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Contrary to the traditional optimization methods, as depicted by figure 2.1 , genetic algorithm is an iterative procedure that starts with an initial fixed set or pool of candidate solutions called population . A candidate solution point is called an individual and represents a possible solution to the problem under co... |
, a chromosome can be a binary bit string or any otherwise representation. The chromosomes in the initial population are usually created randomly or via a simple heuristic construction. During each iteration step, called a generation , a stochastic selection process is applied on the initial population to choose better... |
In order to explore other areas of the search space, the parent chromosomes undergo recombination and/or mutation operations and generate a new set of chromosomes called offspring . The recombination entails exchange of characteristics by merging two parent chromosomes using a crossover operator, while mutation operati... |
A new generation of chromosomes is then formed by selecting from either the combined pool of parents and offspring or the offspring pool based on a prescribed fitness measure. Fitter chromosomes have higher chances of being selected and the average fitness of the population is expected to grow with successive generatio... |
2.2.1 A Generalized Model for Genetic Algorithm Based on the foregoing discussion on GA dynamics, without loss of generality, the evolution processes involved in a typical genetic algorithm can be modelled as shown in Algorithm . For any generation [MATH] , the parameters [MATH] and [MATH] used in this algorithm respec... |
Algorithm 1 A Canonical Model of Genetic Algorithm begin [MATH] initialize [MATH] evaluate [MATH] while not termination do [MATH] select [MATH] [MATH] recombine [MATH] [MATH] mutate [MATH] evaluate [MATH] [MATH] select [MATH] [MATH] |
end while end Because of their simple and stochastic nature, GAs require only the evaluation of the objective function but not its gradients. Such a derivative-free nature relieved GAs of the computational burden of evaluating derivatives especially when dealing with complex objective functions where derivatives are di... |
summarises the following key features of GAs that made them robust optimization search methods. Genetic algorithms search from a population of solutions, not a single solution; |
The genetic operations (i.e. recombination and mutation) work on the encoded solution set, not the solution themselves; The evolution operation (i.e. selection) uses a fitness measure rather than derivative or other auxiliary knowledge; |
The progress of the process relies on probabilistic transition rules, not deterministic rules. 2.3 Simulation of Evolution: Phenotype and |
Genotype Spaces In spite of the simplicity in the informational physics of the processes governing evolutionary system, it has always been an area of misunderstanding to clearly delineate which part of the evolution occurs at what space. Atmar |
argued that it is only possible to adopt and successfully simulate the process of natural evolution for engineering purposes if the physics of evolution is well understood and the sequence of causation is represented appropriately. Formally, evolutionary system inherently runs in two distinct spaces: phenotypic and gen... |
. The phenotype space [MATH] represents the behavioural or physical characteristics of an individual or chromosome, whereas the genotype space [MATH] is the encoding space and represents the exact genetic makeup of a chromosome. Figure 2.2 shows a simple simulation of evolution processes within and across generations d... |
Very often, simulated evolution mimic the natural evolution by creating the initial population as a set of chromosomes encoded in a genotype space. Thus, the process usually begins in a genotype space with [MATH] population that evolve over generations and finally ends with a solution set [MATH] in a phenotype space as... |
i) The first mapping function [MATH] decodes from [MATH] to [MATH] space such that each [MATH] is translated into phenotype [MATH] and get evaluated. Thus, the function shifts the evolution from the genotype to the phenotype space. |
ii) The second mapping function [MATH] describes the selection operation. It is the process of choosing individuals for reproduction and it occurs entirely in the phenotypic space. |
iii) The third mapping function [MATH] describes the genotypic representation. It is the process of encoding the genotype prior to reproduction and shifts the evolution back to genotype space. |
iv) The fourth mapping function [MATH] is the reproduction function. It is where the variation operations such as recombination and mutation take place. It incorporates the rules of random and directed coding alteration during the reproduction process. This process entirely happens in the genotype space and it is where... |
Lewontin stressed that although the distinction between evolutionary spaces [MATH] and [MATH] is sometimes illusory , it is very important to clearly understand which part of the evolutionary process take place at which state space. Failure to clearly delineate the two spaces led to the confusion that surrounds the the... |
2.4 Initialization of Evolutionary Computation Algorithms A number of questions need to be answered in order to properly set up an EC model for any optimization problem under consideration. Of primary importance are the choices of suitable data structure (i.e. chromosome representation and encoding), the method of crea... |
