Blending Problem 1 (Single-period) |
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| What
rock quarries should be used and how much should they produce to meet a
certain |
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| quality
of limestone (calcium and magnesium content) and minimize cost? There are 4
quarries with |
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qualities, capacity and cost to operate. A minimum output of 6000 tons per
year is required. |
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| Information on rock quarries |
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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) |
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| Quarry
1 |
1 |
2.3 |
2000 |
3.5 |
1 |
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| Quarry
2 |
0.7 |
1.6 |
2500 |
4 |
1 |
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| Quarry
3 |
1.5 |
1.2 |
1300 |
4 |
1 |
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| Quarry
4 |
0.7 |
4.1 |
3000 |
2 |
1 |
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| Amounts to produce (tons). |
Avail Prod |
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| Quarry
1 |
0.00 |
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2000 |
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| Quarry
2 |
0.00 |
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2500 |
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| Quarry
3 |
0.00 |
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1300 |
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| Quarry
4 |
0.00 |
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3000 |
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| Totals |
0 |
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| Required |
6000 |
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| Calcium restrictions |
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| Total Amount of Calcium |
0 |
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| Total Amount Required |
0 |
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| Calcium Required per Ton |
0.9 |
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| Magnesium restrictions |
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| Total Amount of Magnesium |
0 |
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| Total Amount Required |
0 |
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| Magnesium Required per Ton |
2.3 |
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| Cost |
$14 |
Million |
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| Problem |
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| A
company owns four rock quarries from which it can extract limestone with
different qualities. Two |
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| qualities
are important, the relative amount of calcium and magnesium in the
stone. The company |
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| must produce a certain total amount of limestone (6000 tons
in this case), and this stone must contain |
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least a certain amount of calcium per ton and a certain amount of magnesium
per ton. There is a |
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| large
fixed cost to keep a quarry operating for extraction purposes each year. Which quarries should |
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| be used to meet the production requirement, and how much
limestone should each one produce? |
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| Solution |
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| 1) The variables are 0-1 or binary integer variables which
determine whether each quarry is open, |
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| and amounts of limestone to be extracted from each
quarry. In worksheet Blend1 these are
given |
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| the names
Quarry_use and Amounts_to_produce. |
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| 2) First,
there are the logical constraints. These are |
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Amounts_to_produce >= 0 via the Assume Non-Negative option |
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Quarry_use = binary |
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| Second, there are contraints on the total production and
the amount that can be produced at each |
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| quarry. These constraints are: |
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Total_produced >= Total_required |
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Amounts_to_produce <= Avail_Production |
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| The right hand
side of the second constraint depends on the binary integer variables. |
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| Third, there are constraints on the qualities (calcium and
magnesium content) of the limestone: |
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Calcium_produced >= Calcium_required |
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Magnesium_produced >= Magnesium_required |
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| Both
the left-hand and right-hand sides of these constraints depend on the
Amounts_to_produce |
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| decision variables. |
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| 3) The objective is to minimize the cost of operating the
quarries. This is defined on the worksheet as |
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| Total_cost. |
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| Remarks |
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| Blending problems are characterized by 'ratio constraints'
where the constraint is often thought of as |
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quality ratio where the numerator and denominator contain decision
variables. These would be |
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| nonlinear,
but the ratios can be expressed as linear constraints by multiplying both
sides by the |
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| denominator of
the ratio. |
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