This image is a graphicalrepresentation of the search space.

We can observe in the image the severallocal minima and the global one. The number of structures in this type ofsearch space is usually between one and thirty.This function will focus on thecrossover operator and his effects on the performance of the genetic algorithmbecause four different types of crossover: single point crossover, two pointcrossover, even odd crossover and uniform crossover, can produce a differentperformance when applied to our population.The single point crossover means thatonly one point is selected for crossover, this means that the binary stringfrom the beginning of the chromosome is copied from one parent until it reachesthe crossover point and the rest from the second parent. An example of singlepoint crossover:11001011+ 11011111 = 11001111The two-point crossover will becopying the first and ending part from he first parent and from the secondparent the middle part of the chromosome. Below we have an example of two-pointcrossover.11001011 + 11011111 = 11011111Uniform crossover it means that bitsfrom the parents are copied randomly.

11001011 + 11011101 = 11011111The even odd crossover happens whenthe genes combined are the even numbered locations from the first parent to theodd numbered locations from the second parent.These operators will be applied to ourpopulation in the beginning to see their effects and after to a biggerpopulation to compare how the size of the population affects this operator.Results andobservations:Using a single point crossover on apopulation of 100 and 100 generations, the generation will converge at the 20generation with a 99% percentage. There will be only one convergence, after10000 selections, 5040 crossovers since the initialization. The maximum score was -6.11 and the minimumwas -2.

10.The two-point crossover gave the exactsame results, except for the number of crossovers which this time was 5012,which mean that two-point crossover in this case was more efficient then thesingle point crossover because it pulled the population towards the globalminimum with a smaller number of crossover since the initialization.The even odd and uniform crossoverperformed the exact same number of crossover (4956) which was less than the two-pointor the single point crossover, which makes both more efficient than the firsttwo, but equal when compared to each other. This means that when the crossoverhas more points to reach, will perform less number of crossovers on the samenumber of population than the crossover with less points to reach, making themless efficient and slower. After these results were obtained, the populationsize was changed to 1000, to see how the number of crossovers will be affected,if they still perform less crossovers if a multi-point crossover is used.

Wewill see the same result, if the population size increases the single pointcrossover will be slower and the multi-point crossovers will still performfaster than the single-point crossover.Replication ofresults:These results can be replicated usinga computer with a Intel Core i7-6700HQ processor, 16 GB of RAM, but should workas well with 8GB of RAM and i5 model of the processor. On the machine theoperating system installed is Windows 10 Pro 64-bit and for the geneticalgorithm code, the Microsoft Visual C++ project was opened and experimentedwith using Visual Studio 2017, the community edition.The algorithm should produce somethinglike this in 5 to 20 seconds depending on the population size and number of generations. Fortunately, no errors or problemsoccurred, and the program worked properly.Conclusion:In this paper four crossover operatorswere compared to see their effect on the performance of the genetic algorithm.

We analysed which crossover operator was the most efficient and if it changedanything else besides on how fast the new population will reach the globalminimum.We observed that nothing else changesbesides the number of crossover performed since the initialization. The exactsame fitness score was produced in all the cases, this means the fitness scoreis not affected by the crossover operator in the initial population. And thisis what happens for the rest of the genetic algorithm operators.

The number of mutations remains thesame through all the generations, the same observation we make for the numberof replacements, number of genome evaluation and number of populationevaluation.The single point crossover is the onethat performs the most numbers of crossovers, this global minimum is reachedslower using this type of crossover.The two-point crossover performed lesscrossovers, this mean it will reach faster the global minimum.The even odd crossover was even fasterthan the two-point, making it more efficient than the first two types ofcrossover. And the uniform crossover had the same efficiency as the even oddcrossover, which makes them equally efficient.The application of these four types ofcrossover to the genetic algorithm makes evident which type of crossover isbetter towards obtaining a better solution to the problem.In conclusion, the multi pointcrossover type is more efficient and faster, the solution will be found in lesstime if the multi-point crossover is used instead of the single pointcrossover.

The same conclusion was reached byother scientists, they compered the single point, two-point, three points,four-point crossover. Wu and Chowcompared the one-point, two-point, three-point, and four-point crossoveroperators and showed that two-point, three-point, and four-point crossoveroperators are better than the one-point crossover (Kaya, 2010). Jenkins (Jenkins, 1991), argues in favour ofmulti-point crossover operator in term of fast progress becomes very slow incase single-point crossover is used (Kaya, 2010).Also, a relationship between the sizeof the population and the crossover was found, the two-point crossover willperform better when the population is large, and the single point crossoverwill perform better on small populations. The uniform and even odd crossoverperforms better then the other two in a small population or a large one. Thisrelationship is clearly visible when experimenting with different sizes of thepopulation.The performance of the geneticalgorithm it is affected by the crossover because as shown above themulti-point crossover is faster, which will make the algorithm show a bettersolution in less time.Future Work:This study can be further developed bycombining the different types of crossovers on the behaviour of the geneticalgorithm to see if the mixed crossover operator can further improve the performanceof the genetic algorithm.

Also, there is no guaranty that the relationshipbetween the crossover and the population size will remain the same or it willchange, when using a mixed crossover operator.