ppt- nsga-ii
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Genetic Algorithm based
Multi-objective OptimizationProf. Ganapati Panda, FNAE, FNASc.
Dean Academic AffairsProfessor, School of Electrical
SciencesIIT Bhubaneswar
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 2
“Multiobjective optimization is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints.”
Multiobjective Optimization
Examples of multi-objective optimization problems:- Maximizing profit and minimizing the cost of a product. Maximizing performance and minimizing fuel consumption of a
vehicle. Minimizing weight while maximizing the strength of a particular
component.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 3
DifferenceSingle Objective Optimization
Optimize only one objective function
Single optimal solution Maximum/Minimum fitness value
is selected as the best solution.
Multiobjective Optimization Optimize two or more than two
objective functions Set of optimal solutions Comparison of solutions by
• Domination• Non-domination
Minimize
where -10 < x < 20
Optimal solution:-
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
f1
f2-10 -5 0 5 10 15 200
100
200
300
400
x
f(x)
f1(x)f2(x)
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 4
Standard Approach :Weighted Sum of Objective Functions
Limitations: Result depends on weights. Some solutions may be missed. Multiple runs of the algorithm are required in order to get the whole
range of solutions. Difficult to select proper combination of weights. Combining objectives loses information and predetermines trade-offs
between objectives.
)()()()( 2211 xfxfxfxg mm
)(,),(),( 2`1 xfxfxfMinimize m
1& 21 m m ,,., 21 where are weights valuesand m represents the number of objective functions.
Formulate as a single objective with weighted sum of all objective functions -
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 5
DefinitionsDomination : One solution is said to dominate another if it is better in all objectives.
Non-Domination [Pareto points] : A solution is said to be non-dominated if it is better than other solutions in at least one objective.
Minimize function
Min
imize
func
tion A
B
C
D
1f
2f
A dominates B (better in both and ) A dominates C (same in but better in ) A does not dominate D (non-dominated points) A and D are in the “Pareto optimal front” These non-dominated solutions are called Pareto optimal solutions. This non-dominated curve is said to be Pareto front.
2f1f
2f1f
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 6
Definitions
Pareto Optimal
A vector variable is Pareto optimal if for every and either or, there is at least one such that
where is the vector of decision variables, is the vector of objective
functions, is the feasible region ,where represents the whole search space.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 7
Definitions Cont….
Pareto Optimal Set For a given MOP the Pareto optimal set is defined as
Pareto Front For a given MOP and Pareto optimal set , the
Pareto front is defined as
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 8
Desirable MOEA features
Best SolutionsLie on true Pareto front
They are uniformly distributed on the front
Aim: To achieve convergence to Pareto optimal front To achieve diversity (representation of the entire Pareto optimal front)
Minimize function
Min
imize
func
tion
1f
2f
Diversity
Convergence
True Pareto front
Possible solutions
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 9
Non Dominated Sorting based Genetic Algorithm II (NSGA- II)
Developed by Prof. K. Deb at Kanpur Genetic Algorithms Laboratory (2002)
Famous for Fast non-dominated search
Fitness assignment - Ranking based on non-domination sorting
Diversity mechanism is based on Crowding distance
Uses Elitism
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 10
Minimize
where
,21 xxf 22 2 xxf
55 x
Initialize Population
• Search space is of single dimension (given). • Objective space is of two dimension (given).
• Let population size = 10
• Initialize population with 10 chromosomes having single dimensioned real value.
• These values are randomly distributed in between [-5,5].
0.46781.73550.8183-0.4143.2105-1.272-1.508-1.832-2.161-4.105
x
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 11
-0.414 0.171 5.8290.467 0.218 2.3470.818 0.669 1.3961.735 3.011 0.073.210 10.308 1.465-1.272 1.618 10.708-1.508 2.275 12.308-1.832 3.355 14.682-2.161 4.671 17.317-4.105 16.854 37.275
x xf1 xf2
Evaluate Fitness values
• Find out all objective functions values (fitness values) for all chromosomes.
0 2 4 6 8 10 12 14 16 180
5
10
15
20
25
30
35
40
f1(x)
f 2(x)
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 12
1111223456
-0.414 0.171 5.8290.467 0.218 2.3470.818 0.669 1.3961.735 3.011 0.073.210 10.308 1.465-1.272 1.618 10.708-1.508 2.275 12.308-1.832 3.355 14.682-2.161 4.671 17.317-4.105 16.854 37.275
x xf1 xf2Rank
Fast Non-domination Sorting • Assigning the rank to each individual of the population.• Rank based on the non-domination sorting (front wise).• It helps in selection and sorting.
2,,,
1,,
1,
0
0
0
0
763217
63216
5435
44
33
22
11
nxxxxx
nxxxx
nxxx
nx
nx
nx
nx
Referencechromosome
Dominatedchromosomes
Counter
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 13
-0.414 0.171 5.829 10.467 0.218 2.347 10.818 0.669 1.396 11.735 3.011 0.07 13.210 10.308 1.465 2-1.272 1.618 10.708 2-1.508 2.275 12.308 3-1.832 3.355 14.682 4-2.161 4.671 17.317 5-4.105 16.854 37.275 6
x xf1 xf2 Rank
0 2 4 6 8 10 12 14 16 180
5
10
15
20
25
30
35
40
f1(x)
f 2(x)
43
5
6
21
Fast Non-domination Sorting
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 14
Crowding Distance Assignment• To get an estimate of density of solutions surrounding a particular solution in population.
• Choose individuals having large crowding distance.
• Help for obtaining uniformly distribution.
where represent objective function value of solution.
and is the maximum value of function in the Pareto front.maxmf mf
)1(,,3,2 liwhere
,1 .... DCDC l
m mm
mmDC ff
ififi minmax..]1[]1[
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 15
0.5 1 1.5 2 2.5 3
0
1
2
3
4
5
6
f1(x)
f 2(x)
i-1
ii+1
1
2
4
3
0 2 4 6 8 10 12 14 16 180
5
10
15
20
25
30
35
40
f1(x)
f 2(x)
43
5
6
21
-0.414 0.171 5.829 10.467 0.218 2.347 1 0.9450.818 0.669 1.396 1 1.3781.735 3.011 0.07 13.210 10.308 1.465 2-1.272 1.618 10.708 2-1.508 2.275 12.308 3-1.832 3.355 14.682 4-2.161 4.671 17.317 5-4.105 16.854 37.275 6
x xf1 xf2 Rank
..DC
Crowding Distance Assignment
• Crowning distance can be calculated for all chromosomes of same Pareto front.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 16
Selection
Selection is the stage of a genetic algorithm in which individual are chosen from a
population for later breeding (recombination or crossover).
The crowding operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto optimal front.
where shows non-domination rank & is crowding
distance of individual.
Crowding operator based sorting
ji n rankrank ji
n
..DCirankithi
.... DCDCrankrank jiandjior
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 17
Runs a ‘tournament’ among a few individuals chosen at random from the population and selects the winner (the one with the best fitness) for crossover.
• In tournament selection, a number Tour size of individuals is chosen randomly from the population and the best individual from this group is selected as parent. (Based on the crowding operator)
Tournament Selection
0.818 0.669 1.396 1 1.378
-1.508 2.275 12.30 3
x xf1 xf2 Rank ..DC
0.818 0.669 1.396 1 1.378
rankrank 21
0.467 0.218 2.347 1 0.945
0.818 0.669 1.396 1 1.378
x xf1 xf2 Rank ..DC
0.818 0.669 1.396 1 1.378
rankrank 21 .... 21 DCDC
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 18
where • is a random number 0,1• is a crossover operator• represent dimension of
individual.
rj
CrossoverCrossover is a genetic operator that combines (mates) two individuals (parents) to produce two new individuals (Childs). • The idea behind crossover is that the new chromosome may be better
than both of the parents if it takes the best characteristics from each of the parents.
5.01*21
5.0*2
11
11
rifr
rifrb
Simulated Binary Crossover
)(*1)(*121)(
)(*1)(*121)(
212
211
jparentbjparentbjchild
jparentbjparentbjchild
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 19
Mutation
Mutation is a genetic operator that alters one ore more gene values in a chromosome from its initial state.• Mutation is an important part of the genetic search as helps to prevent the
population from stagnating at any local optima.
Polynomial Mutation
5.01*21
5.01*2
11
11
rifr
rifrd
djparentjchild )()( where • is a random number 0,1• is a mutation operator• represent dimension of
individual.
rj
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 20
Pt
Qt
Rt =Pt , Qt
F1
Non-dominated sorting (Rank)
Crowding distance sorting
F2
F3
Rejected
Selection for next generation
Pt+1
Elitist Replacement
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 21
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
f1(x)
f 2(x)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
f1(x)
f 2(x)
0 2 4 6 8 10 12 14 16 180
5
10
15
20
25
30
35
40
f1(x)
f 2(x)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
f1(x)
f 2(x)
Initial State After 20 generation
After 10 generation After 40 generation
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 22
Report Final Population and Stop
Begin : Initialize Population (N)
Evaluate objective functions
Non-dominated Sorting
Tournament Selection
Combine parent and child populations , Non-dominating
Sorting
Crossover & Mutation
Evaluate objective functions
Flowchart of NSGA-II
Yes
No
Select N individuals
Stopping criteria met ?
