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# Crossover And Mutation In Genetic Algorithm Pdf

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- Dynamic Crossover and Mutation Genetic Algorithm Based on Expansion Sampling
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- Introduction to Genetic Algorithms — Including Example Code
- Introduction to Genetic Algorithms — Including Example Code

*Sign in. This algorithm reflects the process of natural selection where the fittest individuals are selected for reproduction in order to produce offspring of the next generation. The process of natural selection starts with the selection of fittest individuals from a population.*

Osaba, R. Carballedo, F. Diaz, E. Onieva, I. Since their first formulation, genetic algorithms GAs have been one of the most widely used techniques to solve combinatorial optimization problems. The basic structure of the GAs is known by the scientific community, and thanks to their easy application and good performance, GAs are the focus of a lot of research works annually. Although throughout history there have been many studies analyzing various concepts of GAs, in the literature there are few studies that analyze objectively the influence of using blind crossover operators for combinatorial optimization problems.

For this reason, in this paper a deep study on the influence of using them is conducted. The study is based on a comparison of nine techniques applied to four well-known combinatorial optimization problems. Six of the techniques are GAs with different configurations, and the remaining three are evolutionary algorithms that focus exclusively on the mutation process.

Finally, to perform a reliable comparison of these results, a statistical study of them is made, performing the normal distribution -test. Genetic algorithms GAs are one of the most successful metaheuristics for solving combinatorial optimization problems. Thanks to their easy application and good performance, GAs have been used to solve many complex problems framed in various fields, as, for example, transport [ 1 , 2 ], software engineering [ 3 , 4 ], or industry [ 5 , 6 ].

GAs were proposed in by Holland [ 7 ], in an attempt to imitate the genetic process of living organisms and the law of the evolution of species. Anyway, their practical use to solve complex optimization problems was shown later, by Goldberg [ 8 ] and De Jong [ 9 ].

Throughout history, many researches have focused on the study of genetic algorithms. These studies can be grouped into 3 different categories. These studies focused on the application of GAs for solving specific problems.

Among these three categories, this is the most common in the literature. Two subcategories can be identified in this first group of works: variations of a classic GA [ 10 — 12 ] or hybridization of a GA with some other technique [ 13 — 15 ]. These researches present new specific operators, such as crossover [ 16 , 17 ] or mutation functions [ 18 , 19 ].

Normally, these operators are heuristic, and they are applied to a particular problem, in which they get a great performance. These works focus on the theoretical and practical analysis of GAs. This kind of research analyzes, for example, behavioural characteristics of the algorithm, as the convergence [ 20 ], or the efficiency of certain phases of the algorithm, such as crossover [ 21 , 22 ] or mutation phases [ 23 , 24 ], or the influence of adapting some parameters, as the crossover and mutation probability [ 25 — 27 ].

These works attempt to overcome the drawbacks of traditional genetic algorithms and are the source of new problem-solving techniques, such as the adaptive genetic algorithms [ 28 , 29 ] or the parallel genetic algorithms [ 30 , 31 ].

In this paper a deep study on the influence of using blind crossover operators in GAs for solving combinatorial optimization problems is conducted. This study is developed by means of a comparison between GAs with this kind of operators and EAs based only on mutation operators. Thus, this work could be framed into the third category. Previously, other studies in the literature have had a similar purpose, for example [ 32 ], where the authors tried to validate the hypothesis that the crossover phase of genetic algorithms is not efficient when it is applied to routing problems.

In that work, the authors develop several versions of the basic GA with some blind crossover operators e. Performances of these GAs are compared with the one of an evolutionary algorithm EA based solely on mutations.

The comparison is based on the quality of the solution and the runtime. Furthermore, the comparison also takes into account the percentage of deviation from the average values of each parameter. On the other hand, in [ 22 ] the efficiency of six different versions of the classic GA applied to the degree constrained minimal spanning tree problem [ 36 ] is compared.

