« Previous « Start

7  GENO Algorithmic Details

7.1  Introduction

The Genetic Algorithm (or GA for short) is a recent development in the arena of numerical search methods. GAs belong to a class of techniques called Evolutionary Algorithms, including Evolutionary Strategies, Evolutionary Programming and Genetic Programming. One description of GAs is that they are stochastic search procedures that operate a population of entities; these entities are suitably coded candidate solutions to a specific problem; and the GA solves the problem by artificially mimicking evolutionary processes observed in natural genetics.

Naturally, terminology from the field of natural genetics is used to describe genetic algorithms. In a biological organism, the structure that encodes how the organism is to be constructed is called the chromosome. One or more chromosomes may be required to completely specify the organism; the complete set of chromosomes is called a genotype, and the resulting organism is called a phenotype. Each chromosome comprises a set of genes, each with a specific position or locus on the chromosome. The loci in conjunction with the values of the genes (which are called the alleles) determine what characteristics are expressed by the organism.

In the GA analogy, a problem would first be coded and in this regard, genetic algorithms have traditionally used a binary representation in which each candidate solution is coded as a string of 0's and 1's. The GA's "chromosomes" are therefore the strings of 0's and 1's, each representing a different point in the space of solutions. Normally each candidate solution (or "organism") would have only one chromosome, and so the terms organism, chromosome and genotype are often used synonymously in GA literature. At each position on a chromosome is a gene that can take on the alleles 0 or 1; the phenotype is the decoded value of the chromosome. The population of candidate solutions represent a sample of different points of the search space, and the algorithm's genetic processes (see below) are such that the chromosomes evolve and become better and better approximations of the problem's solution over time.

Before a GA can be run, a fitness function is required: this assigns a figure of relative merit to each potential solution. Before fitness values can be assigned, each coded solution has to be decoded and evaluated, and the module designed to do this is generally called the evaluation function. But whereas the evaluation function is a problem-specific mapping used to provide a measure of how individuals have performed in the problem domain, the fitness function, on the other hand, is a problem-independent mapping that transforms evaluation values into an allocation of reproductive opportunities. For any given problem therefore, different fitness functions may be defined. At each generation, the chromosomes in the current population are rated for their effectiveness as candidate solutions, and a process that emulates nature's survival-of-fittest principle is applied to generate a new population which is then "evolved" using some genetic operators defined on the population. This process is repeated a sufficient number of times until a good-enough solution emerges. The three primary genetic operators focused on in practice are selection, crossover and mutation.

7.2  Selection

This operator is sometimes called Reproduction. The reproduction operation is in fact comprised of two phases: the selection mechanism and the sampling algorithm. The selection mechanism assigns to each individual x, a positive real number, called the target sampling rate (or simply: fitness), which indicates the expected number of offspring reproduced by x at generation t. In the commonly used fitness proportionate selection method, an individual is assigned a target sampling rate equal to the ratio of the individual's evaluation to the average evaluation of all chromosomes in the population. This simple scheme however suffers from the so-called scaling problem where a mere shift of the underlying function can result in significantly different fitness values. A technique that has been suggested to overcome this is to assign target sampling rates according to some form of population ranking scheme. Here, individuals are first assigned a rank based on their performance as determined by the evaluation function; thereafter, the sampling rates are computed as some linear or non-linear function of the ranks.

After fitness values have been assigned, the sampling algorithm then reproduces copies of individuals to form an intermediate mating population. The most common method of sampling the population is by the roulette wheel method in which each individual is assigned a slice of a virtual roulette wheel which is proportional to the individual's fitness. To reproduce a population of size P, the wheel is spun P times. On each spin, the individual under the wheel's maker is selected to be in the mating pool of parents who are to undergo further genetic action. An alternative approach and one which minimizes spread is Stochastic Universal Sampling (SUS). Here, rather than spin the wheel P times to select P parents, SUS spins the wheel once but with P equally spaced pointers which are used to select the P parents simultaneously. Reproduction may also be done by a tournament selection. A typical implementation is as follows. Two individuals are chosen at random from the population and a random number r is chosen between 0 and 1. If r < k (where k is a tuning parameter, say 0.75), the fitter of the two individuals is selected to go into the mating pool; otherwise the less fit individual is chosen. The two are then returned to the original population and can be selected again.

Reproductive processes may be implemented in generational or steady-state mode. Generational reproduction replaces the entire population with a new population, and the GA is said to have a generation gap of 1. Steady-state reproduction on the other hand replaces only a few individuals at a time. Elitism is an addition to selection that forces the GA to retain some number of the best individuals at each generation. Such individuals can be lost if they are not selected to reproduce or if they are operated on by the genetic operators.

The selection operator is the driving force in GAs, and the selection pressure is a critical parameter. Too much selection pressure may cause the GA to converge prematurely; too little pressure makes the GA's progress towards the solution unnecessarily slow.

