introduction to genetic algorithms - iitg.ac.in algorithms.pdf · addison – wesley (1989) john h....

R.K. Bhattacharjya/CE/IITG

Introduction To Genetic Algorithms

Dr. Rajib Kumar BhattacharjyaDepartment of Civil Engineering

IIT GuwahatiEmail: [email protected]

7 November 2013

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References

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� D. E. Goldberg, ‘Genetic Algorithm In Search, Optimization And Machine Learning’, New York: Addison – Wesley (1989)

� John H. Holland ‘Genetic Algorithms’, Scientific American Journal, July 1992.

� Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5.


Introduction to optimization

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Global optimaLocal optima

Local optima

Local optima

Local optima

f

X


Introduction to optimization

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Multiple optimal solutions


Genetic Algorithms

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Genetic Algorithms are the heuristic search

and optimization techniques that mimic theprocess of natural evolution.


Principle Of Natural Selection

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“Select The Best, Discard The Rest”


An Example….

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Giraffes have long necks

� Giraffes with slightly longer necks could feed on leaves of higher branches when all lower ones had been eaten off.

� They had a better chance of survival.

� Favorable characteristic propagated through generations of giraffes.

� Now, evolved species has long necks.


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This longer necks may have due to the effect of mutationinitially. However as it was favorable, this was propagated overthe generations.

An Example….


Evolution of species

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Initial Population of animals

Struggle For ExistenceSurvival Of the Fittest

Surviving Individuals Reproduce, Propagate Favorable Characteristics

Millio

ns Of

Years

Evolved Species


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Thus genetic algorithms implement the optimization strategies by simulating evolution of species through natural selection


Simple Genetic Algorithms

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Start

populationInitialize population

Solution?Optimum Solution?

Evaluate Solutions

YES

Stop

T = 0

T=T+1

Selection

CrossoverCrossover

MutationMutation

NO


Simple Genetic Algorithm

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function sga (){Initialize population;Calculate fitness function;

While(fitness value != termination criteria){Selection;

Crossover;

Mutation;

Calculate fitness function;}

}


GA Operators and Parameters

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� Selection

� Crossover

� Mutation

� Now we will discuss about genetic operators


Selection

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The process that determines which solutions areto be preserved and allowed to reproduce andwhich ones deserve to die out.

The primary objective of the selection operator isto emphasize the good solutions and eliminatethe bad solutions in a population while keepingthe population size constant.

“Selects the best, discards the rest”


Functions of Selection operator

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Identify the good solutions in a population

Make multiple copies of the good solutions

Eliminate bad solutions from the population so that multiple copies of good solutions can be placed in the population

Now how to identify the good solutions?


Fitness function

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A fitness function value quantifies the optimality of a solution. The value is used to rank a particular solution against all the other solutions

A fitness value is assigned to each solution depending on how close it is actually to the optimal solution of the problem

A fitness value can be assigned to evaluate the solutions


Assigning a fitness value

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Considering c = 0.0654

23

d

h


Selection operator

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� There are different techniques to implement selection in Genetic Algorithms.

� They are:

� Tournament selection

� Roulette wheel selection

� Proportionate selection

� Rank selection

� Steady state selection, etc


Tournament selection

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� In tournament selection several tournaments are played among a few individuals. The individuals are chosen at random from the population.

� The winner of each tournament is selected for next generation.

� Selection pressure can be adjusted by changing the tournament size.

� Weak individuals have a smaller chance to be selected if tournament size is large.


