# 117. Genetic Algorithms 2 – a multiple objective genetic algorithm (NSGA-II)

Note: As there is quite a substantial amount of code in this post, you may also copy the code as a single block from here.

If you have not looked at our description of a more simple genetic algorithm, with a single objective, then we advise you to look at that first (here). The example here will assume some basic understand of genetic algorithms.

In our previous example of a genetic algorithm, we looked at a genetic algorithm that optimised a single parameter. But what if we have two or more objectives to optimise?

An example in healthcare modelling is in deciding which hospitals should provide a specialist service in order to 1) minimise travel time for patients who need an emergency hospital admission, while 2) ensuring the hospital has sufficient admissions to maintain expertise and a 24/7 service, and 3) and ensuring no hospital is over-loaded with too many admissions.

One option when considering multiple objectives is to combine different objectives into a single number. This requires standardising the values in some way and giving them weights for their importance. This can simplify the optimisation problem but may be sensitive to how individual objectives are weighted (if following this approach it will probably be sensible to re-run the algorithm using alternative weighting schemes).

## Pareto fronts

The approach we will describe here takes an alternative approach and seeks to explicitly investigate the trade-off between different objectives. When one objective cannot be improved without the worsening of another objective we are on what is known as the ‘Pareto front’. That is easy to visualise with two objectives, but Pareto fronts exist across any number of objectives – we are on the Pareto front when we cannot improve one objective without necessarily worsening at least one other objective (as shown by the red line in the figure below).

## NSGA-II

The algorithm implemented here is based on an algorithm called ‘NSGA-II’. This was published by Deb et al.

Deb et al. (2002) A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation. 6, 182-197.

With NSGA-II we maintain a population of a given size (in the original paper that is fixed; in our implementation we define a range – the population must be larger than a minimum size, but not exceed a given maximum size.

The key process steps of NSGA-II are:

1) Start with a random population of solutions (P), encoded in binary form (‘chromosomes’) 2) Create a child population (Q) through crossover and mutation 3) Combine P & Q and score for all objectives 4) Identify the first Pareto front (F1); that is all solutions where there are no other solutions that are at least equally good in all objectives and better in at least one objective 5) If F1 is larger than the maximum permitted solution then reduce the size of F1 by ‘crowding selection’ (see below). 6) If F1 is smaller than the required population size then repeat Pareto selection (after removal of selected already selected). This new set of solutions is F2. If the total number selected solutions is greater than the permitted maximum population size then reduce the size of just the latest selection (in this case F2) so that the number of all selected solutions is equal to the maximum permitted population size. 7) Repeat Pareto selection until the required population size is reached (and then reduce the last selected Pareto front by ‘crowding selection’ as required to avoid exceeding the maximum permitted population size). 8) The selected Pareto fronts for the new population, P. 9) Repeat from (2) for the required number of generations or until some other ‘stop’ criterion is reached. 10) Perform a final Pareto selection so that the final reported population are just those on the first Pareto front.

A minimum population size is required to maintain diversity of solutions. If we only selected the first Pareto front we may have a very small population which would lack the diversity required to reach good final solutions.

This process is shown diagrammatically below (from Deb et al., 2002):

## Crowding distances/selection

When we are using a Pareto front to select solutions (as described here), all solutions are on the optimal front, that is in each solution there is no other solution that is at least as good in all scores, and better in at least one score. We therefore cannot rank solutions by performance. In order to select solutions, if we need to control the number of solutions we are generating we can use ‘crowding distances’. Crowding distances give a measure of closeness in performance to other solutions. The crowding distance is the average distance to its two neighbouring solutions. we will bias the selection of solutions to those with greater distances to neighbouring solutions.

We will use a ‘tournament’ method to select between two individual solutions. The crowding distances are calculated for all solutions. Two solutions are picked at random and the one with the greater crowding distance will be selected (with the unselected solution being returned back). This process is repeated until the require number of solutions have been selected.

An example is shown below – by using crowding distances we have reduced the number of points in densely packed regions by more than in the sparsely populated areas.

# The Code

We will break the code down into sections and functions.

To enale easier visualisation we will look to identify a Pareto population based on two competing objectives. But all the code will work on as many objectives as you wish (though it is not recommended to try to optimise more than five objectives).

