Improved African Vulture Optimization Algorithm Based on Quasi-Oppositional Differential Evolution Operator

In this study, an improved African vulture optimization algorithm (IAVOA) that combines the African vulture optimization algorithm (AVOA) with both quasi-oppositional learning and differential evolution is proposed to address specific drawbacks of the AVOA, including low population diversity, bad development capability, and unbalanced exploration and development capabilities. The improved algorithm has three parts. First, quasi-oppositional learning is introduced in the population initialization and exploration stages to improve population diversity. Second, a differential evolution operator is introduced in the local search position update of each population to improve exploration capability. Third, adaptive parameters are introduced to the differential evolution operator, thus balancing the algorithm exploration and development. A numerical simulation experiment based on 36 different types of benchmark functions showed that while the IAVOA can enhance the convergence speed and solution accuracy of the basic AVOA and two variants of AVOA, IAVOA outperforms the other 7 swarm intelligence algorithms in the mean and best values of 33 benchmark functions.

is proposed as a new method to solve this kind of prob-23 lems [1], [2], [3], [4]. Approximation algorithms are divided 24 into two categories: heuristic and meta-heuristic. Heuristic 25 The associate editor coordinating the review of this manuscript and approving it for publication was Nikhil Padhi .
algorithms have received less attention because they are 26 targeted at certain optimization problems and are prone to 27 fall into local optima. In recent years, metaheuristic algo- 28 rithms have attracted considerable attention in various fields. 29 Compared with traditional optimization algorithms, meta-30 heuristic algorithms are simple, easy to implement, require 31 no gradient information, and can avoid local optima. Thus, 32 this type of algorithm has been widely used in various dis- 33 ciplines and engineering applications to solve optimization 34 problems [5], [6], [7], [8]. Metaheuristic algorithms mainly 35 include optimization algorithms inspired by swarm intelli- 36 gence, natural evolution, physical laws, and human behav-37 ior [9]. Examples include the particle swarm optimization 38 algorithm (PSO) [10], fruit fly optimization algorithm [11], 39 tioned above, the AVOA has been proven more efficient 48 on some benchmark functions. Currently, the AVOA has 49 been applied in solving various problems of multiple dis-50 ciplines and engineering optimization fields. For example, 51 Yakout et al. [19] applied different swarm intelligence algo-

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[23] used the collective guidance factor and the adaptive 76 learning operator, respectively, to modify the algorithm, thus 77 improving the accuracy of parameter estimation of SOFCs. 78 Kumar et al. [24] improved the AVOA by orthogonal learn-

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(2) In the later development stage, a Lévy flight mech- 96 anism is added to the AVOA algorithm to make the 97 population more capable of avoiding local optima, but 98 it affects the local search capability of the population in 99 the solution space.

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To address these two problems, a new improved African 101 vulture optimization algorithm (IAVOA) is proposed. First, 102 quasi-oppositional learning is applied to the population ini-103 tialization process and exploration stage to improve popula-104 tion diversity and have more comprehensive global search 105 capabilities in the early stage. Second, in the development 106 stage, a differential evolution operator based on adaptive 107 parameters is proposed to balance the algorithm in the explo-108 ration and development stages and enhance convergence 109 speed and result accuracy. To verify the effectiveness of the 110 proposed IAVOA, 36 test functions were used to conduct sim-111 ulation experiments. The performance results were compared 112 with those of the basic AVOA, the AVOA based on quasi-113 oppositional learning, the AVOA based on a differential evo-114 lution operator, and six other swarm intelligence optimization 115 algorithms. The results show that the proposed IAVOA has a 116 better convergence speed and optimization capability.

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The remainder of this article is organized as follows. 118 In Section 2, the AVOA is described, and the IAVOA is intro-119 duced in Section 3. In Section 4, the simulation experiments 120 conducted to analyze the detailed test results are discussed. 121 The conclusions of this study are presented in Section 5, and 122 planned future work is described 123

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The vulture is a bird native to all parts of the world except 126 Antarctica and Australia. According to research, one of the 127 main features of the bird is its bald head, which helps the 128 vulture keep its body temperature low and prevents bacteria 129 from growing and causing illness. Vultures are carnivores that 130 attack weak and sick animals and even eat their carcasses. 131 Among different species of vultures, the African vulture is 132 the only bird on Earth that can fly above elevations of 11,000 133 meters, traveling to places far away in search of food. African 134 vultures rotate and move to obtain food, sometimes leading to 135 fierce battles between populations after finding a food source. 136 The smaller vultures surround the stronger ones, feeding 137 when the stronger ones are tired. The process of African 138 vultures searching for food prompted Abdollahzadeh et al. 139 [18] to propose a new metaheuristic algorithm, the detailed 140 steps of which are described below.

