Abstract
This work, presents a novel optimizer called fertilization optimization algorithm, which is based on levy flight and random search within a search space. It is a biologically inspired algorithm by the fertilization of the egg in reproduction of mammals. The performance of the algorithm was compared with other optimization algorithms on CEC2015 time expensive benchmarks and large scale optimization problems. Remarkably, the fertilization optimization algorithm has overcome other optimizers in many cases and the examination and comparison results are encouraging to use the fertilization optimization algorithm in other possible applications.
1 Introduction
During its history, optimization algorithms have been inspired by natural or humanmade 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.
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
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.
CEC2015 expensive benchmark problems F1 to F9
CEC2015  
Type  No.  Description  f _{min} 
Unimodal functions  F1  Rotated Bent Cigar Function  100 
F2  Rotated Discus Function  200  
Simple Multimodal Functions  F3  Shifted and Rotated Weierstrass Function  300 
F4  Shifted and Rotated Schwefel's Function  400  
F5  Shifted and Rotated Katsuura Function  500  
F6  Shifted and Rotated HappyCat Function  600  
F7  Shifted and Rotated HGBat Function  700  
F8  Shifted and Rotated Expanded Griewank's plus Rosenbrock's Function  800  
F9  Shifted and Rotated Expanded Scaffer's F6 Function  900 
CEC2015 expensive benchmark problems F10 to F15
CEC2015  
Type  No.  Description  f _{min} 
Hybrid functions  F10  Hybrid Function 1 (N = 3)  1,000 
F11  Hybrid Function 2 (N = 4)  1,100  
F12  Hybrid Function 3 (N = 5)  1,200  
Composition Functions  F13  Composition Function 1 (N = 5)  1,300 
F14  Composition Function 2 (N = 3)  1,400  
F15  Composition Function 3 (N = 5)  1,500 
Standard deviation results of the FO algorithm vs. HPSOFF and HFPSO on CEC2015
HPSOFF  HFPSO  FO  
F1  3.4292E+07  6.6375E+06  0E+00 
F2  1.2383E+04  1.5696E+04  1.9569E−06 
F3  1.5636E+00  1.4189E+00  7.0195E−02 
F4  3.0718E+02  3.9950E+02  1.8913E+00 
F5  8.2275E−01  5.7466E−01  2.0303E+02 
F6  1.4097E−01  1.4584E−01  7.1663E−10 
F7  9.3694E−01  2.5433E−01  6.1138E+00 
F8  2.8927E+00  4.0866E+00  4.7333E+04 
F9  2.3372E−01  2.6387E−01  4.6656E−13 
F10  2.9730E+05  3.3036E+05  8.8424E+04 
F11  1.9652E+00  2.6814E+00  0E+00 
F12  9.5565E+01  1.0221E+02  2.9857E−01 
F13  2.5959E+01  2.8341E+01  3.3013E+01 
F14  5.0554E+00  5.8221E+00  2.8957E+02 
F15  1.8650E+02  1.0398E+02  6.8567E+00 
Average solutions results of the FO algorithm vs. HPSOFF and HFPSO on CEC2015
HPSOFF  HFPSO  FO  
F1  4.8387E+07  1.3768E+07  7.0974E+07 
F2  3.8331E+04  3.8542E+04  1.1254E+10 
F3  3.0845E+02  3.0671E+02  3.2049E+02 
F4  1.7084E+03  1.3159E+03  4.8685E+02 
F5  5.0273E+02  5.0250E+02  2.1652E+03 
F6  6.0063E+02  6.0054E+02  1.6116E+06 
F7  7.0087E+02  7.0060E+02  7.5666E+02 
F8  8.0740E+02  8.0773E+02  1.6292E+05 
F9  9.0388E+02  9.0393E+02  1.0413E+03 
F10  3.5402E+05  3.3099E+05  6.8481E+04 
F11  1.1067E+03  1.1074E+03  1.4195E+03 
F12  1.4517E+03  1.3983E+03  1.3391E+03 
F13  1.6333E+03  1.6452E+03  1.3908E+03 
F14  1.6053E+03  1.6021E+03  1.5594E+04 
F15  1.8365E+03  1.9233E+03  2.0528E+03 
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.