2.4.1 Representation in Evolutionary Computations Defining a proper representation scheme for an EC algorithm is crucial to its overall performance with regard to efficiency and robustness. Holland’s original idea |
advocates the use of binary representation and was based on the motive of ensuring the genetic processes operate in a domain (i.e. space) that is distinct from that of the original problem. This will ultimately enhance the robustness of evolutionary algorithms by making them more problem-independent. Furthermore, binar... |
As a crucial part in EC algorithms, categorizes the parameterization aspect of EC algorithms into two groups: structural and numerical and argue that representation constitute the major part of the structural group. The following section will show how the mapping between a genotypic space (encoded in binary) and a phen... |
2.4.2 Real-Binary Encoding and Mapping functions Since our goal is to design a hybrid EC algorithm for solving continuous linear/nonlinear problems where in most cases the design variables are usually real-valued, integer or mixture of the two, a real-coded binary representation will make a good choice. Consider the pr... |
. It is a continuous optimization problem that requires maximizing the function over the search space [MATH] . The optimal solution [MATH] is a real number in the range [MATH] . This problem can adequately be encoded if the range of the design variables and the precision requirement is known. |
[EQUATION] The first step is to encode the problem domain data from the phenotype space into a sensible formulation for the EC process (i.e. the genotype space). For any variable [MATH] , assuming the precision requirement [MATH] is two places after decimal, i.e. [MATH] , then, length for the binary bits [MATH] require... |
[EQUATION] Thus, for the problem in ( 2.1 ), we have: [EQUATION] [EQUATION] [EQUATION] Hence, the required bit length for the variable [MATH] is [MATH] |
Now, for any multidimensional function having [MATH] real variables, if each of these variables is mapped to its corresponding binary variable [MATH] , then, for a population of size [MATH] , every individual binary chromosome [MATH] is obtained by concatenating all the binary variables as follows: |
[EQUATION] Hence, the length of the resulting binary chromosome [MATH] is [MATH] which is equal to the sum of the bit length [MATH] of all the [MATH] binary variables [MATH] , such that |
[EQUATION] It is worth mentioning at this point that the precision requirement for the decision variables may differ from one variable to another within a given problem. Thus, in a general case, if a decision variable is defined in the range [MATH] in order to map it to a binary string of length [MATH] , the precision ... |
[EQUATION] Having successfully encoded the problem into binary (i.e. the genotype space), decoding chromosomes back to the phenotype space is a reverse process and it is necessary for evaluating their fitness before selection. This process entails the following two steps: |
First: Decomposing the binary chromosome [MATH] into its constituent binary variables [MATH] . This requires splitting the [MATH] bits of [MATH] into a chunk of [MATH] bits corresponding to the [MATH] binary variables. Then, the corresponding real variables [MATH] are derived via binary to decimal transformation of the... |
[EQUATION] where [MATH] are the binary bits of [MATH] [MATH] is its length and [MATH] is the total number of these variables. Second: |
Mapping the obtained real variables [MATH] to conform to their originally defined ranges [MATH] , such that: [EQUATION] Hence, for the problem under consideration ( 2.1 ), we have |
[EQUATION] [EQUATION] Example Supposing for the problem in ( 2.1 ), the value of the first variable is [MATH] , and its precision requirement [MATH] , then, based on equation ( 2.2 ), the required bit length to represent [MATH] in binary is determined to be [MATH] . Hence, the 10 bits binary equivalent of [MATH] is: |
[EQUATION] Suppose after the parent chromosome of [MATH] undergoes genetic reproduction (i.e. crossover and mutation), the value of [MATH] get transformed to [MATH] . Then, in order to derive the corresponding phenotype value, this is first converted to a decimal value [MATH] , and then mapped to its prescribed domain/... |
[EQUATION] [MATH] Although this may seldom happen, situations arise where binary representation is not only promising but is also the natural choice. The knapsack problem in operations research is a typical example. The 0-1 knapsack problem consists of a set of [MATH] items to be packed into a knapsack of size [MATH] u... |
[EQUATION] Reeves et al. have shown that the knapsack problem can be reformulated as an integer programming problem and a solution can be represented as a binary string of length [MATH] . In such case, there will be no distinction between the genotype and the phenotype and thus completely eliminating the need for mappi... |
In the past, the general view in the EC community regarding problem’s data structure and the choice of suitable EC algorithm was to match the problem to an appropriate EC algorithm. Evolutionary strategies were designed based on real valued representation and are therefore used for continuous problems. Genetic algorith... |
