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Intro Management Science
472.212
Fall 2011Bruce Duggan
Providence University College
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This Week
Review Cases from ch 1
Linear Programming ch 2 formulas & graphs
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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
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Case 1: Clean Clothes Corner
B. Increase needed to break even?
v =cf
p-cv
v =$16,200.00/12$1.10 - $0.25
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Case 1: Clean Clothes Corner
C. Monthly profit?
Z = vp - cf - vcv
Z = 4,300.00 $1.10
- ($1,700.00 + $1,350.00)
- 4,300 $0.25
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Case 1: Clean Clothes Corner
D. If lower price?
BE?
Z? Z = vp - cf - vcv
v =cf
p-cv
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Case 1: Clean Clothes Corner
E. Which is the better choice?
Z with new equipment?
Z without new equipment?
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Case 2: Ocobee
Which option is better?
make the rafts yourself?
buy them from North Carolina?
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ch 2: Linear Programming
George Dantzighttp://forum.stanford.edu/blog/?p=27
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Linear Programming
Jargon Linear programming
• l.p.
• “figuring stuff out with basic algebra”
Model formulation• Stating our problem in words/math/graphs
Sensitivity analysis• “What happens if…?”
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Linear Programming
Jargon Why is there jargon?
handout
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Applications
Kellogg pg 35
Nutrition Coordinating Center pg 46
Soquimich pg 51
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Example: Maximization
The St. Adolphe Historical Museum We have a group of older volunteers
• The St. Adolphe Craft League
They’ve offered to make toothpick tchochkes to sell at the gift shop
• Red River ox carts
• the first church in St. Adolphe
We can sell everything
they make
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St. Adolphe Craft League
They want to know: How many ox carts? How many churches?
Goal To make the most profit possible for the
museum
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St. Adolphe Craft League
Resource availability 40 hrs of labor 120 boxes of toothpicks
Decision variables x1 = number of ox carts to make
x2 = number of churches to make
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St. Adolphe Craft League
Product resource requirements and unit profit:
41
32
Product
cart
church
Profit ($/unit)
40
50
Material (boxes/unit)
Labour (hr/unit)
Resource Requirements
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St. Adolphe Craft League
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
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Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
St. Adolphe Craft League
Problem definition Complete linear programming model
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Max Z = $40x1 + $50x2
s.t. 1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
St. Adolphe Craft League
Model formulation: l.p.
no computers yet
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St. Adolphe Craft League
words
math graphs
Max Z = $40x1 + $50x2
s.t. 1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
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St. Adolphe Craft League
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
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St. Adolphe Craft League
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
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St. Adolphe Craft League
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
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Max Z = $40x1 + $50x2
s.t. 1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
St. Adolphe Craft League
x2
0 10 20 30 40
10
20
30
40
x1
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St. Adolphe Craft League
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
x1 = 30 ox cartsx2 = 0 churchesZ = $1,200
x1 = 24 ox cartsx2 = 8 churchesZ = $1,360
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Linear Programming
lp has 2 main tools maximization
• most profit
minimization• least cost
Z means profit
Z means cost
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Example: Minimization
Friesen Farms section of land needs at least
• 16 lb nitrogen
• 24 lb phosphate
2 brands of fertilizer available• DeSallaberry Superior
• Carmen Crop
Goal• Meet fertilizer needs at minimum cost
Problem• How much of each brand should you buy?
words
math
graphs
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Friesen Farms
Chemical Contributions
ProductNitrogen (lb/bag)
Phosphate (lb/bag)
Cost ($/bag)
DeSallaberry Superior
Carmen Crop
words
math
graphs
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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
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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
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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
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Min Z = $6x1 + $3x2
s.t. 2x1 + 4x2 ≥ 16
4x1 + 3x2 ≥ 24
x1, x2 0
Friesen Farms
Model formulation: l.p.
words
math
graphs
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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
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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
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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
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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
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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
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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
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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
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On computer
much easier to do
goals up to now the idea the formulas
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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
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Assignment
ch 2 problems in group
• 2
• 38
yourself• 1
• 16
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Max Z = $40x1 + $50x2
s.t. 1x1 + 2x2 40
4x1 + 3x2 120
x1, x2 0
St. Adolphe Craft League
Model formulation: l.p.
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Next Week
review ch 2 problems
ch 3 on the computer sensitivity analysis