deterministic optimization
TRANSCRIPT
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AN INTRODUCTIONTO DETERMINISTICOPTIMIZATION
Boni Sena, 2013
Various types of LP Linear Programming (LP)
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LECTURER
Instructor : Boni Sena
Contact : 0856-9-2423-611 (Only forclassrepresentative by phone or SMS)
Email : [email protected]
Web : binusmaya.binus.ac.id
Text book :
Ernest F. Haussler, Richard S. Paul, Richard J. Wood,Introductory Mathematical Analysis, Prentice Hall
P. Rama Murthy, Operations Research, New Age
International Frederick S. Hiller, Gerald J. Lieberman, Introduction to
operations research, Mc-Graw Hill
Hamdy A. Taha, Operations research : an introduction,
Pearson Prentice Hall Boni Sena, 2013
mailto:[email protected]:[email protected] -
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YOUARE ...
Boni Sena, 2013
Each of you is responsible for your own decision
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CLASSISLIKE..
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We need some regulations to achieve our goal..
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TOACHIEVEOURGOALS.. Our class will be started at 11:10 WIB The last attendance allowed 20 minute after the class is
started
Students are responsible for all announcement,
assignment, quiz and exam Students must show courtesy and respect to your
friends and speakerduring the class
Dont working on other subject, sleeping, any informal
chatter or activity not related to the lectureAny usage ofcell-phone/blackberry is strictly prohibited
during class period
No late assignments accepted
Boni Sena, 2013
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POINTOF EVALUATION Exams (70%)
Exam 1: 35% (midterm) Exam 2: 35% (final)
Assignment (30%) Project: 15% Quizzes: 15% (Quiz will be held every two weeks)
No make up will be given under any circumstances
REGRADE POLICY
No regrade request will be accepted more than one week after a
paper has been returned Do not write on the original document, but turn in a separate
Grade Groveling form with what you want regraded and why
Ifno additional points given, then 20 points will be deducted
Boni Sena, 2013
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Problem ?
Governor of Jakarta want to make MRT. Can you help them to design
the track of MRT ?
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TRAFFIC JAM
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How to solve traffic jam ?
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Car Assembly LineHow to weld all the weld beads within minimum time ?
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BRAIN CANCER TREATMENT [1]
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Where to deploy radiotheraphy to maximize impactOn carcinogenic cells and minimize damage inOther cells ?
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Mathematics
Model/linear programming
Algorithm
Problem
Maximize the benefit/achieve the goal
Minimize the effort
The best decision
The best decision
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MODELOF DETERMINISTIC OPTIMIZATION [6]
Actual system
Simplifiedsystem
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Model
Mathematics
Stochastic
Statistic
LinearProgramming
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LINEAR PROGRAMMING Linear Programming A model which is used for solving optimization
problems under such assumptions as certainty, linearity, fixedtechnology and constantprofit per unit [8]
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Objective function
Constraints
DecisionVariable
Component of
LP [2]
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LP MODEL FORMULATION
Decision variables mathematical symbols representing levels of activity of an
operation
Objective function
a linear relationship reflecting the objective of an operation most frequent objective of business firms is to maximize
profit
most frequent objective of individual operational units(such as a production or packaging department) is tominimize cost
Constraint a linear relationship representing a restriction on decision
making
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TYPESOF LP [2]
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TYPESOF LP (CONT.)
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TYPESOF LP (CONT.)
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MATHEMATICAL MODELOF LP [2]
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Max/min z = c1x1 + c2x2 + ... + cnxn
subject to constraints :
a11x1 + a12x2 + ... + a1nxn(, =, ) b1a21x1 + a22x2 + ... + a2nxn(, =, ) b2
:
am1x1 + am2x2 + ... + amnxn(, =, ) bm
xj = decision variablesbi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
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EXAMPLEOF LP
Z= 4x1 + 7x2
Subject to
9 x1 + 4x2 100
7x1 + x2 60
x1 ,x2 0
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Objective function
Constraints
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STEP TO FORMULATEMODELOF LINEAR PROGRAMMING
1. Identify the decision variables
2. Quantify the decision consequences to bemaximized or minimized through ObjectiveFunction
3. What limits decisions? Formulate the constraints
Main constraints
Variable-type constraints (sign restriction)
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MAXIMIZATION PROBLEM
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Labor Clay RevenuePRODUCT (hr/unit) (lb/unit) ($/unit)
Bowl 1 4 40Mug 2 3 50
There are 40 hours of labor and 120 pounds of clayavailable each day
Decision variablesx1 = number of bowls to produce
x2 = number of mugs to produce
RESOURCE REQUIREMENTS
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LP Formulation: Example of maximization
Maximize Z= 40 x1 + 50 x2
Subject to
x1 + 2x2 40 hr (labor constraint)
4x1 + 3x2 120 lb (clay constraint)
x1 , x20
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MINIMIZATION PROBLEM
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CHEMICAL CONTRIBUTION
Brand Nitrogen (lb/bag) Phosphate (lb/bag)
Gro-plus 2 4
Crop-fast 4 3
Minimize the production cost to buy nitrogen and phosphate every day.The cost of nitrogen $ 6/day and the cost of phosphate $ 3/day.
There are 16 lb of Nitrogen and 24 lb of phosphate available each day
Decision variables
x1 = number of nitrogen
x2
= number of phosphate
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LP FORMULATION : EXAMPLEOFMINIMIZATION
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Minimize Z= $6x1 + $3x2
subject to
2x1 + 4x2 16 lb of nitrogen
4x1 + 3x2 24 lb of phosphatex1, x2 0
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REFERENCES
1.
Andrs Ramos, 2012. Operation Research and OptimizationTechniques. Lecture Notes. Universidad Pontificia Comillas
2. Beni Asllani. 2006. Linear Programming. Lecture Notes.University of Tennessee. John Wiley & Sons, Inc.
3. Ernest F. Haussler, Richard S. Paul, Richard J. Wood,
Introductory Mathematical Analysis, Prentice Hall4. Frederick S. Hiller, Gerald J. Lieberman, Introduction to
operations research, Mc-Graw Hill
5. Hamdy A. Taha. Operations research : an introduction,Pearson Prentice Hall
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REFERENCES6. Jong Jek Siang. 2011. Riset Operasi Dalam Pendekatan
Algoritmis. Yogyakarta, Andi
7. K. Gita Ayu. 2011. An Introduction to deterministicoptimization. Lecture Notes. Bina Nusantara University.
(Available at binusmaya.binus.ac.id)8. P. Rama Murthy, Operations Research, New Age
International
9. Rosihan Asmara. Operation Research : LinearProgramming. Lecture Notes. Brawijaya University
B i S 2013