# One Max Problem¶

This is the first complete example built with DEAP. It will help new users to overview some of the framework’s possibilities and illustrate the potential of evolutionary algorithms in general. The problem itself is both very simple and widely used in the evolutionary computational community. We will create a population of individuals consisting of integer vectors randomly filled with 0 and 1. Then we let our population evolve until one of its members contains only 1 and no 0 anymore.

## Setting Things Up¶

In order to solve the One Max problem, we need a bunch of ingredients. First we have to define our individuals, which will be lists of integer values, and to generate a population using them. Then we will add some functions and operators taking care of the evaluation and evolution of our population and finally put everything together in script.

But first of all, we need to import some modules.

import random

from deap import base
from deap import creator
from deap import tools


### Creator¶

Since the actual structure of the required individuals in genetic algorithms does strongly depend on the task at hand, DEAP does not contain any explicit structure. It will rather provide a convenient method for creating containers of attributes, associated with fitnesses, called the deap.creator. Using this method we can create custom individuals in a very simple way.

The creator is a class factory that can build new classes at run-time. It will be called with first the desired name of the new class, second the base class it will inherit, and in addition any subsequent arguments you want to become attributes of your class. This allows us to build new and complex structures of any type of container from lists to n-ary trees.

creator.create("FitnessMax", base.Fitness, weights=(1.0,))
creator.create("Individual", list, fitness=creator.FitnessMax)


First we will define the class FitnessMax. It will inherit the Fitness class of the deap.base module and contain an additional attribute called weights. Please mind the value of weights to be the tuple (1.0,). This way we will be maximizing a single objective fitness. We can’t repeat it enough, in DEAP single objectives is a special case of multi objectives.

Next we will create the class Individual, which will inherit the class list and contain our previously defined FitnessMax class in its fitness attribute. Note that upon creation all our defined classes will be part of the creator container and can be called directly.

### Toolbox¶

Now we will use our custom classes to create types representing our individuals as well as our whole population.

All the objects we will use on our way, an individual, the population, as well as all functions, operators, and arguments will be stored in a DEAP container called Toolbox. It contains two methods for adding and removing content, register() and unregister().

toolbox = base.Toolbox()
# Attribute generator
toolbox.register("attr_bool", random.randint, 0, 1)
# Structure initializers
toolbox.register("individual", tools.initRepeat, creator.Individual,
toolbox.attr_bool, 100)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)


In this code block we register a generation function toolbox.attr_bool() and two initialization ones individual() and population(). toolbox.attr_bool(), when called, will draw a random integer between 0 and 1. The two initializers, on the other hand, will instantiate an individual or population.

The registration of the tools to the toolbox only associates aliases to the already existing functions and freezes part of their arguments. This allows us to fix an arbitrary amount of argument at certain values so we only have to specify the remaining ones when calling the method. For example, the attr_bool() generator is made from the randint() function that takes two arguments a and b, with a <= n <= b, where n is the returned integer. Here, we fix a = 0 and b = 1.

Our individuals will be generated using the function initRepeat(). Its first argument is a container class, in our example the Individual one we defined in the previous section. This container will be filled using the method attr_bool(), provided as second argument, and will contain 100 integers, as specified using the third argument. When called, the individual() method will thus return an individual initialized with what would be returned by calling the attr_bool() method 100 times. Finally, the population() method uses the same paradigm, but we don’t fix the number of individuals that it should contain.

## The Evaluation Function¶

The evaluation function is pretty simple in our example. We just need to count the number of ones in an individual.

def evalOneMax(individual):
return sum(individual),


The returned value must be an iterable of a length equal to the number of objectives (weights).

## The Genetic Operators¶

Within DEAP there are two ways of using operators. We can either simply call a function from the tools module or register it with its arguments in a toolbox, as we have already seen for our initialization methods. The most convenient way, however, is to register them in the toolbox, because this allows us to easily switch between the operators if desired. The toolbox method is also used when working with the algorithms module. See the One Max Problem: Short Version for an example.

Registering the genetic operators required for the evolution in our One Max problem and their default arguments in the toolbox is done as follows.

toolbox.register("evaluate", evalOneMax)
toolbox.register("mate", tools.cxTwoPoint)
toolbox.register("mutate", tools.mutFlipBit, indpb=0.05)
toolbox.register("select", tools.selTournament, tournsize=3)


The evaluation will be performed by calling the alias evaluate. It is important to not fix its argument in here. We will need it later on to apply the function to each separate individual in our population. The mutation, on the other hand, needs an argument to be fixed (the independent probability of each attribute to be mutated indpb).