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 23
Performance Measures
There are two main goals in a multi-objective optimization: 1) Convergence to the Pareto-optimal set2) Maintenance of diversity in solutions of the Pareto-
optimal set.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2
4
6
8
10
12
function - 1
func
tion
- 2
DISTANCE MEASURE BETWEEN PARETO FRONTS
TRUE PARETO FRONT
PARETO FRONT
min distance
min distance
Convergence metric
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 24
Cont….
Non uniformity in the distribution,
If distance between the solutions is equal to average distance , that gives uniformly distribution.
The parameters and are the Euclidean distances between the extreme solutions of true Pareto front and the boundary solutions of the obtained non-dominated set. The parameter is the average of all distances , , assuming that there are solutions on the best non- dominated front.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.5
1
1.5
2
2.5
3
3.5
4
function -1
func
tion
-2
DIVERSITY PLOT
Pareto front
Extremesolution
Extremesolution
1d
2d
3d
4d
1nd
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 25
Simulation and Results
NSGA II parameters Population (N) = 100 Crossover Probability (Pc)= 0.9 Mutation Probability (Pm) = 0.1 Distribution index for crossover (µ)= 20 Distribution index for mutation (η) = 20 Tour size (selection) = 2
Implementation use real numbers representation. * These parameters were kept in all test functions optimization. * Only changed the total number of fitness function evaluations.
MOPSO parameters Population = 100 particles Repository (Archive) size = 100 particles Mutation rate = 0.5 Divisions for Archive Grid = 30
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 26
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
F1
F2
Nondominated solutions with NSGA-II on SCH
Pareto-optimal FrontNSGA-II
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
F1
F2
Nondominated solutions with MOPSO on SCH
Pareto-optimal Front MOPSO
Test Problem : SCH
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 27
Convergence Metric NSGA-II MOPSOBest 0.0148 0.0093
Worst 0.9578 0.1569Mean 0.2096 0.0259
Diversity Metric NSGA-II MOPSO
Best 0.5104 0.6947Worst 0.7904 1.3575Mean 0.6425 0.8582
Tab. 1: Results of the Convergence Metric for the SCH Test Function
Tab. 2: Results of the Diversity Metric for the SCH Test Function
Comparison for SCH Test Function
* Total number of fitness evaluations was set to 10,000.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 28
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
F1
F2
Nondominated solutions with NSGA-II on DEB-1
Pareto-optimal FrontNSGA-II
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
F1
F2
Nondominated solutions with MOPSO on DEB-1
Pareto-optimal FrontMOPSO
Test Problem : DEB-1
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 29
Convergence Metric NSGA-II MOPSOBest 0.0066 0.0070
Worst 0.5140 0.1664Mean 0.0078 0.0079
Diversity Metric NSGA-II MOPSO
Best 0.3467 0.5112Worst 0.5140 0.7168Mean 0.4243 0.5938
Tab. 1: Results of the Convergence Metric for the DEB-1 Test Function
Tab. 2: Results of the Diversity Metric for the DEB-1 Test Function
Comparison for DEB-1 Test Function
* Total number of fitness evaluations was set to 15,000.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 300.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
F1
F2
Nondominated solutions with MOPSO using DEB-2
Pareto-optimal FrontMOPSO
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
F1
F2
Nondominated solutions with NSGA-II on DEB-2
Pareto-optimal FrontNAGA-II
Test Problem : DEB-2
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 31
Convergence Metric NSGA-II MOPSO
Best 0.0449 0.0515Worst 0.0559 0.0725Mean 0.0516 0.0608
Diversity Metric NSGA-II MOPSOBest 0.7248 0.6800
Worst 0.7939 0.7582Mean 0.7597 0.7193
Tab. 1: Results of the Convergence Metric for the DEB-2 Test Function
Tab. 2: Results of the Diversity Metric for the DEB-2 Test Function
Comparison for DEB-2 Test Function
* Total number of fitness evaluations was set to 25,000.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 32
-20 -19 -18 -17 -16 -15 -14-12
-10
-8
-6
-4
-2
0
2
F1
F2
Nondominated solutions with MOPSO on KUR
Pareto-optimal FrontMOPSO
Test Problem : KUR
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 33
Convergence Metric NSGA-II MOPSOBest 0.0021 0.0021
Worst 0.0041 0.0034Mean 0.0028 0.0026
Diversity Metric NSGA-II MOPSOBest 0.3344 0.4803
Worst 0.7825 0.6413Mean 0.4399 0.5602
Tab.1: Results of the Convergence Metric for the KUR Test Function
Tab.2: Results of the Diversity Metric for the KUR Test Function
Comparison for KUR Test Function
* Total number of fitness evaluations was set to 20,000.
