1/17: dscb-305-50 getting started, linear programming administrative issues –syllabus –calendar...

1/17: DSCB-305-50 Getting Started, Linear Programming

• Administrative Issues– Syllabus– Calendar– Get usernames, email addresses, majors

• Linear Programming

• Small Groups

• Homework

Administrative Issues• Syllabus:

– Available at http://adbook.net/teach022/syl305.htm– Midterm exam: Thursday, February 21, due 2/23– Final exam: Thursday, April 25, 6:30 – 9:00 p.m.

• Calendar:– Available at http://adbook.net/teach022/cal305.htm– PowerPoint viewer available at

http://office.microsoft.com/downloads/2000/Ppview97.aspx

• Grade page:– Available at http://adbook.net/teach022/gra305.htm

Administrative Issues• Grade page:

– Available at http://adbook.net/teach022/gra305.htm – Example at http://adbook.net/teach021/gra310.htm

• I need from you a sheet of paper with:– Your name (as you like to be called)– Your username for the grade page

(don’t be obvious)– Your major, or what major(s)

you are considering

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

– Something to be minimized or maximized– Usually maximizing profit, sometimes minimizing

costs

• Constraints– Limitations on pursuit of the objective

• Alternative Courses of Action– Must have more than one possible course of action,

or there is no need for LP

• Linear Equations (or inequalities)

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 CompanyFormulate 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,

• x2 = dolls,

• x3 = chess sets.

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

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.


• Hand Time: total of 8 hours. -- or 480 min.

• Truck - 10 min.

• Doll - 8 min.


• so 10 x1 + 8 x2 + 3 x3 <= 480


• Machine Time: total of 8 hrs. -- or 480 min.

• Truck - 3 min.

• Doll - 10 min.



• Machine Time: total of 8 hrs. -- or 480 min.

• Truck - 3 min.

• Doll - 10 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.


• 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 ?

S. T. : ?

?


Maximize Z ( profit ) = 7 x1 + 5 x2

S. T. : ?

?



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



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

2 x1 + 1 x2 <= 100



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

2 x1 + 1 x2 <= 100

x1 , x2 >= 0


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

x1 . x2 >= 0


4 x1 + 3 x2 = 240

x1

x2


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

x1

x2


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100FeasibleSolutionRegion

x1

x2


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

x1

x2


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

Max Z = 7 x1 + 5 x2

Z = $400

Z = $410

Z = $350

x1

x2


4 x1 + 3 x2 = 240

2 x1 + 1 x2 = 100

Max Z = 7 x1 + 5 x2

Z = $400

Z = $410

Z = $350

Small Groups: Kelson Sporting Gds.• Kelson Co. makes two types of baseball gloves: a

standard model and a catcher’s mitt. There is 900 hours of production time available in the cutting department, 300 hours in finishing, & 100 hours in packaging.

• Write the LP model for this problem.• Use the graphical solution method to find the optimal

solution. What is the total profit possible?

Model Cutting Finishing Packing $ profit

Regular 1 1/2 1/8 $5

Catcher’s 1 ½ 1/3 ¼ $8

Homework• ch. 1 #11, 18

• ch. 2 #11, 21

• Reminder: should be legible, labeled with your name and the date, and make sense.