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#9: The Brewery The brewery produces beer and ale. Beer sells for $5 per barrel, and ale for $2/barrel. Producing a barrel of beer requires 5 pounds of corn and 2 pounds of hops. Producing a barrel of ale requires 2 pounds of corn and 1 pound of hops. The brewery has 60 pounds of corn and 25 pounds of hops.

#9: The Brewery

#9: The Brewery Max Z ( revenue ) = 5 x1 + 2 x2 S. T. 5 x1 + 2 x2 <= 60 2 x1 + 1 x2 <= 25 x1, x2 >= 0

#9: The Brewery Max Z ( revenue ) = 5 x1 + 2 x2 30 25 Max Z ( revenue ) = 5 x1 + 2 x2 S. T. 5 x1 + 2 x2 <= 60 2 x1 + 1 x2 <= 25 x1, x2 >= 0 0, 25 = $50 10, 5 = $60 3 12, 0 = $60 3 12 ,12.5

#10: The Baker A gourmet cook bakes two types of cake, chocolate and vanilla, to supplement her income. Each chocolate cake can be sold for $12, and each vanilla cake can be sold for $9. Each chocolate cake requires 20 minutes of baking time and uses 4 eggs. Each vanilla cake requires 40 minutes of baking time and uses 1 egg. The baker has 8 hours of baking time and 30 eggs.

#10: The Baker

#10: The Baker Max Z ( profit ) = 12 x1 + 9 x2 S. T. 20 x1 + 40 x2 <= 480 4 x1 + 1 x2 <= 30 x1, x2 >= 0, integers

#10: The Baker 30 To find a point’s coordinates that aren’t obvious, solve simultaneous equations. 0,12 ??, ?? 12 7.5,0 3 3 7.5 24

#10: The Baker 30 20 x1 + 40 x2 <= 480 -5(4) x1 + -5 (1) x2 <= -5 (30) 0,12 ??, ?? 12 7.5,0 3 3 7.5 24

#10: The Baker 20 x1 + 40 x2 = 480 -20 x1 - 5 x2 = -150 0 35 x2 = 330 x2 = 330/35 or = 66/7 0,12 ??, ?? 12 7.5,0 3 3 7.5 24

#10: The Baker 30 Isoprofit lines: Solve the optimization equation for x2 to find the slope of the profit line. 0,12 36/7, 66/7 12 7.5,0 3 3 7.5 24

#10: The Baker Z = 12 x1 + 9 x2 Z - 12 x1 = 9 x2 Z/9 - 12/9 x1 = x2 5, 9 = $141

11. Woodco ...tables and chairs. Each table and chair must be made entirely out of oak or entirely out of pine. A total of 150 board feet of oak and 210 board feet of pine are available. A table requires either 17 board feet of oak or 30 board feet of pine. A chair requires either 5 board feet of oak or 13 board feet of pine. Each table sells for $40, and each chair for $15.

#11: Woodco

#11: Woodco Max Z ( profit ) = 40 x1 + 15 x2 S. T. 17 x1 + 5 x2 <= 150 30x1 + 13 x2 <= 210 x1, x2 >= 0, integers

WRONG - FOLLOW ME #11: Woodco Max Z ( profit ) = 40 x1 + 15 x2 S. T. 17 x1 + 5 x2 <= 150 30x1 + 13 x2 <= 210 x1, x2 >= 0, integers WRONG - FOLLOW ME 0, 210/13 = ~$240 7, 0 = $280

Using MS Excel’s Solver EX: using the Woodco example. Create the table of information. Set up formulas to relate the information to be optimized. Set up solver to list constraints. Run solver.

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EX: Chicken Feed A farmer feeds chickens, maintaining a particular level of nutrition. Minimize the cost, using this information: