1 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.

2 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.

Another experiment has been done to compare the performance of the FO algorithm on large scale optimization problems against Ant Lion Optimizer ALO [13], Butterfly Optimization Algorithm (BOA) [14], GWO [7], PSO, Sine Cosine Algorithm (SCA) optimization [15], Dynamic Differential Annealed Optimization (DDAO) [16], Bat Algorithm (BA) [17], and Tree-Seed Algorithm (TSA) [18]. Tables 7 and 8 illustrate the statistical results for this test in terms of best solution (Best), worst solution (Worst), mean solution (Mean), and STandard Deviation (STD). Four large scale optimization problems are chosen in these experiments, and the run conditions are: variable dimensions 1,000, population size 25, number of iterations 100, and 51 independent runs. The description and formulation of the large scale problems can be written as follows:

• F16: Rastrigin: $f(x)=10n+{\displaystyle \sum }_{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)\right]$ , Range = [−5.12, 5.12], F_min = 0,

• F17: $f(x)={\displaystyle \sum }_{i=1}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(1-x_i\right)}^2\right]$ , Range = [−2.048, 2.048], F_min = 0,

• F18: $\begin{array}{c}f(x)=-a\text{exp}\left(-b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^d{x}_i^2}}\right)-\\ -\text{exp}(-b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^d\cos (c{x}_i)}})+a+\exp (1)\text{,}\end{array}$ Range = [−32.768, 32.768], F_min = 0,

• F19: $f(x)={\displaystyle \sum _{i=1}^d\frac{{\text{x}}_i^2}{4000}}-{\displaystyle \prod }_{i=1}^d\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$ , Range = [−600, 600], F_min = 0.

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.

3 Conclusion

The fertilization optimization algorithm is a powerful biologically inspired algorithm developed for mathematical optimization problems. It mimics the interaction between sperms and uterus in the process of fertilization the egg. The statistical results on 19 test functions; CEC2015 time expensive benchmarks, unimodal, multimodal, small scale, and large scale problems have shown the efficiency of the proposed algorithm compared with many optimization algorithms. During examinations of the FO algorithm, it has been noticed that the performance of the FO algorithm on large scale problems is better than its performance on small scale problems. The statistical results illustrate that FO algorithm is stable with less STD and best solutions than other eight competitive. The FO algorithm has proven its powerful on unimodal functions and it has promising applications on continuous differentiable objective functions and large scale optimization. The FO algorithm is fast and simple and can efficiently skip local points in the search space and go-ahead to the global point.