Fertilization optimization algorithm on CEC2015 and large scale problems

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INTRODUCTION
During its history, optimization algorithms have been inspired by natural or human-made phenomena to introduce mathematical formulation that can solve problems in different fields of sciences.Specifically, optimization algorithms used to find the maximum or minimum of a function, and they have a wide range of applications in the industry [1] and engineering problems like as robotic [2] and structures [3].Developers are more interested in phenomena that could inspire them to develop a new method that can solve new problems or find the best solutions for the existing ones.One of the inspiration engines is flock of animal, birds, and insects that lead to developing swarm intelligence [4,5] methods; this term can be defined as accumulative and shared knowledge among a group of individuals, and this kind of intelligence cannot be reached by one of them alone.Examples of swarm intelligence Particle Swarm Optimization (PSO) [6], Artificial Bee Colony (ABC) [7], and Grey Wolf Optimization (GWO) [8].Not all the biologically inspired algorithms are swarm intelligence; bacteria and invasive weeds optimization do not follow the rules of a swarm.In this article, a biologically inspired algorithm from the fertilization process in the reproductive tract of mammal animals during reproduction is presented.The new algorithm is called Fertilization Optimization (FO) algorithm.Computationally expensive benchmarks CEC2015 [9] are employed during experiments.On these mathematical optimization problems, FO was compared with other meta heuristics.Remarkably, FO has shown great performance and overcome many other algorithms in many cases.The variety and difficulty of the mathematical optimization problems that FO could pass through successfully have proved the reliability of the fertilization algorithm for mathematical optimization.In brief, the FO algorithms can be described as follows.
Each solution have a position (X) and velocity (v) in the search space.For each iteration, the velocity decreased by some value d v tþ1 ¼ dv t ; 0<d<1; (1) where t is the number of iteration in the optimization process.The solutions move in the search space using levy flight L and the solution is updated by the following equation: where i the index of solution components, and n is the total number of variables in the solution (3).The average value of the best X t first , medium best X t middle , and worst solutions X t end can also have effect on the update solution process: The combination of equations ( 1)À(4) give the search engine of the F algorithm: where m is the number of variables in the proposed solution, and the pseudocode can be seen in Code 1.

RESULTS AND DISCUSSION
CEC2015 benchmark functions, which are described in Tables 1 and 2, are used in this study to examine the performance of the FO algorithm.The run conditions on CEC2015 experiment are: variable dimensions 10, population size 10, maximum number of iterations 1,000, and 20 independent runs.Firstly, FO algorithm is compared with Hybrid Particle Swarm Optimization algorithm and FireFly algorithm (HPSOFF) [10], and Hybrid Firefly and Particle Optimization (HFPO) algorithm [11].Tables 3 and 4 show  the results of comparison on mean solutions and standard deviation among FO, HPSOFF, and HFPSO.Table 5 reveals the comparison on standard deviation results among FO, PSO, FFPSO algorithm [12], and FireFly (FF) algorithm while Table 6 reveals the comparison on mean solutions results among the same algorithms in Table 5.
The FO algorithm is less efficient on high-degree multimodal benchmarks, and this behavior can be seen on the statistical results.The experimental results show that the FO algorithm is more effective on large scale optimization than small scale.The behavior on large and small scale problems needs a dedicated study that can be suggested for a future work.In brief, the FO algorithm can be stable and fast convergent on unimodal optimization problems as well as its efficiency on large scale problems.

Code 1 .
The pseudocode Define problem parameters (No. of variables, objective, limits) Define algorithm parameters (population size, max iteration, velocity reduction coefficient, damping) Initialize random positions and velocities for the population Initialize best cost Repeat from 1 to max iteration Define new solution Repeat from 1 to the number of population Use equation (26) to calculate new position of the new solution Stop when the maximum number of population is reached Merge the old solution with the new solution Sort solutions Choose the first solution in the population Choose the solution in the middle of population Choose the last solution in the population The first solution in the sorted group is the best solution The cost of the best solution is the best cost Update best cost Stop when the maximum number of iterations is reached

Table 3 .
Standard deviation results of the FO algorithm vs.

Table 4 .
Average solutions results of the FO algorithm vs.

Table 6 .
Average solutions results of the FO algorithm vs. PSO, FF, and FFPSO on CEC2015

Table 7 .
Results for large scale optimization on F16 and F17

Table 5 .
Standard deviation results of the FO algorithm vs. PSO, FF, and FFPSO on CEC2015

Table 8 .
Results for large scale optimization on F18and F19