Minimize the cost of operating 3 different types of machines while meeting product demand over a | ||||||

week's time. Each machine has a different cost and capacity. There are a certain number of machines | ||||||

available for each type. | ||||||

Information on machines | ||||||

Initial cost per day | Additional cost per product | Products per day (Max) | Number of machines | |||

Alpha-1000 | $200 | $1.00 | 40 | 8 | ||

Alpha-2000 | $275 | $1.80 | 60 | 5 | ||

Alpha-3000 | $325 | $1.90 | 85 | 3 | ||

Number of machines to use | ||||||

Monday | Tuesday | Wednesday | Thursday | Friday | ||

Alpha-1000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-2000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-3000 | 0 | 0 | 0 | 0 | 0 | |

Number of products to make per day | ||||||

Monday | Tuesday | Wednesday | Thursday | Friday | ||

Alpha-1000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-2000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-3000 | 0 | 0 | 0 | 0 | 0 | |

Made | 0 | 0 | 0 | 0 | 0 | |

Carry-over | 0 | -600 | -1400 | -2400 | -3125 | |

Total | 0 | -600 | -1400 | -2400 | -3125 | |

Demand | 600 | 800 | 1000 | 725 | 750 | |

Maximum number of products that can be made | ||||||

Monday | Tuesday | Wednesday | Thursday | Friday | ||

Alpha-1000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-2000 | 0 | 0 | 0 | 0 | 0 | |

Alpha-3000 | 0 | 0 | 0 | 0 | 0 | |

Total | ||||||

Cost | $0.00 | $0.00 | $0.00 | $0.00 | $0.00 | $0.00 |

Problem | ||||||

A company has three different types of machines that all make the same product. Each machine has | ||||||

a different capacity, start-up cost and cost per product. How should the company produce its | ||||||

product with the available machines to meet the demand over a week's time? | ||||||

Solution | ||||||

The solution is very similar in structure to the one found on worksheet Alloc1. | ||||||

1) The variables are the number of machines to use and the number of products to make on each | ||||||

machine. In worksheet Alloc2, these given the names Products_made and Machines_used. | ||||||

2) First, there are the logical constraints. These are | ||||||

Products_made >= 0 via the Assume Non-Negative option | ||||||

Machines_used >= 0 via the Assume Non-Negative option | ||||||

Machines_used = integer. | ||||||

Second, there are the demand and capacity constraints. These are: | ||||||

Alpha1000s_used <= Alpha1000s_available | ||||||

Alpha2000s_used <= Alpha2000s_available | ||||||

Alpha3000s_used <= Alpha3000s_available | ||||||

Products_made <= Maximum_products | ||||||

Total_made >= Demand | ||||||

3) The objective is to minimize cost. This is defined on the worksheet as Total_cost.. | ||||||

Remarks | ||||||

Please see the comments on integer constraints in worksheet Alloc1. In this model we allow for | ||||||

products made on one day to be carried over to the next. This makes it possible to meet a demand | ||||||

for one day that exceeds the capacity of the machines on that particular day. | ||||||