Each version has its own crossover function. In that work, the only data shown for each version of the GA is the average value of the results obtained, so, the comparison is performed based only on this criterion. Moreover, the authors do not perform the comparison of the results obtained by a conventional GA and an EA. For this reason, with this study it is not possible to quantify the real influence of the crossover phase in the optimization capacity of a GA.

Together with the above studies, in the literature there are many others that are not comparable with the study presented in this paper. The main reason is that they are focused on other types of problems [ 21 ] or because they analyzed only the crossover process of a traditional GA [ 37 — 39 ]. The motivation of this work stems from the absence in the literature of a study that proves objectively the efficiency of using blind crossover operators in GAs for combinatorial optimization problems.

Although [ 32 ] focuses on this topic, it is only applicable to routing problems, and it is only tested with one problem, the TSP. In addition, the comparison of the results done in [ 32 ] is not as deep as the one made in the present work. On the other hand, as it has been mentioned, the study presented in the abovementioned [ 22 ] is not truly conclusive to prove the real influence of the crossover process in a GA.

Therefore, the goal of this paper is to perform an objective study on the efficiency of blind crossover operators in basic GAs with respect to blind mutation operators in basic EAs. In order to reach this goal, an exhaustive comparison between different versions of genetic and evolutionary algorithms is presented. This comparison includes the following criteria: quality of the results, runtime, and convergence behavior of each of the techniques reviewed.

Furthermore, to perform a reliable comparison of these results a statistical study is made. For this purpose, the normal distribution -test is performed. The rest of the paper is structured as follows. In Section 2 the description of the experimentation is presented.

In Section 3 , the tests for the TSP are shown. Finally, the work is finished with the conclusions of the study and further work Section 7. In this section a description of the experimentation is made. First, in Section 2. Then, in Section 2. Finally, in Section 2. For this study four different combinatorial problems have been used.

In addition, to verify that the results of this study are valid for other types of problems apart from the routing ones, two constraint satisfaction problems have also been used in the experimentation, the NQP and the BPP. These problems were chosen because they are well known, and easy to implement. In addition, they are easily replicable.

In this way, any researcher can perform these same tests, either to check the results or to perform them with other crossover functions or different parameters.

The first problem used is the TSP. The TSP is one of the most famous and widely studied problems throughout history in operations research and computer science. It has a great scientific interest, and it is used in a large number of studies [ 42 — 44 ].

This problem can be defined on a complete graph where is the set of vertexes which represents the nodes of the system, and is the set of arcs which represents the interconnection between nodes. Each arc has an associated distance cost. The objective of the TSP is to find a route that visits every customer once and only once , that is, a Hamiltonian cycle in the graph , and that minimizes the total distance traveled.

In a formal way, the TSP can be formulated as follows [ 45 ]: where in 2 a binary variable is 1 if the arc is used in the solution. Furthermore, is the set of nodes of the system and is the distance between the nodes and. The objective function, 1 , is the sum of all the arcs in the solution used; that is, it is the total distance of the route. Constraints 3 and 4 indicate that each node have to be visited and abandoned only once, while the formula 5 guarantees the absence of subtours and indicates that any subset of nodes has to be abandoned at least 1 time.

This restriction is vital, because it avoids the presence of cycles. Finally, all the solutions are encoded following the path representation [ 46 ]. In this way, each individual is encoded by a permutation of numbers, which represents the path. Figure 1 a represents a possible 9-node instance of the TSP, and Figure 1 b represents a possible solution. This solution would be encoded as , and its fitness would be. The second selected problem is the CVRP.

Due to its complexity and, above all, its applicability to real life, the CVRP is also used in many researches every year [ 47 , 48 ]. For the TSP, this problem can be defined on a complete graph. In addition, the vertex represents the depot, and the rest are the customers, each of them with a demand. A fleet of vehicles is available with a limited capacity for each vehicle. The objective of the CVRP is to find a number of routes with a minimum cost such that i each route starts and ends at the depot, ii each client is visited exactly by one route, and iii the total demand of the customers visited by one route does not exceed the total capacity of the vehicle that performs it [ 49 ].

