evolutionary of the variable-length multi-objective genetic algorithm

Evolutionary of the Variable-Length Multi-objective Genetic Algorithm

李宗南

國立中山大學資訊工程學系

December 3, 2008

Outline•A tale of single objective optimization and

multi-objective optimization•The single genetic algorithm•The multi-objective genetic algorithm •The variable length genetic algorithm•The variable length multi-objective optimization•Applications - Aircraft routing - Placement of heterogeneous wireless transmitters •Conclusions

2

A Tale of Single Objective Optimization and Multi-Objective Optimization

3

Courtesy of Dr. YoaChu Jin

Single Objective Optimization

Given: a function f : A → R from some set A to the real numbers

Sought: an element x0 in A such that f(x0) ≤ f(x) for all

x in A ("minimization") or such that f(x0) ≥ f(x) for all x

in A ("maximization").

5

Gradient descent aka steepest aka steepest descent or steepest ascent descent or steepest ascent Hill climbing Hill climbing Simulated annealing Simulated annealing Quantum annealing Quantum annealing Tabu search Tabu search Beam search Beam search Genetic algorithms Genetic algorithms Ant colony optimization Ant colony optimization Evolution strategy Evolution strategy Stochastic tunneling Stochastic tunneling Differential evolution Differential evolution Particle swarm optimization Particle swarm optimization Harmony search Harmony search Bees algorithm Bees algorithm Dynamic relaxation Dynamic relaxation

Algorithms for Single Objective Optimization

Multi-Objective Optimization

6

Courtesy of Dr. YoaChu Jin

Multi-Objective Optimization

7

Courtesy of Dr. YoaChu Jinm= 1 , single objective optimization

8

Solutions for Multi-objective Optimization

•Map into a single objective optimization by a weighted sum

•The multi-objective approach (rank-based fitness assignment method) to evaluate each objective individually

8

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Comparison of the single objective approach and the multi-objective approach

SO+ simple- Hard to determine weight for each objective .- Hard to prevent some objectives from dominating others.

MO+ Have the ideal situation where each objective function attains a satisfactory level.+ Have the flexibility to achieve different levels of tradeoff.- Not so easy to solve.

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Single GA

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Single GA Parent A 110011001Parent B 101111011

110011001 …. ….101010101

110011001 101111011 …. 101010101

110011001 101111011 …. 101010101

110011001 110011011 101111011 101111001

110011001 => 110101001

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Task 1:To find a set of solutions as close as possible to the Pareto-optimal front.Task 2:To find a set of solutions as diverse as possible

Introduction of multi-objective genetic algorithm

Minimization Objective 1 f1

f2

Pareto Front

Solution Space

proximityDiversity

Dominated solutionsNondominated solutions

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Why Variable-Length GAs (VLGAs)?

The number of solutions is not fixed.i.e. Fixed Length GAs must know number of variables a priori

Ex: Finding number of base stations for a given regionEx: Finding rules for autonomous agents

The variable length genetic algorithm

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The number of solutions is not fixed.

It is a multi-objective optimization problem

We would like to solve the problem by GA

Evolutionary of the variable length multi-objective optimization

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Nondominated solutions

Minimization Objective 1 f1

f2

Pareto Front

Solution Space

proximityDiversity

Dominated solutionsNondominated solutions

16


Rank-based fitness assignment method

1

1

11 1 1

4

5

2

3

3

3

58

Front 1

Front 2

Front 3

Front 4

f1

f2

17


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MOGA - Aircraft routing

VLMOGA - Placement of heterogeneous wireless transmitters

Applications

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Application 1

Aircraft Routing using Multi- using Multi-objective Genetic Algorithmobjective Genetic Algorithm

Problem Description

•Aircraft routingAircraft routing▫A given set of flights A given set of flights a group a group of aircraftsof aircrafts

▫Available amount of aircraftsAvailable amount of aircrafts

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Aircraft Routing (1/2)

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Flight ID Dep. Time Arv. Time Origin Destination

f1 09:15 09:50 KHH TSAf2 19:00 19:35 TSA KHHf3 14:00 14:35 MZG TSAf4 11:20 11:55 TSA MZGf5 22:00 22:50 TNN TSAf6 16:30 17:20 TSA TNNf7 08:10 09:00 KHH TSAf8 20:35 21:25 KHH TSA

f9 18:00 18:50 TNN TSA

f10 15:30 16:20 TSA TNN

Flight ID Dep. Time Arv. Time Origin Destination

f11 09:30 10:05 HUN TSAf12 19:10 19:45 TSA HUNf13 14:10 14:45 MZG TSAf14 10:20 10:55 TSA MZGf15 18:00 18:50 TXG TSAf16 15:30 16:20 TSA TXGf17 08:20 09:20 KHH TSAf18 18:35 19:25 KHH TSAf19 13:00 13:35 MZG TSAf20 10:20 10:55 TSA MZG

