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8/27: Linear Programming Lecture: LP Small Groups Homework

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Linear Programming What is it? –Synthesizing a problem in words into a series of equations. –A type of modeling tool –Optimizing a linear function subject to several constraints, expressed as inequalities.

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LP - 4 Characteristics Objective Function Constraints Alternative Courses of Action Linear Equations

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EX: Toy Company A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively.

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Toy Company Formulate a linear program set to maximize the company's profit.

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Terminology Z : variable to be optimized. x 1, x 2, x 3,… : decision variables. So we write Max Z ( profit ) = (some combo of x 1...x X ) S. T. ("subject to"): (the constraints)

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Toy Company What are we supposed to maximize? What factors play a part in that? What constraints are there to the profit?

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A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

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A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

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A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

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A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hrs. of hand labor time, 8 hrs. machine time, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

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Toy Company What are we supposed to maximize? –THE PROFIT What factors play a part in that? –PROFIT FROM TRUCKS, DOLLS, and CHESS SETS What constraints are there to the profit? –HAND TIME, MACHINE TIME, and WOOD

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Toy Company Let x 1 = toy trucks, w/ a $7 profit each x 2 = dolls, w/ a $5 profit each x 3 = chess sets w/ a $12 profit each So Max Z (profit) = 7 x 1 + 5 x 2 + 12 x 3

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Toy Company - constraints Hand Time: total of 8 hours. -- or 480 min. Truck - 10 min. Doll - 8 min. Chess Set - 3 min. so 10 x 1 + 8 x 2 + 3 x 3 <= 480

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Toy Company - constraints Machine Time: total of 8 hrs. -- or 480 min. Truck - 3 min. Doll - 10 min. Chess Set - 20 min. so 3 x 1 + 10 x 2 + 20 x 3 <= 480

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Toy Company - constraints Wood: total of 1000 ft. -- or 12,000 in. Truck - 15 in. Doll - 11 in. Chess Set - 31 in. so 15 x 1 + 11 x 2 + 31 x 3 <= 12000

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Toy Company - constraints Other constraints: Integers:x 1, x 2, x 3 must be integers. Positive: x 1, x 2, x 3 >= 0

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Toy Company - total LP Max Z (profit) = 7 x 1 + 5 x 2 + 12 x 3 S. T.: 10 x 1 + 8 x 2 + 3 x 3 <= 480 3 x 1 + 10 x 2 + 20 x 3 <= 480 15 x 1 + 11 x 2 + 31 x 3 <= 12000 x 1, x 2, x 3 >= 0 x 1, x 2, x 3 must be integers.

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EX: Camping Trip. PCF $/lb beef jerky104813.00 dried potatoes 012 22.50 granola mix 4 8 118.50 NutriGrain bars 21459.00 Must have 30 g. protein, 60 g. carbohydrates, and 15 g. of fat. Minimize the cost.

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Graphical Solutions for LP Sparky Electronics 2 products, WalkFM & WristTV profit: $7 $5 machine time 4 3 assembly time 2 1 Total machine time 240 Total assembly time 100

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LP - Graphical Solution Limitation to the method: only TWO decision variables can exist.

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LP - Graphical Solution Maximize Z ( profit ) = 7 x 1 + 5 x 2 S. T. :4 x 1 + 3 x 2 <= 240 2 x 1 + 1 x 2 <= 100 x 1. x 2 >= 0

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LP - Graphical Solution 4 x 1 + 3 x 2 = 240

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LP - Graphical Solution 4 x 1 + 3 x 2 = 240 2 x 1 + 1 x 2 = 100

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LP - Graphical Solution 4 x 1 + 3 x 2 = 240 2 x 1 + 1 x 2 = 100 Feasible Solution Region

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LP - Graphical Solution 4 x 1 + 3 x 2 = 240 2 x 1 + 1 x 2 = 100 Max Z = 7 x 1 + 5 x 2 Z = $400 Z = $410 Z = $350

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