## Import required libraries

``````import random as rn
import numpy as np
import matplotlib.pyplot as plt
# For use in Jupyter notebooks only:
% matplotlib inline``````

## Create reference solutions

In this example we will pick the easiest example possible. We will create two reference chromosomes as ‘ideal solutions’. This function may be used to create more references to mimic optimising against more than two objectives.

We will pick a chromosome length of 25 (this mimics solutions that would be coded as 25 binary ‘genes’).

In real life this binary representation would lead to better or worse solutions. We might, for example, use the binary notation to denote open/closed hospitals in a location problem. Or any number of binary digits might be used to encode the value of a variable.

In real life this binary representation would lead to better or worse solutions. We might, for example, use the binary notation to denote open/closed hospitals in a location problem. Or any number of binary digits might be used to encode the value of a variable.

``````def create_reference_solutions(chromosome_length, solutions):
"""
Function to create reference chromosomes that will mimic an ideal solution
"""
references = np.zeros((solutions, chromosome_length))
number_of_ones = int(chromosome_length / 2)

for solution in range(solutions):
# Build an array with an equal mix of zero and ones
reference = np.zeros(chromosome_length)
reference[0: number_of_ones] = 1

# Shuffle the array to mix the zeros and ones
np.random.shuffle(reference)
references[solution,:] = reference

return references``````

Show an example set of reference solutions:

``````print (create_reference_solutions(25, 2))

Out:
print (create_reference_solutions(25, 2))

[[0. 1. 0. 1. 0. 0. 1. 1. 0. 1. 1. 1. 0. 0. 0. 1. 0. 0. 0. 0. 1. 1. 1. 1.
0.]
[0. 1. 0. 1. 0. 0. 0. 1. 0. 0. 1. 0. 1. 1. 0. 1. 1. 1. 0. 0. 1. 0. 1. 1.
0.]]``````

## Evaluating solutions

Normally this section of the code would be more complex – it decodes the chromosome and applies it to a problem, and assesses the performance. For example we might use the chromosome to store whether hospitals are open/closed and then a piece of code will test travel distances for all patients to the open hospitals, and will also calculate the number of admissions to each hospital.

In our ‘toy’ example here we will simply calculate how many binary digits in each solution are the same as our two reference solutions (so we will have two scores for each solution).

We have two functions. The first will calculate the score (‘fitness’) against a single reference. The second will loop through all references (equivalent to all objectives) and call for the score/fitness to be calculated.

``````def calculate_fitness(reference, population):
"""
Calculate how many binary digits in each solution are the same as our
reference solution.
"""
# Create an array of True/False compared to reference
identical_to_reference = population == reference
# Sum number of genes that are identical to the reference
fitness_scores = identical_to_reference.sum(axis=1)

return fitness_scores``````
``````def score_population(population, references):
"""
Loop through all reference solutions and request score/fitness of
population against that reference solution.
"""
scores = np.zeros((population.shape[0], references.shape[0]))
for i, reference in enumerate(references):
scores[:,i] = calculate_fitness(reference, population)

return scores ``````

## Calculate crowding and select a population based on crowding scores

We have two functions here. The first to calculate the crowding of a population, based on similarity of scores. The second is a Tournament selection method that uses those crowding scores to pick a given number of solutions from a population.

``````def calculate_crowding(scores):
"""
Crowding is based on a vector for each individual
All scores are normalised between low and high. For any one score, all
solutions are sorted in order low to high. Crowding for chromsome x
for that score is the difference between the next highest and next
lowest score. Total crowding value sums all crowding for all scores
"""

population_size = len(scores[:, 0])
number_of_scores = len(scores[0, :])

# create crowding matrix of population (row) and score (column)
crowding_matrix = np.zeros((population_size, number_of_scores))

# normalise scores (ptp is max-min)
normed_scores = (scores - scores.min(0)) / scores.ptp(0)

# calculate crowding distance for each score in turn
for col in range(number_of_scores):
crowding = np.zeros(population_size)

# end points have maximum crowding
crowding[0] = 1
crowding[population_size - 1] = 1

# Sort each score (to calculate crowding between adjacent scores)
sorted_scores = np.sort(normed_scores[:, col])

sorted_scores_index = np.argsort(
normed_scores[:, col])

# Calculate crowding distance for each individual
crowding[1:population_size - 1] = \
(sorted_scores[2:population_size] -
sorted_scores[0:population_size - 2])

# resort to orginal order (two steps)
re_sort_order = np.argsort(sorted_scores_index)
sorted_crowding = crowding[re_sort_order]

