An Improvement of Genetic Algorithm based on Dynamic Operators Rates Controlled by the Population Performance Beatriz Flamia Azevedo1;3 a, Ana I. Pereira1;2 b and Glaucia Maria Bressan3 c 1Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Politecnico´ de Braganc¸a, Braganc¸a, Portugal 2Algoritmi Research Centre, University of Minho, Campus de Gualtar, Braga, Portugal 3Federal University of Technology - Parana,´ Cornelio´ Procopio,´ Parana,´ Brazil [email protected], [email protected], [email protected] Keywords: Genetic Algorithm, Genetic Operators, Dynamic Rates, Hybrid Approach. Abstract: This work presents a hybrid approach of genetic algorithm with dynamic operators rates that adapt to the phases of the evolutionary process. The operator’s rates are controlled by the amplitude variation and standard deviation of the objective function. Besides, a new stopping criterion is presented to be used in conjunction with the proposed algorithm. The developed approach is tested with six optimization benchmark functions from the literature. The results are compared to the genetic algorithm with constant rates in terms of the number of function evaluations, the number of iterations, execution time and optimum solution analysis. 1 INTRODUCTION The several applications of GA led to numerous computational implementations and algorithm varia- Optimization is a mathematics field that studies the tions. Many works propose strategies to improve the identification of functions’ extreme points, either GA and consequently the optimization problem solu- maximal or minimal. In the last decade, the use of tion. However, in most works, these improvements optimization methods has become an essential tool are restricted to specific applications that cannot be for management, decision making, as well as improv- extended to optimization problems in general. ing and developing technologies as it allows gaining Knowing that GA performance depends on the op- competitive advantages (Mitchell, 1998; Haupt and timization problem, this work consists of exploring Haupt, 2004). strategies to automatically adapt the GA to the op- Inspired by Darwin natural selection theory, op- timization problem and proposes a variation of the timization techniques had considerable progress in Genetic Algorithm to be used in general optimization the area of population algorithms. J. H. Holland problems. (Holland, 1992) and his collaborators tightly studied The traditional version of GA uses constant val- the natural optimization mechanisms and mathemati- ues in the genetic operator’s rates for the evolution- cally formalized the natural process of evolution and ary process. In this paper is presented a dynamic GA adaptation of living beings. These researchers devel- that considers three phases, where different operators oped artificial systems inspired in natural optimiza- rates are used and they are dynamically controlled by tion mechanisms (Mitchell, 1998), that can be used the amplitude and standard deviation of the objective to solve real optimization problems that arise from function. Besides this, a new stopping criterion is pro- industrial fields. Genetic Algorithms (GA) are the posed that depends on the algorithm behavior in the most famous example of this methodology and they last phase. This approach is tested by six optimiza- are used in wide fields, such as image processing, tion test function and the results are compared with pattern recognition, financial analysis, industrial opti- the Genetic Algorithm with constant rates. mization, etc (Ghaheri et al., 2015; Haupt and Haupt, 2004; Xu et al., 2018). This paper is organized as follows: in Section 2, GA concepts are introduced, highlighting the behav- a https://orcid.org/0000-0002-8527-7409 ior of the genetic operators. In Section 3 some stud- b https://orcid.org/0000-0003-3803-2043 ies and variations of GA are presented. In Section 4 c https://orcid.org/0000-0001-6996-3129 the dynamic GA proposed in this work is described, while the numerical results are presented in Section 5. Algorithm 1 : Hybrid Genetic Algorithm with Constant Finally, the conclusion and future work proposed are Rates Operators. presented in Section 6. Generates a randomly population of individuals, 0 P , with dimension Npop. Set k = 0. 2 GENETIC ALGORITHM Set the operators rates. while stopping criterion is not met do Set k = k + 1. The Genetic Algorithm is composed of a set of in- 0 P = Apply selection procedure in Npop. dividuals, usually named as chromosomes, that are 00 P = Apply crossover procedure in Nkeep. considered solutions for the optimization problem. 