Genetic Algorithm Modeling

Genetic algorithms are implemented as a computer simulation in which a population of abstract representations (called chromosomes or the genotype or the genome) of candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem evolves toward better solutions. Traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually starts from a population of randomly generated individuals and happens in generations. In each generation, the fitness of every individual in the population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number

30

31

of generations has been produced, or a satisfactory fitness level has been reached for the population. If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached.

Genetic algorithms find application in bioinformatics, phylogenetics, computer science, engineering, economics, chemistry, manufacturing, mathematics, physics and other fields.

A typical genetic algorithm requires two things to be defined:

1. a genetic representation of the solution domain, 2. a fitness function to evaluate the solution domain.

A standard representation of the solution is as an array of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, which facilitates simple crossover operation. Variable length representations may also be used, but crossover implementation is more complex in this case. Tree-like representations are explored in Genetic programming and graph-form representations are explored in Evolutionary programming.

The fitness function is defined over the genetic representation and measures the

quality of the represented solution. The fitness function is always problem dependent.

For instance, in the knapsack problem we want to maximize the total value of objects

32

that we can put in a knapsack of some fixed capacity. A representation of a solution might be an array of bits, where each bit represents a different object, and the value of the bit (0 or 1) represents whether or not the object is in the knapsack. Not every such representation is valid, as the size of objects may exceed the capacity of the knapsack.

The fitness of the solution is the sum of values of all objects in the knapsack if the representation is valid or 0 otherwise. In some problems, it is hard or even impossible to define the fitness expression; in these cases, interactive genetic algorithms are used.

Once we have the genetic representation and the fitness function defined, GA proceeds to initialize a population of solutions randomly, and then improve it through repetitive application of mutation, crossover, inversion, and selection operators.

4.1.1 Individual

The individual represent a candidate solution. In this problem, it represents a legal label placement without crossing. The individuals are stored as real-valued vector. We let the element “0” of the vector representing connecting to left side boundary. On the other side, the element “1” means putting label to the right side boundary. Although it is conceivable that different genetic representations influence the optimization behavior significantly, we choose this representation instinctively. Because we only need two

33

types of groups to represent connecting to the left side or connecting to the right side, it is obviously that binary integer representation is just what we need.

4.1.2 Initialization

At the beginning of the genetic algorithm, the individuals in the population have to be initialized. Initially many individual solutions are randomly generated to form an initial population. The population size depends on the nature of the problem, but typically contains several hundreds or thousands of possible solutions. Traditionally, the population is generated randomly, covering the entire range of possible solutions (the

search space). Occasionally, the solutions may be "seeded" in areas where optimal solutions are likely to be found. In our case, this is done randomly. We generate a label placement with restricted given area (including rectangle R, track routing area, and space for labels) and let bad offspring eliminated by selection.

4.1.3 Evaluation

The choice of the evaluation function plays a crucial role in the design of a genetic algorithm. There is a big advantage of using evaluation in genetic algorithm. One can

34

measure desired criteria on the resulting placement and weight these criteria to suit personal preferences. Because genetic algorithm is always using in multi-purpose problem, we can then analyze how important these criteria are.

Among the criteria we test are

• All the leaders do not cross to each other.

• All the labels do not overlap to each other.

• Total labels should be placed balanced on left side and right side.

• The longer label height on left side and right side is minimum height.

• Total leader length is a minimum length of all combination.

The result presents in Chapter 5 show some cases of legal placement.

4.1.4 Selection

Selection is important to genetic algorithm, since only selection drives the search towards more promising regions of the search space. In our implementation, we select individuals for reproduction (i.e. parents) according to the common linear ranking selection scheme, i.e. individuals are selected according to their rank, with better individuals receiving a higher chance of being selected. So that, the selective pressure of actual fitness values, which may be important, since it is not known beforehand in

35

which range of fitness values the optimal solution is located. The algorithm is of the steady state type, i.e. the offspring is introduced into the population, and at least fit individual is deleted. This way, the best solution so far is never lost. Also, in our problem, when crossing happened, the individual should not be counted in the population.

We define fitness function as follow:

f λ ∑ c

In order to normalize the function, we divide these parameters to their intuitive maximum value. Thus without generality, λ and λ can be chosen between 0 and 1, and satisfy λ λ 1.

4.1.5 Recombination

36

In order to get better result, we combine two good parents into a new offspring which may be brown better or not. The purpose of the crossover operator is to recombining sub-placements of different individuals to produce an offspring. Since, we expect that good parts of a placement are connected, we perform crossover by choosing randomly a connected parts of the placement of two parents and swap the sub-placement.

However, unfortunately, there is a problem with this operator using this method. A combination of two good parents may yield a poor offspring. This poor offspring will be deleted during the natural selection process.

4.1.6 Mutation

Mutation is a crucial step of genetic algorithm. While using recombination, we can only find new combination of individuals that are already at present. We may lose some information forever while it is not in the population. Another method called mutation can introduce new material into the population, i.e. the slight changing of individuals. It is necessary and reasonable to get new materials to increase the probability of getting better answers. In out implementation, mutation is done by randomly changing binary vector with a given small probability. We try to change one leader from the right side to

37

left side (or from the left side to the right side). This way, we will have a probability to get a better individual through the present individual and the result is different from the recombination (or crossover) process.