Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic...
Transcript of Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic...
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R.K. Bhattacharjya/CE/IITG
Introduction To Genetic Algorithms
Dr. Rajib Kumar BhattacharjyaDepartment of Civil Engineering
IIT GuwahatiEmail: [email protected]
7 November 2013
1
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R.K. Bhattacharjya/CE/IITG
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.
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Introduction to optimization
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Global optimaLocal optima
Local optima
Local optima
Local optima
f
X
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Introduction to optimization
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Multiple optimal solutions
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Genetic Algorithms
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Genetic Algorithms are the heuristic search
and optimization techniques that mimic theprocess of natural evolution.
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Principle Of Natural Selection
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“Select The Best, Discard The Rest”
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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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R.K. Bhattacharjya/CE/IITG
7 November 2013
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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….
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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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R.K. Bhattacharjya/CE/IITG
7 November 2013
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Thus genetic algorithms implement the optimization strategies by simulating evolution of species through natural selection
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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
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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;}
}
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GA Operators and Parameters
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� Selection
� Crossover
� Mutation
� Now we will discuss about genetic operators
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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”
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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?
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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
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Assigning a fitness value
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Considering c = 0.0654
23
d
h
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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
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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.
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Tournament selection
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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
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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
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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
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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
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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
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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.
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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
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Encoding Methods
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d
h (d,h) = (8,10) cm
Chromosome = [0100001010]
Defining a string [0100001010]
d h
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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
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Binary Crossover
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Source: Deb 1999
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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
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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
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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
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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
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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
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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
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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
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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
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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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R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
39
� 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.
![Page 40: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/40.jpg)
R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
40
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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R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
41
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
![Page 42: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/42.jpg)
R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
42
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
![Page 43: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/43.jpg)
R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
43
Simple arithmetic crossover
• Parents: (x1,…,xn ) and (y1,…,yn )• Pick random gene (k) after this point mix values
• child1 is created as:
• reverse for other child. e.g. with α = 0.5
))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
![Page 44: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/44.jpg)
R.K. Bhattacharjya/CE/IITG
Real coded Genetic Algorithms
7 November 2013
44
Whole arithmetic crossover
• Most commonly used
• Parents: (x1,…,xn ) and (y1,…,yn )• child1 is:
• reverse for other child. e.g. with α = 0.5
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
![Page 45: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/45.jpg)
R.K. Bhattacharjya/CE/IITG
Simulated binary crossover
7 November 2013
45
� 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
![Page 46: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/46.jpg)
R.K. Bhattacharjya/CE/IITG
Random mutation
7 November 2013
46
Where is a random number between [0,1]
Where, is the user defined maximum perturbation
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R.K. Bhattacharjya/CE/IITG
Normally distributed mutation
7 November 2013
47
A simple and popular method
Where is the Gaussian probability distribution with zero mean
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R.K. Bhattacharjya/CE/IITG
Polynomial mutation
7 November 2013
48
���,��� = ���,��� + �� − ��� �
� = � 2�� �/ ���� − 1 1 − 2 1 − �� �/ ���� If �� < 0.5If �� ≥ 0.5
Deb and Goyal,1996 proposed
![Page 49: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/49.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-modal optimization
7 November 2013
49
Solve this problem using simple Genetic Algorithms
![Page 50: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/50.jpg)
R.K. Bhattacharjya/CE/IITG
After Generation 200
7 November 2013
50
The population are in and around the global optimal solution
![Page 51: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/51.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-modal optimization
7 November 2013
51
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.
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R.K. Bhattacharjya/CE/IITG
Hand calculation
7 November 2013
52
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
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R.K. Bhattacharjya/CE/IITG
Distance table
7 November 2013
53
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
![Page 54: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/54.jpg)
R.K. Bhattacharjya/CE/IITG
Sharing function values
7 November 2013
54
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
![Page 55: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/55.jpg)
R.K. Bhattacharjya/CE/IITG
Sharing fitness value
7 November 2013
55
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
![Page 56: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/56.jpg)
R.K. Bhattacharjya/CE/IITG
Solutions obtained using modified fitness value
7 November 2013
56
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R.K. Bhattacharjya/CE/IITG
Evolutionary Strategies
7 November 2013
57
� 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
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R.K. Bhattacharjya/CE/IITG
Two members ES: (1+1) ES
7 November 2013
58
� In each iteration one parent is used to create one offspring by using Gaussian mutation operator
![Page 59: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/59.jpg)
R.K. Bhattacharjya/CE/IITG
Two members ES: (1+1) ES
7 November 2013
59
� 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
![Page 60: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/60.jpg)
R.K. Bhattacharjya/CE/IITG
Two members ES: (1+1) ES
7 November 2013
60
� 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.