# Record crowding distances
crowding_matrix[:, col] = sorted_crowding

# Sum crowding distances of each score
crowding_distances = np.sum(crowding_matrix, axis=1)

return crowding_distances``````
``````def reduce_by_crowding(scores, number_to_select):
"""
This function selects a number of solutions based on tournament of
crowding distances. Two members of the population are picked at
random. The one with the higher croding dostance is always picked
"""
population_ids = np.arange(scores.shape[0])

crowding_distances = calculate_crowding(scores)

picked_population_ids = np.zeros((number_to_select))

picked_scores = np.zeros((number_to_select, len(scores[0, :])))

for i in range(number_to_select):

population_size = population_ids.shape[0]

fighter1ID = rn.randint(0, population_size - 1)

fighter2ID = rn.randint(0, population_size - 1)

# If fighter # 1 is better
if crowding_distances[fighter1ID] >= crowding_distances[
fighter2ID]:

# add solution to picked solutions array
picked_population_ids[i] = population_ids[
fighter1ID]

# Add score to picked scores array
picked_scores[i, :] = scores[fighter1ID, :]

# remove selected solution from available solutions
population_ids = np.delete(population_ids, (fighter1ID),
axis=0)

scores = np.delete(scores, (fighter1ID), axis=0)

crowding_distances = np.delete(crowding_distances, (fighter1ID),
axis=0)
else:
picked_population_ids[i] = population_ids[fighter2ID]

picked_scores[i, :] = scores[fighter2ID, :]

population_ids = np.delete(population_ids, (fighter2ID), axis=0)

scores = np.delete(scores, (fighter2ID), axis=0)

crowding_distances = np.delete(
crowding_distances, (fighter2ID), axis=0)

# Convert to integer
picked_population_ids = np.asarray(picked_population_ids, dtype=int)

return (picked_population_ids)``````

## Pareto selection

Pareto selection involves two functions. The first will select a single Pareto front. The second will, as necessary, repeat Pareto front selection to build a population within defined size limits, and, as necessary, will reduce a Pareto front by applying crowding selection.

``````def identify_pareto(scores, population_ids):
"""
Identifies a single Pareto front, and returns the population IDs of
the selected solutions.
"""
population_size = scores.shape[0]
# Create a starting list of items on the Pareto front
# All items start off as being labelled as on the Parteo front
pareto_front = np.ones(population_size, dtype=bool)
# Loop through each item. This will then be compared with all other items
for i in range(population_size):
# Loop through all other items
for j in range(population_size):
# Check if our 'i' pint is dominated by out 'j' point
if all(scores[j] >= scores[i]) and any(scores[j] > scores[i]):
# j dominates i. Label 'i' point as not on Pareto front
pareto_front[i] = 0
# Stop further comparisons with 'i' (no more comparisons needed)
break
# Return ids of scenarios on pareto front
return population_ids[pareto_front]``````
``````def build_pareto_population(
population, scores, minimum_population_size, maximum_population_size):
"""
As necessary repeats Pareto front selection to build a population within
defined size limits. Will reduce a Pareto front by applying crowding
selection as necessary.
"""
unselected_population_ids = np.arange(population.shape[0])
all_population_ids = np.arange(population.shape[0])
pareto_front = []
while len(pareto_front) < minimum_population_size:
temp_pareto_front = identify_pareto(
scores[unselected_population_ids, :], unselected_population_ids)

# Check size of total parteo front.
# If larger than maximum size reduce new pareto front by crowding
combined_pareto_size = len(pareto_front) + len(temp_pareto_front)
if combined_pareto_size > maximum_population_size:
number_to_select = combined_pareto_size - maximum_population_size
selected_individuals = (reduce_by_crowding(
scores[temp_pareto_front], number_to_select))
temp_pareto_front = temp_pareto_front[selected_individuals]

# Add latest pareto front to full Pareto front
pareto_front = np.hstack((pareto_front, temp_pareto_front))

# Update unselected population ID by using sets to find IDs in all
# ids that are not in the selected front
unselected_set = set(all_population_ids) - set(pareto_front)
unselected_population_ids = np.array(list(unselected_set))

population = population[pareto_front.astype(int)]
return population``````

## Population functions

There are four functions concerning the population

1) Create a random population

2) Breed by crossover – two children produced from two parents

3) Randomly mutate population (small probability that any given gene in population will switch between 1/0 – applied to the child population)

4) Breed population -Create child population by repeatedly calling breeding function (two parents producing two children), applying genetic mutation to the child population, combining parent and child population, and removing duplicate chromosomes.