have pointed out that this is not the case at the moment as every one of these algorithms is been successfully used with all kinds of representations for various optimization problems. |
2.4.3 Other representations in the Literature Many types of representations for genetic algorithms are echoed in the literature for different problem domains. Special cases arise where the binary representation is inadequate or even unsuitable for the problem under investigation. Greenhalgh et al. |
argue that although Goldberg’s notion of implicit parallelism in genetic processing favours binary representation, practitioners report better performance with non-binary representations in many applications |
. Rees et al. extend their results from binary to alphabets of cardinality of powers of [MATH] (i.e. [MATH] ) and uphold the use of higher cardinality representation by deriving an upper bound for the required number of iterations for such higher cardinality GAs to visit all individuals in a population. |
Thus, in some situations, use of problem dependent representations is necessary. For instance, the rotor stacking problem originally described by |
is a typical discrete non-binary problem that requires higher cardinality ( [MATH] -ary) to be properly represented. For a set of [MATH] rotors having [MATH] holes to be stacked, a straightforward representation is to create a candidate solution with a fixed length of [MATH] -bits with cardinality [MATH] -ary correspon... |
Very often, apparent representation schemes exist that can best suit the problem to be modelled. Optimization of permutation problems is a typical example where there is a natural choice for representation. Here, the representation can directly be defined over the range of all the possible permutations. A typical examp... |
. Flowshop sequencing is a permutation problem in which [MATH] jobs are to be processed on [MATH] machines over a certain time limit. The objective was to find the permutation of jobs that will minimize the total time required to complete all the jobs (i.e. the makespan). For any job [MATH] on machine [MATH] with a job... |
In a similar approach, Man et al. proposed a non-binary representation for a combinatorial optimization problem of scheduling partially ordered tasks in a multiple processor environment. The goal was to schedule an optimal execution of [MATH] set of tasks with each requiring a duration [MATH] on a set of [MATH] process... |
Elsewhere, Chambers proposed a generalized model for scheduling problems in which a scheduling strategy is parameterized and used in matching various loop characteristics to system environment. The various parameters for the generalized loop scheduling strategy are concatenated into a binary chromosome representing a c... |
As noted by , majority of the researches on network configuration and distribution systems have adopted direct representation for the state of the network |
. This entails setting the bits of the chromosomes to the status of the switches (i.e. open or close state) in the network with each chromosome having a length equal to the number of the switches. The advantage is that no extra decoding task is required as the design made the genotype to map directly to the phenotype. ... |
Realizing the critical role of chromosome representation in the overall success of GA, Queiroz et al. suggest a tree-like representation for the network reconfiguration problem of finding a topology that will minimize technical losses throughout a given planning period. They adopted the so-called network random keys (N... |
representation for minimum spanning tree problem. NRK is an arc-based representation that can exploit the sparsity in the distribution networks graphs. The representation defines chromosomes to be of length equal to the number of arcs in the network and to consist of integer weights corresponding to each arc. The autho... |
Worth mentioning at this point is the assertion that designing representation schemes that easily map the genotype to phenotype is very essential as it limits the overhead caused by complex mapping functions |
. Very often, complex encoding functions tend to introduce additional nonlinearities, discontinuities and multimodalities to the optimization problem. This can hinder the search process substantially thereby making the combined objective function more complex than that of the original problem. |
Elsewhere, Radcliff et al. introduced the concept of allelic-representation and described how it distinctively differs from the traditional genetic representation. The authors present formalizations for both the genetic and allelic representations and use it to model a typical travelling salesman problem (TSP). They ar... |
[EQUATION] returns the chromosome in [MATH] that represents it. The representation function is injective such that there is a one-to-one mapping between any solution point in the phenotype [MATH] to any chromosome represented in the genotype [MATH] . In the context of genetic representation, a formal allele is formulat... |
[EQUATION] has alleles [MATH] . Thus, in an allelic representation, instead of being a vector, a chromosome is a set whose elements are drawn from some universal set [MATH] |
A rather recent application of GA on feature selection problem by demonstrates how binary representation can be used to appropriately represent chromosomes. The author noted that feature selection problems have exponential search space making genetic algorithms the natural choice for their optimization. A string of bin... |
