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
2
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
3
A Tale of Single Objective Optimization and
Multi-Objective Optimization
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
Algorithms for Single Objective
Optimization
Gradient descent aka steepest
descent or steepest ascent
Hill climbing
Simulated annealing
Quantum annealing
Tabu search
Beam search
Genetic algorithms
Ant colony optimization
Evolution strategy
Stochastic tunneling
Differential evolution
Particle swarm optimization
Harmony search
Bees algorithm
Dynamic relaxation
6
Multi-Objective Optimization
Courtesy of Dr. YoaChu Jin
7
Multi-Objective Optimization
m= 1 , single objective optimization
Courtesy of Dr. YoaChu Jin
8
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
9
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.
10
Single GA
11
Single GA
Parent A 110011001
Parent B 101111011
110011001
….
….
101010101
110011001
101111011
….
101010101
110011001 => 110101001
110011001
101111011
110011011
101111001
110011001
101111011
….
101010101
12
Introduction of multi-objective genetic algorithm
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
f2
Dominated solutions
Nondominated solutions
Solution Space
Pareto Front
Diversity
proximity
 Minimization
Objective 1
f1
13
The variable length genetic algorithm
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 region
Ex: Finding rules for autonomous agents
14
Evolutionary of the variable length multiobjective optimization
The number of solutions is not fixed.
It is a multi-objective optimization problem
We would like to solve the problem by GA
15
Evolutionary of the variable length multiobjective optimization
Nondominated solutions
f2
Solution Space
Dominated solutions
Nondominated solution
Pareto Front
Diversity
proximity
 Minimization
Objective 1 f1
16
Evolutionary of the variable length multiobjective optimization
Rank-based fitness assignment
method
f2
3
3
5
1
8
3
Front 4
1
5
Front 3
2
4
1
1
1
Front 2
Front 1
1
f1
17
Evolutionary of the variable length multiobjective optimization
18
Applications
MOGA
- Aircraft routing
VLMOGA
- Placement of heterogeneous wireless
transmitters
19
Application 1
Aircraft Routing using Multiobjective Genetic Algorithm
20
Problem Description
• Aircraft routing
▫ A given set of flights  a group of
aircrafts
▫ Available amount of aircrafts
21
Aircraft Routing (1/2)
Timetable  assign to aircrafts
Flight ID Dep. Time Arv. Time
Origin
Destination
f1
f2
f3
f4
f5
f6
f7
f8
09:15
19:00
14:00
11:20
22:00
16:30
08:10
20:35
09:50
19:35
14:35
11:55
22:50
17:20
09:00
21:25
KHH
TSA
MZG
TSA
TNN
TSA
KHH
KHH
TSA
KHH
TSA
MZG
TSA
TNN
TSA
TSA
f9
18:00
18:50
TNN
TSA
f10
15:30
16:20
TSA
TNN
Flight ID Dep. Time Arv. Time
f11
f12
f13
f14
f15
f16
f17
f18
f19
f20
09:30
19:10
14:10
10:20
18:00
15:30
08:20
18:35
13:00
10:20
10:05
19:45
14:45
10:55
18:50
16:20
09:20
19:25
13:35
10:55
Origin
Destination
HUN
TSA
MZG
TSA
TXG
TSA
KHH
KHH
MZG
TSA
TSA
HUN
TSA
MZG
TSA
TXG
TSA
TSA
TSA
MZG
22
Aircraft Routing (2/2)
f2
f1
f3
f4
f12
f14
f11
f8
f9
f6
f17
f5
f15
f7
f16
f18
f13
f10
f19
f20
Flight set F
aircraft 1
f8
f2
f13
f11
aircraft 2
f3
f6
f4
f10
f17
aircraft 3
f18
f16
f19
f9
f5
aircraft 4
f20
f1
f14
f7
f12
f15
Flight
schedule S
23
Notations
• Let α, β, ω, and γ represent
▫
▫
▫
▫
α:
β:
ω:
γ:
number of aircrafts
maximal number of flights per aircraft
number of airports
number of daily flights, respectively.
