This paper introduces the African Bison Optimization (ABO) algorithm, which is based on biological population. ABO is inspired by the survival behaviors of the African bison, including foraging, bathing, jousting, mating, and eliminating. The foraging behavior prompts the bison to seek a richer food source for survival. When bison find a food source, they stick around for a while by bathing behavior. The jousting behavior makes bison stand out in the population, then the winner gets the chance to produce offspring in the mating behavior. The eliminating behavior causes the old or injured bison to be weeded out from the herd, thus maintaining the excellent individuals. The above behaviors are translated into ABO by mathematical modeling. To assess the reliability and performance of ABO, it is evaluated on a diverse set of 23 benchmark functions and applied to solve five practical engineering problems with constraints. The findings from the simulation demonstrate that ABO exhibits superior and more competitive performance by effectively managing the trade-off between exploration and exploitation when compared with the other nine popular metaheuristics algorithms.
OptimizationmetaheuristicsAfrican bison optimizationengineering problemsNational Natural Science Foundation of ChinaU1731128Natural Science Foundation of Liaoning Province2019-MS-174Liaoning Province EducationAdministrationLJKZ0279Introduction
There are lots of complex computational problems in the fields of scientific computing, socioeconomics, and engineering which exhibit a high degree of non-linearity, non-differentiability, discontinuity, and high complexity [1]. In many cases, the conventional optimization algorithms that rely on analysis are inadequate for solving these problems. As a result, many new metaheuristic optimization algorithms are explored to overcome the above problems.
Nature has evolved over billions of years through genetic evolution to perfectly demonstrate its efficiency and magic. Meanwhile, humans have learned a great scale from natural systems to develop new algorithms and models for solving complex problems [2–4]. Nature-inspired metaheuristic optimization algorithms can be categorized into four groups, biological evolution-based, natural physics-based, human-based, and biological swarm-based approaches (See Fig. 1).
Metaheuristics classification
Evolution-based algorithms mimic natural evolutionary processes, including reproduction, mutation, recombination, and selection. Genetic Algorithms [5] stand out as a popular technique that draws inspiration from Darwin’s theory of evolution and simulates its principles. Other popular algorithms are Differential Evolution (DE) [6] and Evolution Programming [7].
Physics-based algorithms are derived from fundamental principles of the natural world. These physical phenomena are represented by objects such as sound, light, force, electricity, magnetism, and heat. The most popular algorithm is the Sine Cosine Algorithm (SCA) [8]. Other popular algorithms are Gravitational Search Algorithm [9], and Magnetic Optimization Algorithm [10].
Human-based algorithms draw inspiration from a variety of human activities, including cognition and physical behaviors. There are teaching-learning based optimization [11], Human Mental Search [12], and Social Group Optimization [13].
Swarm-based algorithms emulate the social behaviors exhibited by animals. Among these techniques, Particle Swarm Optimization (PSO) [14], Whale Optimization Algorithm (WOA) [15], and Gray Wolf Optimizer (GWO) [16] have gained tremendous popularity. In recent years, new algorithms such as Flower Pollination Algorithm (FPA) [17], Chimp Optimization Algorithm (CHOA) [18], Rat Swarm Optimizer (RSO) [19], Tunicate Swarm Algorithm (TSA) [20], Reptile Search Algorithm (RSA) [21], and Artificial Rabbits Optimization (ARO) [22] are proposed.
All swarm-based optimization algorithms share common properties: exploration and exploitation. The tradeoff between exploration and exploitation has been a central issue in the field of evolutionary computation and optimization research. It is crucial for minimizing computational costs and achieving efficient results. Therefore, optimization algorithms must possess the capability to explore the global space and exploit local regions. The No Free Lunch (NFL) states that there is no algorithm that can solve all optimization problems well [23]. In other words, an algorithm can have brilliant results in a specific class of problems but fails to solve other problems.
The above-mentioned facts inspire us to propose a novel algorithm with swarm intelligence characteristics for solving optimization problems. In this paper, the proposed African Bison Optimization (ABO) algorithm is inspired by the survival behaviors of African bison, including foraging, jousting, mating, and eliminating from the herd. As far as we know, these behaviors are proposed for the first time, and there is no similar study in the literature. The African Buffalo Optimization algorithm models the organizational skills of African buffalos by two basic sounds, waaa and maaa when they are finding food sources [24]. Although the African buffalo and bison are related species, ABO proposed in this paper is based on completely different principles.
The main contributions of this paper can be summarized as follows:
A swarm-based African Bison Optimization (ABO) is proposed and the survival strategies of bison are investigated and modeled mathematically.
The proposed ABO is implemented and tested on 23 benchmark test functions.
The performance of ABO is compared with some classical and latest optimizers.
The efficiency of ABO is examined for solving the real-world engineering problems with constraints.
The rest of the paper is arranged as follows: Section 2 shows the biological characteristics of the African bison and introduces the proposed ABO algorithm. In Section 3, the performance and efficiency of ABO are tested by using benchmark test functions. The results of the algorithm on five constrained real-world engineering problems are discussed in Section 4. Section 5 provides a comprehensive conclusion of the study, along with future research directions.
African Bison Optimization AlgorithmAfrican Bison
African bison is a genus of animals in the even-toed ungulates, bovidae, with five subspecies. As shown in Fig. 2, it has characteristics of a broad chest, strong limbs, a large head, and long horns. The African bison is widely distributed throughout most of sub-Saharan Africa. Their habitats are extensive, including open grasslands, savannas with drinking water, and lowland rainforests.
African Bison in nature
African bison are social animals, and only old or injured individuals will be broken away from the herd. The strongest bison becomes the leader and enjoys the right to eat the best grains. African bison cannot live without water, so they are rarely far away from water. They often hide in the shade, or dip in pools or mud to keep their bodies at a cooler temperature. To drink and feed, African bison often live near water. When food and water are plentiful, they will soak their entire body in water and reduce activities significantly to avoid high temperatures. Jousting and Mating take place in the rainy season when the temperature is comfortable and the food is abundant. Parts of the survival behaviors of the African bison described above are shown in Figs. 3–5.
Foraging behavior
Bathing behavior
Jousting behavior
Mathematical Model and Algorithm
This subsection focuses on the mathematical modeling of foraging, jousting, mating, and eliminating of African bison herd, and provides a detailed description of ABO.
Initialization
In the initialization part, a uniformly distributed population is generated by random initialization, and this approach can be implemented by the following equation:
Xi=Xmin+rand∗(Xmax−Xmin)where Xi is the position of i-th individual, rand is a random number in the range from 0 to 1, and Xmin and Xmax are the lower and upper bounds of the problem, respectively.
In Eq. (2), the members of the bison population are represented through a population matrix. In this matrix, each row represents a potential solution, while the columns correspond to the suggested values for the variables of the problem.
X=[X1⋮Xi⋮XN]=[x1,1⋯x1,j⋯x1,d⋮⋱⋮⋮xi,1⋯xi,j⋯xi,d⋮⋮⋱⋮xN,1⋯xN,j⋯xN,d]where X is the population matrix, N and d represent population size and individual dimension, respectively. xi,j is the j-th dimension of the i-th individual.
In ABO, each individual is metaphorically represented as a bison, serving as a potential solution to the given problem. The fitness values are determined by using a vector in Eq. (3).
F=[F1⋮Fi⋮FN]=[F(X1)⋮F(Xi)⋮F(XN)]where Fi is the fitness value of the i-th individual. F is the vector of these fitness values.
Cluster and Finding the Best Individual
In ABO, the population is divided into two subgroups with the same size N/2: male and female (See Fig. 6). In the herd, the position of the best individual also represents the place where the richest food source is located in. Therefore, we define the best position as the location of the richest food source and it is denoted as XGbest (See Fig. 7).
Examples of population grouping
The definition of the best individual
Defined the Satiety Rate and Temperature
African bison are very sensitive to temperature. When the temperature is high and food is plentiful, they soak their body in the water and reduce activities. They tend to breed during the rainy season when the temperature is suitable and food is plentiful. Individuals are competing for the right to mate. Inspired by above behaviors, we use the satiety rate (S) and temperature (Q) to balance the exploration and exploitation in ABO, and defined by Eqs. (4) and (5):
S=2∗sin((3t/2T)∗π)Q=cos(−((T−t)/T)∗π/2)where the S is used for the balance of the exploration and exploitation phase, which has values from −2 to 2. Q is used for the balance of the development phase and it has values from 0 to 1. The current and maximum number of iterations are denoted by t and T, respectively.
Exploration Phase
If |S|< Threshold_1 (Threshold_1 = 0.5), it means that the individuals are hungry. They will randomly update their position to find food. This behavior happens in the male and female subgroups and is modeled by Eqs. (6) and (7):
Xit+1=(Xrandt∗S)+A∗((Xmax−Xmin)∗R1)∗R23A=cos((Frand/Fi)∗π)∗R3where Xrandt is the current position of a random individual, R1 and R2 are random numbers between [−1, 1], respectively. A represents the bison’s ability to find food. Frand and Fi are fitness values of random and current one, respectively. R3 is a random number between [−5, 5].
Exploration Phase
If |S|≥ Threshold_1, it means that food and water are enough. The bison will be bathing, jousting, mating, and eliminating according to the temperature level (Q).
If Q≥ Threshold_2 (Threshold_2 = 0.6), it means that food and water are enough but the temperature is high. The bison will soak their body in water and reduce activities to avoid high temperatures. We define this behavior as bathing behavior. It is represented by Eq. (8) in the male and female subgroups:
Xit+1=2∗(XGbestt−Xit)∗R4+exp(R5∗Q5)∗cos(Q∗2π)where XGbestt is the best position in the herd, R4 and R5 are the random numbers between [0, 2], respectively.
If Q < Threshold_2, it means that food and water are enough and the temperature is suitable for surviving. Then, the herd enters the jousting stage, and the winner has the right to mate.
The bison jousting behavior is represented by Eqs. (9) and (10) in the male and female subgroups:
J=XSbestt−XSbestt∗(rand∗Xit2)∗(cos(Xit)+sin(Xit))∗R65Xit+1=XSbestt−J∗R7where J is the fighting power of bison in the population. R6 is a random number between [−0.01, 0.01], R7 is a random number between [0, 2], and XSbestt denotes the position of the best individual in the male or female subgroup.
The mating behavior of male bison is represented by Eqs. (11) and (12):
MM=cos(Fi_f/Fi_m)Xi_mt+1=Xi_mt+sin(2π∗R8)∗R8∗MM∗(Xi_mt−Xi_ft)where MM is the mating ability of male individuals, Fi_m and Fi_f are the fitness values of the current male and female individual, respectively. Xi_mt and Xi_ft denote the current position in the male and female population, respectively. R8 is a random number between [0, 1].
