《1 引言》

1 引言

$\begin{array}{l}\begin{array}{l}x\left(g\right)=\left[a{}_{22}\left(g\right)-a{}_{21}\left(g\right)\right]/\left[\left(a{}_{11}\left(g\right)+a{}_{22}\left(g\right)\right)-\\ \text{ }\left(a{}_{12}\left(g\right)+a{}_{21}\left(g\right)\right)\right]=\left[1/7,\phantom{\rule{0.25em}{0ex}}3/5\right]\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(1\right)\end{array}\\ \mathrm{max}\phantom{\rule{0.25em}{0ex}}\left\{x\left(g\right)\right\}=\frac{3}{7}|{}_{a{}_{21}=3,a{}_{22}=0},\\ \mathrm{min}\phantom{\rule{0.25em}{0ex}}\left\{x\left(g\right)\right\}=\frac{1}{5}|{}_{a{}_{21}=2,a{}_{22}=1}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(2\right)\end{array}$

《2 标准区间灰数及其运算》

2 标准区间灰数及其运算

$\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}G{}_{i}=g{}_{i}+c{}_{i}\gamma {}_{i}\phantom{\rule{0.25em}{0ex}},\phantom{\rule{0.25em}{0ex}}i=1,\phantom{\rule{0.25em}{0ex}}2,\phantom{\rule{0.25em}{0ex}}\cdots \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(3\right)$

ciγi称为Gi的灰部, 其中ci称为灰系数, γi称为单位灰数 (或灰数单位) , 则称式 (3) 所表示的灰数形式为标准区间灰数[6,7,8,9,10] , 或简称为标准灰数。

$\begin{array}{l}G{}_{i}\phantom{\rule{0.25em}{0ex}}\in \left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]=\left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]+\left[a{}_{i},\phantom{\rule{0.25em}{0ex}}a{}_{i}\right]\phantom{\rule{0.25em}{0ex}}-\phantom{\rule{0.25em}{0ex}}\left[a{}_{i},\phantom{\rule{0.25em}{0ex}}a{}_{i}\right]=\\ \left[a{}_{i},\phantom{\rule{0.25em}{0ex}}a{}_{i}\right]\phantom{\rule{0.25em}{0ex}}+\left[0,\phantom{\rule{0.25em}{0ex}}b{}_{i}-a{}_{i}\right]=a{}_{i}+\left(b{}_{i}-a{}_{i}\right)\cdot \left[0,1\right]‚\end{array}$

ci= (bi-ai) , γi∈[0, 1];则由上式可得

$G{}_{i}=a{}_{i}+c{}_{i}\gamma {}_{i}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(4\right)$

$\begin{array}{l}\mathrm{min}\phantom{\rule{0.25em}{0ex}}\left\{F\left(g\right)\right\}=\mathrm{min}\phantom{\rule{0.25em}{0ex}}g\left(G{}_{1},\phantom{\rule{0.25em}{0ex}}G{}_{2},\phantom{\rule{0.25em}{0ex}}\cdots ,\phantom{\rule{0.25em}{0ex}}G{}_{n}\right)\\ |{}_{\gamma {}_{i}=c{}_{i},c{}_{i}\in \left[0,1\right],i=1,2,\cdots n}‚\\ \mathrm{max}\phantom{\rule{0.25em}{0ex}}\left\{F\left(g\right)\right\}=\mathrm{max}\phantom{\rule{0.25em}{0ex}}g\left(G{}_{1},\phantom{\rule{0.25em}{0ex}}G{}_{2},\phantom{\rule{0.25em}{0ex}}\cdots ,\phantom{\rule{0.25em}{0ex}}G{}_{n}\right)\\ |{}_{\gamma {}_{i}=c{}_{i},c{}_{i}\in \left[0,1\right],i=1,2,\cdots n}‚\end{array}$

《3 第一和第二标准灰数》

3 第一和第二标准灰数

$\begin{array}{l}G{}_{i}^{\left(1\right)}\in \left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]=a{}_{i}-a{}_{i}+\left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]=\\ a{}_{i}+\left[0,\phantom{\rule{0.25em}{0ex}}b{}_{i}-a{}_{i}\right]=a{}_{i}+\left(b{}_{i}-a{}_{i}\right)\left[0,\phantom{\rule{0.25em}{0ex}}1\right]=\\ a{}_{i}+\left(b{}_{i}-a{}_{i}\right)\gamma {}_{i}^{\left(1\right)},\phantom{\rule{0.25em}{0ex}}\left(0\le \gamma {}_{i}^{\left(1\right)}\le 1\right)\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(5\right)\\ G{}_{i}^{\left(2\right)}\in \left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]=b{}_{i}-b{}_{i}+\left[a{}_{i},\phantom{\rule{0.25em}{0ex}}b{}_{i}\right]=\\ b{}_{i}-\left[0,\phantom{\rule{0.25em}{0ex}}b{}_{i}-a{}_{i}\right]=b{}_{i}-\left(b{}_{i}-a{}_{i}\right)\left[0,\phantom{\rule{0.25em}{0ex}}1\right]=\\ b{}_{i}-\left(b{}_{i}-a{}_{i}\right)\gamma {}_{i}^{\left(2\right)},\phantom{\rule{0.25em}{0ex}}\left(0\le \gamma {}_{i}^{\left(2\right)}\le 1\right)\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(6\right)\phantom{\rule{0.25em}{0ex}}\end{array}$