• Set of flights: F = {fi|1 ≤ i ≤ γ}
• Set of airports: P = {pj|1 ≤j ≤ ω}
P= {台北松山機場、 高雄小港、 台中、 台南 、馬公、 金門}
24
,
Associate Information of One Flight
The flight schedule S can be represented as:
si,j: the jth flight assigned to the ith aircraft
: origin of si,j, where
: destination of si,j, where
: departure time from
: arrival time in
25
Definition of a Flight Schedule
Maximal flights assigned to
each aircraft
Number of
aircrafts
Flight Schedule S
Number of
aircrafts
Maximal flights assigned
to each aircraft
s1,1
sα,β
2
6
27
Objectives
 Objectives:
Subject to
• Ground turn-around time objective
• Flow balance objective
28
Ground Turn-around Objective(1)
Legal ground turn-around time: TGH
Taipei
P
Kaohsiung
Q
Makung
Δt
29
Flow Balance Objective(2)
Taipei
Taipei
f1
f1
Kaohsiung
Kaosiung
f2
Makung
f2
Makung
(a)
(b)
Extra cost
30
Encoding Scheme
1
2
…
β-1
β
c1
s1,1
s1, 2
…
s1, β-1
s1, β
c2
s2,1
s2, 2
…
s2, β-1
s2, β
:
:
:
:
:
:
cα
sα, 1
sα, 2
…
sα, β-1
sα, β
s1,1
s1, 2
…
s1, β-1
flights of c1
s1, β
s2,1
s2, 2
…
s2, β-1
flights of c2
s2, β
…
…
…
sα, 1
sα, 2
…
sα, β-1
flights of cα
sα, β
31
Crossover
A B C D E F
Mapping
exchange
relationship
B C E A F D
cutpoint
A B C A F D
mapping
relationship
E
A
A B C E F D
duplicate
genes
B C E D E F 1. change A to E B C E D A F
2. change E to A
32
Reciprocal Mutation
A
B C D E
F
A
B
F
E D C
Experimental Results
• 7 airports, 9 aircrafts, 12 flights one day, 79 flights.
Airports
HUN
KHH
KNH
MZG
TNN
TSA
Parameter
Value
Crossover rate
1
Mutation rate
0.2
No. of generations
5000
Population size
100
Reproduction rate
0.8
TTT
33
34
Experimental Results
35
Symbols in Experimental Results
Gantt chart:
time
Aircraft1
/crew1
Aircraft2
/crew2
departure time
origin
flight ID
682
KNH
TSA
destination
arrival time
36
Scheduling Result of 9 aircrafts
37
Scheduling Result of 8 aircrafts
Example
ε = [ε1, ε2] =[k1 × α × TGH, k2× α]
= [1 × 8 × 25, 1 × 8]
= [200, 8]
Original value
Φ(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]
38
Scheduling Result of 8
aircrafts
Result1
Result2
Result3
39
Scheduling Result of 8 Aircrafts
40
Retiming Process
Retiming
process
Station K
Station K
P
P
Station K+1
Station K+1
Q
Q
Station K+2
Station K+2
Inspection line
Flights P and Q cannot be
assigned to the same aircraft
Inspection line
Flights P and Q can be
assigned to the same aircraft
41
Scheduling Result-Retiming
42
Application 2:
Heterogeneous Wireless Transmitter
Placement with Multiple Constraints based on
the Variable-Length Multi-objective Genetic
Algorithm
43
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
44
Introduction (cont.)
23meters
15meters
45
Parameters
Parameters
Type of transmitters
Value
1
Transmitter cost
Maximum allowed power
loss threshold
Transmitter capacity
2200
63.5dB (radius 15 meters)
Test points
Total 10404 points
Data Rate Demand
21kbps
Generation
3000
54Mb
Wireless transmitter placement problem
47
Problem Definition
• Model
▫ Map, receiver, transmitter
• Receiver
▫ Position, data rate
demand, sensitivity
• Transmitter
▫ Position, type=(cost, power, capacity)
48
Path Loss Propagation Models
1. Free space path loss model
L(r i , t j )  20  log 10(
4d

)
2. Log-distance path loss model with shadowing
effect
o ( r ,t )
L(r i , t j )  20  log 10(
3. ECC-33 model
4d

)
i
j
 P (r , t )  A (r , t )
g 1
g
i
j
g
i
j
49
Objectives
•
•
•
•
Coverage
Cost
Data Rate Demand
Overlap
50
Coverage
Coverage Rate=4/9=44.4%
Uncoverage=5
n
Uncoverage (T )   cT (ri ), {
i 1
cT (ri )  1, if coverage T ( ri )  -1
cT (ri )  0,
otherwise
51
Cost
• Cost=1000+300*2
=1600
Cost=1000
Cost=300
Cost=300
Cost (T )   cost (t j )
t j T
52
Data Rate Demand
• Yellow: 16kbps
• Blue: 128 kbps
• Demand(t1)=128*
4+16*2=544
• Capacity(t1)=1024
• DRD(T)=|5441024|=480
DRD (T )   |
 demand
ri
t j T ri R , power ( t j )  L ( ri ,t j )  sensitivityi
 capacity (t j ) |
Next generation wireless LAN system design, Proceedings on MILCOM, 2002
53
Overlap
• Overlap=2
• Overlap rate=2/9
=22.2%
Overlap (T ) 
 overlap
ri
overlapri  0, if | AS ri | 0,1
,{
overlapri  1, if | AS ri | 1
ri PG
Evolutionary multiobjective optimization for base station transmitter placement with frequency assignment,
