genetic algorithm tutorial

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Introduction to Genetic AlgorithmsTheory and Applications

The Seventh Oklahoma Symposium on ArtificialIntelligence

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

Tutorial Outline

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Part I Introduction and Concepts of Genetic Algorithms

_ 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

Tutorial Outline

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Part II Example Applications of Genetic Algorithms

_ 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

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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.

12.Parallel Problem Solving from Nature 2, Reinhard Manner and Bernard4


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Manderick, Editors, Noth-Holland, 1992.

13.Foundations of Genetic Algorithms, Gregory Rawlins, Editor, Morgan

Kaufmann, 1991.

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.

22.D. Whitley, T. Starkweather, and C. Bogart, "Genetic Algorithm and Neural

Networks: Optimizing Connections and Connectivity", Parallel Computing,5


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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.

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, 14-16,6


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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 SimulatedAnnealing and Parallel Genetic Algorithms", Proceedings of the 1992

ACM/SIGAPP Symposium on Applied Computing, March 1-3, 1992. pp. 1031-

1038, 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.

Genetic Algorithm Definitions7


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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."

Genetic Algorithms Overview8


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_ Developed by John Holland in 1975 [9].

_ Genetic Algorithms (GAs) are searchalgorithms based on the mechanics of the naturalselection process (biological evolution). The mostbasic concept is that the strong tend to adapt andsurvive while the weak tend to die out. That is,optimization is based on evolution, and the "Survivalof the fittest" concept.

_ GAs have the ability to create an initialpopulation of feasible solutions, and then recombinethem in a way to guide their search to only the mostpromising areas of the state space.

_ Each feasible solution is encoded as achromosome (string) also called a genotype, and eachchromosome is given a measure of fitness via afitness (evaluation or objective) function.

_ The fitness of a chromosome determines itsability to survive and produce offspring.

_ A finite population of chromosomes ismaintained.

Genetic Algorithms Overview Continued9


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_ GAs use probabilistic rules to evolve apopulation from one generation to the next. The

generations of the new solutions are developed bygenetic recombination operators:

_ Biased Reproduction: selecting the fittest toreproduce)

_ Crossover: combining parent chromosomes toproduce children chromosomes

_ Mutation: altering some genes in achromosome.

_ Crossover combines the "fittest" chromosomesand passes superior genes to the next generation.

_ Mutation ensures the entire state-space willbe searched, (given enough time) and can lead thepopulation out of a local minima.

_ Most Important Parameters in GAs:_ Population Size_ Evaluation Function

_ Crossover Method_ Mutation Rate

Genetic Algorithms Overview Continued

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_ Determining the size of the population is acrucial factor

_ Choosing a population size too small

increases the risk of converging prematurely to alocal minima, since the population does not haveenough genetic material to sufficiently cover theproblem space.

_ A larger population has a greater chance offinding the global optimum at the expense of

more CPU time.

_ The population size remains constant fromgeneration to generation.

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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. TheFitness Function Drives the Population Toward BetterSolutions and is the most important part of thealgorithm.

_Probability and randomness are essential parts of GA

Use Genetic Algorithms12


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_ When an acceptable solution representation isavailable

_ When a good fitness function is available

_ When it is feasible to evaluate each potential solution

_ When a near-optimal, but not optimal solution isacceptable.

_ When the state-space is too large for other methods

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Genetic Algorithms vs TraditionalAlgorithm

Goldberg [8]

1. The GA works with a coding of the parameter ratherthan the actual parameter.

2. The GA works from a population of strings instead ofa single point.

3. Application of GA operators causes information fromthe previous generation to be carried over to the next.

4. The GA uses probabilistic transition rules, notdeterministic 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

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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 optimumat the expense of more CPU time.

3. Determine the Objective Function to be used in the algorithm.

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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 super-

performers 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 problemspace 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

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Initial Population P0

No. Times

No. Chromosome x f(x) P(x) 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

* Based on Selection Roulette Wheel23


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The Selection Roulette Wheel

Roulette: the classical selection operator for generationalGA as described by Goldberg. Each member of the poolis assigned space on a roulette wheel proportional to itsfitness. The members with the greatest fitness have the

highest probability of selection. This selection techniqueworks only for a GA which maximizes its objectivefunction.

