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Introduction to Genetic Algorithms
Theory and Applications
The Seventh Oklahoma Symposium on Artificial
Intelligence
November 19, 1993
Roger L. Wainwright
Dept. of Mathematical and Computer Sciences
The University of Tulsa
600 South College Avenue
Tulsa, OK 74104-3189
(918) 631-3143
[email protected]
C Copyright RLW 1993
1
Tutorial Outline
Part I Introduction and Concepts of Genetic Algorithms
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Bibliography
GA Definitions
Overview of GAs
When to Use GAs
GA vs Traditional Algorithms
GA vs SA
Applications of GA
The Standard GA (Algorithm)
Population Representation
Reproduction
Example GA Fitness Function
Roulette Wheel
Crossover
Mutation
Schema Theory
Fundamental Theorem of GA
Deception
Genetic Programming
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Tutorial Outline
Part II Example Applications of Genetic Algorithms
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Order Based GAs
PMX Crossover
TSP GA
Additional Applications using GAs
Three Dimensional Bin Packing
Set Covering Problem
Multiple Vehicle Routing
Neural Network GA
Parallel GA Issues
Genetic Algorithm Packages
GATutor
GA Newsletter and How to Join
3
Primary GA Bibliography (1993)
1.Proceedings of the First International Conference on Genetic Algorithms,
John Grefenstette, Editor, Lawrence Erlbaum Assoc., 1985.
2.Proceedings of the Second International Conference on Genetic Algorithms,
John Grefenstette, Editor, Lawrence Erlbaum Assoc., 1987.
3.Proceedings of the Third International Conference on Genetic Algorithms, J.
David Schaffer, Editor, Morgan Kaufmann, 1989.
4.Proceedings of the Fourth International Conference on Genetic Algorithms,
Richard Beler, Editor, Morgan Kaufmann, 1991.
5.Proceedings of the Fifth International Conference on Genetic Algorithms,
Stephanie Forrest, Editor, Morgan Kaufmann, 1993
6.Genetic Algorithms and Simulated Annealing, Lawrence Davis, Editor,
Pitman Publishing, 1987.
7.Handbook of Genetic Algorithms, Lawrence Davis, Editor, Van Nostrand
Reinhold, 1991.
8.Genetic Algorithms in Optimization, Search and Machine Learning, David
Goldberg, Addison Wesley, 1989.
9.Adaptation in Natural and Artificial Systems, John H. Holland, The University
of Michigan Press, Ann Arbor, MI, 1975.
10.Adaptation in Natural and Artificial Systems, John H. Holland, MIT Press,
1992.
11.Genetic Programming, John R. Koza, MIT Press, 1992.
4
12.Parallel Problem Solving from Nature 2, Reinhard Manner and Bernard
Manderick, Editors, Noth-Holland, 1992.
13.Foundations of Genetic Algorithms, Gregory Rawlins, Editor,
Kaufmann, 1991.
Morgan
14.Foundations of Genetic Algorithms 2, Darrell Whitley, Editor, Morgan
Kaufmann, 1993.
15.Nicol N. Schraudolph, "Genetic Algorithm Software Survey", available by
anonymous ftp from cs.ucsd.edu as /pub/GAucsd/GAsoft.txt, August, 1992.
Other GA References (1993)
16.J. Grefenstette, GENESIS, Navy Center for Applied Research in Artificial
Intelligence, Navy research Lab., Wash. D.C. 20375-5000.
17.J. Grefenstette, "Optimization of Control Parameters for Genetic Algorithms",
IEEE Transactions on Systems, Man and Cybernetics, pp. 122-128, 1986.
18.H. Muehlenbein, "Parallel Genetic Algorithms, Population Genetics and
Combinatorial Optimization", Proceedings of the 3rd International Conference
on Genetic Algorithms, Morgan Kaufmann, 1989.
19.R. Tanese, "Distributed Genetic Algorithms, Proceedings of the Third
International Conference on Genetic Algorithms, ed. J.D. Schaffer, Morgan
Kaufmann, pp. 434-439, 1989.
20.T. Starkweather, S. McDaniel, K. Mathias, D. Whitley and C. Whitley, "A
Comparison of Genetic Sequencing Operators", Proceedings of the Fourth
International Conference on Genetic Algorithms, June, 1991.
21.T. Starkweather, D. Whitley, and K. Mathias, "Optimization Using
Distributed Genetic Algorithms," in Parallel Problem Solving from Nature, ed.
