MODELLING FOR DECISION SUPPORT Lecture 2 Introduction to Linear Programming Last Class Introduction to Operations Research Examples of OR in forestry Introduction to mathematical models Objective function decision variables constraints ID: 310350
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WOOD 492 MODELLING FOR DECISION SUPPORT
Lecture 2
Introduction to
Linear ProgrammingSlide2
Last ClassIntroduction to Operations ResearchExamples of OR in forestryIntroduction to mathematical models
Objective function, decision variables, constraints
Sept 7, 2012
Wood 492 - Saba Vahid
2Slide3
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
Sept 7, 2012
Wood 492 - Saba Vahid
3
Department
Capacity
per unit
Available capacity
per week
Pine desk
Alder hutch
Solid wood0.25012Panel00.25Finishing0.250.518Profit per unit $40$50
Objective
Decision variable
ConstraintsSlide4
Maximize:
x
1
=
number
of
desks/week
40 x
1
+ 50 x
2
x
2= number of hutches/week
Formulating the Linear Program (LP)What do you want to maximize or minimize? ProfitsWhat are the constraints? Available capacitySubject to: 0.25x1 <= 12 (Solid Wood Capacity) 0.20x2 <= 5 (Panel Capacity) 0.25x1 + 0.50x2 <= 18 (Finishing Capacity) x1 >= 0 x2 >= 0
Sept 7, 2012
4
Wood 492 - Saba Vahid
LinearSlide5
Note that because of the inequalities there are many feasible solutions. You have to find the best
one.
Maximize
:
$
40x
1
+ $
50x
2
Subject to
:
0.25x
1 <= 12 (Solid Wood Capacity) 0.20x2 <= 5 (Panel Capacity) 0.25x1 + 0.50x2 <= 18 (Finishing Capacity) x1 >= 0 x2 >= 0Solving the LP by trial and errorSept 7, 20125Wood 492 - Saba Vahid
Try x
1 = 10, x
2 =
5Z=$650All constraints are satisfied
Custom Cabinet LP1Slide6
Matrix format for LP
Sept 7, 2012
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Wood 492 - Saba Vahid
# 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 LP2Slide7
Sept 7, 2012
Wood 492 - Saba Vahid
7
Feasible Region
Custom Cabinet LP2Slide8
Sept 7, 2012Wood 492 - Saba Vahid8
Custom Cabinet LP2
Z=1000
Z=2000
x
1
=48
0.25*48 + 0.5* x
2
=18
x
2
=12Slide9
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 ft
2
of glass per day
Each wood-framed window uses 6 ft
2
glass
Each aluminum-framed window uses 8 ft
2
glassHow many windows per day to make in order to maximize profits?
Sept 7, 2012Wood 492 - Saba Vahid9ObjectiveDecision variablesConstraintsWhitt Windows LPSlide10
Next WeekSolving an LP with Excel SolverSimplex AlgorithmSept 7, 2012
Wood 492 - Saba Vahid
10