8/27: linear programming

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 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


• 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


• 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


• 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.


Maximize Z ( profit ) = 7 x1 + 5 x2

S. T. : 4 x1 + 3 x2 <= 240

2 x1 + 1 x2 <= 100

x1 . x2 >= 0


4 x1 + 3 x2 = 240


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100FeasibleSolutionRegion


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

Max Z = 7 x1 + 5 x2

Z = $400

Z = $410

Z = $350

8/27: linear programming

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