Crossover operator

Creation of a new Kbrary (the next generation) by means of evolutionary operators (crossover and mutation) from the best catalysts (identified by their catalytic performance) of all materials of all previous generations. 

The reproduction step includes two operations, crossover and mutation, which are described further, and the adjustment is implied. 

The evolutionary process of a genetic algorithm is accomplished by genetic operators which translate the evolutionary concepts of selection, recombination or crossover, and mutation into data processing to solve an optimization problem dynamically. Possible solutions to the problem are coded as so-called artificial chromosomes, which are changed and adapted throughout the optimization process until an optimrun solution is obtained. 

Figure 9-26 shows a typical GA run in a first step, the original population is created. For each chromosome the fitness is determined and a selection algorithm is applied to choose chromosomes for mating. These chromosomes are then subject to the crossover and the mutation operators, which finally yields a new generation of chromosomes. 

Selection alone cannot achieve an optimization towards the solution With mere scicction performed over a number of generations, one would get a population which comprises only the best chromosome of the original population. Therefore, an operator has to be applied which causes variance within the population, This is achieved by the application of genetic operators such as the crossover and the mutation operators. 

Crossover, which is also called recombijiation, follows the idea that aji offspring in natiu c always holds genes from both its parents. Accordingly, the genetic crossover operator takes parts of two parent chromosomes to create a new offspring. 

The crossover operator is applied to the selected pairs of parents with a probability a typical value being 0.8 (i.e. there is an 80% chance that any of the p/2 pairs will actually undergo this type of recombination). Following the crossover phase mutation is appUed to all individuals in the population. Here, each bit may be inverted (0 to 1 and vice versa) with a probability P. The mutation operator is usually assigned a low probability (e.g. 0.01). 

Figure 2 Genetic operators used to create a population of children chromosomes from a population of parent chromosomes, (a) Single-point mutation. A gene to he mutated is selected at random, and its value is modified, (b) One-point crossover. The crossover point is selected randomly, and the genes are exchanged between the two parents. Two children are created, each having genes from both parents.

Figure 11.11 shows examples of the three basic genetic operations of reproduction, crossover and mutation, as applied to a population of 8-bit chromosomes. Reproduction makes a set of identical copies of a given chromosome, where the number of copies depends on the chromosome s fitness (see below). The crossover operator exchanges subparts of two chromosomes, where the position of the crossover is randomly selected, and is thus a crude facsimile of biological sexual recombination between two single-chromosome organisms. The mutation operator randomly flips one or more bits in the chromosome, where the bit positions are randomly chosen. 

Fig. 11.11 Schematic representation of the basic genetic operations of reproduction, crossover and mutation.

Step 4 Randomly pairing up the chromosomes in the new population, apply the genetic crossover operator to each pair. That is, randomly select a bit-position, say k, for each pair of chromosomes, say and and replace this pair with two new pairs - and - constructed via genetic < rossover C, consists of the first k bits of C,i, and the last N — k) bits of and C, consists of the first k bits of and the last N — k) bits of. ... 

Init Pop Initial Fitness Exp/ Copies Actual Copies Mating Pop Crossover Operation Mutation Operation New Fitness... 

Table 11.3 One pass (read left to right) through the step.s of a basic genetic algorithm scheme to maximize the fitness function f x) = using a population of six 6-bit chromosomes. The crossover notation aina2) means that chromosomes Ca, and Ca2 exchange bits beyond the bit. The underlined bits in the Mutation Operation column are the only ones that have undergone random mutation. See text for other details.

Next, we randomly pair up the new chromosomes, and perform the genetic crossover operation at randomly selected bit-positions -- chromosomes C and C4 exchange their last three bits, C2 and Cg exchange their last four bits, and C3 and C5 exchange their last bit ... 

This basic difference equation - known as the Schema Theorem [holl92] - expresses the fact that the sample representation of schemas whose average fitness remains above average relative to the whole population increases exponentially over time. As it stands, however, this equation addresses only the reproduction operator, and ignores effects of both crossover and mutation. 

A lower bound on the overall effect of crossover, which can both create and destroy instances of a given schema, can be estimated by calculating the probability, Pc S), that crossover leaves a schema S unaltered. Let be the probability that the crossover operation will be applied to a string. Since a schema S will be destroyed by crossover if the operation is applied anywhere within its defining length, the probability that S will be destroyed is equal to Pc x 6 S)/ K — 1), where 6 S) is the defining length of S. Hence, the probability of survival ps = 1 — PcS S)/ K — 1), and equation 11.9 takes the updated form ... 

Finally, in order to also take into account the mutation operator, we note that the probability that a schema S survives under mutation is given by pu S) = (1 — Pm) where pm is the single-bit mutation probability and 0( S) is the number of fixed-bits (i.e. the order) or S. With this we can now express the Schema Theorem that (partially) respects the operations of reproduction, crossover and mutation ... 

Increased demand for softened MU, perhaps caused by an increase in plant capacity without increasing softener capacity installed, or the use of additional steam for processing so that the return of condensate is diminished. Here a larger plant is required. If the existing plant is not too old, a second identical unit usually is installed and the equipment is operated as duty/standby, with water-meter countdown crossover and immediate regeneration of the out-of-service unit. 

When 2M methanol solution is fed to the stack at a flow rate of 2 ml/min and the stack is operated at a constant voltage output of 3.8V, the transient response of the stack current density is shown in Fig. 3 varying the flow rate of air to the cathode. The stack was maintained at a temperature of 50°C throughout the experiment. As shown in the figure, while the stack current is maintained at the air flow rates higher than 2 L/min, the stack current begins unstable at the slower flow rates. A similar result is shown in Fig. 4 for varying methanol flow rate at an air flow rate of 2 lymin. At a methanol flow rate of 8 ml/min, the current density reaches initially a current density value of about 130 mA/cm and then starts to decrease probably due to medianol crossover. As the methanol flow rate decreases, the stack current density increases slowly until the methanol flow rate reaches 3 ml/min because of the reduced methanol crossover. The current density drops rapidly from the methanol flow rate of 2 ml/min. 

---