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 minimum output of 6000 tons per year is required. | |||||||
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 in use (1=yes, 0=no) | |||
Quarry 1 | 1 | 2.3 | 2000 | 3.5 | 1 | ||
Quarry 2 | 0.7 | 1.6 | 2500 | 4 | 1 | ||
Quarry 3 | 1.5 | 1.2 | 1300 | 4 | 1 | ||
Quarry 4 | 0.7 | 4.1 | 3000 | 2 | 1 | ||
Amounts to produce (tons). | Avail Prod | ||||||
Quarry 1 | 0.00 | 2000 | |||||
Quarry 2 | 0.00 | 2500 | |||||
Quarry 3 | 0.00 | 1300 | |||||
Quarry 4 | 0.00 | 3000 | |||||
Totals | 0 | ||||||
Required | 6000 | ||||||
Calcium restrictions | |||||||
Total Amount of Calcium | 0 | ||||||
Total Amount Required | 0 | ||||||
Calcium Required per Ton | 0.9 | ||||||
Magnesium restrictions | |||||||
Total Amount of Magnesium | 0 | ||||||
Total Amount Required | 0 | ||||||
Magnesium Required per Ton | 2.3 | ||||||
Cost | $14 | Million | |||||
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 (6000 tons in this case), and this stone must contain | |||||||
at least a certain amount of calcium per ton and a certain amount of magnesium per ton. There is a | |||||||
large fixed cost to keep a quarry operating for extraction purposes each year. Which quarries should | |||||||
be used to meet the production requirement, and how much limestone should each one produce? | |||||||
Solution | |||||||
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. In worksheet Blend1 these are given | |||||||
the names Quarry_use and Amounts_to_produce. | |||||||
2) First, there are the logical constraints. These are | |||||||
Amounts_to_produce >= 0 via the Assume Non-Negative option | |||||||
Quarry_use = 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_to_produce <= Avail_Production | |||||||
The right hand side of the second constraint depends on the binary integer variables. | |||||||
Third, there are constraints on the qualities (calcium and magnesium content) of the limestone: | |||||||
Calcium_produced >= Calcium_required | |||||||
Magnesium_produced >= Magnesium_required | |||||||
Both the left-hand and right-hand sides of these constraints depend on the Amounts_to_produce | |||||||
decision variables. | |||||||
3) The objective is to minimize the cost of operating the quarries. This is defined on the worksheet as | |||||||
Total_cost. | |||||||
Remarks | |||||||
Blending problems are characterized by 'ratio constraints' where the constraint is often thought of as | |||||||
a quality ratio where the numerator and denominator contain decision variables. These would be | |||||||
nonlinear, but the ratios can be expressed as linear constraints by multiplying both sides by the | |||||||
denominator of the ratio. | |||||||