1. Introduction to Genetic Algorithms and Evolutionary Computing
Randy Ribler
Computer Science Department
Lynchburg College, Lynchburg, VA, USA

2. Evolutionary Computing
Consider the spectacular results of biological evolution
Designs better than any other known
Adaptive to changing requirements
Natural Selection works the same way regardless of the application
Algorithm is relatively unaware of application
Making it “more” aware may improve performance

3. Evolutionary Computing (Limitations)
Simulation of Natural Selection On a Smaller Scale
Can’t wait millions of years for evolution
Computer are fast and getting faster
EC works well on distributed computer architectures
Population sizes are much more limited
Actual evolutionary process is very complex – we can only approximate the process
Process is not fully understood
More accurate approximation requires additional computation resources

4. Simple Genetic Algorithms
General Technique for multi-variable function optimization and machine learning
Example: Design an airplane wing
Variables:
length, width, rotation in x-axis, rotation in y-axis

5. Example – Design an airplane wing
Variables
Wing length (l)
Wing Width (w)
Sweep Angle (s)
Pitch Angle (p)

6. Fitness Function for Airplane Wing
Fitness(l,w,s,p) = Efficiency(l,w,s,p) = lift(l,w,s,p) / drag(l,w,s,p)
This is an extreme simplification, and I don’t know anything about designing airplane wings, but an aerospace engineer can give us the proper input variables and efficiency functions and that is what we would use.
Real case would have many more variables and a complex fitness function – perhaps a wind tunnel simulation.

7. What is the Range of Each Variable?
Length
Range 0 to 16 meters
4-bits provides a 1 meter resolution
5-bits would provides a .5 meter resolution
Width
Range 0 to 8 feet
3-bits provides a 1 foot resolution
Sweep angle
Range 0 to 180 Degrees
4-bits provides a 180/16 degree resolution
Pitch angle
Range -90 to 90 degrees
5-bits provides a 180/32 degree resolution

8. Genetic Representation of Airplane Wing
4 bits for length
3 bits for width
4 bits for sweep angle
5 bits for pitch angle
Total of 16 bits to describe all 4 variables
Any random 16 bit value completely describes particular values for an airplane wing.

9. Create a population of random airplane wing strings
Select a population size.
Let’s use 500 for our airplane wing.
Create 500 random 16-bit values.
Each 16-bit value represents the string for an experimental aircraft wing
The fitness function can provide a measure of the quality of each of or 500 experimental wings
Because the values were generated randomly, most wings will be pretty bad and have low fitness values, but some will be better than others

10. Terminology
String - Sequence of genetic information fully describing an individual in the population.
Gene – location of a bit on a string
Allele – value of a gene

11. Natural Selection As in the natural world
The more fit members of the population have a greater probability of propagating their genes into the next generation

12. Creating the Gene Pool for the Next Generation
More fit members should be more prominent in the gene pool
Less fit members should not be completely excluded
Excluding the less fit might remove important gene subsequences from the gene pool
Converging too quickly to a solution can provide suboptimal results

13. Reproduction
Select 500 members of the gene pool where the probability of the ith member of the population being selected is:

14. Reproduction (continued)
Once the 500 members of the gene pool for the next generation is established, pairs of parents are selected randomly. To distinguish between the two parents we will designate one as the father and the other as the mother, although their functions are identical.

15. Reproduction (continued)
Each pair of parents creates two children using one of two methods
Cloning
One child is an exact copy of the father
One child is an exact copy of the mother
Crossover
Some bits are copied from the mother, some from the father
Copy from one parent until you reach a crossover point, then copy from the other parent.

16. Crossover with Single Crossover Point
Father
Mother
Child 1
Child 2 ↓ 1 1 1

17. Crossover with Three Crossover Points↓ ↓ ↓
Father
Mother
Child 1
Child 2 1 1 1

18. Mutation
Each bit copied to a child has a probability of being changed (from 1 to 0, or 0 to 1).
Usually the probability of mutation is relatively low, but enough to encourage diversity.
In other models, mutation is the principal method of genetic change.

