Problem Solving using genetic algorithm

Example: Suppose there is equality a +2 b +3 c +4 d = 30, genetic algorithm will be used to find the value of a, b, c, and d that satisfy the above equation.

Objective: Minimize the value of function f(x) where f (x) = ((a + 2b + 3c + 4d) - 30). Since there are four variables in the equation, namely a, b, c, and d, we can compose the chromosome as follow:

$\boxed{A|B|C|D}$

To speed up the computation, we can restrict that the values of variables a, b, c, and d are integers between 0 and 30.

Step 1. Initialization

For example we define the number of chromosomes in population are 6, then we generate random value of gene a, b, c, d for 6 chromosomes

Chromosome[1] = [a;b;c;d] = [12;05;23;08]

Chromosome[2] = [a;b;c;d] = [02;21;18;03]

Chromosome[3] = [a;b;c;d] = [10;04;13;14]

Chromosome[4] = [a;b;c;d] = [20;01;10;06]

Chromosome[5] = [a;b;c;d] = [01;04;13;19]

Chromosome[6] = [a;b;c;d] = [20;05;17;01]

Step 2. Evaluation

We compute the objective function value for each chromosome produced in initialization step:

F_obj[1] = Abs(( 12 + 205 + 323 + 4*08 ) - 30)=Abs((12 + 10 + 69 + 32 ) - 30)=Abs(123 - 30)= 93

F_obj[2] = Abs((02 + 221 + 318 + 4*03) - 30)=Abs((02 + 42 + 54 + 12) - 30)= Abs(110 - 30)= 80

F_obj[3] = Abs((10 + 204 + 313 + 4*14) - 30) = Abs((10 + 08 + 39 + 56) - 30)= Abs(113 - 30)= 83

F_obj[4] = Abs((20 + 201 + 310 + 4*06) - 30)= Abs((20 + 02 + 30 + 24) - 30)= Abs(76 - 30)= 46

F_obj[5] = Abs((01 + 204 + 313 + 4*19) - 30) = Abs((01 + 08 + 39 + 76) - 30)= Abs(124 - 30)= 94

F_obj[6] = Abs((20 + 205 + 317 + 4*01) - 30)= Abs((20 + 10 + 51 + 04) - 30)= Abs(85 - 30)= 55

Step 3. Selection

1. The fittest chromosomes have higher probability to be selected for the next generation. To compute fitness probability we must compute the fitness of each chromosome. To avoid divide by zero problem, the value of F_obj is added by 1.

Fitness[1] = 1 / (1+F_obj[1]) = 1 / 94 = 0.0106

Fitness[2] = 1 / (1+F_obj[2]) = 1 / 81 = 0.0123

Fitness[3] = 1 / (1+F_obj[3]) = 1 / 84 = 0.0119

Fitness[4] = 1 / (1+F_obj[4]) = 1 / 47 = 0.0213

Fitness[5] = 1 / (1+F_obj[5]) = 1 / 95 = 0.0105

Fitness[6] = 1 / (1+F_obj[6]) = 1 / 56 = 0.0179

Total = 0.0106 + 0.0123 + 0.0119 + 0.0213 + 0.0105 + 0.0179 = 0.0845

The probability for each chromosomes is formulated by: P[i] = Fitness[i] / Total

P[1] = 0.0106 / 0.0845 = 0.1254

P[2] = 0.0123 / 0.0845 = 0.1456

P[3] = 0.0119 / 0.0845 = 0.1408

P[4] = 0.0213 / 0.0845 = 0.2521

P[5] = 0.0105 / 0.0845 = 0.1243

P[6] = 0.0179 / 0.0845 = 0.2118

From the probabilities above we can see that Chromosome 4 that has the highest fitness, this chromosome has highest probability to be selected for next generation chromosomes. For the selection process we use roulette wheel, for that we should compute the cumulative probability values:

C[1] = 0.1254

C[2] = 0.1254 + 0.1456 = 0.2710

C[3] = 0.1254 + 0.1456 + 0.1408 = 0.4118

C[4] = 0.1254 + 0.1456 + 0.1408 + 0.2521 = 0.6639

C[5] = 0.1254 + 0.1456 + 0.1408 + 0.2521 + 0.1243 = 0.7882

C[6] = 0.1254 + 0.1456 + 0.1408 + 0.2521 + 0.1243 + 0.2118 = 1.0

Having calculated the cumulative probability of selection process using roulette-wheel can be done. The process is to generate random number R in the range 0-1 as follows.