H. Schwefel and R. Maenner, Springer Verlag, Berlin, Germany, 1991.
5
22.D. Whitley, T. Starkweather, and C. Bogart, "Genetic Algorithm and Neural
Networks: Optimizing Connections and Connectivity", Parallel Computing,
14:347-361.
23.D. Whitley and J. Kauth, GENITOR: A Different Genetic Algorithm,
Proceedings of the Rocky Mountain Conference on Artificial Intelligence, Denver,
Co., 1988, pp. 118-130.
24.D. Whitley, T. Starkweather, and D. Fuquat, "Scheduling Problems and
Traveling Salesman: The Genetic Edge Recombination Operator", Proceedings of
the Third International Conference on Genetic Algorithms, June, 1989.
25.D. Whitley and T. Starkweather, "GENITOR II: A Distributed Genetic
Algorithm", Journal of Experimental and Theoretical Artificial Intelligence,
2(1990) 189-214.
University of Tulsa GA References (1993)
26.Abuali, F. N., Schoenefeld, D. A. and Wainwright, R. L. "The Design of a
Multipoint Line Topology for a Communication Network Using Genetic
Algorithms", Seventh Oklahoma Conference on Artificial Intelligence, November,
1993.
27.Abuali, F.N., Schoenefeld, D.A. and Wainwright, R.L., "Terminal Assignment
in a Communication Network Using Genetic Algorithms", to appear in the 22nd
Annual ACM Computer Science Conference - CSC'94, March, 1994.
28.Blanton, J.L. and Wainwright, R.L. "Multiple Vehicle Routing with Time
and Capacity Constraints using Genetic Algorithms", Proceedings of the Fifth
International Conference on Genetic Algorithms (ICGA-93), Stephanie
Forrest, Editor, Morgan Kaufmann Publisher, 1993, pp. 452-459.
29.Corcoran, A. L. and Wainwright, R. L. "A Genetic Algorithm for Packing in
Three Dimensions", Proceedings of the 1992 ACM Symposium on Applied
Computing, March 1-3, 1992, pp. 1021-1030, ACM Press.
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30.Corcoran, A. L. and Wainwright, R. L., "LibGA: A User-Friendly Workbench
for Order-Based Genetic Algorithm Research", Proceedings of the 1993
ACM/SIGAPP Symposium on Applied Computing, February, 14-16, 1993, pp.
111-117, ACM Press.
31.Corcoran, A.L. and Wainwright, R.L. "The Performance of a Genetic
Algorithm on a Chaotic Objective Function", Seventh Oklahoma Conference on
Artificial Intelligence, November, 1993.
32.Knight, L. R. and Wainwright, R. L. "HYPERGEN - A Distributed Genetic
Algorithm on a Hypercube", Proceedings of the 1992 IEEE Scalable High
Performance Computing Conference, Williamsburg, VA., April 26-29, 1992, pp.
232-235, IEEE Press.
33.Mutalik, P. M., Knight, L. R., Blanton, J. L. and Wainwright, R. L.
"Solving Combinatorial Optimization Problems Using Parallel Simulated
Annealing
and Parallel Genetic Algorithms", Proceedings of the 1992
ACM/SIGAPP Symposium on Applied Computing, March 1-3, 1992. pp. 10311038, ACM Press.
34.Prince, C., Wainwright, R.L., Schoenefeld, D.A. and Tull, T., "GATutor: A
Graphical Tutorial System for Genetic Algorithms" to appear in the 25th ACM
SIGCSE Technical Symposium - SIGCSE'94, March, 1994.
35.Sekharan, D. Ansa and Wainwright, R. L., "Manipulating Subpopulations of
Feasible and Infeasible Solutions in Genetic Algorithms", Proceedings of the
1993 ACM/SIGAPP Symposium on Applied Computing, February, 14-16, 1993,
pp. 118-125, ACM Press.
36.Sekharan, D. Ansa and Wainwright, R. L., "Manipulating Subpopulations in
Genetic Algorithms for Solving the k-way Graph Partitioning Problem", Seventh
Oklahoma Conference on Artificial Intelligence, November, 1993.
37.Wu, Yu and Wainwright, R. L., "Near-Optimal Triangulation of a Point Set
using Genetic Algorithm", Seventh Oklahoma Conference on Artificial
Intelligence, November, 1993.