## Evolving the Population¶

Once the representation and the genetic operators are chosen, we will define an algorithm combining all the individual parts and performing the evolution of our population until the One Max problem is solved. It is good style in programming to do so within a function, generally named main().

### Creating the Population¶

First of all, we need to actually instantiate our population. But this step is effortlessly done using the population() method we registered in our toolbox earlier on.

def main():
pop = toolbox.population(n=300)


pop will be a list composed of 300 individuals. Since we left the parameter n open during the registration of the population() method in our toolbox, we are free to create populations of arbitrary size.

The next thing to do is to evaluate our brand new population.

    # Evaluate the entire population
fitnesses = list(map(toolbox.evaluate, pop))
for ind, fit in zip(pop, fitnesses):
ind.fitness.values = fit


We map() the evaluation function to every individual and then assign their respective fitness. Note that the order in fitnesses and population is the same.

Before we go on, this is the time to define some constants we will use later on.

    # CXPB  is the probability with which two individuals
#       are crossed
#
# MUTPB is the probability for mutating an individual
CXPB, MUTPB = 0.5, 0.2


### Performing the Evolution¶

The evolution of the population is the final step we have to accomplish. Recall, our individuals consist of 100 integer numbers and we want to evolve our population until we got at least one individual consisting of only 1 and no 0. So all we have to do is to obtain the fitness values of the individuals

    # Extracting all the fitnesses of
fits = [ind.fitness.values[0] for ind in pop]


and evolve our population until one of them reaches 100 or the number of generations reaches 1000.

    # Variable keeping track of the number of generations
g = 0

# Begin the evolution
while max(fits) < 100 and g < 1000:
# A new generation
g = g + 1
print("-- Generation %i --" % g)


The evolution itself will be performed by selecting, mating, and mutating the individuals in our population.

In our simple example of a genetic algorithm, the first step is to select the next generation.

        # Select the next generation individuals
offspring = toolbox.select(pop, len(pop))
# Clone the selected individuals
offspring = list(map(toolbox.clone, offspring))


This will creates an offspring list, which is an exact copy of the selected individuals. The toolbox.clone() method ensure that we don’t use a reference to the individuals but an completely independent instance. This is of utter importance since the genetic operators in toolbox will modify the provided objects in-place.

Next, we will perform both the crossover (mating) and the mutation of the produced children with a certain probability of CXPB and MUTPB. The del statement will invalidate the fitness of the modified offspring.

        # Apply crossover and mutation on the offspring
for child1, child2 in zip(offspring[::2], offspring[1::2]):
if random.random() < CXPB:
toolbox.mate(child1, child2)
del child1.fitness.values
del child2.fitness.values

for mutant in offspring:
if random.random() < MUTPB:
toolbox.mutate(mutant)
del mutant.fitness.values


The crossover (or mating) and mutation operators, provided within DEAP, usually take respectively 2 or 1 individual(s) as input and return 2 or 1 modified individual(s). In addition they modify those individuals within the toolbox container and we do not need to reassign their results.

Since the content of some of our offspring changed during the last step, we now need to re-evaluate their fitnesses. To save time and resources, we just map those offspring which fitnesses were marked invalid.

        # Evaluate the individuals with an invalid fitness
invalid_ind = [ind for ind in offspring if not ind.fitness.valid]
fitnesses = map(toolbox.evaluate, invalid_ind)
for ind, fit in zip(invalid_ind, fitnesses):
ind.fitness.values = fit


And last but not least, we replace the old population by the offspring.

        pop[:] = offspring


To check the performance of the evolution, we will calculate and print the minimal, maximal, and mean values of the fitnesses of all individuals in our population as well as their standard deviations.

        # Gather all the fitnesses in one list and print the stats
fits = [ind.fitness.values[0] for ind in pop]

length = len(pop)
mean = sum(fits) / length
sum2 = sum(x*x for x in fits)
std = abs(sum2 / length - mean**2)**0.5

print("  Min %s" % min(fits))
print("  Max %s" % max(fits))
print("  Avg %s" % mean)
print("  Std %s" % std)


This evolution will now run until at least one of the individuals will be filled with 1 exclusively.

A Statistics object is available within DEAP to facilitate the gathering of the evolution’s statistics. See the One Max Problem: Short Version for an example.

The complete source code of this example: examples/ga/onemax.