R[1] = 0.201

R[2] = 0.284

R[3] = 0.099

R[4] = 0.822

R[5] = 0.398

R[6] = 0.501

If random number R [1] is greater than P [1] and smaller than P [2] then select Chromosome [2] as a chromosome in the new population for next generation:

NewChromosome[1] = Chromosome[2]

NewChromosome[2] = Chromosome[3]

NewChromosome[3] = Chromosome[1]

NewChromosome[4] = Chromosome[6]

NewChromosome[5] = Chromosome[3]

NewChromosome[6] = Chromosome[4]

Chromosome in the population thus became:

Chromosome[1] = [02;21;18;03]

Chromosome[2] = [10;04;13;14]

Chromosome[3] = [12;05;23;08]

Chromosome[4] = [20;05;17;01]

Chromosome[5] = [10;04;13;14]

Chromosome[6] = [20;01;10;06]

Step 4. Crossover

In this example, we use one-cut point, i.e. randomly select a position in the parent chromosome then exchanging sub-chromosome. Parent chromosome which will mate is randomly selected and the number of mate Chromosomes is controlled using crossover_rate (ρc) parameters. Pseudo-code for the crossover process is as follows:

begin

k← 0;

while(k<population) do="" <="" p="">

R[k] ← random(0-1);

if (R[k] < ρc ) then

select Chromosome[k] as parent;

end;

k = k + 1;

end;

end;

Chromosome k will be selected as a parent if R [k] <ρc. Suppose we set that the crossover rate is 25%, then Chromosome number k will be selected for crossover if random generated value for Chromosome k below 0.25. The process is as follows: First we generate a random number R as the number of population.

R[1] = 0.191

R[2] = 0.259

R[3] = 0.760

R[4] = 0.006

R[5] = 0.159

R[6] = 0.340

For random number R above, parents are Chromosome [1], Chromosome [4] and Chromosome [5] will be selected for crossover.

Chromosome[1] >< Chromosome[4]

Chromosome[4] >< Chromosome[5]

Chromosome[5] >< Chromosome[1]

After chromosome selection, the next process is determining the position of the crossover point. This is done by generating random numbers between 1 to (length of Chromosome – 1). In this case, generated random numbers should be between 1 and 3. After we get the crossover point, parents Chromosome will be cut at crossover point and its gens will be interchanged. For example we generated 3 random number and we get:

C[1] = 1

C[2] = 1

C[3] = 2

Then for first crossover, second crossover and third crossover, parent’s gens will be cut at gen number 1, gen number 1 and gen number 3 respectively, e.g.

Chromosome[1] = Chromosome[1] >< Chromosome[4] = [02;21;18;03] >< [20;05;17;01] = [02;05;17;01]

Chromosome[4] = Chromosome[4] >< Chromosome[5] = [20;05;17;01] >< [10;04;13;14] = [20;04;13;14]

Chromosome[5] = Chromosome[5] >< Chromosome[1] = [10;04;13;14] >< [02;21;18;03] = [10;04;18;03]

Thus Chromosome population after experiencing a crossover process:

Chromosome[1] = [02;05;17;01]

Chromosome[2] = [10;04;13;14]

Chromosome[3] = [12;05;23;08]

Chromosome[4] = [20;04;13;14]

Chromosome[5] = [10;04;18;03]

Chromosome[6] = [20;01;10;06]