Timetable assign to aircrafts

Aircraft Routing (2/2)

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f1

f2 f3f4

f5

f6

f7

f8

f9 f10

f11

f12

f13

f14

f15 f16

f17

f18

f19

f20

aircraft 1

aircraft 2

aircraft 3

aircraft 4 f1

f2

f3 f4

f5

f6

f7

f8

f9

f10

f11

f12

f13

f14

f15f16

f17

f18 f19

f20

Flight set F

Flight schedule S

Notations• Let Let αα, , ββ, , ωω, , andand γγ represent represent

▫ αα:: number of aircraftsnumber of aircrafts▫ ββ: : maximal number of flights per aircraftmaximal number of flights per aircraft▫ ωω:: number of airportsnumber of airports▫ γγ:: number of daily flights, respectively.number of daily flights, respectively.

• Set of flights: Set of flights: F F == {{ffii|1 ≤ |1 ≤ ii ≤ ≤ γ γ}}

• Set of airports: Set of airports: P P = {= {ppjj|1 ≤|1 ≤jj ≤ ≤ ω ω}}

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P= { 台北松山機場、高雄小港、台中、台南、馬公、金門 }

Associate Information of One Flight

,

ssi,ji,j:: the the jjthth flight assigned to the flight assigned to the iithth aircraftaircraft

: origin of : origin of ssi,ji,j, where , where

: destination of : destination of ssi,ji,j, where, where

: departure time from : departure time from

: arrival time in : arrival time in

The flight schedule The flight schedule SS can be represented as: can be represented as:

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Definition of a Flight Schedule

Maximal flights assigned to each aircraft

Number of aircrafts

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Flight Schedule S

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Maximal flights assigned to each aircraft

Number of aircrafts

s1,1

sα,β

ObjectivesObjectives

• Ground turn-around time objectiveGround turn-around time objective• Flow balance objectiveFlow balance objective

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Objectives:Objectives:

Subject to

Ground Turn-around Objective(1)

Legal ground turn-around time: TGH

TaipeiP

Q

Δt

Kaohsiung

Makung

28

Flow Balance Objective(2)

f1

f2

f1

f2

Taipei

Kaohsiung

Makung

Taipei

Kaosiung

Makung

(a) (b)

Extra costExtra cost

29

Encoding SchemeEncoding Scheme

s1,1s2,1:

sα, 1

s1, 2s2, 2:

sα, 2

……:…

s1, β-1s2, β-1:

sα, β-1

s1, βs2, β:

sα, β

c1c2:cα

1 2 … β-1 β

s1,1 s2,1 sα, 1s1, 2 s2, 2 sα, 2… … …s1, β-1 s2, β-1 sα, β-1s1, β s2, β sα, β… …

…flights of c1 flights of c2 flights of cα

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Crossover

E AA B C D E F

B C E A F D

A B C

D E FB C E

A F D A B C

D A FB C E

E F D

exchangeMapping relationship

duplicate genes

cutpoint

mapping relationship

1. change A to E2. change E to A

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Reciprocal MutationReciprocal Mutation

AA BB CC DD EE FF

AA BB EE DD CC FF

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Experimental ResultsExperimental Results

• 7 airports, 9 aircrafts, 12 flights one day, 79 flights.

Airports

HUN

KHH

KNH

MZG

TNN

TSA

TTT

Parameter Value

Crossover rate 1

Mutation rate 0.2

No. of generations

5000

Population size 100

Reproduction rate

0.8

33

Experimental Results

34

Symbols in Experimental Results

682KNHTSA

flight ID

origindestination

departure time

arrival time

35

Gantt chart:

Aircraft1/crew1

Aircraft2/crew2

time

Scheduling Result of 9 aircrafts36

Scheduling Result of 8 aircrafts

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Original valueΦ(S) = [ϕ1(S), ϕ2(S)]

Values in auxiliary vector of performance indices Λ(S, ε)=[λ(S,ε1), λ(S,ε2)]

Result 1 [195,0] [0,0]

Result 2 [190,1] [0,0]

Result 3 [185,2] [0,0]

Example

ε = [ε1, ε2] =[k1 × α × TGH, k2× α]= [1 × 8 × 25, 1 × 8]= [200, 8]

Scheduling Result of 8 aircrafts

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Result1

Result2

Result3

Scheduling Result of 8 AircraftsScheduling Result of 8 Aircrafts39

Retiming Process

Station K

Station K+1

Station K+2

Inspection line

P

Q

Station K

Station K+1

Station K+2

Inspection line

P

Q

Retimingprocess

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Flights P and Q cannot beassigned to the same aircraft

Flights P and Q can beassigned to the same aircraft

Scheduling Result-RetimingScheduling Result-Retiming41

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Application 2:

Heterogeneous Wireless Transmitter Placement with Multiple Constraints based on the Variable-Length Multi-objective Genetic Algorithm

Problem statement

Choose a set of heterogeneous wireless transmitters to place on the designed space to fulfill certain design requirements such as

Position, power, capacity, frequency channel assignment, overlap, data rate demand, population density, cost and coverage

Evolutionary multiobjective optimization for base station transmitter placement with frequency assignment, IEEE Trans. on Evolutionary Computation, 2003

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Introduction (cont.)

