耦合协调度的计算
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流程图
原始系统如下:
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times19}} & -P1 & +P2 & -P3 & -P4 & +B1 & -B2 & -B3 & -B4 & -B5 & -B6 & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
兰州 &163.54 &3189 &21.95 &96.92 &9744 &1112.7 &5977.36 &14861.55 &36.61 &226.19 &47 &76 &68.8 &13.19 &33.14 &85.2 &96.38 &99.9 &6732\\
\hline
白银 &14.54 &317 &21.47 &40.2 &12172.25 &0 &1134 &10144 &7.86 &150 &25 &64 &65.5 &11.56 &36.34 &94 &94.35 &100 &4959\\
\hline
定西 &4.69 &532 &14.17 &53.1 &12808 &0 &2152 &4874 &11.3017 &96.0875 &24 &53 &67.2 &17.69 &30.1 &98.1 &95.34 &100 &2248\\
\hline
临夏 &11.2 &261 &16.07 &49.4 &7859.24 &0 &1687 &7305 &9.28 &120.53 &24 &55 &66.8 &16.51 &33 &88.8 &95.1 &100 &2263\\
\hline
西宁 &74.6 &1920 &12.5 &73.03 &5715 &586.2 &2680.8 &5998.5 &21.15 &62.46 &48 &70 &68.3 &12.82 &40.02 &92.1 &94.87 &98.73 &7008\\
\hline
海东 &0.98 &170 &8.1 &23.69 &12024 &0 &1013 &3520 &4.23 &48.13 &36 &62 &68.7 &5.75 &35.86 &95 &95.81 &96.2 &18581\\
\hline
\end{array} $$
采用的归一方法如下
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:$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times19}} & -P1 & +P2 & -P3 & -P4 & +B1 & -B2 & -B3 & -B4 & -B5 & -B6 & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
兰州 &0 &1 &0 &0 &0.568 &0 &0 &0 &0 &0 &0.042 &0 &0 &0.623 &0.306 &0 &1 &0.974 &0.725\\
\hline
白银 &0.917 &0.049 &0.035 &0.775 &0.91 &1 &0.976 &0.416 &0.888 &0.428 &0.958 &0.522 &1 &0.487 &0.629 &0.682 &0 &1 &0.834\\
\hline
定西 &0.977 &0.12 &0.562 &0.598 &1 &1 &0.771 &0.881 &0.782 &0.731 &1 &1 &0.485 &1 &0 &1 &0.488 &1 &1\\
\hline
临夏 &0.937 &0.03 &0.425 &0.649 &0.302 &1 &0.864 &0.666 &0.844 &0.593 &1 &0.913 &0.606 &0.901 &0.292 &0.279 &0.369 &1 &0.999\\
\hline
西宁 &0.547 &0.58 &0.682 &0.326 &0 &0.473 &0.664 &0.781 &0.477 &0.92 &0 &0.261 &0.152 &0.592 &1 &0.535 &0.256 &0.666 &0.709\\
\hline
海东 &1 &0 &1 &1 &0.889 &1 &1 &1 &1 &1 &0.5 &0.609 &0.03 &0 &0.581 &0.76 &0.719 &0 &0\\
\hline
\end{array} $$正极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times19}} & -P1 & +P2 & -P3 & -P4 & +B1 & -B2 & -B3 & -B4 & -B5 & -B6 & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
\mathbf{Zone^+} &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\
\hline
\end{array} $$负极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times19}} & -P1 & +P2 & -P3 & -P4 & +B1 & -B2 & -B3 & -B4 & -B5 & -B6 & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
\mathbf{Zone^-} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
\hline
\end{array} $$采用的是熵权法(EWM)求权重
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times19}} & -P1 & +P2 & -P3 & -P4 & +B1 & -B2 & -B3 & -B4 & -B5 & -B6 & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
