《1 引言》

1 引言

《2 区间数的距离及其性质》

2 区间数的距离及其性质

$\stackrel{˜}{a}=\left[a{}^{\text{L}},\phantom{\rule{0.25em}{0ex}}a{}^{\text{U}}\right]=\left\{x|a{}^{\text{L}}\le x\le a{}^{\text{U}}\right\}$$\stackrel{˜}{a}$为一个区间数。特别是, 若aL=aU, 则$\stackrel{˜}{a}$退化为一个实数。

$\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}L{}_{p}\left(A,\phantom{\rule{0.25em}{0ex}}B\right)=2{}^{-1/p}\left[\left(a{}_{1}-b{}_{1}\right){}^{p}+\left(a{}_{2}-b{}_{2}\right){}^{p}\right]{}^{1/p}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(1\right)$

p=1时, 记L1 (A, B) =[|a1-b1|+|a2-b2|]/2, 称L1 (A, B) 为区间数AB的海明距离。

p=2时, 记L2 (A, B) =2-1/2[ (a1-b1) 2+ (a2-b2) 2]1/2, 称L2 (A, B) 为区间数AB的欧氏距离。

B点为原点O时, L (A, O) =2-1/p[ (a1) p+ (a2) p]1/p, 称L (A, O) 为区间数A到原点O的距离。

A, B都为实数时, 即a1=a2, b1=b2, 则L2 (A, B) =d (A, B) , d (A, B) 表示实数AB之间的距离。故区间数距离是实数距离的推广。

1) Lp (A, B) ≥0⇔A=B时等式成立;

2) Lp (A, B) =Lp (B, A) ;

3) Lp (A, B) ≤Lp (A, C) +Lp (B, C) ;

4) Lp (kA, kB) =kLp (A, B) 。

$\begin{array}{l}\left[\left(a{}_{1}-b{}_{1}\right){}^{p}+\left(a{}_{2}-b{}_{2}\right){}^{p}\right]{}^{1/p}\le \left[\left(a{}_{1}-c{}_{1}\right){}^{p}\\ +\left(a{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p}+\left[\left(b{}_{1}-c{}_{1}\right){}^{p}+\left(b{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p}\end{array}$

$\begin{array}{l}L{}_{p}\left(A,\phantom{\rule{0.25em}{0ex}}B\right)=2{}^{-1/p}\left[\left(a{}_{1}-b{}_{1}\right){}^{p}+\left(a{}_{2}-b{}_{2}\right){}^{p}\right]{}^{1/p},\\ L{}_{p}\left(A,\phantom{\rule{0.25em}{0ex}}C\right)+L{}_{p}\left(B,\phantom{\rule{0.25em}{0ex}}C\right)=2{}^{-1/p}\left[\left(a{}_{1}-c{}_{1}\right){}^{p}+\\ \left(a{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p}+2{}^{-1/p}\left[\left(b{}_{1}-c{}_{1}\right){}^{p}+\\ \left(b{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p},\end{array}$

$\begin{array}{l}2{}^{-1/p}\left[\left(a{}_{1}-b{}_{1}\right){}^{p}+\left(a{}_{2}-b{}_{2}\right){}^{p}\right]{}^{1/p}\le \\ 2{}^{-1/p}\left[\left(a{}_{1}-c{}_{1}\right){}^{p}+\left(a{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p}+\\ 2{}^{-p/2}\left[\left(b{}_{1}-c{}_{1}\right){}^{p}+\left(b{}_{2}-c{}_{2}\right){}^{p}\right]{}^{1/p}\end{array}$

k2-1/p[ (a1-b1) p+ (a2-b2) p]1/p=kLp (A, B)

