Intro Management Science 472.21 2 Fall 2011 Bruce Duggan Providence University College

Intro Management Science

472.212

Fall 2011Bruce Duggan

Providence University College

This Week

Review Cases from ch 1

Linear Programming ch 2 formulas & graphs

Case 1: Clean Clothes Corner

A. Current volume?

she’s just breaking even

v =cf

p-cv

v =$1,700.00

$1.10 - $0.25


B. Increase needed to break even?

v =cf

p-cv

v =$16,200.00/12$1.10 - $0.25


C. Monthly profit?

Z = vp - cf - vcv

Z = 4,300.00 $1.10

- ($1,700.00 + $1,350.00)

- 4,300 $0.25


D. If lower price?

BE?

Z? Z = vp - cf - vcv

v =cf

p-cv


E. Which is the better choice?

Z with new equipment?

Z without new equipment?

Case 2: Ocobee

Which option is better?

make the rafts yourself?

buy them from North Carolina?

ch 2: Linear Programming

George Dantzighttp://forum.stanford.edu/blog/?p=27

Linear Programming

Jargon Linear programming

• l.p.

• “figuring stuff out with basic algebra”

Model formulation• Stating our problem in words/math/graphs

Sensitivity analysis• “What happens if…?”

Linear Programming

Jargon Why is there jargon?

handout

Applications

Kellogg pg 35

Nutrition Coordinating Center pg 46

Soquimich pg 51

Example: Maximization

The St. Adolphe Historical Museum We have a group of older volunteers

• The St. Adolphe Craft League

They’ve offered to make toothpick tchochkes to sell at the gift shop

• Red River ox carts

• the first church in St. Adolphe

We can sell everything

they make

St. Adolphe Craft League

They want to know: How many ox carts? How many churches?

Goal To make the most profit possible for the

museum


Resource availability 40 hrs of labor 120 boxes of toothpicks

Decision variables x1 = number of ox carts to make

x2 = number of churches to make


Product resource requirements and unit profit:

41

32

Product

cart

church

Profit ($/unit)

40

50

Material (boxes/unit)

Labour (hr/unit)

Resource Requirements


Objective function Maximize Z = $40x1 + $50x2

Resource constraints 1x1 + 2x2 40 hours of labor

4x1 + 3x2 120 boxes of toothpicks

Non-Negativity constraints x1 0; x2 0

Maximize Z = $40x1 + $50x2

subject to:

1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


Problem definition Complete linear programming model

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


Model formulation: l.p.

no computers yet


words

math graphs

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


x2

0 10 20 30 40

10

20

30

40

x1

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


x2

0 10 20 30 40

10

20

30

40

x1

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


x2

0 10 20 30 40

10

20

30

40

x1

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0


x2

0 10 20 30 40

10

20

30

40

x1


x2

0 10 20 30 40

10

20

30

40

x1

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0

x1 = 0 ox cartsx2 = 20 churchesZ = $1,000



Linear Programming

lp has 2 main tools maximization

• most profit

minimization• least cost

Z means profit

Z means cost

Example: Minimization

Friesen Farms section of land needs at least

• 16 lb nitrogen

• 24 lb phosphate

2 brands of fertilizer available• DeSallaberry Superior

• Carmen Crop

Goal• Meet fertilizer needs at minimum cost

Problem• How much of each brand should you buy?

words

math

graphs

Friesen Farms

Chemical Contributions

ProductNitrogen (lb/bag)

Phosphate (lb/bag)

Cost ($/bag)

DeSallaberry Superior

Carmen Crop

words

math

graphs

Friesen Farms

Chemical Contributions

ProductNitrogen (lb/bag)

Phosphate (lb/bag)

Cost ($/bag)

DeSallaberry Superior

2 4 $6

Carmen Crop 4 3 $3

words

math

graphs

Friesen Farms

Objective function Minimize Z = $6x1 + $3x2

Decision variables x1 = bags of DeSallaberry to buy

x2 = bags of Carmen to buy

words

math

graphs

Friesen Farms

Objective function Minimize Z = $6x1 + $3x2

Model constraints 2x1 + 4x2 16 (lb) nitrogen constraint

4x1 + 3x2 24 (lb) phosphate constraint

x1, x2 0 non-negativity constraint

words

math

graphs

Min Z = $6x1 + $3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

Friesen Farms


words

math

graphs

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

Friesen Farms

x2

0 2 4 6 8

2

4

6

8

x1

words

math

graphs

Friesen Farms

x2

0 2 4 6 8

2

4

6

8

x1

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

Friesen Farms

x2

0 2 4 6 8

2

4

6

8

x1

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

x2

0 2 4 6 8

2

4

6

8

x1

Friesen Farms

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

x1 = 0 bags of DeSallaberry x2 = 8 bags of CarmenZ = $24

x1 = 5 DeSallaberryx2 = 2 CarmenZ = $36

x1 = 8 DeSallaberry x2 = 0 CarmenZ = $48

x2

0 2 4 6 8

2

4

6

8

x1

Friesen Farms

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

x2

0 2 4 6 8

2

4

6

8

x1

Friesen Farms

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

x2

0 2 4 6 8

2

4

6

8

x1

Friesen Farms

Min Z = $6x1 + 3x2

s.t. 2x1 + 4x2 ≥ 16

4x1 + 3x2 ≥ 24

x1, x2 0

Surplus Variableswhat’s left over - don’t contribute to - “slack”

x1 = 0 bags of DeSallaberry x2 = 8 bags of Carmens1 = 16 lb of nitrogens2 = 0 lb of phosphateZ = $2400

x1 = 4.8 DeSallaberryx2 = 1.6 Carmens1 = 0 nitrogens2 = 0 phosphateZ = $3360

x1 = 8 DeSallaberry x2 = 0 Carmens1 = 0 nitrogens2 = 8 phosphateZ = $4800

On computer

much easier to do

goals up to now the idea the formulas

l.p.

usual characteristics & limitations clear goal choice amongst alternatives “certainty”

• non-probabilistic

constraints exist

relationships• linear

• slope constant

additivity divisibility for graphical solution

• 2 variables

Assignment

ch 2 problems in group

• 2

• 38

yourself• 1

• 16

Max Z = $40x1 + $50x2

s.t. 1x1 + 2x2 40

4x1 + 3x2 120

x1, x2 0



Next Week

review ch 2 problems

ch 3 on the computer sensitivity analysis

Documents

Intro Management Science 472.21 2 Fall 2011 Bruce Duggan Providence University College