7
Genetic Algorithm Definitions
Grefenstette [16]
"A genetic Algorithm is an iterative procedure maintaining a
population of structures that are candidate solutions to specific domain
challenges. During each temporal increment (called a generation), the
structures in the current population are rated for their effectiveness as
domain solutions, and on the basis of these evaluations, a new
population of candidate solutions is formed using specific genetic
operators such as reproduction, crossover, and mutation."
Goldberg [8]
"They combine survival of the fittest among string structures with
a structured yet randomized information exchange to form a search
algorithm with some of the innovative flair of human search. In
every generation, a new set of artificial creatures (strings) is created
using bits and pieces of the fittest of the old; an occasional new part
is tried for good measure. While randomized, genetic algorithms
are no simple random walk. They efficiently exploit historical
information to speculate on new search points with expected improved
performance."
8
Genetic Algorithms Overview
_
Developed by John Holland in 1975 [9].
_
Genetic Algorithms (GAs) are search
algorithms based on the mechanics of the natural
selection process (biological evolution). The most
basic concept is that the strong tend to adapt and
survive while the weak tend to die out. That is,
optimization is based on evolution, and the "Survival
of the fittest" concept.
_
GAs have the ability to create an initial
population of feasible solutions, and then recombine
them in a way to guide their search to only the most
promising areas of the state space.
_
Each feasible solution is encoded as a
chromosome (string) also called a genotype, and each
chromosome is given a measure of fitness via a
fitness (evaluation or objective) function.
_
The fitness of a chromosome determines its
ability to survive and produce offspring.
_
A finite population of chromosomes is
maintained.
9
Genetic Algorithms Overview Continued
_
GAs use probabilistic rules to evolve a
population from one generation to the next. The
generations of the new solutions are developed by
genetic recombination operators:
_ Biased Reproduction: selecting the fittest to
reproduce)
_ Crossover: combining parent chromosomes to
produce children chromosomes
_ Mutation: altering some genes in a
chromosome.
_ Crossover combines the "fittest" chromosomes
and passes superior genes to the next generation.
_
Mutation ensures the entire state-space
will be searched, (given enough time) and can lead
the population out of a local minima.
_
Most Important Parameters in GAs:
_ Population Size
_ Evaluation Function
_ Crossover Method
_ Mutation Rate
10
Genetic Algorithms Overview Continued
_ Determining the size of the population is a
crucial factor
_ Choosing a population size too small
increases the risk of converging prematurely to a
local minima, since the population does not have
enough genetic material to sufficiently cover the
problem space.
_ A larger population has a greater chance of
finding the global optimum at the expense of
more CPU time.
_ The population size remains constant from
generation to generation.
11
Genetic Algorithms Overview Continued
_ A robust search technique
_ GAs will produce "close" to optimal results in a
"reasonable" amount of time
_ Suitable for parallel processing
_ Some problems are deceptive
_ Can use a noisy fitness function
_ Fairly simple to develop
_ Makes no assumptions about the problem space
_ GAs are blind without the fitness function. The
Fitness Function Drives the Population Toward Better
Solutions and is the most important part of the
algorithm.
_Probability and randomness are essential parts of GA
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Use Genetic Algorithms
_
When an acceptable solution representation is
available
_ When a good fitness function is available
_ When it is feasible to evaluate each potential solution
_ When a near-optimal, but not optimal solution is
acceptable.
_ When the state-space is too large for other methods
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Genetic
Algorithms
Algorithm
vs
Traditional
Goldberg [8]
1. The GA works with a coding of the parameter rather
than the actual parameter.
2. The GA works from a population of strings instead of
a single point.
3. Application of GA operators causes information from
the previous generation to be carried over to the next.
4. The GA uses probabilistic transition rules, not
deterministic rules.
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Genetic Algorithms vs. Simulated Annealing
SA
_ 1 Feasible Solution
_ Perturbation Function
_ Acceptance Function
_ Temperature Parameter
GA
_ Population of Feasible Solutions
_ Evaluation Function
_ Selection Bias
_ Reproduction
_ Mutation
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Applications of Genetic Algorithms
_ Scheduling:
Facility, Production, Job, and Transportation Scheduling
_ Design:
Circuit board layout, Communication Network design,
keyboard layout, Parametric design in aircraft
_ Control:
Missile evasion, Gas pipeline control, Pole balancing
_ Machine Learning:
Designing Neural Networks, Classifier Systems, Learning rules
_ Robotics:
Trajectory Planning, Path planning
_ Combinatorial Optimization:
TSP, Bin Packing, Set Covering, Graph Bisection, Routing,
_ Signal Processing:
Filter Design
_ Image Processing:
Pattern recognition
_ Business:
Economic Forecasting; Evaluating credit risks
Detecting stolen credit cards before customer reports it is stolen
_ Medical:
Studying health risks for a population exposed to toxins
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The Standard Genetic Algorithm
>>>> Use the flowchart I created
>>>> Replace this page with flowchart page
17
GENETIC ALGORITHMS
Preliminary Considerations
1. Determine how a feasible solution should be represented
a) Choice of Alphabet. This should be the smallest alphabet
that permits a natural expression of the problem.