Lp (kA, kB) =kLp (A, B) 。

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}L{}_{p}\left(A,\phantom{\rule{0.25em}{0ex}}B\right)=2{}^{-1/p}\left[\left(a{}_{11}-b{}_{11}\right){}^{p}+\\ \left(a{}_{12}-b{}_{12}\right){}^{p}+\left(a{}_{21}-b{}_{21}\right){}^{p}+\left(a{}_{22}-\\ b{}_{22}\right){}^{p}+\cdots +\left(a{}_{m1}-b{}_{m1}\right){}^{p}+\left(a{}_{m2}-b{}_{m2}\right){}^{p}\right]{}^{1/p}\end{array}$

m维区间数AB之间的距离。

p=1时, 记

$\begin{array}{l}L{}_{1}\left(A,\phantom{\rule{0.25em}{0ex}}B\right)=\left[|a{}_{11}-b{}_{11}|+|a{}_{12}-b{}_{12}|+\\ |a{}_{21}-b{}_{21}|+|a{}_{22}-b{}_{22}|+\cdots +\\ |\phantom{\rule{0.25em}{0ex}}a{}_{m1}-b{}_{m1}|+|a{}_{m2}-b{}_{m2}|\right]/2,\end{array}$

L1 (A, B) 为海明距离。

p=2时, 记

$\begin{array}{l}L{}_{2}\left(A,\phantom{\rule{0.25em}{0ex}}B\right)=2{}^{-1/2}\left[\left(a{}_{11}-b{}_{11}\right){}^{2}+\left(a{}_{12}-b{}_{12}\right){}^{2}+\\ \left(a{}_{21}-b{}_{21}\right){}^{2}+\left(a{}_{22}-b{}_{22}\right){}^{2}+\cdots +\left(a{}_{m1}-b{}_{m1}\right){}^{2}+\\ \left(a{}_{m2}-b{}_{m2}\right){}^{2}\right]{}^{1/2},\end{array}$

L2 (A, B) 为欧氏距离。

A, B都为实数时, 即ai1=ai2, bi1=bi2, (i=1, 2, …, m) , L2 (A, B) =d (A, B) , d (A, B) 表示实数AB之间的欧氏距离。故m维区间数的距离是m维实数距离的推广。

1) Lp (A, B) ≥0⇔A=B时等式成立;

2) Lp (A, B) =Lp (B, A) ;

3) Lp (A, B) ≤Lp (A, C) +Lp (B, C) ;

4) Lp (kA, kB) =kLp (A, B) 。

《3 灰靶决策模型的建立》

3 灰靶决策模型的建立

$\mathbit{X}=\left[\begin{array}{cccc}\left[x{}_{11}^{\text{L}}‚x{}_{11}^{\text{U}}\right]& \left[x{}_{12}^{\text{L}},x{}_{12}^{\text{U}}\right]& \cdots & \left[x{}_{1m}^{\text{L}}‚x{}_{1m}^{\text{U}}\right]\\ \left[x{}_{21}^{\text{L}}‚x{}_{21}^{\text{U}}\right]& \left[x{}_{22}^{\text{L}},x{}_{22}^{\text{U}}\right]& \cdots & \left[x{}_{2m}^{\text{L}}‚x{}_{2m}^{\text{U}}\right]\\ ⋮& ⋮& & ⋮\\ \left[x{}_{n1}^{\text{L}}‚x{}_{n1}^{\text{U}}\right]& \left[x{}_{n2}^{\text{L}},x{}_{n2}^{\text{U}}\right]& \cdots & \left[x{}_{nm}^{\text{L}}‚x{}_{nm}^{\text{U}}\right]\end{array}\right]\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(2\right)$

《3.1决策矩阵规范化处理方法》

3.1决策矩阵规范化处理方法

$r{}_{ij}^{\text{L}}=x{}_{ij}^{\text{L}}/\sum _{i=1}^{n}x{}_{ij}^{\text{U}}‚r{}_{ij}^{\text{U}}=x{}_{ij}^{\text{U}}/\sum _{i=1}^{n}x{}_{ij}^{\text{L}}\text{ }\text{ }\text{ }\left(3\right)$

$r{}_{ij}^{\text{L}}=\frac{1}{x{}_{ij}^{\text{U}}}/\sum _{i=1}^{n}\frac{1}{x{}_{ij}^{\text{L}}}‚r{}_{ij}^{\text{U}}=\frac{1}{x{}_{ij}^{\text{L}}}/\sum _{i=1}^{n}\frac{1}{x{}_{ij}^{\text{U}}}\text{ }\text{ }\text{ }\left(4\right)$

