Typical Variation Operators in cEA
The first step in a cEA generation is fitness assignment, which is done simultaneously for all individuals. This fitness assignment process consists on selecting a value to each individual according to its proximity to the problem solution.
After this first step the three next genetic operators take place locally within a small neighborhood:
Selection is a process in which individuals are selected to reproduce, generating offsprings for the next generation. Usually, the best individuals (those having higher fitness values) will have a greater probability for being selectd. In nature, an individual undergo two different selection pressures before producing its offspring, i.e., survival to adult state and find of its mates. Selection of EAs models these processes. Some "luck" (random effect) is usually involved too. The selection process is the step that guides the evolutionary algorithm towards ever-better solutions.
We will expose some different selection schemes. We focus on cEAs with a single topology, the frequently used N x N grid in which a single individual exists at each grid point and interacts with its neighbors on nearby grid points.. A typical example would be a
Since we are studying just the case of cellular EAs, all the presented selection methods are variants of local selection wherein every individual resides inside a constrained environment called the local . Individuals interact only with individuals inside this region. The neighborhood can be seen as the group of potential mating partners. The selection methods used in cEAs are typically the same that those used in panmictic EAs. The main selection methods used in the literature are:
The linear rank selection algorithm defines the target sampling rate (TSR) of an individual x as:
TSR(x) = Min + (Max - Min) rank(x)/(N - 1)
where rank(x) is the index of x when the neighborhood is sorted in increasing order based on fitness, and N is the neighborhood size. Also, we impose the constraints that 0 TSR(x) , TSR(x) = N, 1 Max 2, and Min + Max = 2.
The TSR is the number of times an individual should be chosen as a parent for every N sampling operations.
The individuals are mapped to contiguous segments of a line, such that each individual's segment is equal in size to its fitness. A random number is generated and the individual whose segment spans the random number is selected. The process is repeated until the desired number of individuals is obtained (called mating population). This technique is analogous to a roulette wheel with each slice proportional in size to the fitness.
In Figure 1 we can see a graphic example with the individuals of Table 1, which shows the selection probability for 11 individuals, linear ranking and selective pressure together with the fitness value. Individual 1 is the most fit individual and occupies the largest interval, whereas individual 10 as the second least fit individual has the smallest interval on the line (see figure 1). Individual 11, the least fit interval, has a fitness value of 0 and get no chance for reproduction.
|Table 1. Selection probability and fitness value|
|Figure 1. Roulette-wheel selection|
For selecting the mating population the appropriate number of uniformly distributed random numbers (uniform distributed between 0.0 and 1.0) is independently generated. In Figure 1 we can see the selection process of the individuals for the example in Table 1 for the sample of these 6 random numbers: 0.81, 0.32, 0.96, 0.01, 0.65, 0.42.
The roulette-wheel selection algorithm provides a zero bias but does not guarantee minimum spread.
For 6 individuals to be selected, the distance between the pointers is 1/6=0.167. Figure 2 shows the selection for the sample of the random number 0.1 in the range [0, 0.167].
|Figure 2. Stochastic universal sampling|
After selection the mating population
consists of the individuals: 1, 2, 3, 4, 6, 8.
Stochastic universal sampling ensures a selection of offspring which is closer to what is deserved than roulette wheel selection.
In truncation selection individuals are sorted according to their fitness. Only the best individuals are selected for parents. These selected parents produce uniform at random offspring. The parameter for truncation selection is the truncation threshold Trunc. Trunc indicates the proportion of the neighborhood to be selected as parents and takes values ranging from 50%-10%. Individuals below the truncation threshold do not produce offspring.
Tournament selection : In tournament selection a number Tour of individuals is chosen randomly from the neighborhood and the best individual from this group is selected as parent. This process is repeated as often as individuals to choose. These selected parents produce uniform at random offspring. The parameter for tournament selection is the tournament size Tour. Tour takes values ranging from 2 - Nind (number of individuals in neighborhood).
Reproduction is the mechanism that allows us to obtain offsprings from one (asexual) or more (sexual) parents. Among reproduction operators the following stand out:
|Table 2. Crossover operator in binary string chromosomes|
In Figure 3 we show graphically the single and multi-point crossover applied to individuals with string shaped chromosomes.
|Figure 3. Single -up- and multi-point -down- crossover|
Replacement schemes are used by evolutionary algorithms to determine how the new individuals will be assimilated into the population. Replace worst and replace most-similar are the only really useful replacement schemes. Sometimes replace-parent can be effective, but usually when the parents are similar to the offspring, and this is just replace-most-similar. Some examples of replacement policies are given below:
Depending on when the replacement policy is applied we can stand out two different kinds of cEA: