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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 - PowerPoint PPT Presentation
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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
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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").
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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
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Courtesy of Dr. YoaChu Jin
Multi-Objective Optimization
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Courtesy of Dr. YoaChu Jinm= 1 , single objective optimization
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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
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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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Evolutionary of the variable length multi-objective optimization
Nondominated solutions
Minimization Objective 1 f1
f2
Pareto Front
Solution Space
proximityDiversity
Dominated solutionsNondominated solutions
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Evolutionary of the variable length multi-objective optimization
Rank-based fitness assignment method
1
1
11 1 1
4
5
2
3
3
3
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Front 1
Front 2
Front 3
Front 4
f1
f2
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Evolutionary of the variable length multi-objective optimization
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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
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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
31
Reciprocal MutationReciprocal Mutation
AA BB CC DD EE FF
AA BB EE DD CC FF
32
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
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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
37
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
38
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
43
Introduction (cont.)
44
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
62
Overall Parameters
Parameters Value
Termination 5000Crossover rate 0.8
Mutation rate Pm 1/length of chromosome
Frequency 2.4GHz
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Indoor Freespace Parameters
Parameters Value Penetration loss Zero
Type of transmitters 1
Maximum allowed power loss threshold
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
Maximum allowed power loss threshold
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
Test points Total 3057 points
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
Maximum allowed power loss threshold
Type 1 – 130dB (500 meter radius)
Type 2 – 115dB (100 meter radius)
Transmitter costType 1 – 8000 Type 2 – 440
Transmitters capacityType 1 – 75Mb Type 2 – 54Mb
Test points Total 5690 points
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
77
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
78
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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