The mating behavior of female bison is achieved by the following expressions:
MF=sin(Fi_m/Fi_f)
Xi_ft+1=Xi_ft+cos(2π∗R9)∗R9∗MF∗(Xi_ft−Xi_mt)where MF is the mating ability of female individuals and R9 is a random number between [0, 1].
The eliminating behavior means the aged or injured individuals will consciously detach themselves from the herd, thus allowing the entire population to remain at the highest level of competence. This behavior is represented by the following Eq. (15):
Xworstt=Xmin+R10∗(Xmax−Xmin)where Xworstt represents the eliminating of the old or injured individuals from the population. R10 is a random number between [0, 1].
The ABO Pseudo-Code
The above behaviors are mathematically modeled in this paper to construct ABO. To show the iterative process of ABO more clearly, Algorithm 1 gives the pseudo-code of the algorithm.
Computational Complexity
The complexity of an algorithm is determined by the population size (n), number of iterations (T), and problem dimensionality (d). The computational complexity of ABO can be summarized into the following three stages: the initialization of the population, the calculation of the fitness, and the position update of the individuals. It can be expressed as: O(ABO) = O(Initialization) + O(Function evaluation) + O(Location update of the individuals) = O(n + Tn + Tnd).
The Threshold_1 and Threshold_2 are the key parameters to balance the exploration and exploitation stages of ABO. This section analyzes the sensitivity of the parameter Threshold_1 and Threshold_2 by setting different values. The results are shown in Table 1, where the Threshold_1 and Threshold_2 are set to (0, 0.4), (0.5, 0.6), (1, 0.8) and (2, 1), respectively. The performance of (0.5, 0.6) is the best compared with others.
Results of different combinations of Threshold_1 and Threshold_2
Function
(0.5, 0.6)
(0, 0.4)
(1, 0.8)
(2, 1)
F1
Mean
0
0
0
0
F2
0
0
0
0
F3
0
0
0
0
F4
0
0
0
7.83439E-84
F5
1.090042539
7.359920014
0.488914301
0.70970151
F6
0.029361478
0.046890454
0.046474359
0.060522219
F7
0.000176978
8.382E-05
0.000136326
0.000210415
F8
−12398.06721
−12050.56684
−12320.76541
−11939.14066
F9
0
0
0
0
F10
8.88178E-16
8.88178E-16
8.88178E-16
8.88178E-16
F11
0
0
0
0
F12
0.004475098
0.032834546
0.020679249
0.014471592
F13
0.046484731
0.122087636
0.108531792
0.078069042
F14
1.233075861
2.676849143
1.330986282
1.033865839
F15
0.000344918
0.000685222
0.000338614
0.000336355
F16
−1.031628435
−1.031368619
−1.031627418
−1.031626074
F17
0.397887361
0.398676955
0.397887864
0.397887515
F18
3.000008251
3.002797211
3.000000625
3.000000074
F19
−3.862781389
−3.861817866
−3.862751303
−3.862769483
F20
−3.319823751
−3.235794903
−3.295627739
−3.31810832
F21
−10.14934325
−10.12086745
−10.14776529
−10.12883157
F22
−10.40009336
−10.36792913
−10.39813134
−10.38037045
F23
−10.5320396
−10.48339948
−10.52951159
−10.50306925
The Survival Behaviors Analysis
ABO is inspired by the survival behaviors of the African bison. The foraging behavior drives the individuals to search for local optimal solutions randomly. The bathing behavior enables the individuals to fully explore near the local optimal solutions, thus improving the quality of the solution. Through jousting and mating behavior, the population can produce better individuals, so that ABO can converge to the optimal value faster. The worst individual in the population is eliminated and replaced by the elimination behavior, so the whole population can always maintain diversity in the iteration.
Experimental Results and Discussion
This section presents the results of experiments conducted on 23 benchmark functions to demonstrate the performance superiority of ABO. The experimental results of ABO are compared with those of nine other metaheuristic optimization algorithms.
Benchmark Test Functions
To comprehensively evaluate the performance of ABO in terms of optimization, we test it on 23 benchmark functions. These functions can be classified into three main categories: Unimodal, Multimodal, and Fixed-dimension multimodal [25]. The presence of a single global optimum in Unimodal (F1–F7) benchmark functions serves as a valuable test for evaluating the exploration capability of the algorithm under consideration. The optimization algorithms’ diversification capabilities are shown by using the Multimodal (F8–F13) and Fixed-dimension multimodal (F14–F23) benchmark functions. Details about these functions can be found in [15].
Experimental Setup
The parameters of the other algorithms (PSO, GWO, CHOA, SCA, RSA, FPA, WOA, RSO, and TSA) are derived from the corresponding papers. The experimentation process and algorithms have been carried out on an i5 processor operating at 2.60 GHz and 16 GB of RAM.
Performance Comparison
To ensure an equitable comparison, we maintain a consistent population size of 30 individuals across all optimization algorithms and perform 500 generations. The mean, std, best result, and ranking after 30 independent runs of each algorithm are presented in Tables 2–4, where the ranking is determined based on the mean.
