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Distributed quasi steady-state genetic algorithm with niches and species.



Abstract: In this paper, we have proposed a new real coded genetic algorithm genetic algorithm - (GA) An evolutionary algorithm which generates each individual from some encoded form known as a "chromosome" or "genome". Chromosomes are combined or mutated to breed new individuals.  with species and sexual selection (GAS3). GAS3 is a distributed quasi [Latin, Almost as it were; as if; analogous to.] In the legal sense, the term denotes that one subject has certain characteristics in common with another subject but that intrinsic and material differences exist between them.  steady-state real-coded genetic algorithm. GAS3 uses sex determination method (SDM SDM - Schematic Data Model ) to determine the sex (male or female) of members in population. Each female member is considered as a niche in population and the species formation takes place around these niches. Sexual selection strategy selects female and required number of male members from the species to perform the recombination recombination, process of "shuffling" of genes by which new combinations can be generated. In recombination through sexual reproduction, the offspring's complete set of genes differs from that of either parent, being rather a combination of genes from both parents.  operation. The Parent-centric recombination operators are used to generate offspring. If species is not performing well, then the merging to the nearby species takes place. Explorative recombination operator is used to explore a wide range of search space in the beginning, while exploitative recombination operator is used in the later stages. The performance of GAS3 is tested on unimodal Adj. 1. unimodal - having a single mode
statistics - a branch of applied mathematics concerned with the collection and interpretation of quantitative data and the use of probability theory to estimate population parameters
 and multi-modal test functions. It got success in solving wide range of problems. Its performance is also compared with the other real-coded genetic algorithms Genetic algorithms

Search procedures based on the mechanics of natural selection and genetics. Such procedures are known also as evolution strategies, evolutionary programming, genetic programming, and evolutionary computation.
.

Keywords: real-coded genetic algorithms, parent-centric crossover Crossover

The point on a stock chart when a security and an indicator intersect. Crossovers are used by technical analysts to aid in forecasting the future movements in the price of a stock. In most technical analysis models, a crossover is a signal to either buy or sell.
 operators, chromosome differentiation, Clustering, sexual selection scheme, distributed genetic algorithm.

I. Introduction

Darwinian biological evolution, i.e. gradual adaptation through natural selection is a process consisting of three component principles: variation, heredity heredity, transmission from generation to generation through the process of reproduction in plants and animals of factors which cause the offspring to resemble their parents. That like begets like has been a maxim since ancient times.  and individual selection. Variety and diversity, whether fully random or not, are essential because without them there can be no evolution whatsoever. The information driving evolution in actual cases is the distribution of variation in a given population. Variation comes forth via mutations and sexual recombination. Heredity means selected units have some degree of durability and resilience resilience (r·zilˑ·yens),
n
, via a mechanism that passes on characteristics to other units. Individual selection is based on competition between individuals in the face of selection pressure (scarce resources, space, mating partners, etc.). Co-evolution in biology can be considered the result of merging (community and population) ecology and evolutionary biology Some U.S. universities are home to degree programs entitled Ecology and Evolutionary Biology, offering integrated studies in the disciplines of ecology and evolutionary biology.  [1]. Genetic algorithms (GAs) are search and optimization optimization

Field of applied mathematics whose principles and methods are used to solve quantitative problems in disciplines including physics, biology, engineering, and economics.
 procedures that are motivated by the principles of natural selection and natural genetics. Some fundamental ideas of genetics are borrowed and used artificially to construct search algorithms In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions.  that are robust and required minimum problem information [2]. In GAs, the role of selection and recombination operators is very well defined. Selection operator controls the direction of search and recombination operator generates new regions for search.

In real-coded GAs (RCGAs), chromosome is a real-parameter decision-variable vector. The recombination operation is a method of sharing information among chromosomes Chromosomes
Spaghetti-like structures located within the nucleus (or central portion) of each cell. Chromosomes contain the genetic information necessary to direct the development and functioning of all cells and systems in the body.
. The recombination operator has always been regarded as the main search operator in GAs as it exploits the available information in previous samples to influence future searches. The detailed study on recombination operators can be found elsewhere [3] [4]. The parent-centric crossover operators (PCCOs) used in RCGAs; in general, use a probability distribution Probability distribution

A function that describes all the values a random variable can take and the probability associated with each. Also called a probability function.


probability distribution 
 for creating offspring in a restricted search space around the region marked by one of the parent, the female parent. The range of this probability distribution is controlled by distribution index. Generation of offspring depends on the distance among the female parent and the other parents involved in the recombination operation, the male parents [5]. The sex of individuals in population can be determined either randomly or based on some problem specific knowledge where individuals with certain trails are chosen to be one sex while the rest are chosen to be of another sex. In most of the PCCO's solutions are selected randomly for mating. The work on non-random mating in GAs, refers to the incest-prevention techniques or assortative mating as·sor·ta·tive mating
n.
Nonrandom mating in which individuals mate preferentially according to phenotype.



assortative mating

sexual reproduction in which the pairing of male and female is not random.
. Lozano et al. [6] proposed the uniform fertility selection method for the selection of a female parent and the negative assortative mating technique for the selection of a male parent. For male and female parents selection, SexualGA [7] uses two different selection operators. In this way the multiple combination of different selection can be realized.

During evolution, the genetic diversity or variety in population makes the search more effective and it also prevents GAs from a premature convergence This loosely means that something has gone wrong. More precisely, that a evolutionary computation population has converged (every individual in the population is identical, see convergence) to a suboptimal solution. Often the term premature convergence is loosely used. . The centralized cen·tral·ize  
v. cen·tral·ized, cen·tral·iz·ing, cen·tral·iz·es

v.tr.
1. To draw into or toward a center; consolidate.

2.
 or distributed methods are used to prevent premature convergence. A centralized one, such as incest incest, sexual relations between persons to whom marriage is prohibited by custom or law because of their close kinship. Ideas of kinship, however, vary widely from group to group, hence the definition of incest also varies.  prevention or assortative mating incurs high overhead in global comparison. A distributed method divides the whole population into small groups or sub-populations and let these sub-populations evolve separately. A migration mechanism exchanges individuals between subpopulations, allowing the new diversity to be injected in·ject·ed
adj.
1. Of or relating to a substance introduced into the body.

2. Of or relating to a blood vessel that is visibly distended with blood.



injected

1. introduced by injection.

2. congested.
 into converging con·verge  
v. con·verged, con·verg·ing, con·verg·es

v.intr.
1.
a. To tend toward or approach an intersecting point: lines that converge.

b.
 subpopulations. The gradual distributed RCGA RCGA St. Louis Regional Chamber & Growth Association
RCGA Royal Canadian Golf Association
 (GD-RCGA) [8] applies different crossover operator to each subpopulation sub·pop·u·la·tion  
n.
A part or subdivision of a population, especially one originating from some other population: microbial subpopulations.

