In other words, an increase of one step in the value of the decision variable corresponds to a change of only a single bit. The advantage of gray coding is that random bit flips in mutation are likely to make small changes and therefore result in a smooth mapping between the real search space and the encoded strings.

To convert binary coding to gray coding, truth table conversion, as shown in Table 3. When converting from binary to gray, the first bit of the binary code remains as it is, and the remaining bits follow the truth table conversion, two bits taken sequentially at a time, giving the next bit in gray coding.

The number of bit positions that differ in two adjacent bit strings of equal length is known as Hamming distance. For example, the Hamming distance between and is 5, since all bit positions differ, and require alteration of 5 bits when converting the number 15 to 16 in binary representation.

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The Hamming distance associated with certain strings, such as and , poses difficulty in transition to a neighboring solution in real space, as it requires the alteration of many bits. In gray coding, this distance between any two adjacent binary strings is always 1. Gray coding has been preferred by several researchers while using GAs in water resource applications Wardlaw and Sharif, Wright claims that the use of real-valued genes in GAs overcomes a number of drawbacks of binary coding.

In real coding, each variable is represented as a vector of real numbers with the same length as that of the solution vector.

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Efficiency of the GA is increased because genotype into phenotype conversion is not required. In addition, less memory is required because efficient floating-point internal computer representations can be used directly; there is no loss in precision due to formation of discreteness to binary or other values; and there is greater freedom to use different genetic operators. Nonetheless, real coding is more applicable and it seems to fit continuous optimization problems better than binary coding.

Eshelman and Schaffer suggested choosing any of these coding mechanisms, whichever is most suitable for the fitness function. Other authors, such as Michalewicz , justify the use of real coding, showing their advantages with respect to the efficiency and precision reached compared to the binary one. Real coding has been the preferred choice for variable representation in most of the applications found in water resources using GA.

Another form of real number representation is integer coding.

In integer coding, the chromosomes are composed of integer values rather than real numbers. The only difference between real coding and integer coding is in the operation of mutation. Population Size The population size is the number of chromosomes in the population.

The size of a population depends on the nature of the problem, but typically a population contains hundreds or thousands of possible solutions. Traditionally, the popula- tion is generated randomly, covering the entire search space. Given upper and lower bounds for each chromosome decision variable , chromosomes are created randomly so as to remain within the given limits.

The principle is to maintain a population of chromosomes, which represents candidate solutions to the problem that evolve over time through a process of competition and controlled variation.

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Each chromosome in the population has an assigned fitness to determine which chromosomes are used to form new ones in the competition process, which is called selection. The new ones are created using genetic operators such as cross- over and mutation. Larger population sizes increase the amount of variation present in the population but require more fitness evaluations Goldberg, Therefore, when the popula- tion size is too large, users tend to reduce the number of generations in order to reduce the computing effort, since the computing effort depends on the multiple of population size and number of generations.

Reduction in the number of generations reduces the overall solution quality.

On the other hand, a small population size can cause the GAs to converge prematurely to a suboptimal solution. Goldberg reported that a population size ranging from 30 to was the general choice of many GA researchers. Furthermore, Goldberg pointed out that the population size was both application dependent and related to string length. For longer chromosomes and challenging optimization problems, larger population sizes were needed to maintain diversity because it allowed better exploration. The selection process determines which chromosomes are preferred for generating the next population, according to their fitness values in the current population.

The key notion in selection is to give a higher priority or preference to better individuals. During each genera- tion, a proportion of the existing population is selected to breed a new generation; therefore, the selection operator is also known as the reproduction operator. All chromosomes in the population, or in a proportion of the existing population, can undergo the selection process using a selection method.

This percentage is known as the generation gap, which is defined by the user as an input in GAs. The selection process emphasizes to copy the chromosomes with better fitness for the next genera- tion than those with lower fitness values. This may lose population diversity or the variation present in the population and could lead to a premature convergence.

Therefore, the method used in the selection process should be able to maintain the balance between selection pressure and population diversity. There are several selec- tion techniques available for GA optimization. Proportional selection, rank selection, and tournament selection Goldberg and Deb, are among the most commonly used selection methods. These are briefly discussed below. Proportional Selection Method The proportional selection method selects chromosomes for reproduction of the next generation with a probability proportional to the fitness of the chromosomes.

This method provides noninteger copies of chromosomes for reproduction.

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Therefore, various methods have been suggested to select the integer number of copies of selected chromosomes for the next generation, including Monte Carlo, roulette wheel, and stochastic universal selection. The roulette wheel selection method is discussed next. Roulette Wheel Selection The most common selection method is roulette wheel selection.

