Midterm 2, 5 problems, 15 points each. Return this page.

Solve linear programs where all

Solutions.

1. We rewrite the LP in standard row tableau:

x_{2} |
x_{3 } |
1 _{} |
Problem 1 |

-1 | 2_{} |
1 | = x_{1} |

-6 | 0 |
31 | =x_{4} |

1 | -3 | 6 | -> min |

2. We simplify the tableau:

x_{2} |
x_{1} |
1_{} |
Problem 2 |

-2* | 4 | -11 | = x_{1} |

-6 | -8 | -31 | =x_{4} |

-1 | -3 | -6 | -> min |

all

3.Combining the columns with constants on top, we obtain

x_{2} |
x_{3} |
1_{} |
Problem 3 |

2 | -4 | 5 | = x_{1} |

6 | 0 |
7 | =- x_{1} |

1 | -3 | 6 | =f -> max |

all

4. We rewrite th LP in standard row tableau:

_{}x_{2} |
_{}x_{3} |
1 | Problem 4 |

-2 | -3 | 3 | = x_{1} |

6* | 5 | -1 | =x_{4} |

1 | 1 | 2 | -> min |

x_{4} |
x_{3} |
1 | Problem 4 |

-1/3 | -4/3 | 8/3 | = x_{1} |

1/6 | -5/6 | 1/6 | =x_{2}_{} |

1/6 | 1/6 | 13/6 | -> min |

min= 13/6 at

5. We drop the column with 0 on top, scale row 1, and switch and scale columns 3 and 4:

x_{2} |
x_{3} |
1_{} |
Problem 5 |

-2 | 4 | 3 | = x_{1} |

6 | -1 | -1 | =x_{4} |

-1 | -3 | 2 | -> min |

Setting

a bad column:

x_{3} |
1 | Problem 5 |

2 | 1 | = x_{1} |

5 | 5 | =x_{4} |

-4 | 1 |
-> min |