Genetic_algorithm

Genetic_algorithm test case

Optimization problem - maximize

f(x, y) = cos((x - 1.14)⁶) - 100 * (y² - x)⁴

0 ≤ x ≤ 2

0 ≤ y ≤ 3

x + y ≥ 1

The composition of chromosomes:

The number of distinct values N = (b - a) / ε

The number of bits n = log₂N = log₂(b - a) / ε

Precision ε = 10^-4 means four places after the decimal point

The domain length for x is 2 (0 ~ 2)

n_x = 14.28 ~ 15 bits

The domain length for y is 2 (0 ~ 3)

n_y = 14.87 ~ 16 bits

For n_y, the number of bits is rounded up to 16 to ensure there are enough binary combinations to represent all possible values. Using the domain length equal to 15 gives us 2¹⁵=32768 values, which is just above 30000, so from safety reasons the domain length equal to 16 will be used (2¹⁶= 65536 values).

Total chromosome length is 31 bits – first 15 bits to encode x, last 16 bits to encode y.

Number of chromosomes and number of generations:

Size of the population is 200 chromosomes

Number of generations is 5000 generations

Mechanisms of selection, crossover and mutation:

Selection mechanism – roulette wheel

Crossover mechanism – two-point crossover

Mutation mechanism – mutation of one random bit (bit flipping)

Constraint handling:

The constraint x + y ≥ 1 is used as a penalty method. If the solution violates the constraint, a large negative fitness value is applied to decrease the possibility of survival to the next generation.

Setting of tuning parameters:

Crossover rate – 0.9

Mutation rate – 0.1

Penalty rate – 10⁶