b) The String Length. A string is a chromosome and each
symbol in the string is a gene.
2. Determine the Population Size.
This will remain constant throughout the algorithm.
Choosing a population size too small increases the risk of
converging prematurely to a local optimum, since the population
does not sufficiently cover the problem space. A larger
population has a greater chance of finding the global optimum
at the expense of more CPU time.
3. Determine the Objective Function to be used in the algorithm.
18
A Genetic Algorithm
1. Determine an Initial Population.
a) Random or
b) by some Heuristic
2. REPEAT
A. Determine the fitness of each member of the population.
(Perform the objective function on each population member)
Fitness Scaling can be applied at this point. Fitness
Scaling adjusts down the fitness values of the superperformers and adjusts up the lower performers, promoting
competition among the strings. As the population matures,
the really bad strings will drop out. Linear Scaling is
an example.
B. Reproduction (Selection)
Determine which strings are "copied" or "selected" for the
mating pool and how many times a string will be "selected"
for the mating pool. Higher performers will be copied more
often than lower performers. Example: the probability of
selecting a string with a fitness value of f is f/ft, where
ft is the sum of all of the fitness values in the population.
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Genetic Algorithm Continued
C. Crossover
1. Mate each string randomly using some crossover technique
2. For each mating, randomly select the crossover position(s).
(Note one mating of two strings produces two strings.
Thus the population size is preserved.)
D. Mutation
Mutation is performed randomly on a gene of a chromosome.
Mutation is rare, but extremely important. As an example,
perform a mutation on a gene with probability .005.
If the population has g total genes (g = string length *
population size) the probability of a mutation on any one
gene is 0.005g, for example. This step is a no-op most of
the time. Mutation insures that every region of the problem
space can be reached. When a gene is mutated it is randomly
selected and randomly replaced with another symbol from the
alphabet.
UNTIL Maximum number of generation is reached.
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Various Population Representations
_ Chromosomes can be represented as
_ Bit Strings (1011 ... 0100)
_ Reals (19.3, 45.1, -12.9, ... 6.2)
_ Integers (1,4,2,7,5,9,3,6,8) Usually Permutations of 1..n
_ Characters (A, G, Q, ... F) Usually Permutations
_ List of rules (R1, R2, ... R20)
_ Chromosomes are all of the same type (Bit Strings)
_ Chromosomes are all the same length
_ The population size remains constant from generation to generation
21
Reproduction (Survival of the fittest)
Parents are SELECTED for REPRODUCTION biased
on the fitness function
Consider the fitness function
f(x) = 4 * cos(x) + x + 2.5
0 <= x >= 31
>>> Graph of P0 data points <<<
22
Initial Population P0
No.
Chromosome
x
f(x)
P(x)
No. Times
Selected*
1
1001 1
19
25.459
.185
2
2
01 010
10
9.144
.066
1
3
1 1001
25
31.465
.229
2
4
00110
6
12.341
.090
0
5
0 1011
11
13.518
.098
1
6
1011 1
23
23.369
.170
1
7
00100
4
3.885
.028
0
8
10 001
17
18.399
.134
1
SUM
115
137.580
1.000
8
AVE
14
17.198
.125
1
MAX
25
31.465
.229
2
f(Pop)= sum f(x)/n = 137.58 / 8 = 17.198
P(x) = f(x) / sum f(x)
= the Probability of Selection
23
* Based on Selection Roulette Wheel
The Selection Roulette Wheel
Roulette: the classical selection operator for generational
GA as described by Goldberg. Each member of the pool
is assigned space on a roulette wheel proportional to
its fitness. The members with the greatest fitness have
the highest probability of selection. This selection
technique works only for a GA which maximizes its
objective function.