Noun 1.
. These operators are differentiated according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 their associated exploration and exploitation properties and the degree thereof. The effect achieved is a parallel multi-resolution with regard to the crossover operator's action. The subpopulations are adequately connected for exploiting this multi-resolution in a gradual way. In the dynamic distributed GA with directed migration [9], the population is divided into sub-populations, but there still exists a central monitor, which observe the performance of each sub-population and adjust their size accordingly after certain generations. The gendered GA [10] divides population in male and female sub-populations. The selection strategy selects male on competitive fitness and female on co-operative fitness basis.

A population of GAs is altered by the four user-defined plans in the evolution process. These are selection, generation, replacement and update plans. The population diversity versus selection pressure problem has been tackled, considering both the parent selection and the replacement plans of steady-state GA (SSGA SSgA State Street Global Advisors
SSGA Saskatchewan Stock Growers Association (Canada)
SSGA Steady State Genetic Algorithm
SSGA System Support Gate Array
SSGA Scottish Salmon Growers Association (UK) 
)[11][12].

In the sexual selection scheme based GAs, an individual in the population is identified as male and female. But to our knowledge in GAs, no strategy (except random) for male and female identification is proposed. Methods proposed for artificial niche formation uses sharing, crowding, or other modifications to the selection or replacement process. These methods introduce one or more parameters such as the sharing radius, in fitness sharing or the crowding factor in crowding. These parameters cannot be set easily without prior knowledge of the fitness landscape. The inter-species migration and mating is proposed in many algorithms, which are uncommon in nature. In this paper, we have proposed Genetic Algorithm with Species and Sexual Selection (GAS3). GAS3 is based on the facts that

1. Nature generally differentiates the individuals of an organism into more than one class.

2. In nature, organisms are living in a group. Formation of groups (i.e. species) takes place based on requirements.

3. In nature, sexually reproducing organisms do not mate indiscriminately.

4. In nature, species migrate to regroup re·group  
v. re·grouped, re·group·ing, re·groups

v.tr.
To arrange in a new grouping.

v.intr.
1. To come back together in a tactical formation, as after a dispersal in a retreat.
 for survival.

GAS3 uses sex determination method (SDM) to determine the sex (male or female) of members in population. Each female member is considered as a niche in the population and the species formation takes place around these niches. Species formation is based on Euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem.  between female and male members. Each species contains one female member and zero or more male members. All species gets equal chances to produce offspring using PCCO PCCO Pagan Community Council of Ohio (Columbus, OH)
PCCO Pictou County Community Orchestra (Stellarton, NS, Canada)
PCCO Protocol Configuration Control Office
 or mutation mutation, in biology, a sudden, random change in a gene, or unit of hereditary material, that can alter an inheritable characteristic. Most mutations are not beneficial, since any change in the delicate balance of an organism having a high level of adaptation to its  operator. The sexual selection strategy selects female and required number of male members, from species to perform recombination operation. After a certain generation the performance of the species is evaluated. If species is not performing well, then the merging to the nearby species takes place. The parent centric self-adaptive multi-parent recombination operators are used to explore the search space. The species formation, merging and sexual selection technique helps to exploit a search space.

This paper is organized as follows: section 2 introduces multi-parent, parent-centric recombination operators MPX MPX - Multiplexor Channel  and MLX MLX Master Electrician (theatrical terminology)
MLX Multiple Listing Exchange (real estate) 
. Section 3 presents the basic framework of GAS3. This section also covers discussion on SDM, species formation scheme, sexual selection strategy and merging of species. The section 4 covers the issues related to experimentation methodology like algorithm, test problems. Section 5 elaborates the discussion on empirical results of proposed algorithm over unimodal and multi-modal test functions. This Section also gives comparison of GAS3 with other real-coded genetic algorithm and evolutionary strategy proposed in literature. Finally, we draw some conclusions in Section 6.

II. Parent-Centric Crossover Operators

PCCOs assign more probability for creating offspring in neighborhood of the female parent than anywhere in the search space. The neighborhood-based crossover operators used in RCGA are based on uniform, polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a , triangular or lognormal log·nor·mal  
adj. Mathematics
Of, relating to, or being a logarithmic function with a normal distribution.



log
 probability distribution. The ranges of these probability distributions Many probability distributions are so important in theory or applications that they have been given specific names. Discrete distributions
With finite support
  • The Bernoulli distribution, which takes value 1 with probability p
 depend on the distance among the genes of the female parent and the genes of the male parents. PCCOs have two important features [5]:

1. PCCOs behave like a mutation operator. PCCOs generate solutions that are close to the female parent. In this way, they may be seen as a special type of mutation. Hence most of the RCGA models based on PCCOs do not use additional mutation operators. The operation of PCCOs may become particularly promising when they are applied to highly fit individuals.

2. PCCOs are the self-adaptive crossover operators. PCCOs define a probability distribution of offspring solutions based on some measure of distance among the parent solutions. Depending upon the current level of diversity in the population they may favor the production of additional diversity (divergence divergence

In mathematics, a differential operator applied to a three-dimensional vector-valued function. The result is a function that describes a rate of change. The divergence of a vector v is given by
) or the refinement of the solutions (convergence). This behavior is achieved without using an external adaptive mechanism. The performance of RCGA on a particular problem strongly determined by the degree of exploration and exploitation associated with the crossover operator being applied. The degree of exploration and exploitation can be controlled by using a proper value of distribution index of probability distribution used by them

Multi-parent recombination operator combines the features of more than two parents to generate offspring. A family of parent centric multi-parent recombination operator includes parent-centric recombination operator (PCX (1) A bitmapped graphics file format that handles monochrome, 2-bit, 4-bit, 8-bit and 24-bit color and uses RLE to achieve compression ratios of approximately 1.1:1 to 1.5:1. Images with large blocks of solid colors compress best under the RLE method. See PC Paintbrush. ) [11], multi-parent polynomial distribution recombination operator (MPX) and multi-parent lognormal distribution Lognormal distribution

Pattern of frequency of occurrence in which the logarithm of the variable follows a normal distribution. Lognormal distributions are used to describe returns calculated over periods of a year or more.
 recombination operator (MLX) [13].

A. MPX and MLX operators

MPX is a multi-parent extension of the simulated binary crossover (SBX SBX Snowboard Cross
SBX Sea-Based X-Band Radar (missile defense)
SBX Sports Bet Express
SBX Sodium Borate (wood preservative treatment) 
) operator. MLX is a multi-parent extension of the simulated binary crossover with lognormal distribution (SBX-l) operator.

A prototype algorithm for MPX or MLX operator is as follows:

a) From the population select the best parent (female parent) and other ([mu]-1) random solutions (male parents). The participation of genes or variables in the offspring generation process depends on the crossover probability. If crossover probability ([p.sub.c]) is 0.5 then only 50% out of n variables, undergo recombination. [mu] is the number of participating parents in recombination operation.

b) For each gene (i=1-n) undergo recombination. Execute the following steps

i) Choose [[mu].sub.i] randomly from the interval [0, 1].

ii) Using polynomial or lognormal probability distribution function, find the ordinate ordinate: see Cartesian coordinates.