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 34
Layout Optimization for a Wireless Sensor Network using NSGA - II
a) Coverageb) Lifetime
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Wireless Sensor Network (WSN)
Example of a WSN where sensor nodes are communicating with the DPU through HECN
Data Processing Unit(DPU)
High Energy Communication Node (HECN)
Node 1
Node 2
Node 3
Node 6
Node 5
Node 8
Node 4
Node 9
Node 7
04/07/2023 35
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 36
Optimization of Coverage
Coverage is defined as the ratio of the union of areas covered by each node and the area of the entire ROI.
AA
C iNi ,...,1 Ai - Area covered by the ith node
N - Total number of nodes A - Area of the ROI
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Optimization of Lifetime
The lifetime of the whole network is the time until one of the participating nodes run out of energy.
In every sensing cycle, the data from every node is routed to HECN through a route of minimum weight
max
failure
TT
Lifetime
Tfailure = maximum number of sensing cycles before failure of any nodeTmax = maximum number of possible sensing cycles
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 38
Competing Objectives
Lifetime Coverage
• try to spread out the nodes for maximizing coverage
• try to arrange the nodes as close as possible to the HECN for maximizing lifetime
HECN
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 39
Simulation Parameters
Number of chromosomes 100Number of generations 50Crossover Probability 0.9Mutation Probability 0.5Distribution index for crossover 20Distribution index for mutation 20Tour size 2
Parameters of NSGA-II
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 40
NSGA-II Results
Pareto Front obtained for a WSN with 10 sensors, 100 chromosomes and 50 generations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Pareto optimal front
Coverage
Life
time
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 41
NSGA-II Results (Cont’d)
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
Coverage = 0.3335 Lifetime = 0.999
HECN
Best Lifetime
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
Coverage = 0.63709
HECN
Initial Disconnect Network
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
Coverage = 0.5353 Lifetime = 0.249Best
Coverage
HECN
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR 42
Ω
Ω
Ω
Ω
Ω
∑
Input Layer Hidden Layer Output Layer
Radial Basis Function Network
Accuracy Complexity of the model
id
1x
2x
3x
4x
1w2w
3w
4w
5w
0w
1
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Structure determination of RBF network can be considered as the multiobjective optimization problem concerning with accuracy and complexity of the model.
Multiobjective Problem Formulation
Mf 1
2
1 10
1
2
2
1
ˆ1
n
i
M
jjiji
n
iii
wcxwdn
ddn
msef
here is total number of basis functions (centers) in RBF network, : Desired output : Estimated output during the training of RBF network. : Weight vector of the RBF network : Center vector of the RBF network : Gaussian Function
M
idid
wjc
.
2
221exp jj cxcx
where is the spread of the Gaussian function.
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Desired Output 0 0 0 +1 0 0 0In the chromosome, the position of gene value “1” indicate the center position of the basis function (selected center) and number of “1” genes in chromosome indicates the number of basis functions (number of centers).
Chromosome 1 0 1 1 0 0 1Input Data Points
Gau
ssia
n D
istr
ibut
ion
Selected centers
Input Data
Points
+1 +1 -1 +1 0 0 0
0 +1 +1 -1 +1 0 0
0 0 +1 +1 -1 +1 0
0 0 0 +1 +1 -1 +1
Structure selection of RBF network
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Pareto Fronts
• This Pareto Front shows that for the different number of centers, MSE changes. • The performance of an RBF network critically depends upon the chosen centers.
2 4 6 8 10 12 14 16 18 20 22 24 2510-40
10-30
10-20
10-10
100
Numbers of Centers
Mea
n Eq
uare
Error
(log
sca
le)
Pareto Front for 13-element Barker Code
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 46
References
1. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan,” A fast and elitist multi-objective genetic algorithm: NSGA-II”, IEEE Transaction on Evolutionary Computation, 6(2), 181-197,2002.
2. K. Deb and R. B. Agrawal, “Simulated binary crossover for continuous search space,” in Complex Syst., vol. 9, pp. 115–148., Apr. 1995.
3. N. Srinivas and K. Deb, “Multiobjective function optimization using nondominated sorting genetic algorithms,” Evol. Comput., vol. 2, no. 3, pp. 221–248, Fall 1995.
4. J. Horn, N. Nafploitis, and D. E. Goldberg, “A niched Pareto genetic algorithm for multiobjective optimization,” in Proceedings of the First IEEE Conference on Evolutionary Computation, Z. Michalewicz, Ed. Piscataway, NJ: IEEE Press, pp. 82–87 , 1994.
5. J. D. Knowles and D.W. Corne, “Approximating the nondominated front using the Pareto archived evolution strategy,” Evol. Comput., vol. 8, pp. 149–172, 2000.
6. Carlos A. Coello Coello, Member, IEEE, Gregorio Toscano Pulido, and Maximino Salazar Lechuga, “Handling multiple objectives with particle swarm optimization”, Evol. Comput., vol. 8, pp. 256–279, No. 3, June 2004
INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR04/07/2023 47
THANK YOU
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