<< wheel >>
24
Crossover
_ Causes an exchange of genetic material between
two parents
_ Crossover point(s) is determined stochastically
_ The Crossover Operator is the most important feature
in a GA
Single Point Crossover Example
Parent 1
100
1001010
Parent 2
001
0110111
Child 1
100
0110111
Child 2
001
1001010
Double Point Crossover Example
Parent 1
110100
1001
011
Parent 2
010110
0010
101
Child 1
110100
0010
011
Child 2
010110
1001
101
25
Mutation
_ The Mutation operator guarantees the entire state-space
will be searched, given enough time.
_ Restores lost information or adds information to the population.
_ Performed on a child after crossover.
_ Performed infrequently (For example, 0.005 probability of
altering a gene in a chromosome)
Child 1
1101 0 00010011
after mutation 1 1 0 1 1 0 0 0 1 0 0 1 1
_ Special Mutation operators may be application dependent (TSP)
_ Adaptive Mutation:
_ Monitor the hamming distance between two parents.
_ The more similar the parents, the more likely mutation
is applied. Whitley, Starkweather [25]
26
Continuation of the fitness function
f(x) = 4 * cos(x) + x + 2.5
0 <= x >= 31
Population P1
(After Crossover.
No.
New
Chromosome
Assume no Mutation)
Parents
(from P0)
Crossover
Point
x
f(x)
1
2
01 001
10 010
(2,8)
(2,8)
2
2
9
18
7.8552
23.141
3
4
1001 1
1011 1
(1,6)
(1,6)
4
4
19
23
25.459
23.369
5
6
1 1011
0 1001
(3,5)
(3,5)
1
1
27
9
28.331
7.855
7
8
100 01
110 11
(1,3)
(1,3)
3
3
17
27
18.399
28.331
SUM
149
162.740
AVE
18
20.343
MAX
27
28.331
f(Pop0) =
f(Pop1) =
17.198
20.343
27
>>> plot of the function showing
the 8 initial points
of Pop0 and the 8 points of Pop1
28
Schema Theory (John Holland)
_ An abstract way to view the complexities of crossover.
_ Consider 6-bit representations where * indicates don't care
0***** represents a subset of 32 strings
1**00* represents a subset of 8 strings
_ Let H represent a schema such as 1**1**
_ Order: o(H)
The number of fixed positions in the schema, H.
o(1*****) = 1,
o(1**1*1) = 3
_ Length: delta(H)
The distance between sentinel fixed positions in H.
delta(1**1**) = (4-1) = 3
delta(1*****) = 0
delta(***1**) = 0
29
Fundamental
Algorithms
Theorem
of
Genetic
(The Schema Theorem)
The expected number of copies, m, of schema H is bounded by:
>>>> Slide from GATUTOR 91
Where
m - number of schemata
H - schema
t - time or generation
f - fitness function
fave - average fitness value
pc - crossover probability
delta - length
l - string length
pm - mutation probability
o - order
30
Consider H = 1**** in the above problem:
The Schema Theorem states that
m(H,P1) >= m(H,P0) f(H)/fave
6 >= 4 * 25.753 / 17.198
6 >= 6
(Note in this case o(1****) = 1 and delta(1****) = 0
and pm = 0 greatly reducing the formula)
In Other Words:
Theorem: The number of representatives of any schema, S,
increases in proportion to the observed relative performance of S.
31
Deception
What Problems are Difficult for GAs
Example: an order-3 deception [25]
"information represented by the schemata in the search space leads the
search away from the global optimum, and instead directs the search
toward the binary string that is the complement of the global optimum.
The search space is order-3 deceptive .. if the following relationships
hold for the [three-bit] schemata:"
0** > 1** and 00* > 11*, 01*, 10*
*0* > *1* and 0*0 > 1*1, 0*1, 1*0
**0 > **1 and *00 > *11, *01, *10
but, 111 > 000, 001, 010, 100, 110, 101, 011
Example: f(000) = 28 f(100) = 14
f(001) = 26 f(101) = 10
f(010) = 22 f(110) = 5
f(011) = 20 f(111) = 30
Chaotic, noisy and "needle in a haystack" functions
GA-easy, GA-hard problems
32
Overview of Genetic Programming
Koza [11]
Manipulate strings of instructions rather than strings of data.
Goal: Allow computers to develop their own software
(Survival of the fittest computer programs)
Crossover and mutation manipulate branches of the program tree.
