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<html><head><meta http-equiv="Content-Type" content="text/html; charset=gb2312"><title>§6.8</title><meta name="GENERATOR" content="Microsoft FrontPage 3.0"></head><body><p align="center"><font size="4"><b>§6.8 退化情形及其他</b></font></p><p>前面讨论的单纯形法的前提，一是非退化，一是右端常数项b的元素都是正数．现补充讨论如下：<br>1．退化情形<br> 可用A的少于m个列向量线性表示的问题称为退化问题．这是一个理论定义，没有 必要，也不可行事先判断一个问题是否退化．当P<sub>0</sub>中出现0元，称为退化现象．即 P可用少于m个的基线性结合表示，亦即超多面体的顶点坐标的零元素树木多于n个．<br>2．退化情况下可能出现循环不已问题.</p><p><strong>例</strong> <br>min z＝－(3/4)x<sub>1</sub>＋20x<sub>2</sub>－(1/2)x<sub>3</sub>＋6x<sub>4</sub><br>(1/4)x<sub>1</sub>－8x<sub>2</sub>－x<sub>3</sub>＋9x<sub>4</sub>≤0<br>(1/2)x<sub>1</sub>－12x<sub>2</sub>－(1/2)x<sub>3</sub>＋3x<sub>4</sub>≤0<br>x<sub>3</sub>≤1<br>x<sub>1</sub>, … , x<sub>4</sub>≥0<br>引入松弛变量x<sub>5</sub>，x<sub>6</sub>，x<sub>7</sub>．解答过程如下：<br></p><div align="left"><table border="1" width="85%"> <tr> <td width="8%">B</td> <td width="8%">C<sub>B</sub></td> <td width="8%">C<sub>B</sub>B<sup>-1</sup></td> <td width="8%">C</td> <td width="8%">-3/4</td> <td width="8%">20</td> <td width="8%">-1/2</td> <td width="8%">6</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">β</td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">P<sub>0</sub></td> <td width="8%">P<sub>1</sub></td> <td width="8%">P<sub>2</sub></td> <td width="8%">P<sub>3</sub></td> <td width="8%">P<sub>4</sub></td> <td width="9%">P<sub>5</sub></td> <td width="9%">P<sub>6</sub></td> <td width="9%">P<sub>7</sub></td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>5</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">(1/4)</td> <td width="8%">-8</td> <td width="8%">-1</td> <td width="8%">9</td> <td width="9%">1</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">0</td> </tr> <tr> <td width="8%">A<sub>6</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">1/2</td> <td width="8%">-12</td> <td width="8%">-1/2</td> <td width="8%">3</td> <td width="9%">0</td> <td width="9%">1</td> <td width="9%">0</td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>7</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%">0</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">1</td> <td width="9%"> </td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">-3/4</td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%">←λ</td> </tr></table></div><p> </p><div align="left"><table border="1" width="85%"> <tr> <td width="8%">B</td> <td width="8%">C<sub>B</sub></td> <td width="8%">C<sub>B</sub>B<sup>-1</sup></td> <td width="8%">C</td> <td width="8%">-3/4</td> <td width="8%">20</td> <td width="8%">-1/2</td> <td width="8%">6</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">β</td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">P<sub>0</sub></td> <td width="8%">P<sub>1</sub></td> <td width="8%">P<sub>2</sub></td> <td width="8%">P<sub>3</sub></td> <td width="8%">P<sub>4</sub></td> <td width="9%">P<sub>5</sub></td> <td width="9%">P<sub>6</sub></td> <td width="9%">P<sub>7</sub></td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>1</sub></td> <td width="8%">-3/4</td> <td width="8%">-3</td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%">-32</td> <td width="8%"> </td> <td width="8%"> </td> <td width="9%">4</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>6</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">(4)</td> <td width="8%"> </td> <td width="8%"> </td> <td width="9%">-2</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>7</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%"> </td> <td width="8%"> </td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">1</td> <td width="9%"> </td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">-4</td> <td width="8%"> </td> <td width="8%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%">←λ</td> </tr></table></div><p> </p><div align="left"><table border="1" width="85%"> <tr> <td width="8%">B</td> <td width="8%">C<sub>B</sub></td> <td width="8%">C<sub>B</sub>B<sup>-1</sup></td> <td width="8%">C</td> <td width="8%">-3/4</td> <td width="8%">20</td> <td width="8%">-1/2</td> <td width="8%">6</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">β</td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">P<sub>0</sub></td> <td width="8%">P<sub>1</sub></td> <td width="8%">P<sub>2</sub></td> <td width="8%">P<sub>3</sub></td> <td width="8%">P<sub>4</sub></td> <td width="9%">P<sub>5</sub></td> <td width="9%">P<sub>6</sub></td> <td width="9%">P<sub>7</sub></td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>1</sub></td> <td width="8%">-3/4</td> <td width="8%">-1</td> <td width="8%">0</td> <td width="8%"> </td> <td width="8%">0</td> <td width="8%">(8)</td> <td width="8%"> </td> <td width="9%">-12</td> <td width="9%">8</td> <td width="9%">0</td> <td width="9%">0</td> </tr> <tr> <td width="8%">A<sub>2</sub></td> <td width="8%">20</td> <td width="8%">-1</td> <td width="8%">0</td> <td width="8%"> </td> <td width="8%">1</td> <td width="8%">3/8</td> <td width="8%"> </td> <td width="9%">-1/2</td> <td width="9%">1/4</td> <td width="9%">0</td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>7</sub></td> <td width="8%">0</td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%"> </td> <td width="8%">0</td> <td width="8%">1</td> <td width="8%"> </td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">1</td> <td width="9%"> </td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">-2</td> <td width="8%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%"> </td> <td width="9%">←λ</td> </tr></table></div><p> </p><div align="left"><table border="1" width="85%"> <tr> <td width="8%">B</td> <td width="8%">C<sub>B</sub></td> <td width="8%">C<sub>B</sub>B<sup>-1</sup></td> <td width="8%">C</td> <td width="8%">-3/4</td> <td width="8%">20</td> <td width="8%">-1/2</td> <td width="8%">6</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">0</td> <td width="9%">β</td> </tr> <tr> <td width="8%"> </td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">P<sub>0</sub></td> <td width="8%">P<sub>1</sub></td> <td width="8%">P<sub>2</sub></td> <td width="8%">P<sub>3</sub></td> <td width="8%">P<sub>4</sub></td> <td width="9%">P<sub>5</sub></td> <td width="9%">P<sub>6</sub></td> <td width="9%">P<sub>7</sub></td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>3</sub></td> <td width="8%">-1/2</td> <td width="8%">2</td> <td width="8%">0</td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">1</td> <td width="8%">-21/2</td> <td width="9%">-3/2</td> <td width="9%">1</td> <td width="9%">0</td> <td width="9%"> </td> </tr> <tr> <td width="8%">A<sub>2</sub></td> <td width="8%">20</td> <td width="8%">-3</td> <td width="8%">0</td> <td width="8%"> </td> <td width="8%"> </td> <td width="8%">0</td> <td width="8%">(3/16)</td>⌨️ 快捷键说明
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