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23meters

15meters

Parameters

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Parameters Value Type of transmitters 1

Transmitter cost 2200

Maximum allowed power loss threshold

63.5dB (radius 15 meters)

Transmitter capacity 54Mb

Test points Total 10404 points

Data Rate Demand 21kbps

Generation 3000

Wireless transmitter placement problem

Problem Definition

•Model▫Map, receiver, transmitter

•Receiver▫Position, data rate

demand, sensitivity•Transmitter

▫Position, type=(cost, power, capacity)

47

Path Loss Propagation Models

1. Free space path loss model

2. Log-distance path loss model with shadowing effect

3. ECC-33 model

),(),()4

(log20),(),(

1

10 trtrtr jig

o

gjigji

AtrP

dL

ji

)4

(log20),( 10

d

L tr ji

48

Objectives

•Coverage•Cost•Data Rate Demand•Overlap

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Coverage

Coverage Rate=4/9=44.4%

Uncoverage=5

otherwise

-1)(coverage if { Uncoverage T

,0)(

,1)(),()(

1

iT

iiTi

n

iT rc

rrcrcT

50

Cost•Cost=1000+300

*2=1600

Cost=300

Cost=300

Cost=1000

Tt

j

j

tT )(cost)(Cost

51

Data Rate Demand•Yellow: 16kbps•Blue: 128 kbps•Demand(t1)=128

*4+16*2=544•Capacity(t1)=10

24•DRD(T)=|544-

1024|=480

Next generation wireless LAN system design, Proceedings on MILCOM, 2002

Tt ysensitivittrLtpowerRr

jr

j ijiji

itdemandT |)(capacity|)(DRD

),()(,

52

Overlap•Overlap=2•Overlap rate=2/9

=22.2%

PGr rr

rr

r

i ii

ii

i overlap

overlapoverlapT

1|AS| if ,1

1,0|AS| if ,0{ ,)(Overlap

Evolutionary multiobjective optimization for base station transmitter placement with frequency assignment, IEEE Trans. on Evolutionary Computation, 2003

53

Objectives•Minimize

•Subject to

ers transmittofnumber theis ,1 where, ,

receivers ofnumber theis ,1 where, ,

mmjPGpositiont

nniPGpositionr

jj

ii

otherwise

-1)(coverage if { Uncoverage T

,0)(

,1)(),()(

1

iT

iiTi

n

iT rc

rrcrcT

PGr rr

rr

r

i ii

ii

i overlap

overlapoverlapT

1|AS| if ,1

1,0|AS| if ,0{ ,)(Overlap

Tt ysensitivittrLtpowerRr

jr

j ijiji

itdemandT |)(capacity|)(DRD

),()(,

Tt

j

j

tT )(cost)(Cost

54

Encoding:Individual and Chromosome Representation

0 1 1 0 1 0 1 0 0 0

X1 Y1 Z1type1

A chromosome

Transmitter resolution= Encoding bits=3

An individual

55

Population Initialization

•Temporary upper bound UB=10•Random number=4 (1~10)