Tournament selection

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20

22

13 +30

22

40

32

32

25

7 +45

25

25

32

25

22

40

22

13 +30

7 +45

13 +30

22

32

25 13 +30

22

25

Selected

Best solution will have two copies

Worse solution will have no copies

Other solutions will have two, one or zero copies


Roulette wheel and proportionate selection

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Chrom # Fitness

1 50

2 6

3 36

4 30

5 36

6 28

186

% of RW

26.88

3.47

20.81

17.34

20.81

16.18

100.00

EC

1.61

0.19

1.16

0.97

1.16

0.90

6

AC

2

0

1

1

1

1

6

121%

23%

321%

418%

521%

616%

Roulet wheel

Parents are selected

according to their

fitness values

The better

chromosomes have

more chances to be

selected


Rank selection

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Chrom # Fitness

1 37

2 6

3 36

4 30

5 36

6 28

Chrom # Fitness

1 37

3 36

5 36

4 30

6 28

2 6

Sort according to fitness

Assign raking

Rank

6

5

4

3

2

1

Chrom #

1

3

5

4

6

2

% of RW

29

24

19

14

10

5

Chrom #

1

3

5

4

6

2

Roulette wheel

6 5 4 3 2 1

EC AC

1.714 2

1.429 1

1.143 1

0.857 1

0.571 1

0.286 0

Chrom #

1

3

5

4

6

2


Steady state selection

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In this method, a few good chromosomes are used for creating new offspring in every iteration.

The rest of population migrates to the next generation without going through the selection process.

Then some bad chromosomes are removed and the new offspring is placed in their places

Good

Bad

New offspring

Good

New offspring


How to implement crossover

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Source: http://www.biologycorner.com/bio1/celldivision-chromosomes.html

The crossover operator is used to create new solutions from the existing solutions available in the mating pool after applying selection operator.

This operator exchanges the gene information between the solutions in the mating pool.

0 1 0 0 1 1 0 1 1 0

Encoding of solution is necessary so that our

solutions look like a chromosome


Encoding

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The process of representing a solution in the

form of a string that conveys the necessary information.

Just as in a chromosome, each gene controls a particular characteristic of the individual, similarly, each bit in the string represents a characteristic of the solution.


Encoding Methods

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� Most common method of encoding is binary coded.Chromosomes are strings of 1 and 0 and each positionin the chromosome represents a particular characteristicof the problem

Decoded value 52 26

Mapping between decimal

and binary value


Encoding Methods

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d

h (d,h) = (8,10) cm

Chromosome = [0100001010]

Defining a string [0100001010]

d h


Crossover operator

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The most popular crossover selects any two solutions strings randomly from the mating pool and some portion of the strings is exchanged between the strings.

The selection point is selected randomly.

A probability of crossover is also introduced in order to give freedom to an individual solution string to determine whether the solution would go for crossover or not.

Solution 1

Solution 2

Child 1

Child 2


Binary Crossover

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Source: Deb 1999


Mutation operator

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Though crossover has the main responsibility to search for theoptimal solution, mutation is also used for this purpose.

Mutation is the occasional introduction of new features in to thesolution strings of the population pool to maintain diversity in thepopulation.

Before mutation After mutation


Binary Mutation

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� Mutation operator changes a 1 to 0 or vise versa, with a mutationprobability of .

� The mutation probability is generally kept low for steady convergence.

� A high value of mutation probability would search here and there like arandom search technique.

Source: Deb 1999


Elitism

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� Crossover and mutation may destroy the best solution of the population pool

� Elitism is the preservation of few best solutions of the population pool

� Elitism is defined in percentage or in number


Nature to Computer Mapping

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Nature ComputerPopulation

Individual

Fitness

Chromosome

Gene

Set of solutions

Solution to a problem

Quality of a solution

Encoding for a solution

Part of the encoding solution


An example problem

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Consider 6 bit string to represent the solution, then000000 = 0 and 111111 =