>

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Crossover

_ Causes an exchange of genetic material between

two parents

_ Crossover point(s) is determined stochastically

_ The Crossover Operator is the most important featurein a GA

Single Point Crossover Example

Parent 1 1 0 0 1 0 0 1 0 1 0

Parent 2 0 0 1 0 1 1 0 1 1 1

Child 1 1 0 0 0 1 1 0 1 1 1

Child 2 0 0 1 1 0 0 1 0 1 0

Double Point Crossover Example

Parent 1 1 1 0 1 0 0 1 0 0 1 0 1 1

Parent 2 0 1 0 1 1 0 0 0 1 0 1 0 1

Child 1 1 1 0 1 0 0 0 0 1 0 0 1 1

Child 2 0 1 0 1 1 0 1 0 0 1 1 0 1

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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 1 1 0 1 0 0 0 0 1 0 0 1 1

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]

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Continuation of the fitness function

f(x) = 4 * cos(x) + x + 2.5

0 = 31

Population P1

(After Crossover. Assume no Mutation)

No. New Parents Crossover x f(x)

Chromosome (from P0) Point

1 01 001 (2,8) 2 9

7.8552

2 10 010 (2,8) 2 18 23.141

3 1001 1 (1,6) 4 19 25.459

4 1011 1 (1,6) 4 23 23.369

5 1 1011 (3,5) 1 27 28.331

6 0 1001 (3,5) 1 9 7.855

7 100 01 (1,3) 3 17 18.399

8 110 11 (1,3) 3 27 28.331

SUM 149 162.740

AVE 18 20.343

MAX 27 28.331

f(Pop0) = 17.198

f(Pop1) = 20.343

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>>> plot of the function showing

the 8 initial points

of Pop0 and the 8 points of Pop1

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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

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Fundamental Theorem of GeneticAlgorithms

(The Schema Theorem)

The expected number of copies, m, of schema H is bounded by:

>>>> Slide from GATUTOR 91

Wherem - number of schemata

H - schema

t - time or generation

f - fitness function

fave - average fitness value

pc - crossover probability

delta - lengthl - string length

pm - mutation probability

o - order

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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.

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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 thesearch 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

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Overview of Genetic ProgrammingKoza [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 theproblem 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 tocontrol 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.

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Consider the following two parent computer

programs given as LISP S-expressions.

>

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.

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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].

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Parent 1 3 7 1 9 | 6 4 5 | 2 8

Parent 2 4 7 8 5 | 3 9 2 | 1 6

( 6 3 ) ( 4 9 ) ( 5 2 )

Child 1 6 7 1 4 | 3 9 2 | 5 8

Child 2 9 7 8 2 | 6 4 5 | 1 3

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

Move one element

* *

9 8 7 6 5 4 3 2 1 ==> 9 8 6 5 4 3 7 2 1

Reorder a sublist

9 8 | 7 6 5 4 3 | 2 1 ==> 9 8 | 5 3 4 6 7 | 2 1

Traveling Salesman Problem

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Example TSP GA executions adjusting pool size:

TSP 1024 Cities.

PoolSize 500 250 125

Length_String 1024

Trials 100000

Bias 1.9

RandomSeed 15394157

MutateRate 0.15

NodeFile cities1024

StatusInterval 5000

RESULT = 116987 88436

90906

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TSP 320 Cities.

PoolSize 2000 1000 500 250 125

Length_String 320

Trials 100000a

Bias 1.9

RandomSeed 15394157

MutateRate 0.15

NodeFile cities320

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

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TSP 105 Cities.

PoolSize 750 500 250 125

Length_String 105

Trials 70000

Bias 1.9

RandomSeed 15394157

MutateRate 0.15

NodeFile cities105

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

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Additional Applications Using GAs

Three Dimensional Bin Packing Using GAs [29]

Set Covering Problem Using GAs [35]

Multiple Vehicle Routing with Time and Capacity

Constraints Using GAs [28]

Genetic Algorithms and Neural Networks Fixed Architecture

Genetic Algorithms and Neural Networks Unknown

Architecture

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]

Encoding: String of integers representing a permutation of the41


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packages

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 ConnectionTopology

Evaluate each architecture

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Parallel Genetic AlgorithmsParallel Issues [32]

Migration Interval1. After 5,10,20,50,100 Generations2. ADAPTIVE

Migration Rate1. 10%, 20%, 50% of the population2. ADAPTIVE

How to pick the migrants1. uniformly2. skewed towards the more fit3. Generate "an over population" during migration

generations so nothing is lost from the sendingpopulation, only new comers are analyzed as they come in

4. Perform an "exchange" of genetic material

5. ADAPTIVE

Topology1. Ring2. Hypercube dimensions alternation through the dimensions

Crossover and Mutation1. The same strategy on all processors

2. Different crossover operators and mutation rates ondifferent processors

3. ADAPTIVE

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Genetic Algorithm Packages (1993)

_Generational GA: the offspring are saved in a separatepool 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 TheUniversity 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 orgenerational, or a combination (generation gap).

_HYPERGEN [32] -- This package was developed at

The University of Tulsa. This GA package runs on ahypercube topology multiprocessor system.

_GATutor [34] -- This package was developed at TheUniversity of Tulsa. It is a self study GA Tutorial Package

that allows the user to grasp the fundamental concepts of

genetic algorithm tutorial

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