000 This set of individuals is known as the population, P = Apply mutation procedure in Nkeep. k+1 k 00 and they have a fixed number of individuals in each P = Npop best individuals of fP [ P [ 000 generation (iteration). The population is represented P g. by Npop individuals distributed in the feasible region, Apply the local search method in the best solution which is the space where each variable can have val- obtained by genetic algorithm. ues (Sivanandam and Deepa, 2008). Thereby, the basic idea of GA is to create an initial 0 2000; Sivanandam and Deepa, 2008). This procedure population P of feasible solutions, to evaluate each is based on the survival probability that depends on individual using the objective function and to select the objective function value for each individual. some individuals to define the optimum subset of in- The crossover operator is used to create new in- dividuals Nkeep, and to modify them by the crossover dividuals from surviving individuals selected through and mutation operators, in order to create new indi- the selection operator. The crossover procedure is re- viduals (offspring). sponsible for recombining the individuals characteris- The value provided by the objective function de- tic during the reproduction process, with this process fines how adapted an individual is to solve the opti- it is possible for the offspring to inherit characteristics mization problem. The most adapted individuals have of previous generations (Pham and Karaboga, 2000; a greater chance of surviving for the next generation, Sivanandam and Deepa, 2008). while the less adapted are eliminated; similar to what The mutation operator is responsible for diversify- is proposed by Darwin’s theory. This is an iterative ing the existing population allowing the search for the process in which at the end of each iteration all in- solution in promising areas and avoiding premature dividuals are evaluated and ordered according to the convergence in local points. This process helps the al- objective function value until a stopping criterion be gorithm to escape from local optimum points because achieved (Haupt and Haupt, 2004; Sivanandam and it slightly modifies the search direction and introduces Deepa, 2008). new genetic structures in the population (Pham and Some genetic algorithms variants are combined Karaboga, 2000; Sivanandam and Deepa, 2008). with a local search method to present the global so- Each genetic operator has a specific rate that de- lution with better precision. These algorithms are termined how many individuals will be used and gen- considered hybrid genetic algorithms. In this work, erated in each genetic procedure. In the traditional the Nelder Mead method (Nelder and Mead, 1965) is GA the rates are constant values for all evolutionary used to improve the solution precision. The Hybrid process. However, there is not a consensus in the lit- Genetic Algorithm is represented in the Algorithm 1. erature about which value should be used on each op- A Genetic Algorithm depends substantially on the erator. On the other hand, many works in literature efficiency of genetic operators. These operators are assert the intuitive idea that crossover and mutation responsible for new individuals creation, diversifi- rates should not be constant throughout the evolution- cation and maintenance of adaptation characteristics ary process (Vannucci and Colla, 2015), but should acquired in previous generations (Sivanandam and rather vary in the different phases of the search. Once Deepa, 2008). The most useful operators are the se- again there is no consensus on the values of the rates lection, crossover and mutation which are used in this and how to calculated them. For these reasons the de- work and described below. termination of operators rates is normally defined by The selection operator selects elements of the cur- individual problem analysis or they are calculated by rent population in order to produce more individuals means of trial-and-error (Lin et al., 2003). (offspring). It has the mission to guide the algorithm for promising areas, where the probability to find the optimum solution is higher (Pham and Karaboga, 3 RELATED WORKS approach four parameters are defined to control the crossover operator: crossover probability, variable- Several versions of genetic operators are described in wise crossover probability, multiplying factor, direc- the literature and many studies have been done to im- tional probability. It was noted the use of the direc- prove these operators’ performance, see for example tional information helps the algorithm to search in (Jano´ sˇ´ıkova´ et al., 2017; Xu et al., 2018; Das and more potential regions of the variable space. Pratihar, 2019). The problem of operator rates calcu- In (Whitley and Hanson, 1989) is proposed an lation is considered a challenging problem in the liter- adaptive mutation through monitoring