EWM所得权重 &0.0367 &0.1364 &0.073 &0.0434 &0.046 &0.0388 &0.035 &0.0399 &0.0378 &0.0406 &0.0707 &0.0491 &0.0924 &0.0396 &0.0518 &0.0462 &0.0526 &0.035 &0.035\\
\hline
权重大小顺序 &16 &1 &3 &10 &9 &14 &17 &12 &15 &11 &4 &7 &2 &13 &6 &8 &5 &19 &18\\
\hline
\end{array} $$
子系统综合值U的计算
总共有3个子系统
子系统1原始数据如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times4}} & -P1 & +P2 & -P3 & -P4\\
\hline
兰州 &163.54 &3189 &21.95 &96.92\\
\hline
白银 &14.54 &317 &21.47 &40.2\\
\hline
定西 &4.69 &532 &14.17 &53.1\\
\hline
临夏 &11.2 &261 &16.07 &49.4\\
\hline
西宁 &74.6 &1920 &12.5 &73.03\\
\hline
海东 &0.98 &170 &8.1 &23.69\\
\hline
\end{array} $$子系统2原始数据如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times6}} & +B1 & -B2 & -B3 & -B4 & -B5 & -B6\\
\hline
兰州 &9744 &1112.7 &5977.36 &14861.55 &36.61 &226.19\\
\hline
白银 &12172.25 &0 &1134 &10144 &7.86 &150\\
\hline
定西 &12808 &0 &2152 &4874 &11.3017 &96.0875\\
\hline
临夏 &7859.24 &0 &1687 &7305 &9.28 &120.53\\
\hline
西宁 &5715 &586.2 &2680.8 &5998.5 &21.15 &62.46\\
\hline
海东 &12024 &0 &1013 &3520 &4.23 &48.13\\
\hline
\end{array} $$子系统3原始数据如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times9}} & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
兰州 &47 &76 &68.8 &13.19 &33.14 &85.2 &96.38 &99.9 &6732\\
\hline
白银 &25 &64 &65.5 &11.56 &36.34 &94 &94.35 &100 &4959\\
\hline
定西 &24 &53 &67.2 &17.69 &30.1 &98.1 &95.34 &100 &2248\\
\hline
临夏 &24 &55 &66.8 &16.51 &33 &88.8 &95.1 &100 &2263\\
\hline
西宁 &48 &70 &68.3 &12.82 &40.02 &92.1 &94.87 &98.73 &7008\\
\hline
海东 &36 &62 &68.7 &5.75 &35.86 &95 &95.81 &96.2 &18581\\
\hline
\end{array} $$子系统1归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times4}} & -P1 & +P2 & -P3 & -P4\\
\hline
兰州 &0 &1 &0 &0\\
\hline
白银 &0.91658 &0.04869 &0.03466 &0.77455\\
\hline
定西 &0.97718 &0.11991 &0.56173 &0.59839\\
\hline
临夏 &0.93713 &0.03014 &0.42455 &0.64891\\
\hline
西宁 &0.54712 &0.57966 &0.68231 &0.32623\\
\hline
海东 &1 &0 &1 &1\\
\hline
\end{array} $$子系统2归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times6}} & +B1 & -B2 & -B3 & -B4 & -B5 & -B6\\
\hline
兰州 &0.56802 &0 &0 &0 &0 &0\\
\hline
白银 &0.91037 &1 &0.97563 &0.41595 &0.88789 &0.42789\\
\hline
定西 &1 &1 &0.77056 &0.88062 &0.7816 &0.73067\\
\hline
临夏 &0.3023 &1 &0.86423 &0.66627 &0.84404 &0.5934\\
\hline
西宁 &0 &0.47317 &0.66405 &0.78147 &0.47746 &0.91952\\
\hline
海东 &0.88947 &1 &1 &1 &1 &1\\
\hline
\end{array} $$子系统3归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{6 \times9}} & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
兰州 &0.04167 &0 &0 &0.62312 &0.30645 &0 &1 &0.97368 &0.72546\\
\hline
白银 &0.95833 &0.52174 &1 &0.4866 &0.62903 &0.68217 &0 &1 &0.83402\\
\hline
定西 &1 &1 &0.48485 &1 &0 &1 &0.48768 &1 &1\\
\hline
临夏 &1 &0.91304 &0.60606 &0.90117 &0.29234 &0.27907 &0.36946 &1 &0.99908\\
\hline
西宁 &0 &0.26087 &0.15152 &0.59213 &1 &0.53488 &0.25616 &0.66579 &0.70857\\
\hline
海东 &0.5 &0.6087 &0.0303 &0 &0.58065 &0.75969 &0.71921 &0 &0\\
\hline
\end{array} $$
子系统的权重分别计算出
子系统1的权重
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times4}} & -P1 & +P2 & -P3 & -P4\\
\hline
权重 &0.126683 &0.47117277 &0.25230284 &0.14984139\\
\hline
权重大小顺序 &4 &1 &2 &3\\
\hline
\end{array} $$子系统2的权重
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times6}} & +B1 & -B2 & -B3 & -B4 & -B5 & -B6\\
\hline
权重 &0.19294475 &0.16309299 &0.14702994 &0.16761788 &0.15868413 &0.17063031\\
\hline
权重大小顺序 &1 &4 &6 &3 &5 &2\\
\hline
\end{array} $$子系统3的权重
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times9}} & -Q1 & -Q2 & -Q3 & +Q4 & +Q5 & +Q6 & +Q7 & +Q8 & -Q9\\
\hline
权重 &0.14971543 &0.10395119 &0.19568187 &0.08387157 &0.10958907 &0.09771394 &0.11127397 &0.07406873 &0.07413423\\
\hline
权重大小顺序 &2 &5 &1 &7 &4 &6 &3 &9 &8\\
\hline
\end{array} $$
子系统综合评价值计算公式
$$期望综合值 U_i = \sum_\limits{j=1}^m{ \omega_{j} \left({ n_{ij}-Min(n_j)} \right)} \quad \quad $$
子系统1的综合值
$$\begin{array}{c|c|c|c|c|c|c}{M_{6 \times1}} &期望综合值U\\
\hline 兰州 &0.471\\
\hline 白银 &0.264\\
\hline 定西 &0.412\\
\hline 临夏 &0.337\\
\hline 西宁 &0.563\\
\hline 海东 &0.529\\
\hline \end{array} $$子系统2的综合值
$$\begin{array}{c|c|c|c|c|c|c}{M_{6 \times1}} &期望综合值U\\
\hline 兰州 &0.11\\
\hline 白银 &0.766\\
\hline 定西 &0.866\\
\hline 临夏 &0.695\\
\hline 西宁 &0.538\\
\hline 海东 &0.979\\
\hline \end{array} $$子系统3的综合值
$$\begin{array}{c|c|c|c|c|c|c}{M_{6 \times1}} &期望综合值U\\
\hline 兰州 &0.329\\
\hline 白银 &0.706\\
\hline 定西 &0.733\\
\hline 临夏 &0.687\\
\hline 西宁 &0.399\\
\hline 海东 &0.362\\
\hline \end{array} $$
两组协调度的计算
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 $$ D=\sqrt{CT} $$
$$ T=\sum \limits_{i=1}^{n}{\omega _i U_i} $$
综合值
$$综合值=\begin{array}{c|c|c|c|c|c|c}{M_{6 \times3}} &子系统1 &子系统2 &子系统3\\
\hline 兰州 &0.471173 &0.109597 &0.329256\\
\hline 白银 &0.263854 &0.765815 &0.705695\\
\hline 定西 &0.411673 &0.865641 &0.732596\\
\hline 临夏 &0.337263 &0.695352 &0.687354\\
\hline 西宁 &0.56346 &0.538454 &0.398626\\
\hline 海东 &0.528827 &0.978673 &0.361952\\
\hline \end{array} $$分配系数采用总系统中的权重分配为0.28948771070346:0.23818238354716:0.47232990574938
CTD值
$$CTD=\begin{array}{c|c|c|c|c|c|c}{M_{6 \times3}} &C &T &D\\
\hline 兰州 &0.847692 &0.31802 &0.519214\\
\hline 白银 &0.903162 &0.592107 &0.731278\\
\hline 定西 &0.953959 &0.671382 &0.800294\\
\hline 临夏 &0.94926 &0.587912 &0.747049\\
\hline 西宁 &0.988704 &0.479648 &0.688643\\
\hline 海东 &0.918208 &0.557153 &0.71525\\
\hline \end{array} $$
扯蛋模型