What rock quarries should be used and how much should they produce to meet a certain | ||||||

quality of limestone (calcium and magnesium content) and minimize cost? There are 4 quarries with | ||||||

different qualities, capacity and cost to operate. A different output and quality is required each year. | ||||||

Due to environmental restrictions, only 3 quarries are allowed to be open each year. | ||||||

Information on rock quarries | ||||||

Calcium contents (relative to required quality) | Magnesium contents (relative to required quality) | Maximum production per year (tons) | Cost to keep quarry open per year ($Million) | |||

Quarry 1 | 1 | 2.3 | 2000 | 3.5 | ||

Quarry 2 | 0.7 | 1.6 | 2500 | 4 | ||

Quarry 3 | 1.5 | 1.2 | 1300 | 4 | ||

Quarry 4 | 0.7 | 4.1 | 3000 | 2 | ||

Quarries to be used (1=yes, 0=no) | ||||||

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | ||

Quarry 1 | 0 | 0 | 0 | 0 | 0 | |

Quarry 2 | 0 | 0 | 0 | 0 | 0 | |

Quarry 3 | 0 | 0 | 0 | 0 | 0 | |

Quarry 4 | 0 | 0 | 0 | 0 | 0 | |

Total | 0 | 0 | 0 | 0 | 0 | |

Amounts to produce (tons). | ||||||

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | ||

Quarry 1 | 0 | 0 | 0 | 0 | 0 | |

Quarry 2 | 0 | 0 | 0 | 0 | 0 | |

Quarry 3 | 0 | 0 | 0 | 0 | 0 | |

Quarry 4 | 0 | 0 | 0 | 0 | 0 | |

Total | 0 | 0 | 0 | 0 | 0 | |

Required | 4500 | 3100 | 3500 | 3700 | 4000 | |

Amounts that can be produced (tons) | ||||||

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | ||

Quarry 1 | 0 | 0 | 0 | 0 | 0 | |

Quarry 2 | 0 | 0 | 0 | 0 | 0 | |

Quarry 3 | 0 | 0 | 0 | 0 | 0 | |

Quarry 4 | 0 | 0 | 0 | 0 | 0 | |

Calcium restrictions | ||||||

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | ||

Total Amount of Calcium | 0 | 0 | 0 | 0 | 0 | |

Total Amount Required | 0 | 0 | 0 | 0 | 0 | |

Calcium Required per Ton (Minimum) | 0.9 | 1.2 | 1 | 1.1 | 0.8 | |

Magnesium restrictions | ||||||

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | ||

Total Amount of Magnesium | 0 | 0 | 0 | 0 | 0 | |

Total Amount Required | 0 | 0 | 0 | 0 | 0 | |

Magnesium Required per Ton (Minimum) | 1.9 | 1.7 | 2.8 | 1.9 | 2.1 | |

Cost | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Total |

$0.00 | $0.00 | $0.00 | $0.00 | $0.00 | $0.00 | |

Problem | ||||||

A company owns four rock quarries from which it can extract limestone with different qualities. Two | ||||||

qualities are important, the relative amount of calcium and magnesium in the stone. The company | ||||||

must produce a certain total amount of limestone, with certain qualities, each year. There is a large | ||||||

fixed cost to keep a quarry operating for extraction purposes each year. Which quarries should be | ||||||

used each year, and how much limestone should each one produce each year? | ||||||

Solution | ||||||

The solution is very similar in structure to the one found in worksheet Blend1. | ||||||

1) The variables are 0-1 or binary integer variables which determine whether each quarry is open, | ||||||

and amounts of limestone to be extracted from each quarry. These variables occur in each year. | ||||||

In worksheet Blend2, these variables the names Quarry_decisions and Amounts_produced. | ||||||

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

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

Quarry_decisions = binary | ||||||

Second, there are contraints on the total production and the amount that can be produced at each | ||||||

quarry. These constraints are: | ||||||

Total_produced >= Total_required | ||||||

Amounts_produced <= Maximum_Production | ||||||

The right hand side of the second constraint depends on the binary integer variables. | ||||||

Third, there are constraints on the quality (calcium and magnesium content) of the limestone: | ||||||

Calcium_production >= Calcium_requirement | ||||||

Magnesium_production >= Magnesium_requirement | ||||||

Both the left-hand and right-hand sides of these constraints depend on the Amounts_produced | ||||||

decision variables. | ||||||

Fourth, there is a constraint that limits the number of quarries that can be open each year: | ||||||

Number_of_open_quarries <= 3 | ||||||

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

Total_cost. | ||||||

Remarks | ||||||

See the comments on worksheet Blend1 about the characteristics of blending problems, which also | ||||||

apply here. | ||||||