6_7.htm

随着各行各业的发展和生产需要

    <td width="7%">1</td>
    <td width="11%">5/7</td>
    <td width="8%">10/21</td>
    <td width="8%">　</td>
    <td width="8%">1</td>
    <td width="8%">　</td>
    <td width="9%">-1/21</td>
    <td width="8%">0</td>
    <td width="9%">-2/21</td>
    <td width="9%">1/7</td>
    <td width="9%">　</td>
  </tr>
  <tr>
    <td width="6%">　</td>
    <td width="7%">　</td>
    <td width="11%">z'=64/21</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="9%">19/21</td>
    <td width="8%">　</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="6%">B</td>
    <td width="7%">C<sub>B</sub></td>
    <td width="11%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="8%">C</td>
    <td width="8%">2</td>
    <td width="8%">1</td>
    <td width="8%">1</td>
    <td width="9%">0</td>
    <td width="8%">0</td>
    <td width="9%">0</td>
    <td width="9%">-6</td>
    <td width="9%">β</td>
  </tr>
  <tr>
    <td width="6%">　</td>
    <td width="7%">　</td>
    <td width="11%">　</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="9%">P<sub>6</sub></td>
    <td width="8%">P<sub>4</sub></td>
    <td width="9%">P<sub>5</sub></td>
    <td width="9%">P<sub>7</sub></td>
    <td width="9%">　</td>
  </tr>
  <tr>
    <td width="6%">A<sub>6</sub></td>
    <td width="7%">0</td>
    <td width="11%">19/42</td>
    <td width="8%">0</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="9%">1</td>
    <td width="8%">1/2</td>
    <td width="9%">0</td>
    <td width="9%">-1</td>
    <td width="9%">　</td>
  </tr>
  <tr>
    <td width="6%">A<sub>1</sub></td>
    <td width="7%">2</td>
    <td width="11%">4/21</td>
    <td width="8%">9/7</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="9%">0</td>
    <td width="8%">3/14</td>
    <td width="9%">1/7</td>
    <td width="9%">-1/7</td>
    <td width="9%">　</td>
  </tr>
  <tr>
    <td width="6%">A<sub>2</sub></td>
    <td width="7%">1</td>
    <td width="11%">-4/21</td>
    <td width="8%">10/21</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="9%">0</td>
    <td width="8%">1/42</td>
    <td width="9%">-2/21</td>
    <td width="9%">2/21</td>
    <td width="9%">　</td>
  </tr>
  <tr>
    <td width="6%">　</td>
    <td width="7%">　</td>
    <td width="11%">z'=64/21</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="8%">　</td>
    <td width="9%">　</td>
    <td width="8%">　</td>
    <td width="9%">　</td>
    <td width="9%">　</td>
    <td width="9%">←λ</td>
  </tr>
</table>
</div>

<p>故解是x<sub>1</sub>＝9/7 , x<sub>2</sub>＝10/21, maxz＝64/21<br>
</p>

<p><strong>例</strong>　<br>
min z＝2x<sub>1</sub>＋x<sub>2</sub>，<br>
5x<sub>1</sub>＋10x<sub>2</sub>≥8<br>
2x<sub>1</sub>＋2x<sub>2</sub>≤1<br>
x<sub>1</sub>，x<sub>2</sub>≥0<br>
<strong>解</strong>　<br>
引入松弛变量x<sub>3</sub>，x<sub>4</sub>及人工变量x<sub>5</sub>，得<br>
<img src="6_7_4.gif" width="202" height="146"><br>
<font color="#FF8040">解１．用二阶段法．</font><br>
第一阶段设minz’＝x<sub>5</sub>，用改善的单纯形法表格求解过程如下：<br>
</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">1</td>
    <td width="10%">1</td>
    <td width="10%">8</td>
    <td width="10%">5</td>
    <td width="10%">10</td>
    <td width="10%">-1</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>4</sub></td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">(2)</td>
    <td width="10%">2</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">1/2</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=8</td>
    <td width="10%">　</td>
    <td width="10%">-5</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>　</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">1</td>
    <td width="10%">1</td>
    <td width="10%">11/2</td>
    <td width="10%">0</td>
    <td width="10%">5</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">-5/2</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>1</sub></td>
    <td width="10%">0</td>
    <td width="10%">-5/2</td>
    <td width="10%">1/2</td>
    <td width="10%">1</td>
    <td width="10%">(1)</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">1/2</td>
    <td width="10%">1/2</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=11/2</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">-5</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>　</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">1</td>
    <td width="10%">1</td>
    <td width="10%">3</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">-5</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>2</sub></td>
    <td width="10%">0</td>
    <td width="10%">-5</td>
    <td width="10%">1/2</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">1/2</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=3</td>
    <td width="10%">　</td>
    <td width="10%">5</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">5</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>Minz’＝3＞0，但P<sub>5</sub>退不出去．故原问题允许解域是空集．<br>
</p>

<p><font color="#FF0000">解２　用大M法．<br>
</font>M先取3，minz’＝2x<sub>1</sub>＋x<sub>2</sub>＋3x<sub>5</sub>．结果如下．<br>
</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">3</td>
    <td width="10%">3</td>
    <td width="10%">8</td>
    <td width="10%">5</td>
    <td width="10%">10</td>
    <td width="10%">-1</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>4</sub></td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">(2)</td>
    <td width="10%">2</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">1/2</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=24</td>
    <td width="10%">　</td>
    <td width="10%">-13</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>　</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">3</td>
    <td width="10%">3</td>
    <td width="10%">11/2</td>
    <td width="10%">0</td>
    <td width="10%">5</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">-5/2</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>1</sub></td>
    <td width="10%">2</td>
    <td width="10%">-13/2</td>
    <td width="10%">1/2</td>
    <td width="10%">1</td>
    <td width="10%">(1)</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">1/2</td>
    <td width="10%">1/2</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=35/2</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">-16</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>　</p>
<div align="left">

<table border="1" width="85%">
  <tr>
    <td width="10%">B</td>
    <td width="10%">C<sub>B</sub></td>
    <td width="10%">C<sub>B</sub>B<sup>-1</sup></td>
    <td width="10%">C</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">0</td>
    <td width="10%">1</td>
    <td width="10%">0</td>
    <td width="10%">β</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">P<sub>0</sub></td>
    <td width="10%">P<sub>1</sub></td>
    <td width="10%">P<sub>2</sub></td>
    <td width="10%">P<sub>3</sub></td>
    <td width="10%">P<sub>5</sub></td>
    <td width="10%">P<sub>4</sub></td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>5</sub></td>
    <td width="10%">3</td>
    <td width="10%">3</td>
    <td width="10%">3</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">-5</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">A<sub>2</sub></td>
    <td width="10%">1</td>
    <td width="10%">-29/2</td>
    <td width="10%">1/2</td>
    <td width="10%">　</td>
    <td width="10%">1</td>
    <td width="10%">　</td>
    <td width="10%">0</td>
    <td width="10%">1/2</td>
    <td width="10%">　</td>
  </tr>
  <tr>
    <td width="10%">　</td>
    <td width="10%">　</td>
    <td width="10%">z'=19/2</td>
    <td width="10%">　</td>
    <td width="10%">16</td>
    <td width="10%">0</td>
    <td width="10%">3</td>
    <td width="10%">0</td>
    <td width="10%">29/2</td>
    <td width="10%">←λ</td>
  </tr>
</table>
</div>

<p>结束时，P<sub>5</sub>未退出，故原问题的允许解域是空集，无解．<br>
</p>
</body>
</html>