CS420 Project V
Experimentation with Genetic Algorithms
Alexander Saites
11/4/2012
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Introduction
In this project, I implemented a genetic algorithm in Matlab which probabilistically chose individuals
from a population (with a higher probability of choosing more fit individuals) and copied them to serve
as children in the next population. With some probability, called the crossover probability, a crossover
point was selected and the individual genes for the organisms were crossed over in the offspring. For
each bit in the offspring, bits within the offspring were flipped with some probability known as the
mutation probability. The following fitness function was used:
௫ ଵ଴
= ଶ೗ ,
where is the integer interpretation of an individual binary bitstring and is the number of genes
(bits) in the bitstring.
I then wrote a driver program which allowed me to specify quickly the number of genes, the population
size (which is static), the mutation probability, the crossover probability, the number of generations
(before completion of the experiment) and a seed for the random number generator. In addition to
these, the driver program allowed me to specify the number of experiments to run, allowing me to run
several experiments with a given set of parameters and graph them on the same figure. In the following
graphs and analysis, 10 experiments with each set of parameters were performed and are graphed on
top of one another.
Finally, the program displays the individual fitness of each individual in the population for each
generation of the experiment. The fitness of the individual is represented as intensity, with greater
intensity corresponding to higher fitness. Each row is a generation, with earlier generations appearing at
the top of the image.
Graphs and Analysis
I started with the values suggested in the project description: 20 genes, a population size of 30, a
mutation probability of 0.033, a crossover probability of 0.6, and 10 generations. I perfomed this
experiment 10 times. The results are shown below:
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Figure 1: Average fitness converged around 75%
Figure 2: After 10 generations, the most fit individual usually has between 10 and 17 correct bits
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Figure 3: This population shows a gradual transistion from strong lack of fitness to a fairly fit population
As figure 1 shows, the average fitness for the populations tended to increase gradually to about 75%
before stopping. Despite this lower average fitness, the best fitness often reached 1. These populations
also show that, occasionally, the best individual would die out, quickly dropping the average. After these
generations, the best individual usually has about 15 bits (out of twenty correct). Inspecting a typical
individual’s genetic code reveals the following:
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
0
0
1
0
Unsurprisingly, many of this individual’s left-most (higher order) bits are ones. Since setting higher order
bits results in a fitter individual, this result is no surprise. Observing the genetic code of other “best
individuals” after 10 generations show similar results.
Since 10 generations is not very many, I decided to do the same experiment over 50 generations. Here
are the results:
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Figure 4: Same as Figure 1, over 50 generations we see an average fitness of about 75%
Figure 5: After 50 generations, more correct bits are set.
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Figure 6: This population shows a lot of variation in fitness, even in later generations
As seen in the above figures, the best and average fitness results are about the same. The population
quickly converges to an average fitness of about 75%. More interestingly, in figure 5 we see that more
bits are set correctly (an average closer to 16). Again, we see a typical “best individual” has more higher
order bits set:
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
0
0
1
In the extra time provided by the other 40 generations, the algorithm resulted in more lower order bits
being set.
Despite this being better, we still see that many individuals in the population are not that fit (figure 6).
This is likely due to the somewhat high mutation probability. In the next experiment, I set the mutation
probability to 0.0033:
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Figure 7: Dropping the mutation probability results in slower, but better convergence
Figure 8: A lower mutation rate also allows a higher number of correct bits
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Figure 9: With a lower mutation probability, the population is more stable at higher generations
These results show that the mutation probability did indeed have a significant effect on the quality of
the population. Figure 7 shows that the population is able to converge to a higher average fitness
(around 90%). This makes sense intuitively, as once an individual has more bits set than unset (i.e., more
1s than 0s), mutations are more likely to lower the individual’s fitness than increase it. Thus, a lower
mutation rate is more desirable, at least after the population has adapted to the environment (i.e., in
later generations).
