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  Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods
  
  from around the world. Two of the stores leading products use the Romans Food Market
  name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of
  Brazilian Natural and Colombian Mild coffee beans, which are purchased from a
  distributor located in New York City. Because Romans purchases large quantities, the
  coffee beans may be purchased on an as-needed basis for a price 10% higher than the
  market price the distributor pays for the beans. The current market price is \$0.47 per
  pound for Brazilian Natural and \$0.62 per pound for Colombian Mild. The compositions
  of each coffee blend are as follows:
  
  .:.
  Romans sells the Regular blend for \$3.60 per pound and the DeCaf blend for \$4.40 per
  pound. Romans would like to place an order for the Brazilian and Colombian coffee
  beans that will enable the production of 1000 pounds of Romans Regular coffee and 500
  pounds of Romans DeCaf coffee. The production cost is \$0.80 per pound for the Regular
  blend. Because of the extra steps required to produce DeCaf, the production cost for the
  DeCaf blend is \$1.05 per pound. Packaging costs for both products are \$0.25 per pound.
  Formulate a linear programming model that can be used to determine the pounds of
  Brazilian Natural and Colombian Mild that will maximize the total contribution to profit.
  What is the optimal solution and what is the contribution to profit?
  SOLUTION:
  Let
  
  BR
  
  BD
  
  CR
  
  CD
  
  Type of Bean
  Brazilian
  Colombian
  
  Cost per pound (\$)
  
  1.10(0.47) = 0.517
  1.10(0.62) = 0.682
  
  Total revenue = 3.60(BR + CR) + 4.40(BD + CD)
  
  Total cost of beans = 0.517(BR + BD) + 0.682(CR + CD)
  Total production cost = 0.80(BR + CR) + 1.05(BD + CD)
  Total packaging cost = 0.25(BR + CR) + 0.25(BD + CD)
  Total contribution to profit = (total revenue) - (total cost of beans) - (total production
  cost)
  Total
  
  contribution to profit = 2.033BR + 2.583BD + 1.868CR + 2.418CD
  
  Regular % constraint
  BR = 0.75(BR + CR)
  0.25BR - 0.75CR = 0
  DeCaf % constraint
  BD = 0.40(BD + CD)
  0.60BD - 0.40CD = 0
  Pounds of Regular: BR + CR = 1000
  Pounds of DeCaf: BD + CD = 500
  The complete linear program is
  Max
  
  + 2.418CD
  
  s.t.
  0.25BR
  
  0.60BD
  
  BR
  
  =
  0
  0.40CD =
  0
  CR
  = 1000
  +
  CD = 500
  BR, BD, CR, CD 0
  -
  
  +
  BD
  
  0.75CR
  
  250, and
  
  CD = 300.