Assume population size of 4

Let us solve this problem by hand calculation


An example problem

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Actual count

2

1

0

1

Sol No Binary String

1 100101

2 001100

3 111010

4 101110

DV

37

12

58

46

x value

0.587

0.19

0.921

0.73

f

0.96

0.56

0.25

0.75

Avg

Max

F

0.96

0.56

0.25

0.75

0.563

0.96

Relative Fitness

0.38

0.22

0.10

0.30

Expected count

1.53

0.89

0.39

1.19

Initialize population

Calculate decoded value

Calculate real value

Calculate objective function value

Calculate fitness value

Calculate relative fitness value

Calculate expected count

Calculate actual count

Selection: Proportionate selectionInitial populationFitness

calculationDecoding


An example problem: Crossover

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Sol No Matting pool

1 100101

2 001100

3 100101

4 101110

f F

0.97 0.97

0.60 0.60

0.75 0.75

0.96 0.96

Avg 0.8223

Max 0.97

CS

3

3

2

2

New Binary String

100100

001101

101110

100101

DV

36

13

46

37

x value

0.57

0.21

0.73

0.59

Matting poolRandom generation of

crossover siteNew population

Crossover: Single point


An example problem: Mutation

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Sol No

1

2

3

4

Population after crossover

100100

001101

101110

100101

Population after mutation

100000

101101

100110

101101

f F

1.00 1.00

0.78 0.78

0.95 0.95

0.78 0.78

Avg 0.878

Max 1.00

DV

32

45

38

45

x value

0.51

0.71

0.60

0.71

Mutation


Real coded Genetic Algorithms

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� Disadvantage of binary coded GA

� more computation

� lower accuracy

� longer computing time

� solution space discontinuity

� hamming cliff



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� The standard genetic algorithms has the following steps

1. Choose initial population

2. Assign a fitness function

3. Perform elitism

4. Perform selection

5. Perform crossover

6. Perform mutation

� In case of standard Genetic Algorithms, steps 5 and 6 require bitwise manipulation.



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8 6 3 7 6

2 9 4 8 9

8 6 4 8 9

2 9 3 7 6

Simple crossover: similar to binary crossover

P1

P2

C1

C2



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Linear Crossover

• Parents: (x1,…,xn ) and (y1,…,yn )• Select a single gene (k) at random

• Three children are created as,

) ..., ,5.05.0 , ..., ,( 1 nkkk xxyxx ⋅+⋅

) ..., ,5.05.1 , ..., ,( 1 nkkk xxyxx ⋅−⋅

) ..., ,5.15.0- , ..., ,( 1 nkkk xxyxx ⋅+⋅

• From the three children, best two are selected for the

next generation



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Single arithmetic crossover

• Parents: (x1,…,xn ) and (y1,…,yn )• Select a single gene (k) at random

• child1 is created as,

• reverse for other child. e.g. with α = 0.5

) ..., ,)1( , ..., ,( 1 nkkk xxyxx ⋅−+⋅ αα

0.1 0.3 0.1 0.3 0.7 0.2 0.5 0.1 0.2

0.5 0.7 0.7 0.5 0.2 0.8 0.3 0.9 0.4

0.1 0.3 0.1 0.3 0.7 0.5 0.5 0.1 0.2

0.5 0.7 0.7 0.5 0.2 0.5 0.3 0.9 0.4



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Simple arithmetic crossover

• Parents: (x1,…,xn ) and (y1,…,yn )• Pick random gene (k) after this point mix values

• child1 is created as:


))1(n

y ..., ,1

)1(1

, ..., ,1

(n

xk

xk

yk

xx ⋅−+⋅+

⋅−++

⋅ αααα

0.1 0.3 0.1 0.3 0.7 0.2 0.5 0.1 0.2

0.5 0.7 0.7 0.5 0.2 0.8 0.3 0.9 0.4

0.1 0.3 0.1 0.3 0.7 0.5 0.4 0.5 0.3

0.5 0.7 0.7 0.5 0.2 0.5 0.4 0.5 0.3



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Whole arithmetic crossover

• Most commonly used

• Parents: (x1,…,xn ) and (y1,…,yn )• child1 is:


yx ⋅−+⋅ )1( αα

0.1 0.3 0.1 0.3 0.6 0.2 0.5 0.1 0.2

0.5 0.7 0.7 0.5 0.2 0.8 0.3 0.9 0.4

0.3 0.5 0.4 0.4 0.4 0.5 0.4 0.5 0.3

0.3 0.5 0.4 0.4 0.4 0.5 0.4 0.5 0.3


Simulated binary crossover

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� Developed by Deb and Agrawal, 1995)

Where, a random number

is a parameter that controls the crossover process. A high value of the parameter will create near-parent solution


Random mutation

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Where is a random number between [0,1]

Where, is the user defined maximum perturbation


Normally distributed mutation

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A simple and popular method

Where is the Gaussian probability distribution with zero mean


Polynomial mutation

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��,�� = ��,�� + �� − ��

� = � 2�� / �� − 1 1 − 2 1 − �� / �� If �� < 0.5If �� ≥ 0.5

Deb and Goyal,1996 proposed


Multi-modal optimization

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Solve this problem using simple Genetic Algorithms


After Generation 200

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The population are in and around the global optimal solution


Multi-modal optimization

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Niche count

Modified fitness

Sharing function

Simple modification of Simple Genetic Algorithms can capture all the optimal solution of the problem including global optimal solutions

Basic idea is that reduce the fitness of crowded solution, which can be implemented using following three steps.