Standard deviation results of the FO algorithm vs. PSO, FF, and FFPSO on CEC2015
PSO  FF  FFPSO  FO  
F1  1.3549E+08  2.8945E+08  4.9786E+09  0E+00 
F2  1.5114E+04  9.7404E+03  4.6261E+08  1.9569E−06 
F3  1.3259E+00  1.2487E+00  1.6124E+00  7.0195E−02 
F4  3.5521E+02  3.2112E+02  2.6203E+02  1.8913E+00 
F5  6.3611E−01  5.9796E−01  9.3430E−01  2.0303E+02 
F6  2.8490E−01  5.8361E−01  1.3580E+00  7.1663E−10 
F7  1.8947E+00  5.8077E+00  3.3329E+01  6.1138E+00 
F8  2.7690E+01  1.7256E+02  2.7423E+05  4.7333E+04 
F9  3.2749E−01  2.3842E−01  1.7883E−01  4.6656E−13 
F10  1.9786E+05  6.5054E+05  7.7896E+07  8.8424E+04 
F11  2.9153E+00  2.4020E+00  6.7179E+01  0E+00 
F12  1.1574E+02  9.1615E+01  4.3894E+02  2.9857E−01 
F13  1.9141E+01  2.9519E+01  9.8971E+02  3.3013E+01 
F14  4.5254E+00  3.5980E+00  4.2292E+01  2.8957E+02 
F15  1.4570E+02  7.4514E+01  1.0567E+02  6.8567E+00 
Average solutions results of the FO algorithm vs. PSO, FF, and FFPSO on CEC2015
PSO  FF  FFPSO  FO  
F1  2.4553E+08  4.3059E+08  1.6287E+10  7.0974E+07 
F2  3.8112E+04  3.3304E+04  1.4957E+08  1.1254E+10 
F3  3.0779E+02  3.0773E+02  3.1455E+02  3.2049E+02 
F4  2.2534E+03  1.5473E+03  3.1120E+03  4.8685E+02 
F5  5.0277E+02  5.0293E+02  5.0350E+02  2.1652E+03 
F6  6.0089E+02  6.0092E+02  6.0673E+02  1.6116E+06 
F7  7.0193E+02  7.0586E+02  8.0586E+02  7.5666E+02 
F8  8.1583E+02  8.6344E+02  2.7632E+05  1.6292E+05 
F9  9.0391E+02  9.0395E+02  9.0451E+02  1.0413E+03 
F10  2.9540E+05  5.3162E+05  5.1186E+07  6.8481E+04 
F11  1.1088E+03  1.1080E+03  1.2198E+03  1.4195E+03 
F12  1.4620E+03  1.3995E+03  2.1953E+03  1.3391E+03 
F13  1.6415E+03  1.6437E+03  3.0005E+03  1.3908E+03 
F14  1.6076E+03  1.6111E+03  1.6770E+03  1.5594E+04 
F15  1.9149E+03  1.9269E+03  2.1840E+03  2.0528E+03 
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 TreeSeed 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:
$f(x)=10n+{\displaystyle \sum}_{i=1}^{n}\left[{x}_{i}^{2}10\text{\hspace{0.17em}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right]$ , Range = [−5.12, 5.12], F_{min} = 0, 
F17:
$f(x)={\displaystyle \sum}_{i=1}^{n1}\left[100{\left({x}_{i+1}{x}_{i}^{2}\right)}^{2}+{\left(1{x}_{i}\right)}^{2}\right]$ , Range = [−2.048, 2.048], F_{min} = 0, 
F18:
$\begin{array}{c}f(x)=a\text{\hspace{0.17em}}\text{exp}\left(b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^{d}{x}_{i}^{2}}}\right)\\ \text{exp}(b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^{d}\mathrm{cos}(c{x}_{i})}})+a+\mathrm{exp}(1)\text{,}\end{array}$ Range = [−32.768, 32.768], F_{min} = 0, 
F19:
$f(x)={\displaystyle \sum _{i=1}^{d}\frac{{\text{x}}_{i}^{2}}{4000}}{\displaystyle \prod}_{i=1}^{d}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ , Range = [−600, 600], F_{min} = 0.