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Vultures are grouped according to this value. The formula for 153 this process is as follows: In Equation (1)  The calculation formula is shown in Equation (2).
where F represents the vulture starvation rate, iteration rep-     determining the strategy to be adopted. The exploratory stage 210 of vultures can be expressed by Equation (5).
where R t i is obtained and solved by Equation (1), rand 2 and 214 rand 3 are random numbers between [0,1], and lb and ub are 215 upper and lower boundaries, respectively, of the test func-216 tion. Here, D(i) is calculated using Equation (6), indicating 217 the distance between the vulture and the current optimal 218 vulture.
Here, distance vector X represents the distance that vul-221 tures migrate randomly to protect food from other vultures, 222 calculated by 2 × rand.
where D t i is obtained according to Equation can be expressed as Equation (10).
where rand 5 and rand 6 are random numbers between   food. This action is modeled by Equation (14).
292 where d represents the dimensions of the function. In many 293 metaheuristic algorithms, the Lévy flight mechanism is used 294 to improve algorithm performance, and its calculation for-295 mula is shown in Equation (15): In the formula, u and v are random numbers between [0,1], 299 and β is a fixed default number 1.5.

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The implementation process of the AVOA is shown in 301 FIGURE 2.

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[32], [33]. Oppositional learning is defined as    dragonfly algorithm [36], and DE-based hybrid glowworm 339 swarm algorithm [37]. Therefore, the well-known DE/best/1 340 mutation operator is used to enhance the AVOA development where α is the adaptive rate, the formula of which is as  All the algorithms were implemented in MATLAB 2020b 397 and run on Windows 10 Home Chinese 64-bit OS. The CPU 398 was a 3.2-GHz AMD Ryzen R7-5800H, and the RAM was 399 8.00G. Maximum iteration times T and population size N 400 of all algorithms were set to 500 and 30, respectively. Then, 401 each optimization algorithm was run independently 30 times 402 on each benchmark function to take the average, and the 403 obtained results were compared. The relevant parameters of 404 all the algorithms compared are shown in Table 4. The IAVOA was compared with the basic AVOA and two 408 AVOA variants to evaluate the advantage of the improved 409 algorithm on benchmark problems. The two AVOA vari-410 ants were based on quasi-oppositional learning (QOAVOA) 411 and differential evolution (DEAVOA). The average results 412 of the three groups of test functions after 30 independent 413 running experiments are recorded in Tables 6-8 Table 5 shows that the performances of 430 the three improved optimization algorithms were better than 431 that of the traditional AVOA, and the IAVOA was superior 432 to the other algorithms in most test functions. For func-433 tions F1 and F3, the theoretical optimal values of functions 434 could all be obtained. For functions F2 and F4, the average 435 value of the IAVOA could be optimal. For functions F5-F7, 436 no optimal solution could be obtained by the four algorithms. 437 However, the average value of the solutions obtained by the 438 VOLUME 10, 2022  As shown in Table 6, for F8 and F13, the three improved Functions F24-F36 cover the hybrid composition rotated and 468 shifted multimodal cases. Because they require algorithms to 469 balance the exploration and development stages to prevent 470 the algorithms from falling into local optima, they are often 471 used to test algorithm performance. Table 7 shows that, in 472 F24-F29, the performance of the QOAVOA was not as good 473 as that of the AVOA, but the performances of the other 474 two improved algorithms were better. Although the standard 475 deviation of the IAVOA was inferior to that of the AVOA, 476 the difference was small, and the average and optimal values 477 were better than those of the AVOA. In F24 and F25, the 478 performance of the DEAOVA was inferior to that of the 479 AVOA. In F25-F29, the average and the optimal values were 480 better than those of the AVOA, but the difference was not 481 significant. In F30-F36, the mean and optimal values of 482 the three improved algorithms were not much different from 483 those of the AVOA but had a smaller standard deviation 484 and better stability. Thus, the IAVOA maintained a higher 485 level of stability and better ability to balance exploration and 486 development capabilities of these benchmark functions.     faster convergence. In the hybrid algorithm F24-F36, the 499 convergence speeds of the three improved algorithms are 500 faster, and the convergence accuracies are higher than those 501 of the AVOA. Therefore, in most benchmark functions, 502   quasi-oppositional learning and differential evolution opera-  Table 9 and FIGURE 9. The 517 relevant parameters of all the algorithms compared are shown 518 in Table 8 519 As Table 8  to the ''no free lunch'' theorem. For example, the convergence 527 accuracy of the IAVOA is slightly lower than that of AO 528 and the AOA in unimodal optimization functions F5 and F7 529 and multimodal optimization function F13. Therefore, the 530 IAVOA has some limitations and requires further testing and 531 application.

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An IAVOA was proposed that combines the standard AVOA 534 with quasi-oppositional learning and adaptive differential 535 evolution operator. To verify the performance of the IAVOA, 536 36 benchmark functions of different types and dimensions 537 were simulated to analyze the exploration capability, devel-538 opment capability, and convergence performance of the algo-539 rithm. The IAVOA outperformed the basic AVOA, two AVOA 540 variants, and six other population intelligence algorithms. 541 The quasi-oppositional learning and differential evolution 542 operator in an ablation experiment improved the develop-543 ment capability of the AVOA and the ability to balance 544 the exploration and development capabilities. Since the time 545 complexity of the algorithm is increased by the addition of 546 quasi-oppositional learning operators, future plans include 547 further reducing the IAVOA time complexity and applying 548 the IAVOA to practical engineering optimization problems, 549 such as parameter optimization, image processing, data min-550 ing, and feature selection.