BLENDING PRACTICE PROBLEMS
Paper Products, Inc.
Paper is made out of wood pulp and cotton rag. You can distinguish between different types of paper based on the stiffness and the weight of the paper. You find it convenient to measure these attributes as percentages rather than in more typical units, and you measure your ingredients and products in gallons, since everything is converted to a paste before mixing and producing the final product. So, measured as a percentage, bond paper must be at least 40% weight and no more than 30% stiff, while plain paper must be no more than 20% weight and at least 50% stiff. Your current supply of wood pulp is 15% weight and 65% stiff, while your current supply of cotton rag is 50% weight and 20% stiff. You have on hand 40,000 gallons of wood pulp and 30,000 gallons of cotton rag, costing $0.50 per gallon and $0.85 per gallon, respectively. Your company produces only to meet sales, and you have orders for 25,000 gallons of bond paper and 35,000 gallons of plain paper.
WB = # of gallons of wood pulp to use in bond paper
CB = # of gallons of cotton rag to use in bond paper
WP = # of gallons of wood pulp to use in plain paper
CP = # of gallons of cotton rag to use in plain paper
Min Z = 0.50 WB + 0.85 CB + 0.50 WP + 0.85 CP
s.t. WB + WP < 40,000 gallons supply - wood pulp
CB + CP < 30,000 gallons supply - cotton rag
WB + CB = 25,000 gallons demand - bond
WP + CP = 35,000 gallons demand - plain
.15 WB + .50 CB > .4(25,000) weight limit for bond
.65 WB + .20 CB < .3(25,000) stiff limit for bond
.15 WP + .50 CP < .2(35,000) weight limit for plain
.65 WP + .20 CP > .5(35,000) stiff limit for plain
WB, CB, WP, CP > 0 non-negativity
Dyna-Aid, Inc.
Your company has decided to get in on the booming market for sports drinks with two new products: Aspen and Pittsburgh. Aspen is targeted to the health-conscious types, so each liter promises to have no more than 15 mg of carbohydrates, at least 35 mg of antioxidants, and no more than 2 g of solids (a catch-all term denoting the weight of the liquid). Pittsburgh is intended as a more robust supplement, so it guarantees at least 30 mg of carbohydrates, no more than 25 mg of antioxidants, and at least 3 g of solids. Both products are made from the same ingredients, water (which you market as “spring water”), a rather generic fruit juice (whatever is available on the market), Vita-Pro (a supplement with a high vitamin content), and Carb-Load (a supplement with a high carbohydrate content). The current batch of fruit juice is 20 mg of carbohydrates per liter, 5 mg of antioxidants per liter, and 4 g of solids per liter. Vita-Pro is 10 mg of carbohydrates per kg, 50 mg of antioxidants per kg, and .5 g of solids per kg. Carb-Load is 45 mg of carbohydrates per kg, 10 mg of antioxidants per kg, and 2.5 g of solids per kg. The water is irrelevant. You have on hand 50,000 liters of fruit juice, 100 kg of Vita-Pro, and 70 kg of Carb-Load. Since you have already purchased these ingredients, your company considers them a sunk cost. You can sell Aspen for $3.25 per liter and Pittsburgh for $4.10. Your process is to combine the three primary ingredients in the correct proportions then hold that mix as a concentrate. This lets you talk about “liters” of concentrate, where a “liter” of concentrate is enough to make 1 liter of product when mixed with water. In the blending process, a kilogram of Vita-Pro becomes 1.5 liters and a kilogram of Carb-Load becomes 2.1 liters. You want to mix enough concentrate to make exactly 10,000 liters of Aspen and 15,000 liters of Pittsburgh.
FA = # of liters of fruit juice to use in Aspen
FP = # of liters of fruit juice to use in Pittsburg
VA = # of kilograms of Vita-Pro to use in Aspen
VP = # of kilograms of Vita-Pro to use in Pittsburg
CA = # of kilograms of Carb-Load to use in Aspen
CP = # of kilograms of Carb-Load to use in Pittsburg
Max Z = 3.25FA + 4.10FP + 3.25*1.5VA + 4.1*1.5VP + 3.25*2.1CA + 4.1*2.1CP
s.t. FA + FP < 50,000 liters juice
VA + VP < 100 kg Vita-Pro
CA + CP < 70 kg Carb-Load
FA + 1.5VA + 2.1CA = 10,000 liters Aspen
FP + 1.5VP + 2.1CP = 15,000 liters Pittsburg
20 FA 10 VA 45 CA < 15*10,000 mg carbs – A
5 FA 50 VA 10 CA > 35*10,000 mg antiox – A
4 FA 0.5 VA 2.5 CA < 2*10,000 g solids – A
20 FP 10 VP 45 CP > 30*15,000 mg carbs – P
5 FP 50 VP 10 CP < 25*15,000 mg antiox – P
4 FP .5 VP 2.5 CP > 3*15,000 g solids – P
FA, FP, VA, VP, CA, CP > 0 non-negativity