Step 5. Mutation

Number of chromosomes that have mutations in a population is determined by the mutation rate parameter. Mutation process is done by replacing the gen at random position with a new value. The process is as follows. First we must calculate the total length of gen in the population. In this case the total length of gen is total_gen = number_of_gen_in_Chromosome * number of population = 4 * 6 = 24

Mutation process is done by generating a random integer between 1 and total_gen (1 to 24). If generated random number is smaller than mutation_rate(ρm) variable then marked the position of gen in chromosomes. Suppose we define ρm 10%, it is expected that 10% (0.1) of total_gen in the population that will be mutated: number of mutations = 0.1 * 24 = 2.4 ≈ 2

Suppose generation of random number yield 12 and 18 then the chromosome which have mutation are Chromosome number 3 gen number 4 and Chromosome 5 gen number 2. The value of mutated gens at mutation point is replaced by random number between 0-30. Suppose generated random number are 2 and 5 then Chromosome composition after mutation are:

Chromosome[1] = [02;05;17;01]

Chromosome[2] = [10;04;13;14]

Chromosome[3] = [12;05;23;02]

Chromosome[4] = [20;04;13;14]

Chromosome[5] = [10;05;18;03]

Chromosome[6] = [20;01;10;06]

Finishing mutation process then we have one iteration or one generation of the genetic algorithm. We can now evaluate the objective function after one generation:

Chromosome[1] = [02;05;17;01]

F_obj[1] = Abs(( 02 + 205 + 317 + 4*01 ) - 30) = Abs((2 + 10 + 51 + 4 ) - 30) = Abs(67 - 30) = 37

Chromosome[2] = [10;04;13;14]

F_obj[2] = Abs(( 10 + 204 + 313 + 4*14 ) - 30) =Abs((10 + 8 + 33 + 56 ) - 30) =Abs(107 - 30)= 77

Chromosome[3] = [12;05;23;02]

F_obj[3] = Abs(( 12 + 205 + 323 + 4*02 ) - 30)= Abs((12 + 10 + 69 + 8 ) - 30)= Abs(87 - 30) = 47

Chromosome[4] = [20;04;13;14]

F_obj[4] = Abs(( 20 + 204 + 313 + 4*14 ) - 30)= Abs((20 + 8 + 39 + 56 ) - 30)= Abs(123 - 30)= 93

Chromosome[5] = [10;05;18;03]

F_obj[5] = Abs(( 10 + 205 + 318 + 4*03 ) - 30)= Abs((10 + 10 + 54 + 12 ) - 30)= Abs(86 - 30)= 56

Chromosome[6] = [20;01;10;06]

F_obj[6] = Abs(( 20 + 201 + 310 + 4*06 ) - 30)= Abs((20 + 2 + 30 + 24 ) - 30)= Abs(76 - 30)= 46

From the evaluation of new Chromosome we can see that the objective function is decreasing, this means that we have better Chromosome or solution compared with previous Chromosome generation. New Chromosomes for next iteration are:

Chromosome[1] = [02;05;17;01]

Chromosome[2] = [10;04;13;14]

Chromosome[3] = [12;05;23;02]

Chromosome[4] = [20;04;13;14]

Chromosome[5] = [10;05;18;03]

Chromosome[6] = [20;01;10;06]

These new Chromosomes will undergo the same process as the previous generation of Chromosomes such as evaluation, selection, crossover and mutation and at the end it produce new generation of Chromosome for the next iteration. This process will be repeated until a predetermined number of generations. For this example, after running 50 generations, best chromosome is obtained:

Chromosome = [07; 05; 03; 01]

This means that:

a = 7, b = 5, c = 3, d = 1

If we use the number in the problem equation

a +2 b +3 c +4 d = 30

7 + (2 * 5) + (3 * 3) + (4 * 1) = 30

We can see that the value of variable a, b, c and d generated by genetic algorithm can satisfy that equality.