Hand calculation

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Maximize

Sol String Decoded

value

x f

1 110100 52 1.651 0.890

2 101100 44 1.397 0.942

3 011101 29 0.921 0.246

4 001011 11 0.349 0.890

5 110000 48 1.524 0.997

6 101110 46 1.460 0.992


Distance table

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dij 1 2 3 4 5 6

1 0 0.254 0.73 1.302 0.127 0.191

2 0.254 0 0.476 1.048 0.127 0.063

3 0.73 0.476 0 0.572 0.603 0.539

4 1.302 1.048 0.572 0 1.175 1.111

5 0.127 0.127 0.603 1.175 0 0.064

6 0.191 0.063 0.539 1.111 0.064 0


Sharing function values

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sh(dij) 1 2 3 4 5 6 nc

1 1 0.492 0 0 0.746 0.618 2.856

2 0.492 1 0.048 0 0.746 0.874 3.16

3 0 0.048 1 0 0 0 1.048

4 0 0 0 1 0 0 1

5 0.746 0.746 0 0 1 0.872 3.364

6 0.618 0.874 0 0 0.872 1 3.364


Sharing fitness value

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Sol String Decoded

value

x f nc f’

1 110100 52 1.651 0.890 2.856 0.312

2 101100 44 1.397 0.942 3.160 0.300

3 011101 29 0.921 0.246 1.048 0.235

4 001011 11 0.349 0.890 1.000 0.890

5 110000 48 1.524 0.997 3.364 0.296

6 101110 46 1.460 0.992 3.364 0.295


Solutions obtained using modified fitness value

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Evolutionary Strategies

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� ES use real parameter value

� ES does not use crossover operator

� It is just like a real coded genetic algorithms with selection and mutation operators only


Two members ES: (1+1) ES

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� In each iteration one parent is used to create one offspring by using Gaussian mutation operator



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� Step1: Choose a initial solution and a mutation strength

� Step2: Create a mutate solution

� Step 3: If , replace with

� Step4: If termination criteria is satisfied, stop, else go to step 2



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� Strength of the algorithm is the proper value of

� Rechenberg postulate

� The ratio of successful mutations to all the mutations should be 1/5. If this ratio is greater than 1/5, increase mutation strength. If it is less than 1/5, decrease the mutation strength.



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� A mutation is defined as successful if the mutated offspring is better than the parent solution.

� If is the ratio of successful mutation over n trial, Schwefel (1981) suggested a factor in the following update rule


Matlab code

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Some results of 1+1 ES

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Optimal Solution isX*= [3.00 1.99]

Objective function value f = 0.0031007

1

22

55

10

10

10

2020

20

20

20

50

50

50

50

50

10

0

100

100100

100

100

20

020

0

200

200200

X

Y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Multimember ES

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ES

Step1: Choose an initial population of solutions and mutation strength

Step2: Create mutated solution

Step3: Combine and , and choose the best solutions

Step4: Terminate? Else go to step 2


Multimember ES

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ES

Through mutation

Through selection


Multimember ES

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ES

Offspring

Through mutation

Through selection


Multi-objective optimization

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Price

Comfort



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Two objectives are

• Minimize weight

• Minimize deflection



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� More than one objectives

� Objectives are conflicting in nature

� Dealing with two search space� Decision variable space

� Objective space

� Unique mapping between the objectives and often the mapping is non-linear

� Properties of the two search space are not similar

� Proximity of two solutions in one search space does not mean a proximity in other search space



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Vector Evaluated Genetic Algorithm (VEGA)

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f1

f2

f3

f4

…

fn

P1

P2

P3

P4

Pn

…

Crossover and

Mutation

Old population Mating pool New population

Propose by Schaffer (1984)


Non-dominated selection heuristic

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Give more emphasize on the non-dominated solutions of the population

This can be implemented by subtracting ∈ from the dominated solution fitness value

Suppose �� is the number of sub-population and �� is the non-dominated solutions. Then total reduction is �� − �� ∈. The total reduction is then redistributed among the non-dominated solution by adding an amount �� − �� ∈/��This method has two main implications

Non-dominated solutions are given more importance

Additional equal emphasis has been given to all the non-dominated solution


Weighted based genetic algorithm (WBGA)

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� The fitness is calculated

� The spread is maintained using the sharing function approach

Niche count Modified fitness

Sharing function


Multiple objective genetic algorithm (MOGA)

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x1

x2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

f1

f2

Solution space Objective space


Maximize f1

Maximize f2




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Fonseca and Fleming (1993) first introduced multiple objective genetic algorithm (MOGA)

The assigned fitness value based on the non-dominated ranking.

The rank is assigned as �� = 1 + ��where �� is the ranking of the �!solution and �� is the number of solutions that dominate the solution.

1

1

1

1

4

4

6


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� Fonseca and Fleming (1993) maintain the diversity among the non-dominated solution using nichingamong the solution of same rank.

� The normalize distance was calculated as,

� The niche count was calculated as,



NSGA

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� Srinivas and Deb (1994) proposed NSGA

� The algorithm is based on the non-dominated sorting.

� The spread on the Pareto optimal front is maintained using sharing function


NSGA II

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� Non-dominated Sorting Genetic Algorithms

� NSGA II is an elitist non-dominated sorting Genetic Algorithm to solve multi-objective optimization problem developed by Prof. K. Deb and his student at IIT Kanpur.

� It has been reported that NSGA II can converge to the global Pareto-optimal front and can maintain the diversity of population on the Pareto-optimal front


Non-dominated sorting

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Objective 1 (Minimize)

Ob

ject

ive

2 (

Min

imiz

e)

1

2

3

4

5

6

Objective 1 (Minimize)

Ob

ject

ive

2 (

Min

imiz

e)

1

2 3

4

5 Infeasible Region

Feasible Region


Calculation crowding distance

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Cd, the crowded distance is the perimeter of the rectangle constituted by the two neighboring solutions

Cd value more means that the solution is less crowded

Cd value less means that the solution is more crowded

− 1

+ 1

Objective 1

Objective 2


Crowded tournament operator

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� A solution wins a tournament with another solution ", � If the solution has better rank than ", i.e. �� < �#� If they have the same rank, but has a better crowding distance than ", i.e. �� = �# and $� > $#.


Replacement scheme of NSGA II

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P

Q

��&�'

�(�)

��&�'Selected

Rejected

Rejected based on crowding distanceNon dominated sorting

Initialize population of size N

Calculate all the objective functions

Rank the population according to non-

dominating criteria

Selection

Crossover

Mutation

Calculate objective function of the

new population

Combine old and new population

Non-dominating ranking on the

combined population

Replace parent population by the better

members of the combined population

Calculate crowding distance of all

the solutions

Get the N member from the

combined population on the basis

of rank and crowding distance

Termination

Criteria?

Pareto-optimal solution

Yes

No

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Constraints handling in GA

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-5

5

15

25

35

45

55

0 2 4 6 8 10 12

f(X

)

X

*(�)



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-5

5

15

25

35

45

55

0 2 4 6 8 10 12

f(X

)

X

*(�) * � + - . �



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-5

5

15

25

35

45

55

0 2 4 6 8 10 12

f(X

)

X

*(�) * � + - . �

*/01 + . �


Constrain handling in GA

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2 = * 3

Minimize * 3Subject to .# � ≤ 0ℎ6 � = 0 " = 1,2,3, … , 9: = 1,2,3, … , ;

= */01 + < .# � + < ℎ6 �=6>�

?#>�

If 3 is feasible

Otherwise

Deb’s approach


THANKS

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introduction to genetic algorithms - iitg.ac.in algorithms.pdf · addison – wesley (1989) john h....

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