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8/27: Linear Programming. Lecture: LP Small Groups Homework. 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. LP - 4 Characteristics. - PowerPoint PPT Presentation
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8/27: Linear Programming
• Lecture: LP
• Small Groups
• Homework
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.
LP - 4 Characteristics
• Objective Function
• Constraints
• Alternative Courses of Action
• Linear Equations
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.
Toy Company
Formulate a linear program set to maximize the company's profit.
Terminology
• Z : variable to be optimized.
• x1, x2, x3,… : decision variables.
So we write
Max Z ( profit ) = (some combo of x1...xX)
S. T. ("subject to"): (the constraints)
Toy Company
• What are we supposed to maximize?
• What factors play a part in that?
• What constraints are there to the profit?
• 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.
• 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.
• 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.
• 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.
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
Toy Company
• Let x1 = toy trucks, w/ a $7 profit each
• x2 = dolls, w/ a $5 profit each
• x3 = chess sets w/ a $12 profit each
• So Max Z (profit) = 7 x1 + 5 x2 + 12 x3
Toy Company - constraints
• Hand Time: total of 8 hours. -- or 480 min.
• Truck - 10 min.
• Doll - 8 min.
• Chess Set - 3 min.
• so 10 x1 + 8 x2 + 3 x3 <= 480
Toy Company - constraints
• Machine Time: total of 8 hrs. -- or 480 min.
• Truck - 3 min.
• Doll - 10 min.
• Chess Set - 20 min.
• so 3 x1 + 10 x2 + 20 x3 <= 480
Toy Company - constraints
• Wood: total of 1000 ft. -- or 12,000 in.
• Truck - 15 in.
• Doll - 11 in.
• Chess Set - 31 in.
• so 15 x1 + 11 x2 + 31 x3 <= 12000
Toy Company - constraints
• Other constraints:
• Integers: x1, x2, x3 must be integers.
• Positive: x1, x2, x3 >= 0
Toy Company - total LP
• Max Z (profit) = 7 x1 + 5 x2 + 12 x3
S. T.: 10 x1 + 8 x2 + 3 x3 <= 480
3 x1 + 10 x2 + 20 x3 <= 480
15 x1 + 11 x2 + 31 x3 <= 12000
x1, x2, x3 >= 0
x1, x2, x3 must be integers.
EX: Camping Trip.
P C F $/lb
beef jerky 10 4 8 13.00
dried potatoes 0 12 2 2.50
granola mix 4 8 11 8.50
NutriGrain bars 2 14 5 9.00
Must have 30 g. protein, 60 g. carbohydrates, and 15 g. of fat. Minimize the cost.
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
LP - Graphical Solution
• Limitation to the method: only TWO decision variables can exist.
LP - Graphical Solution
Maximize Z ( profit ) = 7 x1 + 5 x2
S. T. : 4 x1 + 3 x2 <= 240
2 x1 + 1 x2 <= 100
x1 . x2 >= 0
LP - Graphical Solution
4 x1 + 3 x2 = 240
LP - Graphical Solution
4 x1 + 3 x2 = 240
2 x1 + 1 x2 = 100
LP - Graphical Solution
4 x1 + 3 x2 = 240
2 x1 + 1 x2 = 100FeasibleSolutionRegion
LP - Graphical Solution
4 x1 + 3 x2 = 240
2 x1 + 1 x2 = 100
Max Z = 7 x1 + 5 x2
Z = $400
Z = $410
Z = $350
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