Metaheuristic optimization algorithms present an effective method for solving several optimization problems from various types of applications and fields. Several metaheuristics and evolutionary optimization algorithms have been emerged recently in the literature and gained widespread attention, such as particle swarm optimization (PSO), whale optimization algorithm (WOA), grey wolf optimization algorithm (GWO), genetic algorithm (GA), and gravitational search algorithm (GSA). According to the literature, no one metaheuristic optimization algorithm can handle all present optimization problems. Hence novel optimization methodologies are still needed. The Al-Biruni earth radius (BER) search optimization algorithm is proposed in this paper. The proposed algorithm was motivated by the behavior of swarm members in achieving their global goals. The search space around local solutions to be explored is determined by Al-Biruni earth radius calculation method. A comparative analysis with existing state-of-the-art optimization algorithms corroborated the findings of BER’s validation and testing against seven mathematical optimization problems. The results show that BER can both explore and avoid local optima. BER has also been tested on an engineering design optimization problem. The results reveal that, in terms of performance and capability, BER outperforms the performance of state-of-the-art metaheuristic optimization algorithms.

Metaheuristic and evolutionary optimization algorithms are currently used to solve a variety of problems in a variety of fields, including machine learning [

The simplicity and flexibility of optimization algorithms based on metaheuristics make these algorithms widespread more than the traditional and accurate optimization algorithms such as Local Search and Greedy Search [

Metaheuristics are broad techniques for resolving challenging combinatorial problems. In general, computer scientists have complex time-solving metaheuristic algorithms because they require them to consider many possibilities, typically exponential and with competing purposes [

Nature-inspired metaheuristic algorithms have piqued the curiosity of scientists who want to solve complex real-world challenges. Harmony Search (HS) [

Many researchers in the literature have proposed hybrid optimization algorithms. These algorithms combine the benefits of two or more optimization techniques to overcome the limits of a single algorithm. In [

The Al-Biruni earth radius (BER) optimization methodology, which is inspired by the computation of the earth radius to estimate the search space surrounding the solutions in the cooperative behavior of swarm members to fulfill their global goals, is presented in this paper. BER tries to compromise between guaranteeing quick convergence and minimizing local optima stagnation. This is accomplished by employing the Al-Biruni technique, which aids in improving exploitation performance, striking a healthy balance between exploration and exploitation, expanding search space exploration, and boosting variety among present population individuals. The introduction of BER as a novel optimization technique with fresh insights into tackling various optimization issues is a significant contribution to this research. According to preliminary results, BER is promising, competitive, and capable of surpassing existing evolutionary algorithms based on swarm optimization. Furthermore, the performance of the proposed algorithm has been tested in terms of seven benchmark optimization functions and one design optimization problem from the engineering domain.

In the eleventh century, Al-Biruni worked out the radius of the Earth [

Second, Al-Biruni scaled the mountain’s peak and measured the horizon’s dip. He was able to compute the radius of the Earth using the following equation based on his data, as shown in

The proposed optimization method used the Al-Biruni technique for the cooperative optimization algorithm’s exploration and exploitation tasks, simulating swarm collaboration to achieve a global optimization goal. In real life, swarm live in groups and communicate together to achieve their goals. They usually work together to acquire food and defend themselves from invaders, exchanging responsibilities as required. They split up into sub-groups, with individuals collaborating both inside their sub-group and with members of other groups to achieve their ultimate goal. Swarm collaboration may be seen in ant and bee colonies, for example. Each member of the swarm makes a distinct contribution to the colony. The colony’s warriors are fed by workers who go out and gather food. The concept of BER was inspired by the fact that members of groups are frequently separated into sub-groups to complete various activities at different times and collaborate to reach their goals. The exploration and exploitation activities are usually performed to find the best solution to an optimization problem. In our case, BER divides individuals into two sub-groups, each accounting for one of the two activities. In BER, the exploitation and exploration activities ensure a deep investigation of the search space to avoid the local optima stagnation. The majority of cooperative optimization approaches need everyone to undertake exploitation after iterations, which might cause local optima to stagnate. BER avoids this case by keeping a collection of search agents that explore more areas in the search space through time. BER also swiftly increases the number of individuals investigating the search space if the algorithm’s performance does not improve after three iterations through solution mutation.