type1 type3 type2 type1

type1

type1

type1 type3 type2

type3

Flowchart

56

Types of crossover

•Uniform crossover for chromosome▫Change the position and type of transmitter

•Variable-length one-point crossover▫Change the length of individual

57

Uniform Crossover for chromosome

(x2,y2,z2),type2(x1,y1,z1),type1

(x1,y1,z1),type1 (x2,y2,z2),type2 (x3,y3,z3),type3Individual1

Individual2

58

One-point Crossover

(x1,y1,z1),type1

(x1,y1,z1),type1Individual1

Individual2

Split point

Split point

(x2,y2,z2),type2

(x2,y2,z2),type2 (x3,y3,z3),type3

(x1,y1,z1),type1

(x1,y1,z1),type1Individual1

Individual2 (x2,y2,z2),type2

(x2,y2,z2),type2 (x3,y3,z3),type3

59

Overall CrossoverParents

1-Pc Pc

Offspring

Pc 1-Pc 1

Crossover Rate: Pc

Uniform crossover for chromosome

Variable-length one-

point crossover

Uniform crossover for chromosome

Copy from parents

Flowchart

60

Mutation

0 1 1 0 1 0 1 0 0 0chromosome1

X1 Y1 Z1type1

0 1 1 0 0 0 1 0 0 0chromosome1

X1 Y1 Z1type1

chromosomeoflengthPm

1

Flowchart

61

1

Types of Simulation

IndoorPopulation

Size

Types of Transmitter

s

Free space 100

1

Path loss 100 1

Outdoor

2D path loss 500 2

3D path loss 200 2

Upper bound UB=8,12,15

500 2

Heterogeneous 500 2

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Overall Parameters

Parameters Value

Termination 5000Crossover rate 0.8

Mutation rate Pm 1/length of chromosome

Frequency 2.4GHz

63

Indoor Freespace Parameters

Parameters Value Penetration loss Zero

Type of transmitters 1


66dB (radius 20 meters)

Transmitter capacity 54Mb

Test points Every 3m x 3m, total 231 points

Data Rate Demand 1024kbps

64

Indoor Map

•Floor plan of IC factory

65

Indoor Free Space

•Threshold 66dB (radius 20 meters)

Uncoverage 0

Cost 6600

Data rate

demand

83968

Overlap 13

BS # 3

66

Indoor Path Loss

•Type 1 –cement wall: 3.3dB•Type 2 – thickened cement wall: 6.5dB

Uncoverage 0

Cost 13200

Data rate demand 30720

Overlap 48

BS # 6

67

Outdoor Map

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Two Dimensional Outdoor Path Loss Parameters

Parameters Value Penetration loss Type 1 –concrete wall: 8dB

Type 2–mountain: 99dB Types of transmitters 2


Type 1 – 103dB (1.5KM radius)

Type 2 – 80dB (100 meter radius)

Transmitter costType 1 – 40000 Type 2 – 2200

Transmitters capacityType 1 – 75Mb Type 2 – 54Mb


69

Outdoor 2D-Data Rate Demand•Blue: 16 kbps/

81%•Green:128kbps/

13.7%•Red:1024kbps/

5.2%

70

Result-11 TransmittersUncoverage 25

Cost 480000

Data rate

demand

403376

Overlap 2000

BS 1 # 12

BS 2 # 0

Outdoor 2D Result

71

Solutions

Idv 1 Idv 2 Idv 3 Idv 4 Idv 5 Idv 6 Idv 7 Idv 8

Uncoverage 36 67 1877 2106 1162 875 477 2941

Cost 440000 400000 40000 26400 80000 93200 13100 2200

Data rate 507776 399600 180352 847312 197648 515168 490832 277568

Overlap 1807 1114 0 79 35 16 248 0

BS 1 # 11 10 1 0 2 2 3 0

BS 2 # 0 0 0 12 0 6 5 1

72

Three Dimensional Outdoor Path Loss Parameters

Parameters Value Types of transmitters 2




Transmitter costType 1 – 8000 Type 2 – 440

Transmitters capacityType 1 – 75Mb Type 2 – 54Mb


73

Outdoor 3D- Data Rate Demand

•Blue: 16 kbps/ 57%

•Green:128kbps/ 28%

•Red:512kbps/ 15%

74

Uncoverage 0

Cost 32000

Data rate

demand

1493184

Overlap 3516

BS 1 # 4

BS 2 # 0

Outdoor 3D Result

75

Conclusions

•Have introduced an evolutionary of the variable-length multi-objective genetic algorithm

•Have presented the applications of MOGA and VLMOGA

- Flight scheduling - The multiple constraints heterogeneous

wireless transmitter placement

76

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本投影片研究主要參與者

•周大源博士中山大學資工系•劉東官教授高雄第一科大 •丁川康教授中正大學資工系•吳建興 •張慧君•黃振愷

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1. Chuan-Kang Ting, Chung-Nan Lee, Hui-Jin Chang, and Jain-Shing Wu “Wireless Heterogeneous Transmitter Placement Using Multi-Objective Variable-Length Genetic Algorithm” accepted to appear in IEEE Trans. on SMC, Part B

2. Ta-Yuan Chow, T. K. Liu and Chung-Nan Lee, Chi-Ruey Jeng “Method of Inequality-Based Multiobjective Genetic Algorithm for Domestic Daily Aircraft Routing ", IEEE Trans. on SMC, Part A. Volume: 38, Issue: 2 . March 2008

3. Sibel Yaman and Chin-Hui Lee“ A Multi-Objective Programming Approach to Compromising Classification Performance Metrics”, IEEE International Workshop on Machine Learning for Signal Processing August 27, 2007

4. Yaochu Jin, “Evolutionary Multi-Objective Optimization”, Honda Research Institute Europe

References

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Question & SuggestionQuestion & Suggestion

•Thank you for your attentions

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evolutionary of the variable-length multi-objective genetic algorithm

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