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