7.3  Crossover

The crossover operation is also called recombination. It is generally considered to be the main exploratory device of genetic algorithms. This operator manipulates a pair of individuals (called parents) to produce two new individuals (called offspring or children) by exchanging corresponding segments from the parents' coding. The simplest form of this operator is the single-point crossover, and this is as illustrated below where the crossover point is the position marked by the symbol, |.

 Parent1: (a b c | d e)      Cross over at position 3     Child1: (a b c | 4 5)
                             -------------------------->
 Parent2: (1 2 3 | 4 5)                                   Child2: (1 2 3 | d e)

Other binary-coded crossover operators which are variations of the above scheme have since been defined, e.g., two-point crossover, uniform crossover and shuffle crossover. For real-coded GAs, recombination is usually defined in a slightly different way. We mention three crossover operators that are employed by GENO:

• ARITHMETIC CROSSOVER. This operator produces two offspring that are convex combinations of the parents. If the chromosomes c_k^v = (x_k¹, x_k²,...,x_k^N) and c_k^w = (x_k¹, x_k²,...,x_k^N) are selected for crossover, the offspring are defined as:
  
  c_k¹ = α *c_k^v+(1−α)*c_k^w and c_k² = α *c_k^w+(1−α)*c_k^v, where α [x_L, x_U]
  
• HEURISTIC CROSSOVER. This operator combines two chromosomes and produces one offspring as follows: if c_k^v and c_k^w are two parent chromosomes such that the fitness of c_k^v is not worse than that of c_k^w, then the offspring is:
  
  c_k^x = c_k^v+α *(c_k^v−c_k^w), where α [x_L, x_U]
  
  Here, the idea is to use the "quasi-gradient" of the evaluation function as a means of directing the search process.
  
• DIFFERENTIAL CROSSOVER. This operator uses three parents: one parent is taken as the "base", and the other two are used to generate the search direction. Thus, if u_T^B, u_T^V and u_T^W, are the parent chromosomes with u_T^B as the "base", then the offspring are:
  
  u_T¹ = u_T^B+α *(u_T^W−u_T^V) and u_T² = u_T^B+α *(u_T^V−u_T^W), where α [x_L,x_U]
  
  In GENO, the factor α is pre-selected from the unit interval, although random variable may also be used.
  

7.4  Mutation

By modifying one or more of the gene values of an existing individual, mutation creates new individuals and thus increases the variability of the population. The mutation operator ensures that the probability of reaching any point in the search space is never zero. The mutation operator is applied to each gene of the chromosome depending on whether a random deviate drawn from a certain probability distribution is above a certain threshold. Usually the uniform or normal distribution is assumed. Again, depending on the representation adopted, variations of the basic mutation operator may be defined.

7.5  Why Genetic Algorithms Work

Currently, there are several competing theories that attempt to explain the macroscopic behavior of GAs. The original description of GAs as schema processing algorithms by John Holland (1975) has underpinned most of the theoretical results derived to date. However, other descriptive models based on Equivalence Relations, Walsh functions, Markov Chains and Statistical Mechanics have since been developed. A survey of these models is beyond the scope of this introductory exposition. Instead, we provide a sketch of the logic leading up to one of the main explanatory models, namely The Building Block Hypothesis. We begin by stating some definitions.

• DEFINITION A1: [Schema; Schemata] A schema is a template that defines the similarities among chromosomes which is built by introducing the don't care symbol (*) in the alphabet of genes. It represents all chromosomes which match it at every locus other than the '*' positions. For example, the schema (1 0 * * 1) represents of four chromosomes, i.e.: (1 0 1 1 1), (1 0 1 0 1), (1 0 0 1 1) and (1 0 0 0 1). A collection of schema is a schemata.
• DEFINITION A2: [Order] The order of a schema S (denoted by o(S)) is the number of non-don't care positions in the schema. For example, the schemata S1 = (1 0 * * 1), S2= (* 0 * * 1), S3= (* * 1 * *) are of orders 3, 2 and 1, respectively.
• DEFINITION A3: [Defining Length] The defining length of a schema S (denoted by δ(S) ) is the positional distance between the first and last fixed positions (i.e., the non-don't care sites) in the schema. It defines the compactness of the information contained in the schema. For example, defining lengths of the three schemata S1 = (1 0 * * 1), S2= (* 0 * * 1), S3= (* * 1 * *) are δ(S1) = 4, δ(S2) = 3 and δ(S3) = 0, respectively.
• DEFINITION A4: [Schema Fitness] The schema fitness is the average of the fitness of all chromosomes matched by the schema in a given population. That is, given the evaluation function eval(.) defined on a population of chromosomes x_j of size P, the fitness of schema S at generation t is:
  