The mean, std, and best values on the unimodal functions
Functions
ABO
PSO
WOA
RSO
CHOA
FPA
GWO
RSA
SCA
TSA
F1
Mean
0
2.739504
4.96E-74
2E-259
1.62E-08
1863.119
6.53E-28
0
8.847723
3.1E-195
Std
0
1.381861
1.66E-73
0
6.87E-08
645.9023
9.15E-28
0
13.03789
0
Best
0
1.059479
1.53E-91
0
9.48E-28
1135.756
7.44E-29
0
0.097024
6.6E-205
Rank
1
8
5
3
7
10
6
2
9
3
F2
Mean
0
4.677332
2.97E-51
1.4E-139
8.22E-07
30.14067
1.24E-16
0
0.026963
1.5E-100
Std
0
1.170234
8.8E-51
5.1E-139
1.95E-06
5.886052
1.39E-16
0
0.042914
5.5E-100
Best
0
3.016678
8.08E-60
0
6.14E-10
22.85537
1.56E-17
0
7.77E-05
1.3E-106
Rank
1
9
4
3
7
10
6
2
8
3
F3
Mean
0
187.8865
45969.67
5.1E-255
102.1504
1456.598
1.28E-05
0
8696.327
1.4E-182
Std
0
51.43875
12211.16
0
214.6503
348.5603
2.6E-05
0
5048.532
0
Best
0
98.23863
16083.88
0
0.335248
885.9155
9.73E-09
0
1579.356
1.8E-187
Rank
1
7
10
3
6
8
5
2
9
3
F4
Mean
0
1.924234
44.24386
4.69E-90
0.013704
23.59319
5.63E-07
0
35.77706
4.15E-92
Std
0
0.229176
29.14158
2.57E-89
0.013756
3.252222
5.81E-07
0
14.17494
1.56E-91
Best
0
1.49975
1.268282
0
0.000243
17.58065
2.58E-08
0
11.50938
2E-100
Rank
1
7
10
4
6
8
5
2
9
3
F5
Mean
1.090043
806.674
27.79136
28.76958
28.93083
301659
26.98143
22.19044
24182.83
28.52099
Std
4.29301
342.0505
0.443458
0.210932
0.065806
155153.4
0.796159
12.45145
61679.71
0.365193
Best
0.000384
359.691
26.91447
28.14204
28.73378
78722.21
25.8111
2.76E-27
30.88447
28.05151
Rank
1
8
5
6
7
10
4
2
9
6
F6
Mean
0.029361
2.29193
0.448323
3.278986
3.138063
1884.918
0.720921
6.946221
15.91793
5.941678
Std
0.036941
0.828593
0.257496
0.602236
0.388782
293.0147
0.37812
0.676677
22.48097
0.845228
Best
1.19E-06
0.712171
0.097252
1.762142
2.336074
1262.862
7.1E-05
3.932309
4.294536
3.941324
Rank
r1
5
2
6
6
10
3
8
9
7
F7
Mean
0.000177
18.89928
0.003832
0.00051
0.001878
0.324021
0.002313
0.000137
0.176878
6.71E-05
Std
0.000166
15.2891
0.004263
0.000505
0.001714
0.087986
0.001023
0.000126
0.261323
6.2E-05
Best
6E-06
1.562643
1.94E-05
3.59E-05
4.35E-05
0.180599
0.000859
4.17E-06
0.003958
2.19E-06
Rank
3
10
7
4
5
9
6
2
8
1
The mean, std, and best values on the multimodal functions
Functions
ABO
PSO
WOA
RSO
CHOA
FPA
GWO
RSA
SCA
TSA
F8
Mean
−12398.1
−6368.57
−10310.7
−5819.19
−5712.53
−6393.11
−5871.21
−5498.44
−3708.51
−3327.75
Std
269.4774
1300.051
1587.465
897.3912
83.868
233.5901
719.9325
132.336
208.6469
461.9023
Best
−12569.1
−8956.26
−12568.8
−6924.98
−6026.69
−6957.85
−7152.02
−5722.93
−4054.57
−4447.49
Rank
1
5
3
6
7
4
6
8
9
10
F9
Mean
0
171.5526
5.68E-15
0
1.082867
183.3753
1.683353
0
35.99174
20.11077
Std
0
32.82081
1.73E-14
0
2.984753
12.67538
2.497436
0
27.08043
42.84416
Best
0
100.3091
0
0
3.98E-13
159.8033
5.68E-14
0
0.473392
0
Rank
1
9
3
2
4
10
5
2
8
7
F10
Mean
8.88E-16
2.583473
4.91E-15
1.36E-15
19.96288
7.321006
1.09E-13
8.88E-16
10.44769
4.56E-15
Std
0
0.347243
2.91E-15
1.23E-15
0.001377
1.265033
2.72E-14
0
9.624672
6.49E-16
Best
8.88E-16
1.876971
8.88E-16
8.88E-16
19.9601
5.059116
7.55E-14
8.88E-16
0.075379
4.44E-15
Rank
1
7
4
3
10
8
5
2
9
3
F11
Mean
0
0.130369
0
0
0.015213
18.85045
0.005716
0
0.812483
0.002264
Std
0
0.060892
0
0
0.024037
4.806672
0.009628
0
0.379886
0.006266
Best
0
0.053012
0
0
3.15E-13
11.20103
0
0
0.019209
0
Rank
1
8
2
3
6
10
5
3
9
4
F12
Mean
0.004475
0.055503
0.022159
0.350741
0.315031
41.34128
0.043116
1.355855
19106.87
1.090577
Std
0.005746
0.046017
0.010872
0.13331
0.106485
69.48961
0.01846
0.286829
71302.46
0.319172
Best
1.37E-07
0.014218
0.007322
0.183195
0.148244
12.0622
0.013064
0.742027
0.920505
0.511953
Rank
1
4
2
6
6
9
3
8
10
7
F13
Mean
0.046485
0.529532
0.558913
2.866925
2.809285
68964.71
0.717013
0.093335
307141.1
2.561741
Std
0.051307
0.213334
0.248765
0.045647
0.10323
95611.47
0.30797
0.47412
882041.5
0.296405
Best