Figure 3 shows that the best individuals were able to hold a higher number of set bits. This, again, makes
sense with a lower mutation rate. Finally, we can see in figure 9 that the overall population is much
fitter than in the other experiments. This, too, holds up for the same reasons.
Now having a good mutation probability, I decided to see what effect changing the crossover probability
had on the population. In the following experiments, I dropped the crossover probability to 0.1.
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Figure 10: The population converges more slowly, but to a greater average, after lowering the crossover probability
Figure 11: The number of correct bits in later populations does not change significantly
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Figure 12: The population takes longer to get fitter, but stays fit better
The results are not significantly different from in the previous set of experiments, but there does seem
to be a trend toward a more fit population that takes longer to converge. This makes sense, as the
crossover rate is not likely to have a strong effect, given the nature of the problem. To show this, in the
following two experiments, I increased the crossover rate to 1 (every child performs crossover), then
dropped it to zero (no crossover). The results are not significantly different from the above:
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Figure 13: Best and average with 100% crossover
Figure 14: Total number of correct bits with 100% crossover
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Figure 15: Sample population with 100% crossover
Figure 16: Best and average with 0 crossover
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Figure 17: Correct bits with 0 crossover
Figure 18: Population with 0 crossover
Although 100% crossover results in more correct bits in later populations, the results overall are not
significantly different. Again, this mainly has to do with the nature of the problem (i.e., the fitness
function is not extremely sensitive to single-point crossover).
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For fun, I pumped the crossover probability back to .6 and dropped the mutation probability to 0. Now
the only way for the population to improve is via crossover:
Figure 19: 0 mutation
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Figure 20: 0 mutation
Figure 21: 0 mutation
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In these results, we see the effect crossover really can have. Without mutation, the populations stay
approximately where they started; however, since more fit individuals are more likely to reproduce,
later generations do still get more fit. This allows the average fitness to increase to the best individual’s
fitness. Although crossover does sometimes increase an individual’s fitness, the lack of sensitivity this
problem has still shows in these results.
I did run a set of experiments in which both crossover probability and mutation probability were zero,
but I did not show the results here because they are really boring: the population is as good as the initial
best individual. Breeding just makes the rest of the population genetic copies of this individual. On
occasion, the fittest individual dies before it is able to spread its genes, stagnating the population at
some less fit individual’s fitness.
For a more interesting experiment, I decreased the population size to 10 while using a mutation
probability of 0.0033 and a crossover probability of .6. Here are the results:
Figure 22: With a smaller population, average fitness is more sensitive to small changes
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Figure 23: The number of correct bits in best individuals still sits around the same number
Figure 24: A representative population shows the same sensitivity
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With this smaller population, the individuals are more sensitive to small changes. With fewer individuals
to carry strong genes, the population takes longer to develop better fitness. As a result, the death of a fit
individual can devastate the average population.
We should expect to see the opposite in a large population: with more individuals, there is a greater
probability of individuals being very fit. Thus, the average fitness should increase quickly. Indeed, it
does:
Figure 25: With a large population, there are many good individuals
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Figure 26: It is more likely in this large population for the best individual to have greater number of correct bits
Figure 27: This huge population can probably benefit from a lower mutation rate
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Finally, I changed the population size back to 30 and started playing with the number of genes. First, I
pumped it up to 200 (Matlab can handle this natively – isn’t that great?)
Figure 28: With more genes, the population fitness it noisier
This population shows a lot more noise. This implies that the population size is not as large as it should
be, so I increased it to 50 individuals:
Figure 29: With 50 individuals, the population is able to adjust to such a high number of genes.
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However, the number of correct genes shows an interesting story:
Figure 30: Only about 55% of the total genes are correct
Again, this just shows that higher order bits are much more important that lower order.
Conclusions
Overall, these results show that the fitness function is much more sensitive to higher order bits,
resulting in low sensitivity to crossover probability. Furthermore, the results show that a higher number
of genes necessitates a greater population, as with too small a population does not allow much genetic
information to move before individuals die.