``````def create_population(individuals, chromosome_length):
"""
Create random population with given number of individuals and chroosome
length.
"""
# Set up an initial array of all zeros
population = np.zeros((individuals, chromosome_length))
# Loop through each row (individual)
for i in range(individuals):
# Choose a random number of ones to create
ones = rn.randint(0, chromosome_length)
# Change the required number of zeros to ones
population[i, 0:ones] = 1
# Sfuffle row
np.random.shuffle(population[i])

return population``````
``````def breed_by_crossover(parent_1, parent_2):
"""
Combine two parent chromsomes by crossover to produce two children.
"""
# Get length of chromosome
chromosome_length = len(parent_1)

# Pick crossover point, avoding ends of chromsome
crossover_point = rn.randint(1,chromosome_length-1)

# Create children. np.hstack joins two arrays
child_1 = np.hstack((parent_1[0:crossover_point],
parent_2[crossover_point:]))

child_2 = np.hstack((parent_2[0:crossover_point],
parent_1[crossover_point:]))

# Return children
return child_1, child_2``````
``````def randomly_mutate_population(population, mutation_probability):
"""
Randomly mutate population with a given individual gene mutation
probability. Individual gene may switch between 0/1.
"""
# Apply random mutation
random_mutation_array = np.random.random(size=(population.shape))

random_mutation_boolean = \
random_mutation_array <= mutation_probability

population[random_mutation_boolean] = \
np.logical_not(population[random_mutation_boolean])

# Return mutation population
return population``````
``````def breed_population(population):
"""
Create child population by repetedly calling breeding function (two parents
producing two children), applying genetic mutation to the child population,
combining parent and child population, and removing duplicatee chromosomes.
"""
# Create an empty list for new population
new_population = []
population_size = population.shape[0]
# Create new popualtion generating two children at a time
for i in range(int(population_size/2)):
parent_1 = population[rn.randint(0, population_size-1)]
parent_2 = population[rn.randint(0, population_size-1)]
child_1, child_2 = breed_by_crossover(parent_1, parent_2)
new_population.append(child_1)
new_population.append(child_2)

# Add the child population to the parent population
# In this method we allow parents and children to compete to be kept
population = np.vstack((population, np.array(new_population)))
population = np.unique(population, axis=0)

return population``````

## Main algorithm code

Now let’s put it all together!

``````# Set general parameters
chromosome_length = 50
starting_population_size = 5000
maximum_generation = 250
minimum_population_size = 500
maximum_population_size = 1000

# Create two reference solutions
# (this is used just to illustrate GAs)
references = create_reference_solutions(chromosome_length, 2)

# Create starting population
population = create_population(
starting_population_size, chromosome_length)

# Loop through the generations of genetic algorithm

for generation in range(maximum_generation):
if generation %10 ==0:
print ('Generation (out of %i): %i '%(maximum_generation, generation))

# Breed
population = breed_population(population)

# Score population
scores = score_population(population, references)

# Build pareto front
population = build_pareto_population(
population, scores, minimum_population_size, maximum_population_size)

# Get final pareto front
scores = score_population(population, references)
population_ids = np.arange(population.shape[0]).astype(int)
pareto_front = identify_pareto(scores, population_ids)
population = population[pareto_front, :]
scores = scores[pareto_front]``````

# Plot final Pareto front

This is the only part of the code that works only for two objectives.

You will see in this example we have a well-described trade-off between achieving the two objectives. With the population sizes used and the number of generations we achieved 96% maximum score on each objective (a larger population and/or more generations would get us to 100% on this quite simple example, though Pareto algorithms should not be expected to find the perfect solution for all objectives, but rather they should identify good solutions).

``````# Plot Pareto front (for two scores only)
x = scores[:, 0]/chromosome_length*100
y = scores[:, 1]/chromosome_length*100
plt.xlabel('Objective A - % maximum obtainable')
plt.ylabel('Objective B - % maximum obtainable')

plt.scatter(x,y)
plt.savefig('pareto.png')
plt.show()``````

# 95: Crowding distances: selecting solutions when too many multi-objective solutions exist

Sometimes in multi-objective algorithms we need to thin out the number of solutions we have.

When we are using a Pareto front to select solutions (as described here), all solutions are on the optimal front, that is in each solution there is no other solution that is at least as good in all scores, and better in at least one score. We therefore cannot rank solutions by performance. In order to select solutions, if we need to control the number of solutions we are generating we can use ‘crowding distances’. Crowding distances give a measure of closeness in performance to other solutions. The crowding distance is the average distance to its two neighbouring solutions.

# 94: Genetic algorithms 1. A simple genetic algorithm

Note: For core code only, without explanation or test code sections see this link: code_only

For more discussion on the general concepts of genetic algorithms, which are only presented briefly here (as we will focus on how to code a simple example in Python), see Wikipedia article.

In this example we will look at a basic genetic algorithm (GA). We will set up the GA to try to match a pre-defined ‘optimal. solution. Often with GAs we are using them to find solutions to problems which 1) cannot be solved with ‘exact’ methods (methods are are guaranteed to find the best solution), and 2) where we cannot recognise when we have found the optimal solution. GAs therefore fall into a collection of algorithms called heuristic (from Greek for ‘search’) algorithms that hunt down good solutions, without us knowing how far off the theoretical optimal solution they are.

In this case however we will test the algorithms against a solution we know we are looking for.

## Principles of genetic algorithms (GAs)

GAs are iterating algorithms, that is they repeatedly loop through a progress until a target is reached or a maximum number of iterations (called ‘generations’ in GAs) is reached.

The basic steps of a genetic algorithm are:

1) Create a population of randomly generated solutions, coded as binary arrays, and score population for performance (or ‘fitness’) of each individual.

2) Loop (until target performance is reached or a maximum number of generations is reached):

• Select two parents to ‘breed’. Selection is based on performance (or ‘fitness’) – better performing parents are more likely to be selected.
• Generate two child solutions from two parents by mixing the genes from each parent and by applying a chance of random mutation.
• Repeat child generation until a required new population size is reached.
• Score new population

## The solution to ‘hunt down’

We code most GAs to work with binary strings (or a binary NumPy array, as we will use here), so that the solution may be represented as a series of 0 and 1.

A real-life example in healthcare is that the array of 0s and 1s may represents the choices of closed or open hospital units providing a given service. We then evaluate each solution against predetermined criteria.

Here we will define a known solution based on a string of 70 0s or 1s. The number of possible combinations for this is 2^18, or 1.2 x 10^18 – that is 1 followed by eighteen zeros. Or, to put it another way (as these large numbers are difficult to imagine) …… the universe is about 15 billion (15 x 10^9) years old, or 5 x 10^17 seconds old. If we could evaluate 1,000 solutions per second, then a computer would need to run for twice the current age of the universe in order to evaluate all possible combinations. Let’s see how close to the perfect solution we can get in reasonable time!

In GA we will call each potential solution to be evaluated a ‘chromsome’. Each element (a 0 or 1) in that chromsome is a ‘gene’.

```import random
import numpy as np

def create_reference_solution(chromosome_length):

number_of_ones = int(chromosome_length / 2)

# Build an array with an equal mix of zero and ones
reference = np.zeros(chromosome_length)
reference[0: number_of_ones] = 1

# Shuffle the array to mix the zeros and ones
np.random.shuffle(reference)

return reference
```

Let’s test the function and show an example 70-gene reference chromosome.

```# Print an example target array
print (create_reference_solution(70))
```

```OUT:

[1. 0. 0. 0. 1. 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 1. 0. 1. 1. 1. 0.
1. 0. 1. 1. 1. 1. 0. 0. 0. 1. 1. 0. 1. 0. 0. 1. 1. 1. 1. 0. 1. 0. 1. 1.
0. 1. 1. 1. 1. 0. 1. 0. 1. 1. 1. 0. 1. 0. 0. 0. 0. 0. 1. 0. 1. 1.]
```

## Create a starting random population

We will use NumPy to store our population. Each row will represent one solution, which will contain a random sequence of zeros and ones. The number of rows will represent to number of ‘individuals, in our population.

```def create_starting_population(individuals, chromosome_length):
# Set up an initial array of all zeros
population = np.zeros((individuals, chromosome_length))
# Loop through each row (individual)
for i in range(individuals):
# Choose a random number of ones to create
ones = random.randint(0, chromosome_length)
# Change the required number of zeros to ones
population[i, 0:ones] = 1
# Sfuffle row
np.random.shuffle(population[i])

return population

```

Let’s test by showing a random population of 4 individuals with a gene length of 10.