$\begin{array}{l}G{}_{i}^{\left(1\right)}=G{}_{i}^{\left(2\right)},\phantom{\rule{0.25em}{0ex}}a{}_{i}+\left(b{}_{i}-ai\right)\gamma {}_{i}^{\left(1\right)}=\\ b{}_{i}-\left(b{}_{i}-a{}_{i}\right)\gamma {}_{i}^{\left(2\right)},\phantom{\rule{0.25em}{0ex}}\gamma {}_{i}^{\left(1\right)}+\gamma {}_{i}^{\left(2\right)}=1\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(7\right)\end{array}$

《图1》

Fig.1 Relationship between γ (1) i and γ (2) i

《4 标准灰数的大小判定》

4 标准灰数的大小判定

G1-G2≥0, 则G1G2;若G1-G2<0, 则G1<G2

1) 当b<c时, 称[a, b]<[c, d], 即灰数[a, b] (ab) 小于灰数 [c, d] (cd) ;

2) 当bc时, 称 [a, b]≤[c, d], 即灰数 [a, b] (ab) 小于等于灰数 [c, d] (cd) ; 且仅当b=c 时, 等号成立;

3) 当a=c, b=d, 且两灰数的取数一致 (γ1=γ2) 时[1], 即满足

$\begin{array}{l}\begin{array}{l}G{}_{1}\in \left[a{}_{1},\phantom{\rule{0.25em}{0ex}}b{}_{1}\right]=\gamma {}_{1}a{}_{1}+\left(1-\gamma {}_{1}\right)\phantom{\rule{0.25em}{0ex}}b{}_{1},\\ \phantom{\rule{0.25em}{0ex}}\gamma {}_{1}=\phantom{\rule{0.25em}{0ex}}\left[0,\phantom{\rule{0.25em}{0ex}}1\right],\phantom{\rule{0.25em}{0ex}}G{}_{2}\in \left[a{}_{2},\phantom{\rule{0.25em}{0ex}}b{}_{2}\right]=a{}_{2}+b{}_{2}\gamma {}_{2},\phantom{\rule{0.25em}{0ex}}\\ \gamma {}_{2}=\left[0,\phantom{\rule{0.25em}{0ex}}1\right],\phantom{\rule{0.25em}{0ex}}\text{且}\phantom{\rule{0.25em}{0ex}}\gamma {}_{1}=\phantom{\rule{0.25em}{0ex}}\gamma {}_{2}‚\end{array}\\ \text{则}\text{称}\phantom{\rule{0.25em}{0ex}}\left[a,\phantom{\rule{0.25em}{0ex}}b\right]=\left[c,\phantom{\rule{0.25em}{0ex}}d\right],\phantom{\rule{0.25em}{0ex}}\text{即}\text{灰}\text{数}\phantom{\rule{0.25em}{0ex}}\left[a,\phantom{\rule{0.25em}{0ex}}b\right]\phantom{\rule{0.25em}{0ex}}\left(a\le b\right)\end{array}$

4) 当a= c, b= d, 且两灰数的取数不一致 (γ1γ2) 时, 若 (γ1 < γ2) , 则G1 > G2;若γ1 > γ2, 则G1 <G2

$\begin{array}{l}G{}_{\text{Κ}}=f\left(g{}_{ij}\right)\gamma {}_{ij}=\left[\underset{\gamma {}_{ij}}{\mathrm{min}}f\left(\cdot \right),\mathrm{max}{}_{\gamma {}_{ij}}f\left(\cdot \right)\right]\phantom{\rule{0.25em}{0ex}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\\ \left(0\le \gamma {}_{ij}\le 1,\phantom{\rule{0.25em}{0ex}}i,\phantom{\rule{0.25em}{0ex}}j=1,\phantom{\rule{0.25em}{0ex}}2,\phantom{\rule{0.25em}{0ex}}\cdots \phantom{\rule{0.25em}{0ex}}\right)。\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(8\right)\end{array}$

$\begin{array}{l}G{}_{12}^{\left(1\right)}=G{}_{1}+G{}_{2}=\\ a+\left(b-a\right)\gamma {}_{1}+c+\phantom{\rule{0.25em}{0ex}}\left(d-c\right)\gamma {}_{2}=\phantom{\rule{0.25em}{0ex}}\\ \left(a+c\right)+\left(\left(b-a\right)\gamma {}_{1}+\left(d-c\right)\gamma {}_{2}\right)\text{ }\text{ }\text{ }\text{ }\left(9\right)\\ G{}_{12}^{\left(2\right)}=G{}_{1}-G{}_{2}=\\ a+\left(b-a\right)\gamma {}_{1}-c-\left(d-c\right)\gamma {}_{2}=\\ \left(a-c\right)+\left(\left(b-a\right)\gamma {}_{1}-\left(d-c\right)\gamma {}_{2}\right)\text{ }\text{ }\text{ }\text{ }\left(10\right)\end{array}$