  

  eval(S,t) =

  P Σ j=1

  eval(xj/P)     (3)

  
  

The evolutionary process of GAs consists of four basic steps which are consecutively repeated, namely:

  t <- t+1
  select P(t) from P(t-1);
  recombine and mutate P(t);
  evaluate P(t)

The main evolutionary process takes place in the select, recombine and mutate phases. After the selection step, we can expect to have ξ(S,t+1) chromosomes matched by the schema S in the mating pool. For an average chromosome matched by the schema S, the probability of its selection is equal to eval(S,t)/F(t) where F(t) is the total fitness for the whole population. Since the number of chromosomes matched by schema S before selection is ξ(S,t), and the number of single chromosome selections is P, it follows that:


or, in terms of the average fitness of the population, F(t):


In other words, the number of chromosomes grows as the ratio of the fitness of the schema to the average fitness of the population. This means that an above-average schema receives an increasing number of matching chromosomes in the next generation, a below-average schema receives a decreasing number of chromosomes, and an average schema remains the same.

The schema growth equation 5 however has to be modified to take into account the destructive effects of recombination and mutation. For chromosomes of length m, the crossover site is selected uniformly from among (m - 1) possible sites. A schema S would be destroyed if the crossover site is located within its defining length. The probability of this happening is p_d(S)=δ(S)/(m−1) and, hence, the probability of a schema surviving the crossover operation is,


The crossover operator is however only applied selectively according to some probability (p_s, say). Furthermore, even when the crossover site is within the defining length there is always a finite chance that the schema may survive. These considerations dictate the modification of 6 to,


Thus, the combined effects of selection and recombination are summarized by:


The mutation operator changes a single gene with probability p_m. It is clear that all the fixed positions of a schema must remain intact if the schema is to survive mutation. Since each mutation is independent of all others, the probability of a schema S surviving mutation is therefore:


And for p_m ≪ 1, p_s(S) may be approximated by the first two terms of its binomial expansion, i.e.:


Therefore, ignoring higher-order terms involving products of p_c and p_m, the combined effects of selection, crossover and mutation is summarized by the following reproductive schema growth inequality:


Clearly, the disruptive effects of mutation and crossover are greater, the higher the order, and the longer the defining length of the schema. One can therefore expect that later generations of chromosomes would increasingly be comprised of short, low-order schemata of above-average fitness. This observation is captured by the Schema Theorem which states:

• THEOREM A1 [Schema Theorem] Short, low-order, above-average schemata receive exponentially increasing trials in subsequent generations of a genetic algorithm.

An immediate result of this theorem is the assertion that genetic algorithms explore the search space by short, low-order schemata which are used for information exchange during recombination. This observation is expressed by the Building Block Hypothesis which states:

• HYPOTHESIS A1 [Building Block Hypothesis] A genetic algorithm seeks near-optimal performance through the juxtaposition of short, low-order, high-performance schemata called building blocks.

Over the years, many GA applications which support the building block hypothesis have been developed in many different problem domains. However, despite this apparent explanatory power, the hypothesis is not universally valid. In particular, it is easily violated in the so-called deceptive problems.

7.6  Setting GA Parameters

Before one can use a GA, one needs to specify some parameter values namely the selection pressure, the population size, and the crossover and mutation rates. Both theoretical and empirical studies show that "optimal" values for these parameters depend on how difficult the problem at hand is. And since prior determination of the degree of difficulty a particular problem poses is hard, there are no generally accepted recipes for choosing effective parameter values in every case. However, many researchers have developed good heuristics for these choices on a variety of problems, and this section outlines some their recommendations.

7.6.1  Experimental Studies

De Jong (1975)

Kenneth A. De Jong tested various combinations of GA parameters on five functions with various characteristics including continuous and discontinuous, convex and non-convex, uni-modal and multi-modal, deterministic and noisy for his PhD Thesis. His function suite has since been adopted by many researchers as the standard test bed for assessing GA designs.

De Jong used a simple GA with roulette wheel selection, one-point cross-over and simple mutation to investigate the effects of four parameters namely: population size, crossover rate, mutation rate and the generation gap. His main conclusions were that:

• Increasing the population size resulted in better long-term performance, but smaller population sizes responded faster and therefore exhibited better initial performance.
  
  
• Mutation is necessary to restore lost alleles but this should be kept low at a low rate for otherwise the GA degenerates into a random search.
  
  
• A cross-over probability of around 0.6 worked best. But increasing the number of cross-over points degraded performance.
  
  
• A non-overlapping population model worked better in general.
  
  
• In summary, he concluded that the following set of parameters were efficient (at least for the functions that he studied): population size - 50 - 100; crossover probability - 0.6; mutation probability - 0.001; generation gap - 1.