2.66E-05
0.200496
0.099758
2.802804
2.586901
662.1132
0.101023
1.45E-30
2.805047
1.841649
Rank
1
4
3
8
8
9
5
2
10
7
The mean, std, and best values on the fixed-dimension multimodal functions
Functions
ABO
PSO
WOA
RSO
CHOA
FPA
GWO
RSA
SCA
TSA
F14
Mean
1.233076
2.770934
2.960349
2.381829
1.062489
0.998144
4.128514
3.671247
2.122922
10.36112
Std
0.514499
2.623468
3.32548
1.847044
0.239231
0.000495
4.305079
2.755154
0.999311
5.126076
Best
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
1.01025
0.998004
1.992031
Rank
3
6
7
5
2
1
9
8
5
10
F15
Mean
0.000345
0.000846
0.000683
0.001603
0.004891
0.000617
0.005057
0.001475
0.000941
0.00462
Std
4.5E-05
0.000133
0.000423
0.003677
0.016176
0.000124
0.008592
0.000693
0.000353
0.008114
Best
0.00031
0.000525
0.000309
0.000399
0.001234
0.000358
0.000308
0.00066
0.000322
0.000313
Rank
1
4
3
7
9
2
10
7
5
8
F16
Mean
−1.03163
−1.03163
−1.03163
−1.03149
−1.02849
−1.03163
−1.03163
−1.02976
−1.03157
−1.02423
Std
5.6E-08
4.7E-16
2.11E-09
0.00018
0.009536
9.51E-08
2.35E-08
0.002971
4.2E-05
0.013599
Best
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03161
−1.03163
−1.03163
Rank
4
1
3
7
9
6
5
8
7
10
F17
Mean
0.397887
0.397887
0.397906
1.180402
0.400341
0.397891
0.397906
1.121332
0.400458
0.399368
Std
1.05E-08
0
4.59E-05
1.215561
0.00516
4.89E-06
8.89E-05
1.276393
0.003794
0.003143
Best
0.397887
0.397887
0.397887
0.398106
0.39789
0.397887
0.397887
0.399553
0.397951
0.397944
Rank
2
1
6
10
8
4
5
10
9
7
F18
Mean
3.000008
3
3.00009
3.000051
3.000185
3
3.000044
5.811115
3.000176
12.26838
Std
4.49E-05
6.32E-15
0.000185
0.000117
0.000193
4.12E-09
4.55E-05
8.577847
0.000516
22.61218
Best
3
3
3
3
3.000007
3
3.000001
3
3
3.000001
Rank
3
1
5
9
7
2
4
8
6
10
F19
Mean
−3.86278
−3.86278
−3.85399
−3.49464
−3.8547
−3.86278
−3.86162
−3.79781
−3.85453
−3.85952
Std
1.19E-06
1.49E-14
0.013546
0.298841
0.002898
1.26E-08
0.002318
0.061135
0.002967
0.002388
Best
−3.86278
−3.86278
−3.86278
−3.84748
−3.86221
−3.86278
−3.86278
−3.85726
−3.86177
−3.86265
Rank
3
1
9
10
7
3
5
10
8
6
F20
Mean
−3.31982
−3.25859
−3.22231
−1.75724
−2.60286
−3.29848
−3.23749
−2.71951
−2.85248
−3.137
Std
0.002542
0.060328
0.115741
0.492095
0.492071
0.015875
0.080339
0.382472
0.347177
0.124958
Best
−3.32177
−3.322
−3.32106
−2.55937
−3.23866
−3.31623
−3.32199
−3.17246
−3.17695
−3.29692
Rank
1
4
6
10
9
3
5
10
8
7
F21
Mean
−10.1493
−7.29673
−8.52957
−0.88619
−3.17522
−10.1399
−9.06434
−5.0552
−2.66714
−7.13858
Std
0.00572
3.005022
2.529567
0.695632
2.041282
0.015535
2.520117
3.32E-07
1.688837
1.51868
Best
−10.1532
−10.1532
−10.1523
−3.37099
−5.00764
−10.1531
−10.153
−5.0552
−5.11201
−9.70639
Rank
1
6
5
10
9
2
4
8
10
7
F22
Mean
−10.4001
−9.60625
−7.33244
−0.93678
−3.06208
−10.2934
−10.2238
−5.08767
−2.95511
−6.26446
Std
0.004205
1.956164
3.184573
0.513169
2.048768
0.151153
0.970056
7.99E-07
1.72394
2.116332
Best
−10.4029
−10.4029
−10.4014
−2.07786
−5.03666
−10.4025
−10.4027
−5.08767
−6.8096
−9.97328
Rank
1
5
6
10
9
3
4
8
10
7
F23
Mean
−10.532
−9.95552
−7.56613
−1.2381
−4.17562
−10.3441
−10.3559
−5.12847
−3.79545
−5.06172
Std
0.010876
1.783603
3.098342
0.666509
1.652114
0.285731
0.978592
1.79E-06
1.473324
2.732771
Best
−10.5364
−10.5364
−10.5362
−3.62608
−5.82866
−10.5233
−10.5363
−5.12848
−6.50974
−9.163
Rank
1
5
6
10
9
4
3
7
10
8
According to Table 2, the mean, std, and best of ABO on F1–F4 are all zero, which indicates that the algorithm can accurately find the global optimum of these functions each time. For F5 and F6, ABO can achieve significantly better optimal solutions than other algorithms. On F7, ABO is similarly competitive, with optimization results next only to TSA and RSA. The results demonstrate that ABO has significant advantages in exploitation compared to other algorithms.