(mathematics) ordinate - The y-coordinate on an (x,y) graph; the output of a function plotted against its input.

x is the "abscissa".

See Cartesian coordinates.
 [[beta].sub.i] so that the area under the probability curve from 0 to bi is equal to the chosen random number [u.sub.i]. [eta] is a probability distribution index. Compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer.  [[beta].sub.i] for MPX

[MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (1)

Or Compute [[beta].sub.i] for MLX using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where z ~ N (0, 1) is standard normal variable. iii) Calculate

D = ([[mu].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (k = 1)] ([[mu].summation over (j = 1)] |[x.sup.j.sub.i] - [x.sup.j.sub.i]|)/[mu])/[mu] (3)

iv) Generate two genes around the gene of female parent (say [x.sub.1]) using

[y.sub.i] = [x.sup.1.sub.i] [+ or -] ([[beta].sub.i] * D) (4)

It is shown that the MPX operator is more exploitative in nature and the exploitation range decreases with increase in eta. It is also shown [13] that the MLX is more explorative in nature i.e. it is capable to generate genes in the wider range around the parent gene. Its exploration range increases with increase in [eta]. Also it is also observed that the probability of creating genes near the parent gene is almost zero in MLX.

III. Genetic Algorithm with Species and Sexual Selection (GAS3)

GAS3 is a distributed quasi steady-state genetic algorithm with species and sexual selection scheme. Each iteration One repetition of a sequence of instructions or events. For example, in a program loop, one iteration is once through the instructions in the loop. See iterative development.

(programming) iteration - Repetition of a sequence of instructions.
 generates offspring in each species and updates subpopulation. There is a generation gap but not as wide as in generational GAs that makes GAS3 quasi steady state GA. The pseudo-code for GAS3 is as follows

1. Create initial population

2. Use sex determination method to determine the sex of an

individual

3. Species evolution phase creates many species around niches

4. Repeat {

4.1 For certain number of evolution with each species in population do {

4.1.1 Selection Plan: Choose one female parent and [mu] male parents from species S (the set M).

4.1.2 Generation Plan: Create the offspring set C from M.

4.1.3 Replacement Plan: Combine solutions in C and M to form set R. Arrange members in R according to their fitness.

4.1.4 Update Plan: if one of the offspring is better than the female parent then replace that female parent with offspring. Otherwise compare offspring with male members and use replace worst strategy for updating.

4.1.5 Increment To add a number to another number. Incrementing a counter means adding 1 to its current value.  the performance count of species if offspring replaces female parent. }

4.2 Examine the performance of each species and merge species.

} Until (termination condition)

This is a generalized gen·er·al·ized
adj.
1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain.

2. Not specifically adapted to a particular environment or function; not specialized.

3.
 framework for GAS3, where one can use his own sex determination method, species evolution scheme, four plans for evolution and scheme for merging of species.

A. Sex Determination Method (SDM)

Nature generally classifies individuals of an organism, into more than one group. The sexual differentiation sexual differentiation See Hermaphroditism, hirsutism, Müllerian ducts, Precocious puberty, Pseudoprecocious puberty, Tanner staging, Testis-determining factor, Virilization, Wolffian ducts, XXX, XXY, XXXY, XYY syndromes, Y Chromosome.  is a typical example, where the individual generally belongs to either male or female group. The prevalence of this form of differentiation indicates an associated advantage, which appears to be in terms of cooperation between two dissimilar individuals, who can at the same time specialize spe·cial·ize
v.
1. To limit one's profession to a particular specialty or subject area for study, research, or treatment.

2. To adapt to a particular function or environment.
 in their own fields [2]. Unlike the natural genetics in GAs, the sex of an individual in population is not known. The sex identification method determines the sex of an individual in the initial population.

Bandyopadhyay et al. [14] developed GA with chromosome differentiation that distinguishes chromosomes in two categories (M and F). The two populations are generated in such a way that the hamming distance (data) Hamming distance - The minimum number of bits that must be changed in order to convert one bit string into another.

Named after the mathematician Richard Hamming.
 between them is maximized. A crossover takes place between species from different categories. Goh et al. [15] used random selection mechanism for SDM. Garcia et al. [5] proposed female male differentiation method where group of female consists of [N.sub.F] best individuals in population and group of male consists of NM best individuals in population ([N.sub.F]<N and [N.sub.M]<N, N is the population size). In addition, it should be ensured that either [N.sub.F]=N or [N.sub.M]=N is fulfilled. Hence individual in population is either female or male, or both.

The method proposed here gives equal chances to all members in initial population to produce offspring. If an individual produces better offspring, then its performance (or fertility) count is incremented. After certain generation of evolution, the performance count of each member is examined. The individuals, who performed better than the average performance count, are treated, as female members and rest of the individuals are male members. The fertility (more fertile is a female) is the criterion used for sex identification. Number of chances (equal for all members) given to each individual to reproduce offspring is a mechanism to handle selection pressure in SDM. The method is outline as follows:

Let N is population size. Let R refers to the parameter responsible for setting selection pressure (S) in SDM.

1. S=N/R N/R Not Required
N/R Not Received
 

2. For i= 1 to S do

For j=1 to N do

a. Take [j.sup.th] member and select four members randomly form population.

b. Perform recombination operation to produce offspring.

c. If offspring is better than [j.sup.th] member then it will replace [j.sup.th] member in population and also increments performance count of [j.sup.th] member by one.

3. For i=1 to N do

If (fertility count of [i.sup.th] member > average fertility count) then

[i.sup.th] Member is female

Else

[i.sup.th] Member is male

Every member in population gets S number of chances to prove its fertility by producing offspring. High value of S (i.e. R is low) gives more chances to an individual to produce offspring. This put a more pressure on the individual to prove its fertility and makes SDM more exploitative. Low value of S makes SDM mild. SDM is deterministic 1. (probability) deterministic - Describes a system whose time evolution can be predicted exactly.

Contrast probabilistic.
2. (algorithm) deterministic - Describes an algorithm in which the correct next step depends only on the current state.
 in nature and each female represent an elite solution in the population.

B. Species evolution phase

The specialization A career option pursued by some attorneys that entails the acquisition of detailed knowledge of, and proficiency in, a particular area of law.

As the law in the United States becomes increasingly complex and covers a greater number of subjects, more and more attorneys are
 permitted by sexual differentiation is carried further in nature through species and niche exploitation. A niche is an organism's job or role in an environment, and a species is a class of organism with the common characteristics. Each female member is considered as a niche in population and its job is to form the species around it. In the proposed algorithm, species formation is based on Euclidean distance between female and male members. Each species contains one female member and zero or more male members. The proximity is the measure used to quantify the degree of similarity between members. A low value of the proximity between two members means these members are similar. The sub-population formation algorithm is described as follows:

1. Population P of N members is divided into subpopulations S[P.sub.1], S[P.sub.2] ... S[P.sub.f], where f is number of female members in P. Each sup-population contains one female member.

2. For each male member in P do

Find the nearest female by calculating Euclidean distance between male and each female

3. The male members are added to the sub-population related to the nearest female.

Distributed Quasi Steady-State Genetic Algorithm 159 The time complexity of this algorithm is O ([f.sup.*](N-f)). Many species are constructed dynamically using distance-based clustering algorithm. This method permits us to create clusters of varying size and shape without any parameters.

C. Sexual Selection Scheme

Incest-prevention technique and assortative mating plan are restricted mating strategies. In the work [16], a new selection scheme inspired by the animal mating behavior is proposed. This inspiration causes a new idea to apply dissimilar measurement to the pair of individual. The member with a higher fitness value has more chance to select as first mate. The second mate

Main article: Seafarer's professions and ranks


A Second Mate (2/M) or Second Officer is a licensed member of the deck department of a merchant ship.
 is selected by consideration of another feature, which can be depended on the first partner (this process is called seduction Seduction
See also Flirtatiousness.

Selfishness (See CONCEIT, STINGINESS.)

Armida

modern Circe; sorceress who seduces Rinaldo. [Ital. Lit.: Jerusalem Delivered]

Aurelius Dorigen’s

nobleminded would-be seducer.
). Subsequently, in the work [17], the chance to be selected as the second partner is affected from the combination of the fitness value and the difference from the first partner with predefined weights of these two values. Since, the mating procedure depends on the difference in each pair of individual; it can maintain the population diversity in an indirect way. Lozano et al. [6] proposed the uniform fertility selection scheme for female parent and negative assortative mating for the selection of male parent. In gendered GA [10], selection of males based on competitive fitness and female is selected co-operative fitness. In the work [15], male and female populations are formed by selecting members randomly. For each female member in female population mating male is decided by playing a tournament between two members selected from male population randomly.

It is observed that the inter-species mating tends to generate low performance offspring. Taking into account that, the species in different ecological niches Noun 1. ecological niche - (ecology) the status of an organism within its environment and community (affecting its survival as a species)
niche

bionomics, environmental science, ecology - the branch of biology concerned with the relations between organisms
 don't compete for the same resources but evolve independently of each other. However, it is possible to generate fruitful fruit·ful  
adj.
1.
a. Producing fruit.

b. Conducive to productivity; causing to bear in abundance: fruitful soil.

2.
 offspring in species by maintaining some selection pressure, when similar individuals do mate with one another. In GAS3, we restricted mating in species only and no inter-species mating is allowed. The male members are selected randomly from sub-population and mate with the female member of the same sub-population. If the species do not contain any male member, then the female undergo mutation operation instead of recombination operation.

D. Merging of Sub-populations

The distributed paradigm enables the searching to be in progress simultaneously in more than one direction, and prevent best fitting individuals to dominate their species. In distributed GAs, migration mechanism exchanges individuals between subpopulations, thus allowing new diversity to be injected into converging subpopulations. For a constant number of generations, these sub-populations evolve independently, evaluating their respective individuals and breeding new ones. At the end of these generations, a central monitor compares the performance of all these subpopulations. The subpopulations isolated for a certain time keeps the diversity of the population high. After migration new promising areas can be discovered by crossing over.

Merging of species that converge con·verge  
v. con·verged, con·verg·ing, con·verg·es

v.intr.
1.
a. To tend toward or approach an intersecting point: lines that converge.

b.
 on the same niche will have the effect that niches with a bigger basin of attraction will be populated pop·u·late  
tr.v. pop·u·lat·ed, pop·u·lat·ing, pop·u·lates
1. To supply with inhabitants, as by colonization; people.

2.
 by the species with more members. But this mechanism also guarantees, that if an algorithm is converged there exists at most one species per niche. This enables an algorithm to identify global/local optima op·ti·ma  
n.
A plural of optimum.
 directly, by returning the best individual of each species. The clustered based niching method (CBN CBN - call-by-name ) for evolutionary algorithm evolutionary algorithm - (EA) An algorithm which incorporates aspects of natural selection or survival of the fittest. An evolutionary algorithm maintains a population of structures (usually randomly generated initially), that evolves according to rules of selection, recombination,  [18], splits the sub-populations for dynamic specialization and merge the species to allow one subpopulation per niche. The distributed algorithm with migration and/or merging permits each sub-population to change their size in response its performance. The better species are encouraged and poorer ones are marginalized

Merging of two or more similar species set a synergy The enhanced result of two or more people, groups or organizations working together. In other words, one and one equals three! It comes from the Greek "synergia," which means joint work and cooperative action. , in which hybridization hybridization /hy·brid·iza·tion/ (hi?brid-i-za´shun)
1. crossbreeding; the act or process of producing hybrids.

2. molecular hybridization

3.
 of species may produce better offspring. GAS3 evaluates performance of each species after certain number of generations and the species not performing well are merged into nearest species (similar in phenotype phenotype (fē`nətīp'): see genetics.
phenotype

All the observable characteristics of an organism, such as shape, size, colour, and behaviour, that result from the interaction of its genotype (total genetic makeup) with
). Phenotype similarity between species is calculated by measuring the Euclidean distance between female of the species not performing well and other female members in population. The age-based strategy can also use to reduce gravity of marginalization mar·gin·al·ize  
tr.v. mar·gin·al·ized, mar·gin·al·iz·ing, mar·gin·al·iz·es
To relegate or confine to a lower or outer limit or edge, as of social standing.
 i.e. young species may get some chances to survive. Merging of species wipe out its separate existence in the population. Thus its male and female members are become male members in merged species. The duration (number of evolution) is a key feature in merging. If duration is short then species may not get sufficient chance to prove its performance and loss of such species may prevent algorithm to reach to the true optimum. The high value of duration gives non-performing species unnecessary chance to produce offspring that leads to more number of function evaluations.

E. Replacement Plan

The fitness value of the evenly produced offspring is compared with the fitness value of its parents in order to decide whether or not the evenly produced offspring is accepted as a member of the next generation. The offspring is accepted as a candidate for the further evolutionary process if and only if the reproduction operator was able to produce an offspring that could outperform Outperform

An analyst recommendation meaning a stock is expected to do slightly better than the market return.

Notes:
Exact definitions vary by brokerage, but in general this rating is better than neutral and worse than buy or strong buy.
 the fitness of its parents. This strategy guarantees that evolution is presumed mainly with crossover results that were able to mix the properties of their parents in an advantageous way. Hence, via these means we are in a position to attack one of the reasons for a premature convergence, namely the loss of relevant genetic information due to improper crossover operation. Furthermore, this strategy has proven to act as a precise mechanism for self-adaptive selection pressure steering [7]. An offspring is better than the parents, if it surpasses the fitness of the parent in mating pool While evolutionary computation mainly treats the population as a whole, an equivalent approach is to separate from the current population those individuals that will have children. These are placed into a mating pool. . In update plan, it replaces the parent whose fitness is just less than offspring.