```print (create_starting_population(4, 10))
```

```OUT:

[[0. 1. 1. 1. 1. 0. 0. 1. 1. 0.]
[1. 1. 1. 1. 0. 1. 1. 0. 1. 1.]
[1. 0. 0. 0. 0. 0. 0. 1. 1. 0.]
[1. 0. 0. 0. 0. 0. 1. 0. 0. 0.]]
```

## Calculate fitness of population

In GAs we refer to how good each individual in the population is, as ‘fitness’. The calculate_fitness function will be the evaluation procedure you wish to apply in your algorithm. In this example we are going to return the number of genes (elements) in a potential solution (chromosome) that match our f=reference standard.

```def calculate_fitness(reference, population):
# Create an array of True/False compared to reference
identical_to_reference = population == reference
# Sum number of genes that are identical to the reference
fitness_scores = identical_to_reference.sum(axis=1)

return fitness_scores
```

Let’s test what we have so far:

```reference = create_reference_solution(10)
print ('Reference solution: \n', reference)
population = create_starting_population(6, 10)
print ('\nStarting population: \n', population)
scores = calculate_fitness(reference, population)
print('\nScores: \n', scores)
```

```OUT:

Reference solution:
[1. 1. 0. 0. 0. 0. 1. 0. 1. 1.]

Starting population:
[[0. 0. 1. 1. 1. 0. 0. 0. 0. 1.]
[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
[1. 1. 1. 1. 0. 1. 1. 1. 1. 1.]
[0. 0. 0. 0. 0. 0. 0. 0. 1. 0.]
[1. 1. 1. 1. 0. 0. 1. 1. 0. 1.]
[1. 1. 0. 0. 0. 0. 0. 1. 1. 1.]]

Scores:
[3 5 6 6 6 8]
```

## Choosing individuals to breed with tournament selection

Genetic algorithms mimic biology in that the individuals with the best fitness cores are most likely to breed and pass on their genes. But we do not simply take all the best individuals from our population to breed, as this might risk ‘in-breeding’. Rather, we use a method that means better individuals are moire likely to breed, but low fitness individuals at times may be chosen to breed.

In tournament selection we first choose two individuals at random from our population (it is possible that two low fitness individuals may be chosen). We then pass those individuals to a ‘tournament’ where the individual with the highest fitness will be chosen.

It is possible to further modify this so that the highest fitness individual will win with a given probability, but we will keep it simple here and have the highest fitness individual always winning.
It is also possible to have more than two individuals in a tournament. The more individuals in a tournament the more the picked population will be biased towards the highest fitness individuals.

```def select_individual_by_tournament(population, scores):
# Get population size
population_size = len(scores)

# Pick individuals for tournament
fighter_1 = random.randint(0, population_size-1)
fighter_2 = random.randint(0, population_size-1)

# Get fitness score for each
fighter_1_fitness = scores[fighter_1]
fighter_2_fitness = scores[fighter_2]

# Identify undividual with highest fitness
# Fighter 1 will win if score are equal
if fighter_1_fitness >= fighter_2_fitness:
winner = fighter_1
else:
winner = fighter_2

# Return the chromsome of the winner
return population[winner, :]
```

Let’s test selection of parents:

```# Set up and score population
reference = create_reference_solution(10)
population = create_starting_population(6, 10)
scores = calculate_fitness(reference, population)

# Pick two parents and dispplay
parent_1 = select_individual_by_tournament(population, scores)
parent_2 = select_individual_by_tournament(population, scores)
print (parent_1)
print (parent_2)
```

```OUT:

[1. 0. 1. 1. 0. 1. 0. 1. 1. 0.]
[1. 0. 1. 1. 0. 1. 0. 1. 1. 0.]
```

## Producing children from parents – crossover

When two individuals are chosen, the next step is to produce ‘children’ from them. We produce these children by ‘crossover’ mix of their two chromosomes. We choose a random point within the chromosome, and then one ‘child’ will take the left portion (up to, but not including, the crossover point) from parent 1 and the corresponding right portion from parent 2. The result is a mix of genes from each parent. The second ‘child’ will be the opposite of this – portion (up to, but not including) the crossover point) from parent 2 and the corresponding right portion from parent 1.

It is possible to have more than one crossover point, but we will keep it simple and have a single crossover point.

In [9]:
```def breed_by_crossover(parent_1, parent_2):
# Get length of chromosome
chromosome_length = len(parent_1)

# Pick crossover point, avoding ends of chromsome
crossover_point = random.randint(1,chromosome_length-1)

# Create children. np.hstack joins two arrays
child_1 = np.hstack((parent_1[0:crossover_point],
parent_2[crossover_point:]))

child_2 = np.hstack((parent_2[0:crossover_point],
parent_1[crossover_point:]))

# Return children
return child_1, child_2
```

And let’s test it so far, creating a population, scoring it, picking two ‘parents’ and producing ‘two children’.