Results for large scale optimization on F16 and F17
Function  F16  F17  
ALO  Best  1.2463E+04  9.6145E+04 
Worst  1.5088E+04  2.6251E+05  
Mean  1.3352E+04  1.2072E+05  
STD  5.7343E+02  2.5922E+04  
BOA  Best  0.0000E+00  9.9873E+02 
Worst  1.1460E−10  9.9894E+02  
Mean  3.3526E−12  9.9885E+02  
STD  1.5967E−11  4.6350E−02  
GWO  Best  6.3429E+03  3.3616E+03 
Worst  7.4800E+03  7.2573E+03  
Mean  6.9049E+03  4.8800E+03  
STD  2.6947E+02  8.9692E+02  
PSO  Best  1.4774E+04  4.2109E+05 
Worst  1.7302E+04  4.6887E+05  
Mean  1.6192E+04  4.5065E+05  
STD  6.4158E+02  1.0330E+04  
SCA  Best  5.1110E+02  1.2307E+05 
Worst  4.6834E+03  3.0827E+05  
Mean  1.7431E+03  2.3237E+05  
STD  9.1819E+02  3.8524E+04  
DDAO  Best  2.2573E−01  9.9897E+02 
Worst  6.4277E+03  1.4620E+03  
Mean  5.7423E+02  1.0382E+03  
STD  1.0834E+03  8.9853E+01  
BA  Best  1.2667E+04  5.4067E+04 
Worst  1.7947E+04  4.3425E+05  
Mean  1.4585E+04  1.8689E+05  
STD  1.3142E+03  7.7892E+04  
TSA  Best  6.3653E+03  1.2015E+04 
Worst  1.4767E+04  5.2552E+04  
Mean  1.0217E+04  2.7948E+04  
STD  2.1618E+03  9.2913E+03  
FO  Best  0.0000E+00  9.9890E+02 
Worst  0.0000E+00  9.9899E+02  
Mean  0.0000E+00  9.9896E+02  
STD  0.0000E+00  2.2892E−02 
Results for large scale optimization on F18and F19
Function  F18  F19  
ALO  Best  1.9469E+01  9.8149E+03 
Worst  2.0307E+01  1.4948E+04  
Mean  1.9750E+01  1.1345E+04  
STD  2.4107E−01  1.4212E+03  
BOA  Best  1.2957E−08  2.0416E−06 
Worst  6.1917E−07  5.9821E−04  
Mean  5.9824E−08  5.5767E−05  
STD  9.7536E−08  1.2535E−04  
GWO  Best  9.1437E+00  6.8521E+02 
Worst  1.2941E+01  1.4847E+03  
Mean  1.0128E+01  9.7278E+02  
STD  9.8562E−01  1.8400E+02  
PSO  Best  1.7001E+01  8.1418E+02 
Worst  1.7812E+01  1.0013E+03  
Mean  1.7356E+01  9.2068E+02  
STD  1.6680E−01  4.4017E+01  
SCA  Best  7.0434E+00  2.0794E+03 
Worst  1.9671E+01  1.2949E+04  
Mean  1.6601E+01  8.3566E+03  
STD  3.2924E+00  2.5246E+03  
DDAO  Best  2.4380E−02  1.3322E+00 
Worst  9.5690E+00  6.0214E+02  
Mean  3.3104E+00  8.4837E+01  
STD  2.1546E+00  1.5768E+02  
BA  Best  1.9362E+01  1.0246E+04 
Worst  2.1148E+01  2.8739E+04  
Mean  2.0312E+01  1.7242E+04  
STD  4.7706E−01  4.9752E+03  
TSA  Best  6.4128E+00  3.1366E+02 
Worst  1.2601E+01  3.3672E+03  
Mean  9.0431E+00  1.0870E+03  
STD  1.5954E+00  5.1949E+02  
FO  Best  8.9617E−13  0.0000E+00 
Worst  2.4986E−07  7.2283E−07  
Mean  5.0366E−09  1.4257E−08  
STD  3.4971E−08  1.0120E−07 
The FO algorithm is less efficient on highdegree 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 goahead to the global point.
Acknowledgments
The research was supported by the Hungarian National Research Development and Innovation OfficeNKFIH under the project number K 134358.
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