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
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?
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
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.