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WOOD 492 WOOD 492

WOOD 492 - PowerPoint Presentation

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WOOD 492 - PPT Presentation

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

capacity wood vahid 492 wood capacity 492 vahid saba 2012 sept framed constraints week custom maximize profits cabinet decision

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Slide1

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

6

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