Goldberg reported that it is also the simplest method. In other words, the fitter a member is, the bigger slice of the wheel it gets. The fitness of each chromosome, fti, and their sum i51 fti are calculated, where the population size is N. A minimal k is determined such that s ki51 fti ; and the kth chromosome is selected for P the next generation. Steps 2 and 3 are repeated until the number of selected chromosomes becomes equal to the population size, N. This process is continued until the required number of chromosomes is selected for the next generation.

Selection pressure can be easily adjusted by changing the tournament size. If the tournament size is larger, weak chromosomes have less chance to be selected.

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In general, in tournament selection, N chromosomes are selected at random and the fittest is selected. The most common type of tournament selection is binary tournament selection, where just two chromosomes are selected. Rank Selection In the rank-selection approach, each population is sorted in order of fitness, assigning a numerical rank to each chromosome based on fitness, and the chromosomes are selected based on this ranking rather than the fitness value using the proportionate selection operator.

Roulette wheel selection, tournament selection, and rank selection are considered to be the most common and popular selection techniques and have been used frequently in many studies. However, there are many other selection techniques, namely, elitist selection, generational selection, steady-state selection, and hierarchical selection.

These techniques may be used independently or in combination. Brief intro- duction of those selection techniques are given next. A detailed review of selection techniques used in GAs is presented by Shivraj and Ravichandran Elitist Selection The fittest chromosomes from each generation are selected for the next generation, a process known as elitism.

Elitism can be combined with any other selection technique. Generational Selection The offspring of the chromosomes selected from each generation become the entire next generation. No chromosomes are retained between generations. Steady-State Selection The offspring of the chromosomes selected from each generation go back into the previous generation and replaces some of the less fit members.

This process helps to keep some chromosomes between generations. Hierarchical Selection Chromosomes go through multiple rounds of selection each generation. Lower-level evaluations are faster and less discriminating, while those that survive to higher levels are evaluated more rigorously. The advantage of this method is that it reduces overall computation time by using faster, less selective evaluation to weed out the majority of chromosomes that show little or no promise, and subjecting only those who survive this initial test to more rigorous and more computationally expensive fitness evaluation.

Crossover The crossover operator is used to create new chromosomes for the next generation by combining randomly two selected chromosomes from the current generation. Crossover helps to transfer the information between successful candidates— chromosomes can benefit from what others have learned, and schemata can be mixed and combined, with the potential to produce an offspring that has the strengths of both its parents and the weaknesses of neither. However, some algorithms use an elitist selection strategy, which ensures that the fittest chromosome from one genera- tion is propagated into the next generation without any disturbance.

The crossover rate is the probability that crossover reproduction will be performed and is an input to GAs. For example, a crossover rate of 0. A higher crossover rate encourages better mixing of the chromosomes. There are several crossover methods available for reproducing the next genera- tion. In general, crossover methods can be classified into two groups depending on the chromosomes representation i. A number of crossover methods are discussed by Herrera et al. The choice of crossover method primarily depends on the application. Crossover Operators for Binary Coding In bit string coding, crossover is performed by simply swapping bits between the crossover points.

Different types of bit string crossover methods Davis, ; Goldberg, are discussed next. Single-Point Crossover Two parent chromosomes are combined randomly at a randomly selected crossover point somewhere along the length of the chromosome, and the sections on either side are swapped. For example, consider the following two chromosomes, each having 6 binary bits. After crossover, the new chromo- somes i.

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Multipoint Crossover In multipoint crossover, the number of crossover points are chosen at random, with no duplicates, and sorted in ascending order. Then, the bits between successive crossover points are exchanged between the two parents to produce two new chromosomes. The section between the first bit and the first crossover point is not exchanged between chromosomes.

For example, consider the same example of two chromosomes used in a single crossover. If the randomly chosen crossover points are 2 and 4, the new chromosomes are created as shown in Figure 3. The two-point crossover is a subset of the multipoint crossover. The disruptive nature of multipoint crossover appears to encourage the exploration of the search space, rather than favoring the convergence to highly fit chromosomes early in the search, thus making the search more robust.

Uniform Crossover Single-point and multipoint crossover define crossover points between the first and last bit of two chromosomes to exchange the bits between them.

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Uniform crossover generalizes this scheme to make every bit position a potential crossover point. In uniform crossover, one offspring is constructed by choosing every bit with a probability P from either parent, as shown next using the same example, by exchanging bits at the first, third, and fifth position between the parents Figure 3. Crossover Operators for Real Coding In real coding, crossover is simply performed by swapping real values of the genes between the crossover points.

Different types of real-value crossover methods have been used.