```# Set up and score population
reference = create_reference_solution(15)
population = create_starting_population(100, 15)
scores = calculate_fitness(reference, population)

# Pick two parents and dispplay
parent_1 = select_individual_by_tournament(population, scores)
parent_2 = select_individual_by_tournament(population, scores)

# Get children
child_1, child_2 = breed_by_crossover(parent_1, parent_2)

# Show output
print ('Parents')
print (parent_1)
print (parent_2)
print ('Children')
print (child_1)
print (child_2)
```

```OUT:

Parents
[1. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 0.]
[0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 1. 1. 1. 0.]
Children
[1. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 1. 1. 1. 0.]
[0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.]
```

## Random mutation of genes

In evolution sometimes genes are copied incorrectly. This change may be harmful or beneficial. We mimic this by having a certain probability of that a gene (which is either a 0 or a 1) becomes switched.

Typically this probability is low (e.g. 0.005), though it can be made to be flexible (e.g. increase mutation rate if progress has stalled)

```def randomly_mutate_population(population, mutation_probability):

# Apply random mutation
random_mutation_array = np.random.random(
size=(population.shape))

random_mutation_boolean = \
random_mutation_array <= mutation_probability

population[random_mutation_boolean] = \
np.logical_not(population[random_mutation_boolean])

# Return mutation population
return population
```

Let’s test our function with a high mutation rate (0.25) to see the effects. You can change the mutation rate and see what happens (a mutation rate of 1.0 will invert all genes).

```# Set up and score population
reference = create_reference_solution(15)
population = create_starting_population(100, 15)
scores = calculate_fitness(reference, population)

# Pick two parents and display
parent_1 = select_individual_by_tournament(population, scores)
parent_2 = select_individual_by_tournament(population, scores)

# Get children and make new population
child_1, child_2 = breed_by_crossover(parent_1, parent_2)
population = np.stack((child_1, child_2))

# Mutate population
mutation_probability = 0.25
print ("Population before mutation")
print (population)
population = randomly_mutate_population(population, mutation_probability)
print ("Population after mutation")
print (population)
```

```OUT:

Population before mutation
[[1. 0. 0. 0. 0. 1. 0. 1. 1. 0. 0. 0. 0. 1. 1.]
[0. 0. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. 1. 1. 1.]]

Population after mutation
[[1. 1. 0. 1. 0. 1. 0. 1. 1. 1. 1. 1. 0. 1. 1.]
[0. 0. 1. 1. 0. 0. 0. 1. 1. 0. 0. 0. 1. 0. 1.]]
```

## Putting it all together

We’ve defined all the functions we need. Now let’s put it all together.

```# Set general parameters
chromosome_length = 75
population_size = 500
maximum_generation = 200
best_score_progress = [] # Tracks progress

# Create reference solution
# (this is used just to illustrate GAs)
reference = create_reference_solution(chromosome_length)

# Create starting population
population = create_starting_population(population_size, chromosome_length)

# Display best score in starting population
scores = calculate_fitness(reference, population)
best_score = np.max(scores)/chromosome_length * 100
print ('Starting best score, percent target: %.1f' %best_score)

# Add starting best score to progress tracker
best_score_progress.append(best_score)

# Now we'll go through the generations of genetic algorithm
for generation in range(maximum_generation):
# Create an empty list for new population
new_population = []

# Create new popualtion generating two children at a time
for i in range(int(population_size/2)):
parent_1 = select_individual_by_tournament(population, scores)
parent_2 = select_individual_by_tournament(population, scores)
child_1, child_2 = breed_by_crossover(parent_1, parent_2)
new_population.append(child_1)
new_population.append(child_2)

# Replace the old population with the new one
population = np.array(new_population)
# Apply mutation    mutation_rate = 0.002    population = randomly_mutate_population(population, mutation_rate)
# Score best solution, and add to tracker
scores = calculate_fitness(reference, population)
best_score = np.max(scores)/chromosome_length * 100
best_score_progress.append(best_score)

# GA has completed required generation
print ('End best score, percent target: %.1f' %best_score)

# Plot progress
import matplotlib.pyplot as plt
%matplotlib inline
plt.plot(best_score_progress)
plt.xlabel('Generation')
plt.ylabel('Best score (% target)')
plt.show()
```