Finding the optimal solution to a problem with a set of constraints is the target of optimization algorithms. In BER, an individual from the population can be represented by a vector,

As the population is divided into subgroups in the proposed algorithm, the number of individuals in each group is modified dynamically to improve the balance between the tasks of exploitation and exploration. The process starts with dividing the population into two groups for exploration and exploitation [

Exploration is in charge of not only locating exciting locations in the search space but also of avoiding local optima stagnation through the movement towards the best solution, as discussed next.

The individual in the exploration group uses this strategy to seek prospective regions around its current position in the search space. This is accomplished by repeatedly searching among surrounding feasible alternatives for a superior option in terms of fitness value. The following equations are utilized in the BER investigation for this purpose:

The exploitation team is in charge of improving existing solutions. The BER calculates all individuals’ fitness values at each cycle and distinguishes the best individual. The BER employs two different approaches to achieve exploitation, as detailed in the sections below.

The following equations are used to move the search agent towards the best solution:

Most promising is the region surrounding the best solution (leader). As a result, some individuals hunt in the vicinity of the best solution with the potential of finding a better solution. To realize this operation, the BER utilizes the following equation.

where

The mutation is another method used by the BER to do research. It is a genetic operator utilized to create and sustain population variety. It may be considered a probabilistic local random disruption of one or more components in individuals. It prevents early convergence by helping to avoid local optima—such a shift in the search field functions as a springboard to another interesting topic. Indeed, the mutation is significant for the BER’s excellent exploration potential.

The BER chooses the best solution for the next iteration to ensure the quality of discovered solutions. Although the elitism method enhances algorithm efficiency, it can cause premature convergence in multimodal functions. It is worth noting that the BER offers impressive exploration capabilities by employing a mutation technique and searching around individuals in the exploration group. The BER can avoid early convergence due to its strong exploration capabilities. Algorithm 1 shows the BER pseudo-code. First, we give the BER some input parameters: number of iterations, population size, and mutation rate. The individuals are then separated into two groups by the BER, namely exploration and exploitation groups. The BER algorithm dynamically controls the number of individuals in each group during the search iterations for the best solution. To complete their responsibilities, each group employs two distinct strategies. To ensure diversity and high exploration, the BER randomly orders solutions after each iteration. For example, a solution from the exploration group in one iteration can become a member of the exploitation group in the next. The elitism method used in the BER helps avoid the loss of the leader throughout the iterations.

In this section, the evaluation of the proposed algorithm is performed in terms of two scenarios on experiments. In the first case, seven typical benchmark mathematical functions were utilized to find their minimum values in a specified domain of the search space to evaluate the performance of the proposed optimization technique. These functions are commonly used to benchmark optimization algorithms in the literature. In addition, the second scenario of experiments is based on solving one of the well-known constrained design engineering problems. These two scenarios of experiments are presented and discussed in the following sections.

On the other hand, the literature has several optimization techniques. In this research, the performance of the proposed approach is compared to five well-known optimization algorithms in the literature to show the superiority and effectiveness of the proposed algorithm. These algorithms are GWO, PSO, GA, WOA, and GSA. The selection of these three algorithms is based on their usefulness and popularity. Unique collaboration to attain global goals is a common theme in the proposed BER, WOA, GWO, and PSO. The GA and GSA are chosen as an example of evolutionary algorithms influenced by natural evolution theory.