![Page 61: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/61.jpg)
R.K. Bhattacharjya/CE/IITG
Two members ES: (1+1) ES
7 November 2013
61
� 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
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R.K. Bhattacharjya/CE/IITG
Matlab code
7 November 2013
62
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7 November 2013
63
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R.K. Bhattacharjya/CE/IITG
Some results of 1+1 ES
7 November 2013
64
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
![Page 65: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/65.jpg)
R.K. Bhattacharjya/CE/IITG
Multimember ES
7 November 2013
65
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
![Page 66: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/66.jpg)
R.K. Bhattacharjya/CE/IITG
Multimember ES
7 November 2013
66
ES
Through mutation
Through selection
![Page 67: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/67.jpg)
R.K. Bhattacharjya/CE/IITG
Multimember ES
7 November 2013
67
ES
Offspring
Through mutation
Through selection
![Page 68: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/68.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-objective optimization
7 November 2013
68
Price
Comfort
![Page 69: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/69.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-objective optimization
7 November 2013
69
Two objectives are
• Minimize weight
• Minimize deflection
![Page 70: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/70.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-objective optimization
7 November 2013
70
� 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
![Page 71: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/71.jpg)
R.K. Bhattacharjya/CE/IITG
Multi-objective optimization
7 November 2013
71
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R.K. Bhattacharjya/CE/IITG
Vector Evaluated Genetic Algorithm (VEGA)
7 November 2013
72
f1
f2
f3
f4
…
fn
P1
P2
P3
P4
Pn
…
Crossover and
Mutation
Old population Mating pool New population
Propose by Schaffer (1984)
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R.K. Bhattacharjya/CE/IITG
Non-dominated selection heuristic
7 November 2013
73
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
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R.K. Bhattacharjya/CE/IITG
Weighted based genetic algorithm (WBGA)
7 November 2013
74
� The fitness is calculated
� The spread is maintained using the sharing function approach
Niche count Modified fitness
Sharing function
![Page 75: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/75.jpg)
R.K. Bhattacharjya/CE/IITG
Multiple objective genetic algorithm (MOGA)
7 November 2013
75
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
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R.K. Bhattacharjya/CE/IITG
Maximize f1
Maximize f2
Multiple objective genetic algorithm (MOGA)
![Page 77: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/77.jpg)
R.K. Bhattacharjya/CE/IITG
Multiple objective genetic algorithm (MOGA)
7 November 2013
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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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R.K. Bhattacharjya/CE/IITG
7 November 2013
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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,
Multiple objective genetic algorithm (MOGA)
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R.K. Bhattacharjya/CE/IITG
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
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R.K. Bhattacharjya/CE/IITG
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
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R.K. Bhattacharjya/CE/IITG
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
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R.K. Bhattacharjya/CE/IITG
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
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R.K. Bhattacharjya/CE/IITG
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 $� > $#.
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R.K. Bhattacharjya/CE/IITG
Replacement scheme of NSGA II
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P
Q
���&�'
�(�)
���&�'Selected
Rejected
Rejected based on crowding distanceNon dominated sorting
![Page 85: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/85.jpg)
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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R.K. Bhattacharjya/CE/IITG
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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R.K. Bhattacharjya/CE/IITG
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
*(�) * � + - . �
![Page 88: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/88.jpg)
R.K. Bhattacharjya/CE/IITG
Constraints handling in GA
7 November 2013
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-5
5
15
25
35
45
55
0 2 4 6 8 10 12
f(X
)
X
*(�) * � + - . �
*/01 + . �
![Page 89: Introduction To Genetic Algorithms - blackquest...Kalyanmoy Deb, ‘An Introduction To Genetic Algorithms’, Sadhana, Vol. 24 Parts 4 And 5. R.K. Bhattacharjya/CE/IITG Introduction](https://reader034.fdocuments.net/reader034/viewer/2022043000/5f73a6ff2db5de4925016877/html5/thumbnails/89.jpg)
R.K. Bhattacharjya/CE/IITG
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
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R.K. Bhattacharjya/CE/IITG
THANKS
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