According to the experimental results in Tables 3 and 4, ABO achieves very competitive results. On F8–F13, F15, and F20–F23, the optimization results of ABO are better than other algorithms, and the ranking is 1. On the other functions, the proposed algorithm also achieves satisfactory results, and the optimization performance is better than most algorithms. These results illustrate that ABO has excellent performance in terms of exploration capability.
Convergence Analysis
Fig. 8 shows the convergence curves of ABO with the other algorithms on benchmark test functions. It can be seen that ABO has three different convergence behaviors. In the early stages, it explores fully the given search space, allowing the highest convergence efficiency. In the second stage, the algorithm initiates iterative processes aimed at approaching the optimal solution until reaching the maximum allowed number of iterations. The last stage is the iterative process of rapid convergence of ABO, where it finds the optimal solution to the benchmark test problem through full exploration and exploitation. The above analysis show that ABO achieves a good balance of exploration and exploitation than other metaheuristic optimization algorithms.
Convergence plots for various benchmark functions
Statistical Testing
In addition to utilizing standard statistical measures such as mean and standard deviation, we have also employed the Wilcoxon sum rank test for further analysis and comparison [26]. It is used to assess whether ABO’s results statistically significantly differ from that of other competing algorithms. In this paper, the sample size of the Wilcoxon sum rank test is 30 and the confidence interval is 95%. The significance of an algorithm can be determined by examining its p-value. If the p-value of a algorithm is lower than 0.05, it signifies that the algorithm exhibits statistical significance when compared to the other algorithms under consideration. Table 5 shows the Wilcoxon sum rank test for the 23 benchmark test functions. From the experimental results of the table, it can be obtained that the p-value obtained from the comparison of ABO with the other algorithms is much less than 0.05 in general for all the benchmark functions. It can be shown that ABO has a significant difference compared with other algorithms.
<italic>p</italic>-values of the Wilcoxon rank-sum test with 5% significance for F1–F23
Functions
PSO
WOA
RSO
CHOA
FPA
SCA
GWO
RSA
TSA
F1
1.21E-12
1.21E-12
1.46E-04
1.21E-12
1.21E-12
1.21E-12
1.21E-12
NaN
1.21E-12
F2
1.21E-12
1.21E-12
1.27E-05
1.21E-12
1.21E-12
1.21E-12
1.21E-12
NaN
1.21E-12
F3
1.21E-12
1.21E-12
5.58E-03
1.21E-12
1.21E-12
1.21E-12
1.21E-12
NaN
1.21E-12
F4
1.21E-12
1.21E-12
2.21E-06
1.21E-12
1.21E-12
1.21E-12
1.21E-12
NaN
1.21E-12
F5
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.99E-04
3.02E-11
F6
3.02E-11
3.69E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
4.62E-10
3.02E-11
3.02E-11
F7
3.02E-11
1.03E-06
1.03E-03
1.20E-08
3.02E-11
3.02E-11
3.02E-11
4.02E-01
1.06E-03
F8
3.02E-11
6.53E-07
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
F9
1.21E-12
8.14E-02
NaN
1.21E-12
1.21E-12
1.21E-12
1.08E-12
NaN
2.93E-05
F10
1.21E-12
3.08E-10
4.18E-02
1.21E-12
1.21E-12
1.21E-12
1.18E-12
NaN
2.71E-14
F11
1.21E-12
NaN
NaN
1.21E-12
1.21E-12
1.21E-12
6.62E-04
NaN
4.19E-02
F12
8.99E-11
1.41E-09
3.02E-11
3.02E-11
3.02E-11
3.02E-11
6.07E-11
3.02E-11
3.02E-11
F13
3.02E-11
4.08E-11
3.02E-11
3.02E-11
3.02E-11
3.02E-11
4.08E-11
6.53E-08
3.02E-11
F14
8.30E-01
7.84E-01
1.45E-01
3.95E-01
7.62E-03
5.26E-04
4.73E-01
7.77E-09
8.15E-11
F15
3.02E-11
7.60E-07
4.98E-11
3.02E-11
1.61E-10
9.76E-10
1.91E-01
3.02E-11
7.06E-07
F16
2.36E-12
4.23E-03
3.02E-11
3.02E-11
1.75E-05
3.02E-11
7.96E-03
3.02E-11
3.34E-11
F17
1.21E-12
7.09E-08
3.02E-11
3.02E-11
3.02E-11
3.02E-11
4.08E-11
3.02E-11
3.02E-11
F18
2.97E-11
6.52E-08
4.69E-08
2.87E-10
4.21E-02
6.12E-10
6.12E-10
2.37E-10
3.47E-10
F19
2.16E-11
1.46E-10
3.02E-11
3.02E-11
9.76E-10
3.02E-11
4.20E-10
3.02E-11
3.02E-11
F20
6.63E-01
7.60E-07
3.02E-11
3.02E-11
8.15E-11
3.02E-11
3.79E-01
3.02E-11
3.02E-11
F21
8.30E-01
6.05E-07
3.02E-11
3.02E-11
8.15E-05
3.02E-11
1.12E-01
3.02E-11
3.02E-11
F22
9.51E-06
1.07E-09
3.02E-11
3.02E-11
2.03E-09
3.02E-11
1.96E-01
3.02E-11
3.02E-11
F23
6.53E-07
1.07E-07
3.02E-11
3.02E-11
1.33E-10
3.02E-11
1.02E-01
3.02E-11
3.02E-11
Engineering Problems
In this section, ABO is tested also with five constrained engineering design problems: Pressure vessel design, Rolling element bearing design, Tension/compression spring design, Cantilever beam design, and Gear train design. Mathematical models of these engineering design problems can be found in the corresponding literature. For the constraint problem, this paper uses the widely used penalty function method to deal with the constraints [22]. To ensure an equitable comparison, we maintain a consistent population size of 30 individuals across all optimization algorithms and perform 500 generations.