This problem could be formulated as follows [ 40 ]:. The formula 6 is the objective function, which is the total distance traveled by all the routes. The variable 11 is a binary variable which is 1 if the vehicle satisfies the demand of the client , and 0 otherwise.

The binary variable 12 is 1 if the arc is used in the solution. Formulas 8 and 9 ensure that every customer is visited by one route only and exactly once.

Finally, clause 9 serves to eliminate subtours, where is the number of customers and the minimum number of vehicles to serve all. Finally, the restriction 10 ensures that the sum of all the demands of a route does not exceed the maximum vehicle capacity.

In the case of CVRP, the path representation is also used for the individuals encoding [ 50 ]. In this case, the routes are also represented as a permutation of nodes. To distinguish the routes of one solution, they are separated by zeros. On the other hand, in Figure 2 b a solution composed by three different routes is depicted. On this occasion, this solution would be encoded as , and its fitness would be.

The third problem is the NQP. This problem is a generalization of the problem of putting eight nonattacking queens on a chessboard [ 51 ], which was introduced by Bezzel in [ 52 ]. The NQP consists of placing queens on a chess board, in order that they cannot attack each other; that is, on every row, column, and diagonal, only one queen can be placed.

This problem is a classical combinatorial design problem constraint satisfaction problem , which can also be formulated as a combinatorial optimization problem [ 53 ].

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In genetic algorithms and evolutionary computation , crossover , also called recombination , is a genetic operator used to combine the genetic information of two parents to generate new offspring. It is one way to stochastically generate new solutions from an existing population, and is analogous to the crossover that happens during sexual reproduction in biology. Solutions can also be generated by cloning an existing solution, which is analogous to asexual reproduction. Newly generated solutions are typically mutated before being added to the population. Different algorithms in evolutionary computation may use different data structures to store genetic information, and each genetic representation can be recombined with different crossover operators.

Osaba, R. Carballedo, F. Diaz, E.

Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Adaptive probabilities of crossover and mutation in genetic algorithms Abstract: In this paper we describe an efficient approach for multimodal function optimization using genetic algorithms GAs. We recommend the use of adaptive probabilities of crossover and mutation to realize the twin goals of maintaining diversity in the population and sustaining the, convergence capacity of the GA.

Items in Shodhganga are protected by copyright, with all rights reserved, unless otherwise indicated. Shodhganga Mirror Site. Show full item record. Operations Research models help to solve optimization problems. Nonetheless, these newlinemodels are suited when the pattern can be seen by the person newlinemodeling.

In computer science and operations research , a genetic algorithm GA is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms EA. Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems by relying on biologically inspired operators such as mutation , crossover and selection. In a genetic algorithm, a population of candidate solutions called individuals, creatures, or phenotypes to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties its chromosomes or genotype which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually starts from a population of randomly generated individuals, and is an iterative process , with the population in each iteration called a generation.

The traditional genetic algorithm gets in local optimum easily, and its convergence rate is not satisfactory. So this paper proposed an improvement, using dynamic cross and mutation rate cooperate with expansion sampling to solve these two problems. The expansion sampling means the new individuals must compete with the old generation when create new generation, as a result, the excellent half ones are selected into the next generation. Whereafter several experiments were performed to compare the proposed method with some other improvements. The results are satisfactory. The experiment results show that the proposed method is better than other improvements at both precision and convergence rate. Unable to display preview.

Sign in. This algorithm reflects the process of natural selection where the fittest individuals are selected for reproduction in order to produce offspring of the next generation.

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PDF | Genetic algorithms (GA) are stimulated by population genetics and evolution at the population level where crossover and mutation comes.

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