In all scenarios of the conducted experiments, we executed 30 runs of the optimization algorithms for both scenarios due to the random initialization of the individuals in the first population. Each run was composed of 500 iterations. One of the algorithm’s parameters is the population size. In this research, it has been set at 30 individuals. The initial parameters of the proposed algorithm are listed in

Parameter name | Initial value |
---|---|

Size of population | 30 |

Iterations count | 500 |

Mutation probability | 0.5 |

Exploration percentage | 70 |

K (decreases from 2 to 0) | 1 |

Number of runs | 30 |

_{min}

Benchmark Function | Range | D | |
---|---|---|---|

0 | [−100, 100] | 30 | |

0 | [−10, 10] | 30 | |

0 | [−100, 100] | 30 | |

0 | [−100, 100] | 30 | |

0 | [−30, 30] | 30 | |

0 | [−100, 100] | 30 | |

0 | [−1.28, 1.28] | 30 |

Function | Algorithm | BER | PSO | WOA | GWO | GA |
---|---|---|---|---|---|---|

F1 | Mean | 0 | 0.000136 | 1.41E-30 | 6.59E-28 | 4.6E-172 |

StDev | 0 | 0.000202 | 4.91E-30 | 6.34E-05 | 0 | |

F2 | Mean | 0 | 0.042144 | 1.06E-21 | 7.18E-17 | 3.44E-90 |

StDev | 0 | 0.045421 | 2.39E-21 | 0.029014 | 6.13E-90 | |

F3 | Mean | 0 | 70.12562 | 5.39E-07 | 3.29E-06 | 1.7E-127 |

StDev | 0 | 22.11924 | 2.93E-06 | 79.14958 | 8.6E-127 | |

F4 | Mean | 0 | 1.086481 | 0.072581 | 5.61E-07 | 1.15E-75 |

StDev | 0 | 0.317039 | 0.39747 | 1.315088 | 2.45E-75 | |

F5 | Mean | 0 | 96.71832 | 27.86558 | 26.81258 | 28.37287 |

StDev | 0 | 60.11559 | 0.763626 | 69.90499 | 0.582802 | |

F6 | Mean | 0 | 0.000102 | 3.116266 | 0.816579 | 3.932626 |

StDev | 0 | 8.28E-05 | 0.532429 | 0.000126 | 0.431755 | |

F7 | Mean | 0 | 0.122854 | 0.001425 | 0.002213 | 0.022992 |

StDev | 0 | 0.044957 | 0.001149 | 0.100286 | 0.021966 |

A one-tailed t-test with a significance threshold of 50% is used for each benchmark function to ensure that the findings generated by the BER have statistical significance. The

Function | BER |
BER |
BER |
BER |
---|---|---|---|---|

F1 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F2 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F3 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F4 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F5 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F6 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F7 | 1.07E-06 | 1.07E-06 | 1.07E-06 | 1.07E-06 |

F1 | F2 | ||||||
---|---|---|---|---|---|---|---|

ANOVA | Treatment | Residual | Total | ANOVA | Treatment | Residual | Total |

SS | 2.42E-07 | 7.3E-07 | 0 | SS | 269.7 | 1265 | 1535 |

DF | 4 | 145 | 149 | DF | 4 | 145 | 149 |

MS | 6.05E-08 | 5E-09 | MS | 67.43 | 8.723 | ||

F (4, 145) | 12.04 | F (4, 145) | 7.73 | ||||

F3 | F4 | ||||||

SS | 4.25E+10 | 8.24E+09 | 5.07E+10 | SS | 61550 | 24147 | 85697 |

DF | 4 | 145 | 149 | DF | 4 | 145 | 149 |

MS | 1.06E+10 | 56830905 | MS | 15387 | 166.5 | ||

F (4, 145) | 186.8 | F (4, 145) | 92.4 | ||||

F5 | F6 | ||||||

SS | 59376 | 50691 | 110067 | SS | 355.2 | 10.21 | 365.4 |

DF | 4 | 145 | 149 | DF | 4 | 145 | 149 |

MS | 14844 | 349.6 | MS | 88.79 | 0.07039 | ||

F (4, 145) | 42.46 | F (4, 145) | 1261 | ||||

F7 | |||||||

SS | 566.5 | 1221 | 1788 | ||||

DF | 4 | 145 | 149 | ||||

MS | 141.6 | 8.422 | |||||

F (4, 145) | 16.82 | ||||||

The proposed BER optimization algorithm is used to solve one of the well-known design engineering problems, referred to as tension/compression spring design, to prove the effectiveness of the proposed algorithm in real-world handling challenges in engineering constrained problems. Based on the components of optimization problems, such as inequality constraints, equality constraints, and upper and lower limits, the formulation of this engineering problem is expressed by the following equations.