Nevertheless, a common point of consensus in the field of hybrid metaheuristics is that use of problem specific representation is viewed by many |
as an act of hybridizing genetic algorithms. This kind of hybridization is believed to crucially affect the performance of the EC algorithm. Details on hybridization techniques will be presented in later sections. |
Table 2.1 compares and summarises the various representation techniques reviewed from various domains of the evolutionary computation. It highlights the cases when phenotype-genotype mapping functions is necessary and pinpoints a suitable category of the genetic operators to adopt. |
Ultimately, although problem specific representations have received wide acceptance in the EC community, they still have both their merits and demerits and should be used with great caution. This is because, although using them may improve the performance of an EC algorithm, the improvement is usually limited to only t... |
2.4.4 EC Population: Creation and Sizing Evolutionary computation algorithms enjoy global search capabilities mainly due to their population based nature. The initial population in a typical genetic algorithm is mainly created randomly and of fixed size. For some problems where domain knowledge is cheaply available, si... |
If we consider the initial population as representing a set of points in the search space of all possible populations, then, evolving over one generation effectively shifts the initial population to a different set of points in the search space. Thus, this action of evolution can be seen as a dynamic process that build... |
At low population sizes, a GA makes many decision errors and the quality of convergence suffers, but larger population sizes allow GA to easily discriminate between good and bad building blocks. And as suggested by |
, it is the parallel processing and recombination of these building blocks that lead to deriving quick solution of even large and deceptive problems. Empirical investigations by De Jong |
have shown that for a standard GA having binary representation, population sizes of [MATH] are sufficient for wide range of optimization problems. |
In spite of the several theoretical viewpoints to the choice of population size, the underlying trade off between efficiency and effectiveness remains. For a given string length, larger population sizes facilitate exploration of the problem’s search space but can impair the efficiency of the search process. On the othe... |
In an attempt to establish the relationship between population size and string length, using the idea of schemata Goldberg had earlier suggested an exponential growth in population size with respect to string length. This was later denounced after a number of empirical investigations by Schaffer |
and Grefensette which show that a linear relation is sufficient. Since string length significantly increases with even a slight increase in problem size and/or parameter precision, a point of further argument remains what could be regarded as the minimum population size for a realistic evolutionary search. |
An interesting finding by reveals that at the very least, there should be one instance of every allele at each locus in the whole population of strings. This sets a minimum requirement for every point in the search space to be reachable from the initial population by a recombinative genetic algorithm (i.e. a GA having ... |
For a typical binary representation, the probability that at least one allele is present at each locus was found by to be [EQUATION] |
where [MATH] is population size, [MATH] is the strength length. Thus, for [MATH] confidence interval, i.e. [MATH] , the minimum population size [MATH] can be evaluated as: |
[EQUATION] Worth noting is that the expression for [MATH] here does not set the optimum value for the population size. It however prescribed a threshold value below which the population may not guarantee adequate exploration of the problem space by a genetic algorithm. Thus, in the experimentations presented in the lat... |
and an interesting plot for the threshold values of the minimum population sizes [MATH] for higher [MATH] ary representations is shown in figure 2.3 |
As can be seen from the family of curves in figure 2.3 , the minimum population size required when a binary representation (i.e. when [MATH] ) is used with up to a string length of [MATH] bits is not more than [MATH] individuals. However, this threshold grows as the cardinality of the problem’s representation increases... |
The study of evolutionary properties of a typical population under the influence of genetic operators resulted into an interesting development of a dynamical system model for the space of all possible population. A population sizing equation was proposed by |
that facilitates accurate statistical decision making among competing building blocks in a population-based search methods like GA. Parallel to that Rowe |
proposes a mathematical model for a population and use it to analyse the effect of selection, mutation and crossover operators. The model permits investigation of the probability distribution of the next population, predicting the expected next population and analysing the long-term behaviour of the population. |