"Genetic Programming starts with an initial population of randomly generated
computer programs composed of functions and terminals appropriate to the
problem domain. The functions may be standard arithmetic operations, standard
programming operations, standard mathematical functions, logical functions,
or domain-specific functions. Depending on the particular problem, the
computer program may be Boolean-valued, integer-valued, real-valued,
complex-valued, vector-valued, symbolic valued, or multiple-valued. The
creation of this initial random population is, in effect, a blind random search of
the search space of the problem. Each individual computer program in the
population is measured in terms of how well it performs in the particular
problem environment. This measure is called the fitness measure. The nature of
the fitness measure varies with the problem" [11].
Koza's initial problem: Given a set of initial predicates and possible actions,
develop (evolve) a computer program (in Lisp) to control the movement of an
ant searching for food. The chromosome is a variable sized Lisp program where
the leaf nodes are actions (left, right, move, etc.), and the internal nodes are
predicates or logic controls (if found food), etc. Each chromosome (program)
is used to control the actions of a simulated ant in searching for food. The
evaluation function for a given chromosome is the amount of food gathered by
an ant in a fixed amount of time.
33
Consider the following two parent computer
programs given as LISP S-expressions.
<<<<Fig 6.5 page 101>>>>>
These two parents are equivalently represented as:
(OR (NOT D1) (AND D0 D1)) and
(OR (OR D1 (NOT D0)) (AND (NOT D0) (NOT D1))).
The first parent has 6 nodes (points) in its S-expression, and the second
parent has 10 points in its S-expression as shown above.
Randomly select any one of the 6 points in parent 1 as its crossover
point, say node "NOT".
Randomly select any one of the 10 points in parent 2 as its crossover
point, say node "AND".
The Selected S-expressions are shown below.
34
<<< Fig 6.6 page 102 >>
The above crossover fragments are exchanged at node "NOT" in the first
parent, and node "AND" in the second parent to produce the following
two children S-expressions [11].
<<<< Fig 6.7 page 102 >>>
35
Part II
Example Applications of Genetic Algorithms
Order-Based Genetic Algorithms
_An order-based GA is where all chromosomes are a
permutation of the list.
_Order-based GAs greatly reduce the size of the search
space by pruning solutions that we do not want to consider.
_Order-based GAs can be applied to a number of
classic combinatorial optimization problems such as:
TSP,
Bin Packing,
Package
Placement, Job
Scheduling, Network Routing, Vehicle Routing, various
layout problems, etc.
_Crossover functions for order-based GAs include
Edge Recombination,
Order
Crossover #1, Order
Crossover
#2, PMX,
Cycle
Crossover, Position
Crossover, etc. Whitley and Starkweather [20,25].
36
PMX (Partially Matched Crossover)
Parent 1
3719 | 645 | 28
Parent 2
4785 | 392 | 16
( 6 <--> 3 )
( 4 <--> 9 )
( 5 <--> 2 )
Child 1
6714 | 392 | 58
Child 2
9782 | 645 | 13
Mutation functions for order-based GAs include
Swap two elements
*
*
9 8 7 6 5 4 3 2 1
==>
*
*
9 3 7 6 5 4 8 2 1
==>
*
9 8 6 5 4 3 7 2 1
Move one element
*
9 8 7 6 5 4 3 2 1
Reorder a sublist
98 | 76543 | 21
==> 9 8 | 5 3 4 6 7 | 2 1
37
Traveling Salesman Problem
Example TSP GA executions adjusting pool size:
TSP 1024 Cities.
PoolSize
Length_String
Trials
Bias
RandomSeed
<same>
MutateRate
NodeFile
StatusInterval
500
1024
100000
1.9
RESULT =
116987
250
15394157
125
<same>
0.15
cities1024
5000
88436
38
90906
TSP 320 Cities.
PoolSize
2000
Length_String
320
Trials
100000a
Bias
1.9
RandomSeed
15394157
MutateRate
0.15
NodeFile
cities320
1000
500
<same>
<same>
250
<same>
125
<same>
StatusInterval 1000
RESULTS:
Best:
30,761b
25,708
21,366
18,676c
23,760d
Worst:
35,102
28,366
23,235
18,676
23,760
Average:
34,209
27,863
22,880
18,676
23,760
a
A poolsize of 2000 for 205,000 trials yielded best of 22,777
b
CPU time on a SPARC 1+ was approximately 100 minutes.
c
Converged after 72,000 trials
d
Converged prematurely after 33,000 trials
39
TSP 105 Cities.