$\begin{array}{l}\mathbit{R}=\left[\begin{array}{cccc}\left[r{}_{11}^{\text{L}}‚r{}_{11}^{\text{U}}\right]& \left[r{}_{12}^{\text{L}},r{}_{12}^{\text{U}}\right]& \cdots & \left[r{}_{1m}^{\text{L}}‚r{}_{1m}^{\text{U}}\right]\\ \left[r{}_{21}^{\text{L}}‚x{}_{21}^{\text{U}}\right]& \left[r{}_{22}^{\text{L}},x{}_{22}^{\text{U}}\right]& \cdots & \left[r{}_{2m}^{\text{L}}‚r{}_{2m}^{\text{U}}\right]\\ ⋮& ⋮& & ⋮\\ \left[r{}_{n1}^{\text{L}}‚r{}_{n1}^{\text{U}}\right]& \left[r{}_{n2}^{\text{L}},r{}_{n2}^{\text{U}}\right]& \cdots & \left[r{}_{nm}^{\text{L}}‚r{}_{nm}^{\text{U}}\right]\end{array}\right]=\\ \left(\mathbit{r}{}_{1}‚\mathbit{r}{}_{2}‚\cdots ‚\mathbit{r}{}_{n}\right){}^{\text{Τ}}\phantom{\rule{0.25em}{0ex}}\end{array}$

《3.2多指标灰靶决策模型》

3.2多指标灰靶决策模型

$\begin{array}{l}\mathbit{r}{}_{0}=\left\{\mathbit{r}{}_{1}^{0},\phantom{\rule{0.25em}{0ex}}\mathbit{r}{}_{2}^{0},\phantom{\rule{0.25em}{0ex}}\cdots ,\phantom{\rule{0.25em}{0ex}}\mathbit{r}{}_{m}^{0}\right\}=\\ \left\{\left[\mathbit{r}{}_{i{}_{0}1}^{\text{L}}‚\mathbit{r}{}_{i{}_{0}1}^{\text{U}}\right]‚\left[\mathbit{r}{}_{i{}_{0}2}^{\text{L}}‚\mathbit{r}{}_{i{}_{0}2}^{\text{U}}\right]‚\cdots ‚\left[\mathbit{r}{}_{i{}_{0}m}^{\text{L}}‚\mathbit{r}{}_{i{}_{0}m}^{\text{U}}\right]\right\}\text{ }\text{ }\text{ }\left(5\right)\end{array}$

$\begin{array}{l}\mathbit{R}{}^{\left(m\right)}=\left\{\left(\left[\mathbit{r}{}_{i1}^{\text{L}},\phantom{\rule{0.25em}{0ex}}\mathbit{r}{}_{i1}^{U}\right],\phantom{\rule{0.25em}{0ex}}\left[\mathbit{r}{}_{i2}^{\text{L}},\phantom{\rule{0.25em}{0ex}}\mathbit{r}{}_{i2}^{\text{U}}\right],\cdots ,\phantom{\rule{0.25em}{0ex}}\\ \left[\mathbit{r}{}_{im}^{\text{L}},\phantom{\rule{0.25em}{0ex}}\mathbit{r}{}_{im}^{\text{U}}\right]\right)\phantom{\rule{0.25em}{0ex}}|2{}^{-1/2}\left[\left(\mathbit{r}{}_{i1}^{\text{L}}-\mathbit{r}{}_{i{}_{0}1}^{\text{L}}\right){}^{2}+\left(\mathbit{r}{}_{i1}^{\text{U}}-\mathbit{r}{}_{i{}_{0}1}^{\text{U}}\right){}^{2}+\\ \cdots +\left(\mathbit{r}{}_{im}^{\text{L}}-\mathbit{r}{}_{i{}_{0}m}^{\text{L}}\right){}^{2}+\left(\mathbit{r}{}_{im}^{\text{U}}-\mathbit{r}{}_{i{}_{0}m}^{\text{U}}\right){}^{2}\right]{}^{1/2}=R{}^{2}\end{array}$