IV. Experimental Setup

GAS3 is a generalized model; hence one can use his own sex determination method, species evolution scheme, four plans for evolution and scheme for merging of species. For the experimentation, fertile female scheme is used for SDM. Different values of R test effect of selection pressure on performance of the algorithm. Explorative recombination operator MLX (with [eta] = 4 is used with five parents ([mu]=5) to generate two offspring. MLX explores wide range of search space in the beginning. Proximity based species evolution scheme has constructed, the species around female parents. The parents, selected by sexual selection scheme undergo recombination or mutation operation. The exploitative recombination operator MPX (with [eta]=1) is used with the male parents selected randomly from species ([mu]<5) and merging is performed after ([N.sup.*]N/R) number of evolution. Any good search algorithm must explore a large search space in the beginning and the search should then narrow down as it converges to the solution. To support this property explorative operator MLX is used in SDM and exploitative operator MPX is used in further evolution. 50 independent runs are executed for all possible combination of parameters R, N and [p.sub.c]. Other parameters in GA are set as given in Table 1.

A. Test functions

The minimization experiments have been performed on unconstrained unimodal and multi-modal functions with or without epistasis e·pis·ta·sis
n. pl. e·pis·ta·ses
1. A film that forms over the surface of a urine specimen.

2. An interaction between nonallelic genes, especially an interaction in which one gene suppresses the expression of
 among n-variables as shown in Table 2. Using skewed skewed

curve of a usually unimodal distribution with one tail drawn out more than the other and the median will lie above or below the mean.

skewed Epidemiology adjective Referring to an asymmetrical distribution of a population or of data
 initialization in·i·tial·ize  
tr.v. in·i·tial·ized, in·i·tial·iz·ing, in·i·tial·iz·es Computer Science
1. To set (a starting value of a variable).

2. To prepare (a computer or a printer) for use; boot.

3.
 these functions are evaluated for global minima at 0.

V. Experimental Results and discussion

The experimental results of GAS3 with test function show that, its performance on unimodal and multi-modal functions strongly depends on value of R. It is observed that, for unimodal functions In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for xm and monotonically decreasing for xm.  as the value of R increases, number of function evaluations (FES) decreases, and for multi-modal functions as the value of R increases, FES also increases. GAS3 has performed better, with unimodal functions for high value of R and with multi-modal function for low value of R. Since the parameter R plays important role in SDM and merging of species of the algorithm. The low value of R means the population members gets more chance to prove its fertility to become a female member. This creates a high selection pressure that results into well-defined niches in population to solve multi-modal functions efficiently. Also the low value of R lengthens the duration (i.e. number of evolutions) to perform the merging, helps the species to prove their survival by exploring/exploiting search space. The high value of R sets a low selection pressure and narrow down the duration of merging that helps in solving unimodal functions. Table 3 shows performance of GAS3 on multimodal Two or more modes of operation. The term is used to refer to a myriad of functions and conditions in which two or more different methods, processes or forms of delivery are used. On the Web, it refers to asking for something one way and receiving the answer another; for example requesting  functions against R. Table 4 shows performance of GAS3 on unimodal functions against R.

The characteristic parameter R allows GAS3 to design two different kinds of specialization algorithms. GAS3 with R=10 (U-GAS3) is capable to solve unimodal functions very accurately. GAS3 with R=1 (M-GAS3) is capable to solve multi-modal functions very efficiently and reliably. The disruptiveness of chromosome depends upon the number of genes changed due to the recombination operation. This disruptiveness on the one-hand leads to more diverse exploration that can prevent a premature convergence, but on the other hand, it slows down the convergence speed at the same time. From empirical results it is observed that [p.sub.c]=0.3 or 0.4 is suitable for multi-modal function optimization and [p.sub.c]= 0.4 or 0.5 is suitable for unimodal function optimization.

The choice of population size has a strong interacting effect on the results. When the population size is too small for the complexity of a particular search space, it lacks the information capacity to provide accurate sampling. However, a larger population results in better solutions but the GA must be run for a greater number of generations.

With smaller populations, more disruptive crossover operators are likely to yield better results because they help to overcome the limited information capacity of smaller population. However, with the larger population, providing sufficient sampling accuracy, less disruptive crossover operators are more likely to work better. GAS3 with high value of R, moderated [p.sub.c] and a small population size has shown better results on unimodal test functions. The test result of U-GAS3 with pc =0.5 and M-GAS3 with [p.sub.c] =0.3, on test functions against different population size is shown in Table 5 and 6. It is observed that the small population size failed to achieve 100% success in solving some of the test functions. Also for most of the functions, increase in population size increases the AFES AFES Armed Forces Emergency Services
AFES Agricultural and Forestry Experiment Station
AFES Australian Fellowship of Evangelical Students (see IFES)
AFES Automated Financial Entitlements System
AFES Army Frequency Engineering Software
. Hence, we have selected N=100 for further experimentations.

Table 7 displays test result of U-GAS3 and M-GAS3 on test functions. U-GAS3 achieved almost 100% success in solving all unimodal test functions. [f.sub.Ros] is a continuous, non-separable and unimodal function, with the optimum located in a steep parabolic par·a·bol·ic   also par·a·bol·i·cal
adj.
1. Of or similar to a parable.

2. Of or having the form of a parabola or paraboloid.
 valley with a flat bottom. U-GAS3 struck to strong local minima at 3.98662 in 2 runs out of 50. M-GAS3 achieved almost 100% success in solving all multimodal test functions.

Garcia et al. [5] proposed Female and Male Differentiation (FMD FMD

foot-and-mouth disease.
) method. The Uniform Fertility Selection (UFS UFS Unix File System
UFS Universal Fighting System (Sabertooth games)
UFS United Feature Syndicate
UFS Unite for Sight
UFS Uncoated Free Sheet (paper grade)
UFS Universal Frame System
) is used to select a female parent and the Negative Assortative Mating (NAM) is to select a male parent. The parent centric crossover operator is used to generate offspring from the selected parent. They proposed Local-RCGA to solve unimodal functions accurately and Global-RCGA to offer reliable solution in solving multimodal functions. Finally, they proposed hybridization technique to put together above two specialization algorithms. GL-25 is a hybrid model that runs Global-RCGA during the 25% of the available evolution and then it performs Local-RCGA. For experimentation, stopping criterion is maximum of [10.sup.5] FES. The performance comparison of U-GAS3, M-GAS3 with Local-RCGA, Global-RCGA and GL-25 is shown in table 8. M-GAS3 and Global-RCGA are particularly designed to solve multimodal problems. It is observed that M-GAS3 has outperformed Global-RCGA in almost all functions. Local-RCGA has shown better performance than U-GAS3 in solving uni-modal test functions. M-GAS3 performed better than GL-25 in solving [f.sub.rast] and [f.sub.grie]. M-GAS3 has solved unimodal function except [f.sub.ros], upto accepted level of precision.