Pressure Vessel Design Problem
The pressure vessel design is an engineering problem initially presented by Kannan and Kramer [22]. As shown in Fig. 9, this design challenge involves the optimization of a pressure vessel’s manufacturing cost, encompassing expenses related to materials, forming processes, and welding. The primary objective is to minimize the overall cost of producing the pressure vessel while ensuring it meets all design constraints and performance requirements.
Pressure vessel design
We use ABO to optimize the problem and compare the experimental results with other algorithms. The results presented in Table 6 illustrate the statistical outcomes of the algorithms in terms of the optimal and variables. By utilizing ABO, it is possible to obtain the optimal function value of f1= 6285.942, with corresponding structure variables x= (0.8140874, 0.4987349, 41.91206, 179.0077). The results show that ABO can find the optimal design of the pressure vessel problem with the lowest manufacturing cost compared with other algorithms.
Comparison of optimum results for pressure vessel design
Algorithm
x1
x2
x3
x4
Optimal value
CHOA
1.25639
0.770405
64.3998
25.8942
9131.5308
ABO
0.8140874
0.4987349
41.91206
179.0077
6285.942
WOA
1.22615
0.660536
63.476
17.7501
7570.0419
GWO
1.25018
0.779287
58.5525
43.3575
8756.1309
SCA
0.8680526
0.6206409
43.22946
186.2539
7503.0508
TSA
1.22625
0.691567
63.2978
20.2611
7890.503
L-SHADE [23]
0.8525
0.5775
56.3105
65.7572
7672.4972
LSHADE-EpSin [23]
0.9330
0.6982
59.9952
47.5678
6854.5191
TEO [23]
2.5816
1.4787
47.1647
148.7692
32593.2941
Rolling Element Bearing Design
The design of rolling element bearings is a well-known engineering benchmark problem introduced by Rao and Tiwari in 2007. This problem aims to maximize the dynamic load-carrying capacity of rolling bearings, as illustrated in Fig. 10. The problem is framed by ten constraints and ten design variables.
Rolling element bearing design
Table 7 shows the optimal values and optimal variables obtained by these algorithms on this problem. ABO can obtain the optimal function value f2=−85380.5266 with the structure variables x= (125.6417, 21.40225, 11, 0.515001, 0.5154961, 0.4021139, 0.6181131, 0.3026727, 0.03466326, 0.6025443). The outcomes reveal that ABO can identify the optimal configuration for the design of rolling element bearings, maximizing their dynamic load-carrying capacity.
Comparison of optimum results for rolling element bearing design
Algorithm
Dm
Db
z
fi
go
KDmin
KDmax
ε
e
ζ
Optimal value
CHOA
125
21.62214
10
0.515
0.6
0.5
0.7
0.3
0.02
0.651701
−81549.3006
ABO
125.6417
21.40225
11
0.515001
0.5154961
0.4021139
0.6181131
0.3026727
0.03466326
0.602544
−85380.5266
PSO
125
21
9
0.515
0.515
0.5
0.7
0.4
0.1
0.6
−35585.8093
WOA
125
21
9
0.515
0.515
0.4
0.6
0.3
0.053251
0.6
−72214.8943
GWO
125
21.09474
5
0.515
0.5693567
0.4246166
0.671365
0.325135
0.020982
0.610848
−49186.3383
Chaotic GWO [3]
125.68089
21.4174
11
0.5153
0.56367
0.49617
0.67858
0.3000001
0.05734287
0.616924
-81803. 645
SCA
125
20.82613
11
0.515
0.515
0.4
0.6
0.3
0.02
0.607731
−81343.9831
TSA
125
21.13977
11
0.515
0.5485597
0.4
0.627163
0.3
0.026647
0.6
−83519.8965
DE [22]
125.7192
21.2508
10.8654
0.5153
0.56021
0.4199
0.6197
0.3012
0.0472
0.6740
−83629.26366
PVS [27]
125.7191
21.4256
11
0.515
0.515
0.4004
0.6802
0.3
0.08
0.7
−81859.74121
Tension/Compression Spring Design Problem
The design problem introduced by Arora focuses on tension/compression spring design. This design problem, as depicted in Fig. 11, provides a visual representation of the optimization process for the spring design. The primary goal is to minimize the weight of the spring while ensuring it meets specific constraints, including minimum deflection, vibration frequency, and shear stress.
