ppt- nsga-ii

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.


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)


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 -


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


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.


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


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


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


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


-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)


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


-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


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[


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.


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


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


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


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


Pt

Qt

Rt =Pt , Qt

F1

Non-dominated sorting (Rank)

Crowding distance sorting

F2

F3

Rejected

Selection for next generation

Pt+1

Elitist Replacement


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


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 ?


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


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 nondominated 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


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


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


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.


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



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


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


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


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



-20 -19 -18 -17 -16 -15 -14-12

-10

-8

-6

-4

-2

0

2

F1

F2

Nondominated solutions with MOPSO on KUR


Test Problem : KUR



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



Layout Optimization for a Wireless Sensor Network using NSGA - II

a) Coverageb) Lifetime

INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR

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


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

INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR

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


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


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


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


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


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.


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


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


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


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ppt- nsga-ii

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