Blending Problem 1 (Single-period)

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.

Blending Problem 1 (Single-period)

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