009=0.204409 2001=0.165051 2012=0.156565 2013=0.121196 2000=0.08171 |
| 7 | 2009=0.335571 2007=0.321616 2013=0.117201 2000=0.087208 2001=0.075284 2012=0.06312 |
| 8 | 2009=0.46002 2011=0.148034 2000=0.127931 2001=0.104088 2012=0.077096 2003=0.049882 2013=0.032949 |
| 9 | 2013=0.24351 2003=0.236262 2004=0.161432 2000=0.148034 2001=0.127076 2012=0.049882 2011=0.033804 |
| 10 | 2004=0.652557 2001=0.100779 2003=0.093997 2011=0.085232 2013=0.067435 |
| 11 | 2013=0.376127 2003=0.19099 2004=0.186011 2011=0.161799 2001=0.050119 2014=0.03094 2010=0.004014 |
| 12 | 2010=0.397839 2014=0.278945 2011=0.169278 2003=0.118984 2001=0.03094 2013=0.004014 |
| 13 | 2010=0.316145 2011=0.277424 2014=0.265172 2003=0.141259 |
| 14 | 2014=0.424943 2010=0.282002 2003=0.168626 2011=0.124429 |
| 15 | 2016=1 |
| 16 | 2017=0.929383 2015=0.070617 |
| 17 | 2019=0.889288 2017=0.070617 2015=0.040095 |
| 18 | 2018=0.596125 2015=0.293163 2019=0.110712 |
| 19 | 2015=0.596125 2018=0.403875 |
| 排名 | 最优妥协解 |
|---|---|
| 劣汰情境 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2000 \prec 2007 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
| 优胜情境 | $2002 \prec 2005 \prec 2012 \prec 2008 \prec 2006 \prec 2000 \prec 2007 \prec 2001 \prec 2009 \prec 2013 \prec 2004 \prec 2003 \prec 2011 \prec 2010 \prec 2014 \prec 2016 \prec 2017 \prec 2019 \prec 2018 \prec 2015$ |
优胜情境的排序为聚类排序中的第20个
取中间值聚类中间值为：(0.51171424948088+0.54937159125727)/2 =0.53054292036907
$$\begin{array}{c|c|c|c|c|c|c}{M_{20 \times3}} &Dw1 &Dw2 &最优妥协解\\ \hline 2000 &0.2815 &0.6584 &0.4815\\ \hline 2001 &0.2819 &0.6625 &0.4838\\ \hline 2002 &0.1937 &0.3703 &0.2874\\ \hline 2003 &0.3166 &0.6815 &0.5102\\ \hline 2004 &0.3258 &0.6586 &0.5023\\ \hline 2005 &0.2809 &0.5564 &0.4271\\ \hline 2006 &0.2957 &0.6225 &0.4691\\ \hline 2007 &0.3103 &0.6353 &0.4827\\ \hline 2008 &0.3078 &0.6095 &0.4679\\ \hline 2009 &0.3139 &0.6399 &0.4869\\ \hline 2010 &0.3456 &0.6756 &0.5207\\ \hline 2011 &0.3672 &0.6435 &0.5138\\ \hline 2012 &0.3215 &0.5878 &0.4628\\ \hline 2013 &0.3458 &0.6339 &0.4986\\ \hline 2014 &0.3648 &0.6598 &0.5213\\ \hline 2015 &0.7113 &0.7837 &0.7497\\ \hline 2016 &0.4829 &0.7078 &0.6022\\ \hline 2017 &0.5068 &0.7993 &0.662\\ \hline 2018 &0.6086 &0.8533 &0.7384\\ \hline 2019 &0.5142 &0.8083 &0.6702\\ \hline \end{array} $$劣汰情境的排序为聚类排序中的第20个
取中间值聚类中间值为：(0.51171424948088+0.54937159125727)/2 =0.53054292036907
$$\begin{array}{c|c|c|c|c|c|c}{M_{20 \times3}} &Dw1 &Dw2 &最优妥协解\\ \hline 2000 &0.2815 &0.6584 &0.4815\\ \hline 2001 &0.2819 &0.6625 &0.4838\\ \hline 2002 &0.1937 &0.3703 &0.2874\\ \hline 2003 &0.3166 &0.6815 &0.5102\\ \hline 2004 &0.3258 &0.6586 &0.5023\\ \hline 2005 &0.2809 &0.5564 &0.4271\\ \hline 2006 &0.2957 &0.6225 &0.4691\\ \hline 2007 &0.3103 &0.6353 &0.4827\\ \hline 2008 &0.3078 &0.6095 &0.4679\\ \hline 2009 &0.3139 &0.6399 &0.4869\\ \hline 2010 &0.3456 &0.6756 &0.5207\\ \hline 2011 &0.3672 &0.6435 &0.5138\\ \hline 2012 &0.3215 &0.5878 &0.4628\\ \hline 2013 &0.3458 &0.6339 &0.4986\\ \hline 2014 &0.3648 &0.6598 &0.5213\\ \hline 2015 &0.7113 &0.7837 &0.7497\\ \hline 2016 &0.4829 &0.7078 &0.6022\\ \hline 2017 &0.5068 &0.7993 &0.662\\ \hline 2018 &0.6086 &0.8533 &0.7384\\ \hline 2019 &0.5142 &0.8083 &0.6702\\ \hline \end{array} $$