Assuming binary representation with string length [MATH] having a search space [MATH] , a population [MATH] of size [MATH] can be represented as a vector comprising of the proportions of each element [MATH] in the search space [MATH] such that: |
[EQUATION] Supposing a population of size [MATH] contains one copy of [MATH] , three copies of [MATH] , two copies of [MATH] and four copies of [MATH] , then, the corresponding population vector [MATH] can be represented as |
[EQUATION] Rowe argued that the population vector [MATH] which is an element of vector space [MATH] satisfies the following three important properties that qualifies it as a simplex and can be denoted by [MATH] . First, a number of population vectors are added together will yield another vector which is also in the sea... |
[EQUATION] As GA progresses from one generation to another, in spite of the randomness induced by stochastic evolutionary operators, the dynamical model makes it possible to predict the expected next population since the probability distribution of the population vector [MATH] is always a member of the set [MATH] . Con... |
2.5 The Selection Process in Evolutionary Algorithms As argued by many , selection is the driving force in evolutionary algorithms. Although evolution as a whole is seen as a set of processes that operate on chromosomes represented in the genotype space, selection process works on the original form of the encoded probl... |
, so long as the genes (encoding space) and the behaviour (phenotype space) are separate entities, as they are almost always are, selection optimizes the functional behaviour not the underlying code. Thus, selection methods in simulated evolution also imitate the process of natural selection by choosing chromosomes tha... |
in the area of evolutionary computation do take issue with this idea and raise questions like, “does natural selection always favours those behavioural strategies that seek to minimize expected loss?” |
In practice, there are varieties of selection methods in evolutionary computing, but not all of them directly use the fitness as the only criteria for carrying out the selection operation. The choice of appropriate selection method can be difficult as it involves deciding on crucial parameters like selection pressure, ... |
asserts, selection is critical to the overall success of an EC algorithm. 2.5.1 Fitness Proportionate Selection methods Roulette wheel selection (RWS) is the simplest and most commonly used fitness proportionate (FPS) means of selection in EC. In this technique, an individual is selected in proportion to its fitness on... |
For a simple construction of the roulette wheel, consider a population of size [MATH] consisting of a set of chromosomes [MATH] . Let the fitness evaluation function be [MATH] , then, the total fitness of the population [MATH] is |
[EQUATION] Therefore, the proportion of the fitness of chromosome [MATH] corresponds to its probability, [MATH] [EQUATION] and its cumulative probability [MATH] is: |
[EQUATION] Selection of [MATH] chromosomes requires spinning the wheel [MATH] -times which corresponds to [MATH] -times sampling (with replacement) from a pseudo-random sequence [MATH] . After every spin, chromosome [MATH] is selected if the random number falls within the interval of its cumulative probability and that... |
Although RWS scheme enjoys a great level of simplicity, it suffers from at least three critical problems: (i) Sampling error : Because RWS uses pseudo-random numbers to choose an individual for reproduction, the expected number of times an individual is selected may significantly differ from the actual value as a resul... |
(ii) Scaling : In fitness proportionate selection methods, for any given evaluation function, if an individual has a fitness value of [MATH] and another has [MATH] , the method will allocate twice chances of reproduction to the latter individual than the former. A mere shift of their fitness values by 1 (i.e. to fitnes... |
(iii) Selection pressure : For the reason described in (ii) above, as the average fitness of the entire population increases, selection pressure falls. Thus, chances of selecting the best fit individual over the average (or worst) individuals dramatically drop. This may lead to stagnation of the entire search process. |
In summary, it is easier for FPS schemes to distinguish between values like ( [MATH] and [MATH] ) than ( [MATH] and [MATH] ). Thus, while too high genetic variation among individuals within a population results in excessive selection pressure and risk premature convergence, too little of this genetic variation collapse... |
A proposal by Baker to address the sampling error caused by the excessive stochastic variability inherent in the RWS method is to use a multi-armed roulette wheel having equally spaced [MATH] -arms. Spinning the wheel ones will corresponds to sampling [MATH] pseudo-random numbers thereby allowing [MATH] selections at o... |
argue that this is a systematic random sampling and is superior from the statistical point of view over the traditional RWS method. Another variety of RWS method can be found in |
where RWS is used in job scheduling in a setup that favours lower fit individuals against those with higher fitness values. The commonly used measures |
to tackle the lack of scaling in RWS include windowing and scaling (linear, sigma). Indeed these techniques can to some degree serve as a remedy to the scaling problem in RWS as empirically proven by |
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