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. | |||||||