wood 492 modelling for decision support lecture 2 introduction to linear programming
TRANSCRIPT
WOOD 492 MODELLING FOR DECISION SUPPORT
Lecture 2
Introduction to Linear Programming
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Last Class
• Introduction to Operations Research
• Examples of OR in forestry
• Introduction to mathematical models– Objective function, decision variables, constraints
Sept 7, 2012
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Example: Custom Cabinets company
• Use excess capacity for 2 new products: Pine desks & Alder hutches
• Has three departments that are partially committed to producing existing products
• Wants to determine how many units of each new product can be produced each week by using the excess capacity of departments to generate the highest profits
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Department Capacity per unit Available capacity per week
Pine desk Alder hutch
Solid wood 0.25 0 12
Panel 0 0.2 5
Finishing 0.25 0.5 18
Profit per unit $40 $50
Objective
Decision variableConstraints
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Maximize: x1 = number of desks/week
40 x1 + 50 x2 x2= number of hutches/week
Formulating the Linear Program (LP)
• What do you want to maximize or minimize? Profits
• What are the constraints? Available capacity
Subject to:
0.25x1 <= 12 (Solid Wood Capacity)
0.20x2 <= 5 (Panel Capacity)
0.25x1 + 0.50x2 <= 18 (Finishing Capacity)
x1 >= 0
x2 >= 0
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Linear
Note that because of the inequalities there are many feasible solutions. You have to find the best one.
Maximize:
$40x1 + $50x2
Subject to:
0.25x1 <= 12 (Solid Wood Capacity)
0.20x2 <= 5 (Panel Capacity)
0.25x1 + 0.50x2 <= 18 (Finishing Capacity)
x1 >= 0
x2 >= 0
Solving the LP by trial and error
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Try x1 = 10, x2 = 5Z=$650All constraints are satisfied
Custom Cabinet LP1
Matrix format for LP
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# Desks # Hutches
Answer x1 x2
Objective (profits) $40.00 $50.00 sign RHS
Solid Wood 0.25 0 <= 12
Panel 0 0.2 <= 5
Finishing 0.25 0.5 <= 18
non-negative 1 0 >= 0
non-negative 1 >= 0
Custom Cabinet LP2
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0
10
20
30
40
50
60
70
80
0 20 40 60 80
hu
tch
es p
rod
uce
d (
x2)
desks produced (x1)
wood
panel
f inish
Prof it
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Feasible Region
Custom Cabinet LP2
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0
10
20
30
40
50
60
70
80
0 20 40 60 80
hu
tch
es p
rod
uce
d (
x2)
desks produced (x1)
wood
panel
f inish
Prof it
Custom Cabinet LP2
Z=1000
Z=2000
x1 =480.25*48 + 0.5* x2 =18x2 =12
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Example: Whitt Window Company (prob. 3.1-7)
• Has three employees• Makes two types of windows: wood-framed and aluminium-framed• Profits per frame: $180 for wood-framed, $90 for aluminum-framed• Dough makes a maximum of 6 wood frames per day• Linda makes a maximum of 4 aluminium frames per day• Bob forms and cuts a maximum of 48 ft2 of glass per day• Each wood-framed window uses 6 ft2 glass• Each aluminum-framed window uses 8 ft2 glass• How many windows per day to make in order to maximize profits?
Sept 7, 2012
Objective
Decision variables
Constraints
Whitt Windows LP
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Next Week
• Solving an LP with Excel Solver• Simplex Algorithm
Sept 7, 2012