IEEE Trans. on Evolutionary Computation, 2003
54
Objectives
• Minimize
n
Uncoverage (T )   cT (ri ), {
i 1
cT (ri )  1, if coverageT ( ri )  -1
cT (ri )  0,
otherwise
Cost (T )   cost (t j )
t j T
DRD (T )   |
 demand
ri
t j T ri R , power ( t j )  L ( ri ,t j )  sensitivityi
Overlap (T ) 
 overlap
ri PG
ri
 capacity (t j ) |
overlapri  0, if | AS ri | 0,1
,{
overlapri  1, if | AS ri | 1
• Subject to
ri , positioni  PG, where 1  i  n, n is the number of receivers
t j , position j  PG, where 1  j  m, m is the number of transmitt ers
55
Encoding:
Individual and Chromosome
Representation
An individual
(x1,y1,z1),type1
(x2,y2,z2),type2
……
(xm,ym,zm),typem
A chromosome
X1
0
1
Y1
1
0
1
type1
Z1
0
1
0
Transmitter resolution= Encoding bits=3
0
0
56
Population Initialization
• Temporary upper bound UB=10
• Random number=4 (1~10)
type1
type1
type1
type1
type3
type3
type3
type2
type2
type1
Flowchart
57
Types of crossover
• Uniform crossover for chromosome
▫ Change the position and type of transmitter
• Variable-length one-point crossover
▫ Change the length of individual
58
Uniform Crossover for chromosome
Individual1
(x1,y1,z1),type1
(x2,y2,z2),type2
Individual2
(x1,y1,z1),type1
(x2,y2,z2),type2
(x3,y3,z3),type3
59
One-point Crossover
Split point
Individual1
(x1,y1,z1),type1
(x2,y2,z2),type2
Individual2
(x1,y1,z1),type1
(x2,y2,z2),type2
Split point
(x3,y3,z3),type3
60
Overall Crossover
Parents
Crossover Rate: Pc
1-Pc
Pc
Variable-length
one-point
crossover
Uniform
crossover for
chromosome
1-Pc
Pc
Uniform
crossover for
chromosome
1
Copy from
parents
Offspring
Flowchart
61
Mutation
X1
chromosome1
0
1
Y1
1
0
X1
chromosome1
0
1
1
0
1
Y1
1
0
0
type1
Z1
0
0
type1
Z1
0
1
0
1
Pm 
length of chromosome
0
0
0
Flowchart
62
Types of Simulation
Population
Size
Types of
Transmitters
Free space
100
1
Path loss
100
1
2D path loss
500
2
3D path loss
200
2
Upper bound
UB=8,12,15
500
2
Heterogeneous
500
2
Indoor
Outdoor
63
Overall Parameters
Parameters
Value
Termination
Crossover rate
5000
0.8
Mutation rate Pm
1/length of chromosome
Frequency
2.4GHz
64
Indoor Freespace Parameters
Parameters
Penetration loss
Type of transmitters
Value
Zero
1
Maximum allowed power
loss threshold
Transmitter capacity
66dB (radius 20 meters)
Test points
Every 3m x 3m, total 231
points
1024kbps
Data Rate Demand
54Mb
65
Indoor Map
• Floor plan of IC factory
66
Indoor Free Space
• Threshold 66dB (radius 20 meters)
Uncoverage
0
Cost
6600
Data rate
83968
demand
Overlap
13
BS #
3
67
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
68
Outdoor Map
69
Two Dimensional Outdoor Path Loss
Parameters
Parameters
Penetration loss
Types of transmitters
Maximum allowed power loss
threshold
Transmitter cost
Transmitters capacity
Test points
Value
Type 1 –concrete wall: 8dB
Type 2–mountain: 99dB
2
Type 1 – 103dB (1.5KM radius)
Type 2 – 80dB (100 meter radius)
Type 1 – 40000
Type 2 – 2200
Type 1 – 75Mb
Type 2 – 54Mb
Total 3057 points
70
Outdoor 2D-Data Rate Demand
• Blue: 16 kbps/ 81%
• Green:128kbps/
13.7%
• Red:1024kbps/
5.2%
71
Outdoor 2D Result
Result-11 Transmitters
Uncoverage
25
Cost
480000
Data rate
403376
demand
Overlap
2000
BS 1 #
12
BS 2 #
0
72
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
73
Three Dimensional Outdoor Path Loss
Parameters
Parameters
Types of transmitters
Value
2
Maximum allowed power loss
threshold
Type 1 – 130dB (500 meter radius)
Type 2 – 115dB (100 meter radius)
Type 1 – 8000
Type 2 – 440
Type 1 – 75Mb
Type 2 – 54Mb
Total 5690 points
Transmitter cost
Transmitters capacity
Test points
74
Outdoor 3D- Data Rate Demand
• Blue: 16 kbps/ 57%
• Green:128kbps/
28%
• Red:512kbps/ 15%
75
Outdoor 3D Result
Uncoverage
0
Cost
32000
Data rate
1493184
demand
Overlap
3516
BS 1 #
4
BS 2 #
0
76
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
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
References
1.
2.
3.
4.
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
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
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
Yaochu Jin, “Evolutionary Multi-Objective Optimization”, Honda Research
Institute Europe
79
Question & Suggestion
80
• Thank you for your attentions