《5 案例研究》

5 案例研究

$\begin{array}{l}A\left(g\right)=\left(\begin{array}{cc}\left[0‚1\right]& \left[2‚3\right]\\ 4& 2\end{array}\right)=\left(\begin{array}{cc}\gamma {}_{11}& 2+\gamma {}_{12}\\ 4& 2\end{array}\right)‚\\ \left(\gamma {}_{11}\in \left[0‚1\right]\right)‚\left(\gamma {}_{12}\in \left[0‚1\right]\right)\end{array}$

$\begin{array}{l}x{}_{1}^{*}\left(g\right)=\\ \frac{a{}_{22}\left(g\right)-a{}_{21}\left(g\right)}{\left(a{}_{11}\left(g\right)-a{}_{22}\left(g\right)\right)-\left(a{}_{12}\left(g\right)-a{}_{21}\left(g\right)\right)}=\\ \left\{\begin{array}{l}\frac{2}{3},if,\gamma {}_{11}=1,\gamma {}_{12}=0\\ \frac{2}{5},if,\gamma {}_{11}=0,\gamma {}_{12}=1\end{array}\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}}\text{即}:x{}_{1}^{*}\left(g\right)=\left[\frac{2}{5},\frac{2}{3}\right]\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}}\left(11\right)\\ x{}_{2}^{*}\left(g\right)=\\ \frac{a{}_{11}\left(g\right)-a{}_{12}\left(g\right)}{\left(a{}_{11}\left(g\right)-a{}_{22}\left(g\right)\right)-\left(a{}_{12}\left(g\right)-a{}_{21}\left(g\right)\right)}=\\ \left\{\begin{array}{l}\frac{1}{3},if,\gamma {}_{11}=1,\gamma {}_{12}=0\\ \frac{3}{5},if,\gamma {}_{11}=0,\gamma {}_{12}=1\end{array}\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}}\text{即}:x{}_{2}^{*}\left(g\right)=\left[\frac{1}{3},\frac{3}{5}\right]\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}}\left(12\right)\\ y{}_{1}^{*}\left(g\right)=\\ \frac{a{}_{22}\left(g\right)-a{}_{12}\left(g\right)}{\left(a{}_{11}\left(g\right)-a{}_{22}\left(g\right)\right)-\left(a{}_{12}\left(g\right)-a{}_{21}\left(g\right)\right)}=\\ \left\{\begin{array}{l}0,if,\gamma {}_{12}=0\\ \frac{1}{4},if,\gamma {}_{11}=1,\gamma {}_{12}={1}^{\prime }\end{array}\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}}\text{即}:y{}_{1}^{*}\left(g\right)=\left[0,\frac{1}{4}\right]\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}}\left(13\right)\\ y{}_{2}^{*}\left(g\right)=\\ \frac{a{}_{11}\left(g\right)-a{}_{21}\left(g\right)}{\left(a{}_{11}\left(g\right)-a{}_{22}\left(g\right)\right)-\left(a{}_{12}\left(g\right)-a{}_{21}\left(g\right)\right)}=\\ \left\{\begin{array}{l}1,if,\gamma {}_{12}=0\\ \frac{4}{5},if,\gamma {}_{11}=0,\gamma {}_{12}={1}^{\prime }\end{array}\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}}\text{即}:y{}_{2}^{*}\left(g\right)=\left[\frac{4}{5},1\right]\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}}\left(14\right)\\ v{}_{\text{G}}^{*}\left(g\right)=\\ \frac{a{}_{11}\left(g\right)\cdot a{}_{22}\left(g\right)-a{}_{12}\left(g\right)\cdot a{}_{21}\left(g\right)}{\left(a{}_{11}\left(g\right)-a{}_{22}\left(g\right)\right)-\left(a{}_{12}\left(g\right)-a{}_{21}\left(g\right)\right)}=\\ \left\{\begin{array}{l}2,if,\gamma {}_{12}=0\\ \frac{5}{2},if,\gamma {}_{11}=1,\gamma {}_{12}={1}^{\prime }\end{array}\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}}\text{即}:v{}_{\text{G}}^{*}\left(g\right)=\left[2,\frac{5}{2}\right]\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}}\left(15\right)\end{array}$

Table 1 Compare data of two arithmetic

《表1》

 指标 原灰数计算方法 标准灰数计算方法 计算结果 灰度 计算结果 灰度 x*1 (g) [5/2, 2/3] 0.5000 [5/2, 2/3] 0.5000 x*2 (g) [1/5, 1] 1.330 [1/3, 3/5] 0.5700 y*1 (g) [0, 1/3] 2.0000 [0, 1/4] 2.0000 y*2 (g) [3/5, 4/3] 0.4889 [5/4, 1] 0.2222 v*G (g) [6/5, 4] 1.0769 [2, 5/2] 0.4444

《6 结语》

6 结语