PoolSize
750
Length_String
105
Trials
70000
Bias
1.9
RandomSeed
15394157
MutateRate
0.15
NodeFile
cities105
500
<same>
250
125
<same>
<same>
StatusInterval 1000
RESULTS:
Trials at
Convergence:
109,000
61,000
32,000
9,000
Best:
16,503
17,193
24,079
32,370
Worst:
16,503
17,193
24,079
32,370
Average:
16,503
17,193
24,079
32,370
40
Additional Applications Using GAs
Three Dimensional Bin Packing Using GAs [29]
Set Covering Problem Using GAs [35]
Multiple
Vehicle
Routing with Time and
Constraints Using GAs [28]
Capacity
Genetic Algorithms and Neural Networks Fixed Architecture
Genetic Algorithms
Architecture
and Neural Networks
Unknown
Parallel Genetic Algorithms [32]
k-way Graph Partitioning Algorithm Using GAs [36]
Graph Bisectioning Problem Using GAs [36]
Triangulation of a Point Set Using GAs [37]
The Package Placement Problem Using GAs [33]
Three Dimensional Bin Packing Using GAs [29]
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Encoding:
packages
String of integers representing a permutation of the
Evaluation: Height returned by the Next Fit of First Fit Heuristic
Crossover: Order2, Cycle, PMX, and Random Swap
Mutation: Adaptive, swap 2 packages and rotate on one axis.
RESULTS (in % Fill)
Rotated
Presorted
Without GA
Using a GA
Using a GA
Next Fit Random
31-35%
41-55%
57-66%
Next Fit Contrived
48-50%
56-71%
60-71%
First Fit Random
36%
41-49%
53-63%
First Fit Contrived
41-53%
56-59%
67-77%
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Tables from ART page 12 and 14
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Set Covering Problem Using GAs [35]
Page 6 and FIG 1 of D. Ansa's Slides
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Multiple Vehicle Routing with Time
and Capacity Constraints Using GAs [28]
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Genetic Algorithms and Neural Networks
Fixed Architecture
Given: A Fixed Connection Topology
Goal: Optimize Connection Weights in a Forward-feed Neural
Network [22]
Example Representation:
Each weight ranges -127 to +127 (8-bits)
Each Chromosome is the concatenated binary weights of the net.
Example Evaluation Function:
Run the Network in a feed-forward fashion for each training
pattern just as if one were going to use back propagation.
Accumulate the sum of the squared error as the fitness value
Crossover results in new weight values to try
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Genetic Algorithms and Neural Networks
Unknown Architecture
Use GAs to determine a network architecture
Each Chromosome Depicts a Possible Connection
Topology
Evaluate each architecture
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Parallel Genetic Algorithms
Parallel Issues [32]
Migration Interval
1. After 5,10,20,50,100 Generations
2. ADAPTIVE
Migration Rate
1. 10%, 20%, 50% of the population
2. ADAPTIVE
How to pick the migrants
1. uniformly
2. skewed towards the more fit
3. Generate "an over population" during migration
generations so nothing is lost from the sending
population, only new comers are analyzed as they come in
4. Perform an "exchange" of genetic material
5. ADAPTIVE
Topology
1. Ring
2. Hypercube dimensions alternation through the dimensions
Crossover and Mutation
1. The same strategy on all processors
2. Different crossover operators and mutation rates on
different processors
3. ADAPTIVE
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Genetic Algorithm Packages (1993)
_Generational GA: the offspring are saved in a separate
pool until the pool size is reached. Then the children pool
replaces the parent pool for the next generation.
_Steady-State GA: the offspring and parents occupy the
same pool. Each time an offspring is generated it is placed
into the pool, and the weakest chromosome is "dropped off"
the pool.
_GENITOR [23] -- A Steady-state GA Package
_GENESIS [16] -- A Generational GA Package
_LibGA [30] -- This package was developed at The
University of Tulsa and offers the best of GENESIS and
GENITOR including the ability to use several additional
features including the ability to use either a steady-state or
generational, or a combination (generation gap).
_HYPERGEN [32] -- This package was developed at
The University of Tulsa. This GA package runs on a
hypercube topology multiprocessor system.
_GATutor [34] -- This package was developed at The
University of Tulsa. It is a self study GA Tutorial Package
that allows the user to grasp the fundamental concepts of
genetic algorithms.
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