We have compared the performance M-GAS3 with G3-MHX [19] and NAM-UFS-PBX-0.8 [6] algorithms. The results of NAM, UFS & PBX (Private Branch eXchange) An inhouse telephone switching system that interconnects telephone extensions to each other as well as to the outside telephone network (PSTN).  based algorithm are taken from literature. Table 9 shows that M-GAS3 performed better than both the algorithms in solving [f.sub.rast] and [f.sub.grie].

The Griewangk function has a product term that introduces strong interdependence in·ter·de·pen·dent  
adj.
Mutually dependent: "Today, the mission of one institution can be accomplished only by recognizing that it lives in an interdependent world with conflicts and overlapping interests" 
 among the variables. The optima of the Griewangk function are regularly distributed. It is considered as one the difficult multi-modal problem to solve. The remarkable achievement of M-GAS3 is it has reached to global minima of this function where other algorithms failed to do it.

VI. Conclusions

This paper presented genetic algorithm with species and sexual selection (GAS3), a basic framework for distributed quasi steady-state RCGA. The major contribution of this study is the introduction of Sex Determination Method (SDM), species-formation (clustering) scheme without parameters and species-merging scheme. SDM is based on fertile female strategy, to identify sex of an individual in the population. A cluster around each female solution is formed in the species formation scheme using distance-based clustering algorithm. The species migration is a common method used in distributed algorithm for diversity preservation. The proposed species-merging scheme sets a synergy in which hybridization of species may produce better offspring and inject in·ject
v.
1. To introduce a substance, such as a drug or vaccine, into a body part.

2. To treat by means of injection.
 diversity into population. The explorative recombination operator, MLX is used to explore a wide range of search space in the beginning and exploitative recombination operator; MPX is used in the later stages. We have proposed two different kinds of specialization algorithms, based on characteristic parameter R. GAS3 with R=10 (U-GAS3) is capable in solving unimodal functions very accurately. GAS3 with R=1 (M-GAS3) is capable in solving multi-modal functions very efficiently and reliably. The test result of U-GAS3 has shown that it achieved almost 100% success in solving unimodal test functions. The test result of M-GAS3 has shown that it achieved almost 100% success in solving multi-modal test functions. We have compared performance of U-GAS3 and M-GAS3 with recent RCGA. We have proposed M-GAS3 as a robust GA that is capable to solve a wide range of unimodal and multi-modal problems, very efficiently.

GAS3 is a generalized model; hence one can use his own sex determination method, species evolution scheme, four plans for evolution and scheme for merging of species. In extended study we are intended to work on: 1. Design of adaptive population size algorithm, 2. Design of self-adaptive mechanism to adjust the value of parameter R, 3. We have used random selection scheme for selecting male parents for recombination operation, deterministic selection method may offer some selection pressure, 4. Age of species may be considered as another parameter while merging species, 5. Use of self-adaptive mechanism to select recombination operator to generate offspring during evolution.

References

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[2] D. E. Goldberg, "Genetic Algorithms in Search, Optimization and Machine Learning", Pearson Education Pearson Education is an international publisher of textbooks and other educational material, such as multimedia learning tools. Pearson Education is part of Pearson PLC. It is headquartered in Upper Saddle River, New Jersey.  Asia, 1989.

[3] K. Deb, "Multi-Objective Optimization using Evolutionary Algorithms", John Wiley John Wiley may refer to:
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  • John C. Wiley, American ambassador
  • John D. Wiley, Chancellor of the University of Wisconsin-Madison
  • John M. Wiley (1846–1912), U.S.
 & Sons, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
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[4] F. Herrera, M. Lozano and A.M. Sanchez, "A taxonomy taxonomy: see classification.
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In biology, the classification of organisms into a hierarchy of groupings, from the general to the particular, that reflect evolutionary and usually morphological relationships: kingdom, phylum, class, order,
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[5] C. Garcia and M. Lozano, "Chromosome Differentiation for the Application of Parent-Centric Real-Parameter Crossover Operators", Technical report, DECSAI DECSAI Department of Computer Science and Artificial Intelligence , E.T.S. Ingenieria Informatica University of Granada Coordinates:  The University of Granada is a university at Granada, Spain, first founded by the Moors in 1349 and then officially founded in 1531 by the Emperor Carlos V, with  2005.

[6] M. Lozano and F. Herrera, "Diversification techniques as a way to improve the performance of real-parameter crossover operators: the case of the parent-centric BLX BLX Business Line Expert
BLX Basic Launch Complex
BLX British Legion of Xbox (gaming clan) 
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[7] S. Wagner and M. Affenzeller, "SexualGA: Gender-Specific Selection for Genetic Algorithm", Proceedings of the 9th World Multi-Conference on Systemics, Cybernetics cybernetics [Gr.,=steersman], term coined by American mathematician Norbert Wiener to refer to the general analysis of control systems and communication systems in living organisms and machines.  and Informatics Same as information technology and information systems. The term is more widely used in Europe.  (WMSCI WMSCI World Multi-Conference on Systemics, Cybernetics and Informatics ), vol. 4, pp. 76-81, 2005.

[8] F. Herrera and M. Lozano, "Gradual Distributed Real-Coded Genetic Algorithms", IEEE Transactions On Evolutionary Computation The IEEE Transactions on Evolutionary Computation (TEC) is a monthly journal published by the Computational Intelligence Society of the IEEE Computer Society. It contains peer-reviewed articles and other contribitions in the area of Evolutionary Computation and Natural Computation. , Vol. 4, No. 1,pp. 43-63, 2000.

[9] W. Yi, Q. Liu and Y. He, "Dynamic Distributed Genetic Algorithm", IEEE Congress on Evolutionary Computation The IEEE Congress on Evolutionary Computation (CEC) is one of the largest and most important conferences within Evolutionary computation (EC), the other conferences of similar importance being Genetic and Evolutionary Computation Conference (GECCO) and Parallel Problem Solving from  2000, San Diego San Diego (săn dēā`gō), city (1990 pop. 1,110,549), seat of San Diego co., S Calif., on San Diego Bay; inc. 1850. San Diego includes the unincorporated communities of La Jolla and Spring Valley. Coronado is across the bay. , CA, USA, pp. 1132-1136, July 2000.

[10] J. Sanchez-Velazco and J. A. Bullinaria, "A Gendered Selection Strategies in Genetic Algorithms for Optimization", In: J.M. Rossiter & T.P. Martin (Eds), Proceedings of the UK Workshop on Computational Intelligence Computational intelligence (CI) is a successor of artificial intelligence. As an alternative to GOFAI it rather relies on heuristic algorithms such as in Fuzzy systems, Neural networks and Evolutionary computation. : UKCI-2003, 217-223. Bristol, UK: University of Bristol, 2003.

