Page 23 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

1. Artificial Intelligence 11

The above operation defined as the simulation for rotating the roulette
wheel and selecting the sector is repeated ‘L’ times for selecting ‘L’ values
for the next generation. Note that the number corresponding to the big sector
is having more chance for selection.
The process of generating new set of numbers (ie) next generation
population from existing set of numbers (ie) current generation population is
repeated for many iteration.
The Best value is chosen from the last generation corresponding to the
maximum fitness value.

2.2 Example

Let us consider the optimization problem for maximizing the function f(x) =
x+10*sin (5*x) +7*cos (4*x) +sin(x) ,subject to the constraint x varies from
0 to 9.
The numbers between 0 and 9 with the resolution of 0.001 are treated as
the chromosomes used in the Genetic Algorithm. (ie) Float chromosomes are
used in this example. Population size of 10 chromosomes is survived in
every generation. Arithmetic cross over is used as the genetic operator.
Mutation operation is not used in this example Roulette Wheel selection is
made at every generation. Algorithm flow is terminated after attaining the
maximum number of iterations. In this example Maximum number of
iterations used is 100.
The Best solution for the above problem is obtained in the thirteenth
generation using Genetic algorithm as 4.165 and the corresponding
fitness function f(x) is computed as 8.443. Note that Genetic algorithm
ends up with local maxima as shown in the figure 1-5. This is the
drawback of the Genetic Algorithm. When you run the algorithm again, it
may end up with Global maxima. The chromosomes generated randomly
during the first generation affects the best solution obtained using genetic
algorithm.
Best chromosome at every generation is collected. Best among the
collection is treated as the final Best solution which maximizes the
function f(x).

2.2.1 M-program for genetic algorithm

The Matlab program for obtaining the best solution for maximizing the
function f(x) = x+10*sin (5*x) +7*cos (4*x) +sin(x) using Genetic
Algorithm is given below.