De Jong's work was very important in that it provided practical guidelines for subsequent applications of GA's. His recommendations for the various parameters have been so widely adopted that they are sometimes referred to as "the standard settings". But subsequence research revealed that applying De Jong's parameter values cases can be a serious mistake in some cases.

Schaffer, Caruana, Eshelman and Das (1989)

Recognizing that parameter values can have a significant impact on the performance of a GA and that a more thorough investigation was needed, Schaffer et al. (1989) expanded De Jong's experimental setup. In addition to the five functions that he had studied, they introduced five more and performed a more exhaustive assessment of the direct and cross effects of the various parameters on a GA's performance. A notable observation they made was that good GA performance results from an inverse relationship between population size and the mutation rate, i.e. high mutation rates were better for small population sizes and low mutation rates were good for large populations. Whilst recognizing that their results may not generalize beyond the 10 functions in their test suite, they recommend the following parameter values:

• Population size: 20 - 30
• Mutation rate: 0.005 - 0.1
• Cross-over rate: 0.75 - 0.95

De Jong's work was very important in that it provided practical guidelines for subsequent applications of GA's. His recommendations for the various parameters have been so widely adopted that they are sometimes referred to as the "standard settings".

7.6.2  Theoretical Studies

Several researchers have theoretically analysed the dynamics of GAs. In his survey, Lobo (2000: chapter 3) reports the most notable of these as being the work on selection by Goldberg and Deb (1991); the work on mutation by Mûhlenbein (1992) and Bäck (1993); the research on population sizing by Goldberg, Deb and Clark (1992) and Harik et al. (1997); and the work on control maps by Goldberg, Deb and Thierens (1993) and Thierens and Goldberg (1993). The insights and practical implications afforded by these studies are summarized below.

• On Selection. In the absence of all other operators, repeated use of the selection operator would eventually result in a population comprised of the single chromosome with the highest fitness. Goldberg and Deb define the takeover time as the time it takes (as measured by the number of generations elapsed) for this event to occur. They derived takeover time formulae for various selection schemes and validated these using computer simulations. For fitness proportionate selection schemes, the takeover time depends on the fitness function distribution; for order-based selection, the takeover time is independent of the fitness function and is of the order O (log P), where P is the population size. Obviously the takeover time is increases in the presence of cross-over and mutation, but one must be careful not to exert too much selection pressure to cancel the diversifying effects of these operators.
  
  
• On Mutation. Independently of each other Mûhlenbein (1992) and Bäck (1993) analyzed the effects of mutation on a simple (1 + 1) evolutionary algorithm. They both concluded that for a chromosome of length L, the optimal fixed mutation rate is L⁻¹. Intuitively, it is easy to see why there should be this inverse relationship between chromosome length and mutation rate. Besides exploring the search space, the mutation (and to so extent, cross-over operation) can disrupt building blocks during the course of a GA run. And obviously, this is more likely to occur for long chromosomes than short ones since the operator is applied (with probability) to each gene. So in order to minimize building block disruption, one should decrease the mutation rate for relatively longer chromosomes.
  
  
• On Population Size. Studies on population size attempt to formulate schema growth equations similar to equation 5 that have population size as a variable. Unfortunately, population sizing equations are difficult to use in practice. Lobo notes:
  
  "In order to apply [the equation] the user has to know or estimate the selective advantage that a building block has over its closest competitors; he has to estimate the building block's fitness variance, he has to estimate the maximum level of deception in the problem at hand; and of course he has to hope that the building blocks are going to mix well, which may not occur in practice" (paraphrased from p.34)
  
  It is difficult to see how these population sizing studies can be used to further inform the choice of parameter values suggested by the empirical work of De Jong (1975) or Schaffer et al. (1989).
  
  
• On Selection. Increasing the population size resulted in better long-term performance, but smaller population sizes responded faster and therefore exhibited better initial performance.

7.6.3  Parameter Adaptation

We mention, in passing, parameter adaptation techniques. These methods change GA parameters as the search progresses. There are three main approaches: (1) centralized methods change parameter values based on a central learning rule; (2) decentralized methods encode the parameters values into the chromosomes themselves; (3) meta-GA's attempt to optimize parameter values by evolving these values for the actual GA that is run at a lower level using the parameters identified by the meta-level GA. The main advantage of a GA so designed is that the user is no longer required to specify parameter values prior to executing the search.

7.7  Concluding Remarks

Genetic Algorithms are simple and yet powerful search and optimization procedures that are widely applicable. Unfortunately, our current knowledge is such that one cannot rigorously predict whether a GA is going to efficient in any given instance due to the difficult in choosing the parameters of the algorithm. Nevertheless, the parameters recommended for GENO are efficient, at least on the examples reported. These parameters were arrived at after extensive "trial and error" experimentation guided by the empirical and theoretical work outlined above: they are summarized in Section 3.1.

« Previous « Start