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[12] P. J. Ballester and J. N. Carter, "An Effective Real-Parameter Genetic Algorithm with Parent Centric Normal Crossover for Multimodal Optimisation", In proceedings of the Genetic and Evolutionary Computation Conference, Springer springer

a North American term commonly used to describe heifers close to term with their first calf.
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[13] M. M. Raghuwanshi and O. G. Kakde, "Multi-parent Recombination operator with Polynomial or Lognormal Distribution for Real Coded Genetic Algorithm" 2nd Indian International Conference on Artificial Intelligence (IICAI IICAI Indian International Conference on Artificial Intelligence ), pp. 3274-3290, 2005.

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[17] K. Matsui, "New Selection Method to Improve the Population Diversity in Genetic Algorithms", Proc. IEEE (Institute of Electrical and Electronics Engineers, New York, www.ieee.org) A membership organization that includes engineers, scientists and students in electronics and allied fields.  Int. Conf. Systems, Man, and Cybernetics, pp. 625-630, 1999.

[18] F. Streichert, G. Stein, H. Ulmer and A. Zell, "A Clustering Based Niching EA for Multimodal Search Spaces", Evolution Artificielle 6th International Conference (EA 2003), Marseille Marseille
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[19] M.M. Raghuwanshi and O.G. Kakde, "Multi-parent Recombination Operators with Multiple Probability Distributions for Real Coded Genetic Algorithm", 10th on-line world conference on soft computing This article or section is in need of attention from an expert on the subject.
Please help recruit one or [ improve this article] yourself. See the talk page for details.
 in industrial application (WSC WSC Winter Symposium on Chemometrics
WSC Winter Simulation Conference
WSC Wayne State College
WSC Westfield State College (Westfield, MA)
WSC Western State College (Colorado) 
10), 2005.

M.M. Raghuwanshi (1) and O.G. Kakde (2)

(1) RCERT RCERT Regional Computer Emergency Response Team , Chandrapur. (M.S.), India. m_raghuwanshi@rediffmail.com

(2) VNIT VNIT Visvesvaraya National Institute of Technology (Nagpur, India) , Nagpur. (M.S.), India. ogkakde@vnitnagpur.ac.in

Author Biographies

M.M. Raghuwanshi received M. Tech. (Computer Sc. &DP) degree from Indian Institute The Indian Institute in central Oxford, England is located at the north end of Catte Street on the corner with Holywell Street and faching down Broad Street from the east.[1]  of Technology, Kharagpur in 1991. He is pursuing Ph.D. from VNIT, Nagpur under the guidance of Prof. O.G. Kakde. He is an Assistant Professor in the Department of Computer Technology at RCERT, Chandrapur (INDIA). His research interests include Real-coded Genetic Algorithms and Programming languages.

O.G. Kakde received the M. Tech. degree in Computer Science in 1988 from Indian Institute of Technology Mumbai, and the Ph.D. degree in 2004, from Nagpur University INDIA. He is Professor of Computer Scienc, and Head of the Department of Electronics and Computer Science Engg., Visvesvaraya National Institute of Technology (deemed university Deemed University is a status of autonomy granted to high performing institutes and departments of various universities in India. It is granted by the University Grants Commission (UGC) of India. ) Nagpur, INDIA.
Table 1. GA setup used for experimentation

Parameter                       Values

Population size (N)             50,75,100,150,200
Crossover probability           0.3 - 0.8 in step of 0.1
  parameter ([p.sub.c])
R (Related to selection         1 - 10 in step of 1
  pressure)
Stopping criteria               Maximum [10.sup.6] function
                                  evaluations or an objective value
                                  of [10.sup.-10]
Results average over            50 independent runs
Parameters for performance        1. Number of function evaluation for
evaluation                          best run and worst run
                                  2. Average number of function
                                    evaluation (AFES)
                                  3. Best fitness, Average fitness and
                                    Worst fitness
                                  4. Number of runs converged to global
                                    minima
Initialization of variables     Skewed initialization within [-10, -5]
Number of children ([lambda])   2

Table 2. Test functions

Function Name      Function

Sphere             [f.sub.sph ere] = [[summation].sup.n.sub.i=1]
                     [x.sup.2.sub.i]
Tablet             [f.sub.tablet] = [10.sup.6][x.sup.2.sub.1] +
                     [[summation].sup.n.sub.i=2][.sup.2.sub.i]
Cigar              [f.sub.cigar] = [x.sup.2.sub.1] +
                     [[summation].sup.n.sub.i=2][10.sup.6]
                     [x.sup.2.sub.i]
Ellipsoidal        [f.sub.elp] (x) = [n.summation over (i=1)]
                     [ix.sup.2.sub.i]
Schwefel           [f.sub.sch] (x) = [n.summation over (i=1)]
                     [([i.summation over (j=1)] [x.sub.j]).sup.2]
Rosenbrock         [f.sub.ros] (x) = [n - 1.summation over (i=1)]
                     [(100 ([x.sup.2.sub.i] - [x.sub.i + 1]).sup.2]
                     + ([x.sub.i - 1]).sup.2]
Rastrigin          [f.sub.rast (x) = 10 n + [n.summation over (i=1]
                     ([x.sup.2.sub.i] - 10 cos(2 [pi] [x.sub.i]))
Scaled Rastrigin   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Skew Rastrigin     [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Griewangk          [f.sub.gri] = 1/4000 [n.summation over (i=1)]
                     [x.sup.2.sub.i] - [[product].sup.n.sub.i=1]
                     cos([x.sub.i]/[square root of i]) + 1
Ackley             [f.sub.ack](x) = -20 x exp(-0.2 x [square root of
                     (1/n [n.summation over (i=1)] [x.sup.2.sub.i])-
                     exp(1/n [n.summation over (i=1)] cos([2.0.sup.*]
                     [3.14].sup.*][x.sub.i]))+20+[e.sup.1]
Bohachevsky        [f.sub.Bohachevaky] (x)=[[summation]
                     .sup.n-1.sub.i=1] ([x.sup.2.sub.i]+2
                     [x.sup.2.sub.i+1] -0.3 cos(3[pi] [x.sub.i])
                     -0.4 cos(4[pi] [x.sub.i+1])+0.7)
Two-Axes           [f.sub.two ax] = [[summation].sup.[n/2].sub.i=1]
                     [10.sup.6] [x.sup.2.sub.i] +
                     [[summation].sup.n.sub.i=[n/2]] [x.sup.2.sub.i]

Table 3. Performance of GAS3 on multi-modal functions against R
(with [p.sub.c] =0.3 and N=100)

R      [f.sub.rast]       [f.sub.gri]         [f.sub.ack]

