multi objective ga
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Elitist Non-dominated Sorting
Genetic Algorithm: NSGA-II
Tushar Goel
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Multi-objective optimization problem Problems with more than one objectives typically
conflicting objectives Cars: Luxury vs. Price
Mathematical formulation
Minimize F(x),
where F(x) = {fi: i = 1, M},
x= {xj: j = 1, N}
Subject to:
C(x) 0, where C = {Ck: k = 1, P}
H(x) = 0, where H = {Hl: l = 1, Q}
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Pareto optimal front Many optimal solutions
Usual approaches:weighted sum strategy,
-constraint modeling,
Multi-objective GA
Algorithm requirements:
Convergence
Spread
M
in
f2
Min f1
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Terminology
Non-domination
criterion
Ranking
f2
f1
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Terminology
Niching parametric
Crowding distance
c = a + b
Ends have infinite
crowding distance
f2
f1
a
b
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Elitism
Elitism: Keep
the bestindividuals
from the
parent andchild
population
f2
f1
Parent
Child
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Flowchart of NSGA-II
Begin: initialize
population (size N)
Evaluate objective
functions
Selection
Crossover
Mutation
Evaluate objective
functionStopping
criteriamet?Yes
No
Childpopulationcreated
Rank
population
Combine parent and
child populations,
rank population
Select N
individuals
Elitism
Report final
population and
Stop
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Elitism Process
Rank 1
Rank 2
Rank 3
Rank 4
Rank 1
Rank 2
Rank 3
Rank 5+
Rank 4
Child
population
Parent
po
pulation
Rank 1
Rank 2
Rank 3
Rank 4
Rank 5Rank 6
Rank 7+
Combined
population
Rank 1
Rank 2
Rank 3
Elitist selection
New
population
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Example: Bicycle Frame Design Objectives
Minimize area
Minimize max. deflection Constraints
Component should be a
valid geometry
Maximum stress < Yield
stress
Maximum deflection