```OUT:

Starting best score, percent target: 64.0
End best score, percent target: 100.0
```

# 94. Genetic Algorithms 1. A simple genetic algorithm (code only)

For explanation see: 94: Genetic algorithms 1. A simple genetic algorithm

```import random
import numpy as np

def create_reference_solution(chromosome_length):

number_of_ones = int(chromosome_length / 2)

# Build an array with an equal mix of zero and ones
reference = np.zeros(chromosome_length)
reference[0: number_of_ones] = 1

# Shuffle the array to mix the zeros and ones
np.random.shuffle(reference)

return reference

def create_starting_population(individuals, chromosome_length):
# Set up an initial array of all zeros
population = np.zeros((individuals, chromosome_length))
# Loop through each row (individual)
for i in range(individuals):
# Choose a random number of ones to create
ones = random.randint(0, chromosome_length)
# Change the required number of zeros to ones
population[i, 0:ones] = 1
# Sfuffle row
np.random.shuffle(population[i])

return population

def calculate_fitness(reference, population):
# Create an array of True/False compared to reference
identical_to_reference = population == reference
# Sum number of genes that are identical to the reference
fitness_scores = identical_to_reference.sum(axis=1)

return fitness_scores

def select_individual_by_tournament(population, scores):
# Get population size
population_size = len(scores)

# Pick individuals for tournament
fighter_1 = random.randint(0, population_size-1)
fighter_2 = random.randint(0, population_size-1)

# Get fitness score for each
fighter_1_fitness = scores[fighter_1]
fighter_2_fitness = scores[fighter_2]

# Identify undividual with highest fitness
# Fighter 1 will win if score are equal
if fighter_1_fitness >= fighter_2_fitness:
winner = fighter_1
else:
winner = fighter_2

# Return the chromsome of the winner
return population[winner, :]

def breed_by_crossover(parent_1, parent_2):
# Get length of chromosome
chromosome_length = len(parent_1)

# Pick crossover point, avoding ends of chromsome
crossover_point = random.randint(1,chromosome_length-1)

# Create children. np.hstack joins two arrays
child_1 = np.hstack((parent_1[0:crossover_point],
parent_2[crossover_point:]))

child_2 = np.hstack((parent_2[0:crossover_point],
parent_1[crossover_point:]))

# Return children
return child_1, child_2

def randomly_mutate_population(population, mutation_probability):

# Apply random mutation
random_mutation_array = np.random.random(
size=(population.shape))

random_mutation_boolean = \
random_mutation_array <= mutation_probability

population[random_mutation_boolean] = \
np.logical_not(population[random_mutation_boolean])

# Return mutation population
return population

# *************************************
# ******** MAIN ALGORITHM CODE ********
# *************************************

# Set general parameters
chromosome_length = 75
population_size = 500
maximum_generation = 200
best_score_progress = [] # Tracks progress

# Create reference solution
# (this is used just to illustrate GAs)
reference = create_reference_solution(chromosome_length)

# Create starting population
population = create_starting_population(population_size, chromosome_length)

# Display best score in starting population
scores = calculate_fitness(reference, population)
best_score = np.max(scores)/chromosome_length * 100
print ('Starting best score, % target: ',best_score)

# Add starting best score to progress tracker
best_score_progress.append(best_score)

# Now we'll go through the generations of genetic algorithm
for generation in range(maximum_generation):
# Create an empty list for new population
new_population = []

# Create new popualtion generating two children at a time
for i in range(int(population_size/2)):
parent_1 = select_individual_by_tournament(population, scores)
parent_2 = select_individual_by_tournament(population, scores)
child_1, child_2 = breed_by_crossover(parent_1, parent_2)
new_population.append(child_1)
new_population.append(child_2)

# Replace the old population with the new one
population = np.array(new_population)

# Apply mutation
mutation_rate = 0.002
population = randomly_mutate_population(population, mutation_rate)

# Score best solution, and add to tracker
scores = calculate_fitness(reference, population)
best_score = np.max(scores)/chromosome_length * 100
best_score_progress.append(best_score)

# GA has completed required generation
print ('End best score, % target: ', best_score)

# Plot progress
import matplotlib.pyplot as plt
plt.plot(best_score_progress)
plt.xlabel('Generation')
plt.ylabel('Best score (% target)')
plt.show()
```