     AFES     Success   AFES      Success   AFES      Success

1    187978   100.00%   46323.2   100.00%   62702.8   100.00%
2    238721    95.56%   52318.8    97.78%   38410.6   100.00%
3    336292    82.22%   93745.5    93.33%   30351.6   100.00%
4    354303    82.22%   65795.4    95.56%   27768.5   100.00%
5    383453    77.78%   23049.2   100.00%   22949.6   100.00%
6    624075    55.56%    193446    82.22%   62592.5    97.78%
7    606035    53.33%   56333.2    95.56%   24987.2   100.00%
8    602072    53.33%    236330    77.78%   42287.6    97.78%
9    548431    53.33%    171609    84.44%     28569   100.00%
10   594357    51.11%    200258    82.22%   62186.3    95.56%

Table 4. Performance of GAS3 on uni-modal functions against R
(with [p.sub.c] =0.5 and N=100)

R       [f.sub.elp]         [f.sub.sch]         [f.sub.ros]

     AFES      Success   AFES      Success   AFES     Success

1    37936.1   100.00%   84279.9   100.00%   914960    37.78%
2    22810     100.00%   72139     100.00%   827883    62.22%
3    16860.6   100.00%   80689.4   100.00%   688279    86.67%
4    15473.5   100.00%   56446.3   100.00%   659736    80.00%
5    14136.1   100.00%   53149.3   100.00%   599986    93.33%
6    11668.8   100.00%   49047.1   100.00%   571940    88.89%
7    10866.6   100.00%   50131.8   100.00%   536157    86.67%
8     7825.8   100.00%   36070.6   100.00%   508573    93.33%
9     8672     100.00%   55152.8   100.00%   564036    77.78%
10   7331.84   100.00%   36570.7   100.00%   434632    93.33%

Table 5. Performance of U-GAS3 on unimodal functions against N

N        [f.sub.elp]         [f.sub.sch]         [f.sub.ros]

      AFES      Success   AFES      Success   AFES     Success

50    5158.13   100.00%   44060.9   100.00%   401162    86.67%
75    7101.64   100.00%   55176.6   100.00%   541917    66.67%
100   7331.84   100.00%   36570.7   100.00%   434632    93.33%
150   15853.6   100.00%   58657.4   100.00%   608751    91.11%
200   25023.3   100.00%   82566.2   100.00%   856712    48.89%

Table 6. Performance of M-GAS3 on multi-modal functions
against N

N       [f.sub.rast]       [f.sub.gri]         [f.sub.ack]

      AFES     Success   AFES      Success   AFES      Success

50    374398    71.11%   38604.7    97.78%   22308.9   100.00%
75    229172    91.11%   30346.1   100.00%   41447.2   100.00%
100   187978   100.00%   46323.2   100.00%   62702.8   100.00%
150   224996   100.00%     87217   100.00%    118166   100.00%
200   322114   100.00%    138785   100.00%    183287   100.00%

Table 7. Test results of GAS3 on unimodal and multi-modal test
functions

                         No Of Function Evaluations     Fitness

Function              Best Run   Avg. Run   Worst Run      Best

                                        U-GAS3
[f.sub.elp]               4071    7331.84       19546   2.57E-11
[f.sub.sch]              17029    36570.7      137041   4.68E-11
[f.sub.sphere]            3427    7786.53       16154   4.53E-11
[f.sub.cigar]             3758    11630.8       25248   5.15E-11
[f.sub.tablet]            4305    13251.4       26751   6.73E-11
[f.sub.twoAxes]           7735    14523.9       46451   4.94E-11
[f.sub.ros]             195748    434632      1000002   9.90E-11
                                        M-GAS3
[f.sub.rast]             68125    187978       412180   1.18E-11
[f.sub.rastScaled]       59384    145779       294441   5.91E-12
[f.sub.rastSkewed]       62895    216585      1000002   4.43E-12
[f.sub.ack]              54738    62702.8       70121   5.92E-11
[f.sub.gri]              40376    46323.2       49366   3.78E-11
[f.sub.bohachevsky]      42828    50335.9       54937   3.85E-11

                             Fitness

Function              Average      Worst    Success

                                U-GAS3
[f.sub.elp]           7.43E-11   9.97E-11   100.00%
[f.sub.sch]           8.62E-11   9.97E-11   100.00%
[f.sub.sphere]        8.84E-11   9.98E-11   100.00%
[f.sub.cigar]         8.46E-11   9.96E-11   100.00%
[f.sub.tablet]        8.75E-11   9.86E-11   100.00%
[f.sub.twoAxes]       8.76E-11   9.93E-11   100.00%
[f.sub.ros]             0.0886   3.98662     93.33%
                                M-GAS3
[f.sub.rast]          7.48E-11   9.98E-11   100.00%
[f.sub.rastScaled]    6.72E-11   9.98E-11   100.00%
[f.sub.rastSkewed]    0.04422    0.99495     95.56%
[f.sub.ack]           8.96E-11   9.99E-11   100.00%
[f.sub.gri]           8.30E-11   1.00E-10   100.00%
[f.sub.bohachevsky]   8.28E-11   9.99E-11   100.00%

Table 8. Comparison of GAS3 with GL-25

Algorithm              U-GAS3           Local-RCGA    M-GAS3

                       Fitness                        Fitness

                                         Average
Functions          Best     Average      Fitness        Best

[f.sub.sphere]      0       5.03E-51    9.98E-187     8.02E-49
[f.sub.sch]      1.11E-86   0.58967     1.74E-09      4.44E-11
[f.sub.ros]      0.00081    5.43726       1.56        22.0982
[f.sub.rast]        0       33.2557      290.00          0
[f.sub.gri]         0       0.004262     0.0127          0

Algorithm        M-GAS3     Global-RCGA     GL-25

                 Fitness

                             Average      Average
Functions        Average     Fitness      Fitness

[f.sub.sphere]   2.66E-32   2.95E-18      3.17E-147
[f.sub.sch]      0.002221     31.2        1.61E-07
[f.sub.ros]      23.0984      19.1        7.61E-01
[f.sub.rast]     3.10518      19.2          3.30
[f.sub.gri]         0       4.93E-04      2.22E-17

Table 9. Comparison of M-GAS3 with G3-MHX and NAM-UFS-PBX-0.8
algorithm PBX-0.8 algorithm

Operator             [f.sub.elp]   [f.sub.sch]   [f.sub.ros.]

M-GAS3                2.66E-32       0.00418        23.098
G3-MHX(.5,5,70:30)    1.63E-114     5.00E-23        3.904
NAM-UFS-PBX-0.8       1.44E-81      6.00E-08         14.7

Operator             [f.sub.rast]   [f.sub.grie.]

M-GAS3                 3.10518           0.0
G3-MHX(.5,5,70:30)     3.85569         0.0424
NAM-UFS-PBX-0.8          30.8          0.0068
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Author:Raghuwanshi, M.M.; Kakde, O.G.
Publication:International Journal of Computational Intelligence Research
Geographic Code:9INDI
Date:Apr 1, 2007
Words:7361
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