Subject to the constraints:

where the objective function, equality, and in-equality constraints are denoted by

In this equation, the new objective function is referred to as

The spring design problem has the following mathematical model:

Subject to the constraints:

A Comparison between the best solutions discovered by the proposed BER and the other competing algorithms is presented in

Algorithm | Design variables | Optimal cost | ||
---|---|---|---|---|

L | d | W | ||

PSO | 11.244543 | 0.357644 | 0.051728 | 0.0126747 |

GSA | 13.525410 | 0.323680 | 0.050276 | 0.0127022 |

WOA | 12.004032 | 0.345215 | 0.051207 | 0.0126763 |

Algorithm | Function evaluation | Standard deviation | Average | Optimal solution |
---|---|---|---|---|

PSO | 5460 | 0.0033 | 0.0139 | 0.012674 |

GSA | 4980 | 0.0026 | 0.0136 | 0.127022 |

WOA | 3600 | 0.0006 | 0.0134 | 0.012668 |

BER | PSO | GSA | GWO | |
---|---|---|---|---|

Number of values | 20 | 20 | 20 | |

Actual mean | 0.0129 | 0.01282 | 0.01288 | |

Theoretical mean | 0 | 0 | 0 | |

Significant (alpha = 0.05)? | Yes | Yes | Yes | Yes |

One sample t-test | ||||

t, df | t = 93.84, df = 19 | t = 186.4, df = 19 | t = 131.1, df = 19 | |

<0.0001 | <0.0001 | <0.0001 | ||

SEM of discrepancy | 0.0001374 | 0.00006876 | 0.00009827 | |

R squared (partial eta squared) | 0.9978 | 0.9995 | 0.9989 | |

SD of discrepancy | 0.0006146 | 0.0003075 | 0.0004395 | |

Discrepancy | 0.0129 | 0.01282 | 0.01288 | |

95% confidence interval | 0.01261 to 0.01318 | 0.01267 to 0.01296 | 0.01268 to 0.01309 |

On the other hand, a visual analysis of the results achieved by the optimization algorithms is depicted in the plots shown in

Al-Biruni Earth Radius (BER) search optimization algorithm is a new optimization algorithm that we proposed in this research. The proposed algorithm is based on exploring the area around the local solutions using the Al-Biruni method, which can help individuals to reach the global solution. Individuals are divided into two groups: exploration and exploitation. The exploration group employs the search strategy to examine the search space for potential solutions thoroughly. On the other hand, the exploitation group uses two approaches to identify better solutions from excellent ones (Move toward the leader and Search around the leader). To ensure finding the best solution, the BER permits mutation of the solutions that do not achieve progress in terms of the fitness function for three iterations. To test the effectiveness of the proposed algorithm in discovering the optimum points of each function and studying its convergence behavior, a comparison was conducted on seven typical benchmark mathematical optimization functions. The findings indicated that the BER is competitive with current metaheuristic and evolutionary optimization algorithms. Due to its promising exploration and exploitation capabilities and its ability to avoid local optima, the proposed BER algorithm has a quick convergence characteristic. In addition, it has been employed to solve a typical engineering design optimization problem referred to as the tension/compression spring design optimization problem. These findings emphasize the effectiveness of the proposed BER algorithm in tackling constraint optimization problems optimally. The future perspective of this research is to include more mathematical optimization functions and engineering problems in the conducted experiments.

Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2022R308), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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