**Example:** Consider the CPM project network in the figure, where time is measured in days.

Activity | Usual time | Crash time | Cost per day |
---|---|---|---|

(0,1) | 4 | 2 | $500 |

(2,3) | 10 | 6 | $700 |

(2,4) | 7 | 4 | $600 |

(5,6) | 5 | 3 | $300 |

**Meeting a project deadline:**If the durations on the edges of the network are the usual times, and the crash times and the costs of a day’s reduction for four of the activities are given in the table, set up a linear program to determine how to reduce the length of the project to 19 days as cheaply as possible.**Minimizing total project cost:**If there is a fixed cost involved with the project of $1,000 a day, set up the LP to minimize total project cost.**Staying within budget:**If a budget of $3,000 is available for reduction in time, set up the LP to determine the shortest time.

**Solution: **Let t_0, t_1, \dots, t_6 indicate the time of the event of the nodes, and let t_{01}, t_{23}, t_{24}, t_{56} indicate the time of the required to complete activity (0,1), (2,3), (2,4), (5,6).

Meeting a project deadline:

Minimize | 500(4-t_{01}) + 700(10-t_{23})+600(7-t_{24})+300(5-t_{56}) |
---|---|

Subject to | t_1 - t_0 \ge t_{01} |

t_2 - t_0 \ge 5 | |

t_3 - t_1 \ge 8 | |

t_3 - t_2 \ge t_{23} | |

t_4 - t_2 \ge t_{24} | |

t_5 - t_3 \ge 3 | |

t_5 - t_4 \ge 2 | |

t_6 - t_5 \ge t_{56} | |

4 \ge t_{01} \ge 2 | |

10 \ge t_{23} \ge 6 | |

7 \ge t_{24} \ge 4 | |

5 \ge t_{56} \ge 3 | |

t_{6} - t_0 \le 19 |

Minimizing total project cost:

Minimize | 500(4-t_{01}) + 700(10-t_{23})+600(7-t_{24})+300(5-t_{56})+1000(t_6-t_0) |
---|---|

Subject to | t_1 - t_0 \ge t_{01} |

t_2 - t_0 \ge 5 | |

t_3 - t_1 \ge 8 | |

t_3 - t_2 \ge t_{23} | |

t_4 - t_2 \ge t_{24} | |

t_5 - t_3 \ge 3 | |

t_5 - t_4 \ge 2 | |

t_6 - t_5 \ge t_{56} | |

4 \ge t_{01} \ge 2 | |

10 \ge t_{23} \ge 6 | |

7 \ge t_{24} \ge 4 | |

5 \ge t_{56} \ge 3 |

Staying within budget:

Minimize | t_6-t_0 |
---|---|

Subject to | t_1 - t_0 \ge t_{01} |

t_2 - t_0 \ge 5 | |

t_3 - t_1 \ge 8 | |

t_3 - t_2 \ge t_{23} | |

t_4 - t_2 \ge t_{24} | |

t_5 - t_3 \ge 3 | |

t_5 - t_4 \ge 2 | |

t_6 - t_5 \ge t_{56} | |

4 \ge t_{01} \ge 2 | |

10 \ge t_{23} \ge 6 | |

7 \ge t_{24} \ge 4 | |

5 \ge t_{56} \ge 3 | |

500(4-t_{01}) + 700(10-t_{23})+600(7-t_{24})+300(5-t_{56})\le 3000 |