intro management science 472.21 2 fall 2011 bruce duggan providence university college
TRANSCRIPT
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
Case 1: Clean Clothes Corner
B. Increase needed to break even?
v =cf
p-cv
v =$16,200.00/12$1.10 - $0.25
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 $0.25
Case 1: Clean Clothes Corner
D. If lower price?
BE?
Z? Z = vp - cf - vcv
v =cf
p-cv
Case 1: Clean Clothes Corner
E. Which is the better choice?
Z with new equipment?
Z without new equipment?
Case 2: Ocobee
Which option is better?
make the rafts yourself?
buy them from North Carolina?
ch 2: Linear Programming
George Dantzighttp://forum.stanford.edu/blog/?p=27
Linear Programming
Jargon Linear programming
• l.p.
• “figuring stuff out with basic algebra”
Model formulation• Stating our problem in words/math/graphs
Sensitivity analysis• “What happens if…?”
Linear Programming
Jargon Why is there jargon?
handout
Applications
Kellogg pg 35
Nutrition Coordinating Center pg 46
Soquimich pg 51
Example: Maximization
The St. Adolphe Historical Museum We have a group of older volunteers
• The St. Adolphe Craft League
They’ve offered to make toothpick tchochkes to sell at the gift shop
• Red River ox carts
• the first church in St. Adolphe
We can sell everything
they make
St. Adolphe Craft League
They want to know: How many ox carts? How many churches?
Goal To make the most profit possible for the
museum
St. Adolphe Craft League
Resource availability 40 hrs of labor 120 boxes of toothpicks
Decision variables x1 = number of ox carts to make
x2 = number of churches to make
St. Adolphe Craft League
Product resource requirements and unit profit:
41
32
Product
cart
church
Profit ($/unit)
40
50
Material (boxes/unit)
Labour (hr/unit)
Resource Requirements
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
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
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
St. Adolphe Craft League
words
math graphs
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
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
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
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
St. Adolphe Craft League
x2
0 10 20 30 40
10
20
30
40
x1
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 = $1,360
Linear Programming
lp has 2 main tools maximization
• most profit
minimization• least cost
Z means profit
Z means cost
Example: Minimization
Friesen Farms section of land needs at least
• 16 lb nitrogen
• 24 lb phosphate
2 brands of fertilizer available• DeSallaberry Superior
• Carmen Crop
Goal• Meet fertilizer needs at minimum cost
Problem• How much of each brand should you buy?
words
math
graphs
Friesen Farms
Chemical Contributions
ProductNitrogen (lb/bag)
Phosphate (lb/bag)
Cost ($/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
Model formulation: l.p.
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
St. Adolphe Craft League
Model formulation: l.p.
Next Week
review ch 2 problems
ch 3 on the computer sensitivity analysis