19. Summary of Simple Genetic Algorithm
Create a population of random genes
For a specified number of generations
Apply the fitness function to each member of the population
Biasing toward the more fit individuals, create a pool of parents
While there are not a number of children equal to the original population size
Randomly select two parents and create two children either using cloning or crossover
Apply mutation
Completely replace the current generation with the children

20. Basic Parameters for a Genetic Algorithm
Population size
Gene Length
Number of crossover points
Probability of using crossover vs. cloning
Number of generations
Scaling Factors
More Complex/ Powerful applicatoin specific modifications

21. Schema
A Schema is a similarity template describing a subset of strings with similarities at certain string positions
Consists of series of 1’s, 0’s, and *’s (for don’t care)
Examples:
1**
Matches (100, 101, 110, 111)
*10**
Matches (01000, 01001, 01010, 01011,
11000, 11001, 11011, 11011)
The plural of the word schema is schemata.

22. Counting Schemata
For a binary schema of length k, there are 3k possible schemata
Each position may contain 0, 1, or *
A binary string of length k can have membership in 2k different schemata
Each position can contain the actual string digit, or *
Example
101 has membership in schemata (101, 10*, 1*1, 1**, *01, *0*, **1, ***)
Schemata per populations
Range from 2k (all strings are the same) to n*2k (all strings have different schemata)

23. Schema Terms Schema order o(H) is the number of fixed values
Number of 1’s and 0’s
0**10 has an order of 3
0*1*11* has an order of 4
Schema defining length δ(H) is the distance between first and last fixed value
Number of positions where crossover can disrupt the schema
0**10 has a defining length of 4
0*1*11* has defining length of 5
**1*1*1 has a defining length of 4
00**01*0 has a defining length of 7

24. Effect of Reproduction on Expected Number of Schemata in Population
A particular schema grows as the ratio of average fitness of the schema to average fitness of the population
m examples of a particular schema H at time t is m(H, t) f H t m ) ( , 1 = +

25. Reproduction of Fit Schemata
Suppose a particular schema H remains above average an amount c. ) , ( * 1 t H m c f + =
Starting at t=0 and assuming a stationary value of c, we obtain
Reproduction allocates exponentially increasing and decreasing numbers
of schemata to future generations!

26. Schema Disruption Due to Crossover
Ps = Probability of surviving crossover
Pd = Probability of being destroyed by crossover
Pc = Probability of crossover versus cloning
l is the length of the string
δ(H) is the defining length of the schema 1 ) ( * - l H P c s d

27. Schema Disruption Due to Mutation
Pm = Probability of mutating each bit
o(H) is the schema order (number of fixed values)
Probability a particular schema is disrupted by mutation is
(1-pm)o(H)
For very small values of pm, we can approximate this to 1 – o(H)pm

28. Fundamental Theorem of Genetic Algorithms
Short, low-order, above-average schemata receive exponentially increasing trials in subsequent generations.

29. Position of Bits on the Gene is Important
Allow this to evolve as well
Dynamically change the location of each bit of the gene
Inversion
Select subsequences of the gene and invert the data and the interpretation
Has no effect on the genetic content, just the way the information is stored.
Requires additional meta-information to be stored that describes the location of each bit.

30. Inversion Example 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 9 8 7 6 5 10 11 12 13 14 15
Randomly select two points in the sequence for inversion. This example, uses 5 and 9. The bit numbers and the data are inverted. Now bits 4 and 9 are adjacent and less likely to be disrupted with crossover.

31. Inversion Complicates Crossover4 5 9 11 12 1 13 14 10 15 2 3 6 7 8 1 2 3 4 9 8 7 6 5 10 11 12 13 14 15
How do we combine these two genes with different bit orders?
Insist that parents have same organization (not very good)
Discard if crossover yields duplicate bit numbers
Reorder one parent, chosen at random, to match the other
Reorder the less fit of the two parents to match the other

32. When Might GA provide a Good Solution?
Highly multimodal functions
Discrete or discontinuous functions
High-dimensional functions, including many combinatorial ones
Nonlinear dependence on parameters (interactions among parameters) – epistasis makes it hard for others
Often used for approximate solutions to NP-complete combinatorial problems
From Introduction to Genetic Algorithms Tutorial, Erik D. Goodman,
Gecco 2005

33. Flavors of Evolutionary Computing
Genetic Algorithms (GA)
Evolutionary Programming (EP)
Classifier Systems
Evolving Neural Networks
Genetic Programming (GP)
Artificial Life (AL)

34. Acknowledgement
Schema theory equations from David E. Goldberg’s, Genetic Algorithms in Search, Optimization and Machine Learning.