Tension spring design
The comparison results between ABO and other algorithms are given in Table 8. The superiority of ABO is evident as it attains the optimal solution with the variables set at x= (0.0518975, 0.36175, 11), resulting in an objective function value of f3= 0.012666.
Comparison of optimum results for tension/compression spring design
Algorithm
d
D
N
Optimal value
CHOA
0.05
0.310598
15
0.0132
ABO
0.0518975
0.36175
11
0.012666
PSO
0.064277
0.74196
3
0.015327
CPSO [22]
0.051728
0.357644
11
0.0126747
WOA
0.056439
0.4746
7
0.013606
GWO
0.057758
0.52053
6
0.013892
CGWO [3]
0.05169
0.356717
11
0.012665
CDE [3]
0.051609
0.354714
11
0.0126702
SCA
0.05
0.310549
15
0.013198
TSA
0.057676
0.51037
6
0.013582
Cantilever Beam Design
The cantilever beam design is given in Fig. 12. The design objective of the problem is to minimize the weight of the cantilever beam and the constraint is to satisfy a vertical displacement constraint [22]. It consists of five hollow members. Each unit is defined by distinct variables with a specific thickness, thus rendering this design problem to encompass five decision variables.
Cantilever beam design
Table 9 presents the optimized outcomes obtained from ABO as well as other competing methods. Notably, ABO achieves the optimal function value of f4=1.3422 by utilizing the structure variables x= (5.8506, 5.4175, 4.4818, 3.6739, 2.0867). The conclusion that can be drawn is that ABO finds the optimal set of structural parameters for the cantilever beam design.
Comparison of optimum results for cantilever beam design
Algorithm
x1
x2
x3
x4
x5
Optimal value
CHOA
6.5777
5.4091
5.2894
3.4565
1.7035
1.4
ABO
5.8506
5.4175
4.4818
3.6739
2.0867
1.3422
WOA
8.0555
3.9969
4.6084
3.4613
2.0585
1.5281
GWO
6.0888
5.5787
4.3128
3.3401
2.2174
1.344
SCA
6.7659
4.833
4.9445
2.9159
3.5721
1.4372
TSA
6.0608
5.7167
4.3507
3.3217
2.1229
1.3461
AOA [23]
6.114257
5.352915
4.085839
3.181646
1.72922
1.3866
ACO [28]
5.0311
4.3266
2.0812
1.7568
3.3254
1.9311
Gear Train Design
The last engineering design problem is the gear train design [11]. As shown in Fig. 13, this problem has four integer decision variables, where Ta, Tb, Td, and Tf represent the number of teeth of four different gears, respectively. Its objective is to minimize the transmission ratio cost of the gear train. The transmission ratio is given as Tb/Ta⋅Td/Tf.
Gear train design
The comparison results obtained by ABO and other optimizers are given in Table 10. The optimal gear parameters obtained by the ABO optimization for this problem are x= (34, 20, 13, 53) and the transmission ratio is f5= 2.3078E−11. The results demonstrate that ABO outperforms other algorithms in identifying the optimal design for the gear train, effectively minimizing the cost of transmission ratio.
Comparison of optimum results for gear train design
Algorithm
Ta
Tb
Td
Tf
Optimal cost
ARO
55
17
21
45
1.36E-09
CHOA
60
25
18
52
2.36E-09
ABO
34
20
13
53
2.3078E-11
PSO
60
13
40
60
2.73E-08
WOA
60
34
14
55
1.36E-09
GWO
59
14
28
46
2.46E-08
SCA
39
23
12
40
2.18E-08
TSA
26
12
15
48
2.36E-09
DE [22]
47
19
15
42
5.5209E-09
Conclusion
In this paper, a new metaheuristic optimization algorithm based on biological population, named ABO, is proposed. ABO mimics the foraging, jousting, mating, and eliminating behaviors of the African bison herd while introducing the parameters S and Q to balance the exploration and exploitation phases. To test the ability of ABO in global exploration and local exploitation, it is tested on 23 benchmark test functions and five constrained real-world engineering problems.
The experimental results of ABO demonstrate its superiority and the ability to solve real-world engineering problems, especially when compared to other existing algorithms. In future work, ABO could be further developed and modified to explore other aspects. One direction is to combine ABO with other algorithms to extend it to more fields, such as neural networks, image processing, etc. Another direction is to expand ABO into a multi-objective optimization tool. Incorporating multi-objective optimization capabilities could greatly enhance the utility of ABO, making it well-suited for solving complex problems with multiple objectives.
None.
Funding Statement
This work was supported in part by the National Natural Science Foundation of China (Grant No. U1731128); the Natural Science Foundation of Liaoning Province (Grant No. 2019-MS-174); the Foundation of Liaoning Province Education Administration (Grant No. LJKZ0279); the Team of Artificial Intelligence Theory and Application for the financial support.
Author Contributions
The authors confirm contribution to the paper as follows: study conception and design: Jian Zhao, Kang Wang; data collection: Jian Zhao; analysis and interpretation of results: Jian Zhao, Kang Wang, Jiacun Wang; draft manuscript preparation: Kang Wang, Jian Zhao, Xiwang Guo, Liang Qi. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
Not applicable.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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