| 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. | ||||||