$\begin{array}{l}\epsilon {}_{i}=|\mathbit{r}{}_{i}-\mathbit{r}{}_{0}|=2{}^{-1/2}\left[\left(\mathbit{r}{}_{i1}^{\text{L}}-\mathbit{r}{}_{i{}_{0}1}^{\text{L}}\right){}^{2}+\\ \left(\mathbit{r}{}_{i1}^{\text{U}}-\mathbit{r}{}_{i{}_{0}1}^{\text{U}}\right){}^{2}+\cdots +\left(\mathbit{r}{}_{im}^{\text{L}}-\mathbit{r}{}_{i{}_{0}m}^{\text{L}}\right){}^{2}+\\ \left(\mathbit{r}{}_{im}^{\text{U}}-r{}_{i{}_{0}m}^{\text{U}}\right){}^{2}\right]{}^{1/2}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(6\right)\end{array}$

《3.3区间数多指标灰靶决策算法》

3.3区间数多指标灰靶决策算法

1) 根据区间数多指标决策问题构造效果样本决策矩阵X= ([xLij, xUij]) m×n, 利用区间数规范化方法把效果样本决策矩阵X化为规范化决策矩阵R= ([rLij, rUij]) m×n;

2) 由规范化决策矩阵R, 根据式 (5) 求出灰靶靶心r0;

3) 利用式 (6) 求出效果向量ri的靶心距εi (i=1, 2, …, n) , ri按从小到大的顺序排列, 即可得到各方案的最优排序;

4) 结束。

《4 应用实例》

4 应用实例

A1 投资净产值率 (净产值与投资额之比) ;A2 投资利税率 (净利税与投资额之比) ;A3 内部收益率;A4 环境污染程度 (环保部门历时检测并模糊量化) 。

《表1》

 Si A1 A2 A3 A4 S1 [1.8, 2.2] [1.2, 1.8] [1.8, 2.2] [5.4, 5.6] S2 [2.3, 2.7] [2.4, 3.0] [1.6, 2.0] [6.4, 6.6] S3 [1.6, 2.0] [1.7, 2.3] [1.9, 2.3] [4.4, 4.6] S4 [2.0, 2.4] [1.5, 2.1] [1.8, 2.2] [4.9, 5.1]

S3, S4的上述指标, 具体数据如表1所示;其中指标A1, A2, A3为乐观准则指标, A4为悲观准则指标。

$\begin{array}{l}\mathbit{r}{}_{0}=\left\{\left[0.2470,\phantom{\rule{0.25em}{0ex}}0.3510\right],\phantom{\rule{0.25em}{0ex}}\left[0.2609,\phantom{\rule{0.25em}{0ex}}0.4412\right],\\ \phantom{\rule{0.25em}{0ex}}\left[0.2814,\phantom{\rule{0.25em}{0ex}}0.3239\right],\phantom{\rule{0.25em}{0ex}}\left[0.2813,\phantom{\rule{0.25em}{0ex}}0.3507\right]\right\}。\end{array}$

《表2》

 Si A1 A2 A3 A4 S1 [0.1940, 0.2857] [0.1304, 0.2647] [0.2069, 0.3098] [0.2311, 0.2491] S2 [0.2470, 0.3510] [0.2609, 0.4412] [0.1839, 0.2817] [0.1960, 0.2102] S3 [0.1720, 0.2597] [0.1848, 0.3382] [0.2814, 0.3239] [0.2813, 0.3057] S4 [0.2151, 0.3120] [0.1630, 0.3088] [0.2059, 0.3098] [0.2537, 0.2747]

$\begin{array}{l}\epsilon {}_{1}=0.2583,\phantom{\rule{0.25em}{0ex}}\epsilon {}_{2}=0.1664,\phantom{\rule{0.25em}{0ex}}\\ \epsilon {}_{3}=0.1742,\phantom{\rule{0.25em}{0ex}}\epsilon {}_{4}=0.1907。\end{array}$

《5 结语》

5 结语