An Improved Equilibrium Optimizer with Application in Unmanned Aerial Vehicle Path Planning

Multi-strategy enterprise development optimizer for numerical optimization and constrained problems

Enterprise Development Optimizer (EDO) is a meta-heuristic algorithm inspired by the enterprise development process with strong global search capability. However, the analysis of the EDO algorithm shows that it suffers from the defects of rapidly decreasing population diversity and weak exploitation ability when dealing with complex optimization problems, while its algorithmic structure has room for further enhancement in the optimization process. In order to solve these challenges, this paper proposes a multi-strategy enterprise development optimizer called MSEDO based on basic EDO. A leader-based covariance learning strategy is proposed, aiming to strengthen the quality of search agents and alleviate the weak population diversity of the EDO algorithm in the later search stage through the guiding role of the dominant group and the modifying role of the leader. To dynamically improve the local exploitation capability of the EDO algorithm, a fitness and distance-based leader selection strategy is proposed. In addition, the structure of EDO algorithm is reconstructed and a diversity-based population restart strategy is presented. The strategy is utilized to assist the population to jump out of the local optimum when the population is stuck in search stagnation. Ablation experiments verify the effectiveness of the strategies of the MSEDO algorithm. The performance of the MSEDO algorithm is confirmed by comparing it with five different types of basic and improved metaheuristic algorithms. The experimental results of CEC2017 and CEC2022 show that MSEDO is effective in escaping from local optimums with its favorable exploitation and exploration capabilities. The experimental results of ten engineering constrained problems show that MSEDO has the ability to competently solve real-world complex optimization problems.

EDECO: An Enhanced Educational Competition Optimizer for Numerical Optimization Problems

The Educational Competition Optimizer (ECO) is a newly proposed human-based metaheuristic algorithm. It derives from the phenomenon of educational competition in society with good performance. However, the basic ECO is constrained by its limited exploitation and exploration abilities when tackling complex optimization problems and exhibits the drawbacks of premature convergence and diminished population diversity. To this end, this paper proposes an enhanced educational competition optimizer, named EDECO, by incorporating estimation of distribution algorithm and replacing some of the best individual(s) using a dynamic fitness distance balancing strategy. On the one hand, the estimation of distribution algorithm enhances the global exploration ability and improves the population quality by establishing a probabilistic model based on the dominant individuals provided by EDECO, which solves the problem that the algorithm is unable to search the neighborhood of the optimal solution. On the other hand, the dynamic fitness distance balancing strategy increases the convergence speed of the algorithm and balances the exploitation and exploration through an adaptive mechanism. Finally, this paper conducts experiments on the proposed EDECO algorithm with 29 CEC 2017 benchmark functions and compares EDECO with four basic algorithms as well as four advanced improved algorithms. The results show that EDECO indeed achieves significant improvements compared to the basic ECO and other compared algorithms, and performs noticeably better than its competitors. Next, this study applies EDECO to 10 engineering constrained optimization problems, and the experimental results show the significant superiority of EDECO in solving real engineering optimization problems. These findings further support the effectiveness and usefulness of our proposed algorithm in solving complex engineering optimization challenges.

A Novel Hybrid Improved RIME Algorithm for Global Optimization Problems

The RIME algorithm is a novel physical-based meta-heuristic algorithm with a strong ability to solve global optimization problems and address challenges in engineering applications. It implements exploration and exploitation behaviors by constructing a rime-ice growth process. However, RIME comes with a couple of disadvantages: a limited exploratory capability, slow convergence, and inherent asymmetry between exploration and exploitation. An improved version with more efficiency and adaptability to solve these issues now comes in the form of Hybrid Estimation Rime-ice Optimization, in short, HERIME. A probabilistic model-based sampling approach of the estimated distribution algorithm is utilized to enhance the quality of the RIME population and boost its global exploration capability. A roulette-based fitness distance balanced selection strategy is used to strengthen the hard-rime phase of RIME to effectively enhance the balance between the exploitation and exploration phases of the optimization process. We validate HERIME using 41 functions from the IEEE CEC2017 and IEEE CEC2022 test suites and compare its optimization accuracy, convergence, and stability with four classical and recent metaheuristic algorithms as well as five advanced algorithms to reveal the fact that the proposed algorithm outperforms all of them. Statistical research using the Friedman test and Wilcoxon rank sum test also confirms its excellent performance. Moreover, ablation experiments validate the effectiveness of each strategy individually. Thus, the experimental results show that HERIME has better search efficiency and optimization accuracy and is effective in dealing with global optimization problems.

AMBWO: An Augmented Multi-Strategy Beluga Whale Optimization for Numerical Optimization Problems

Beluga whale optimization (BWO) is a swarm-based metaheuristic algorithm inspired by the group behavior of beluga whales. BWO suffers from drawbacks such as an insufficient exploration capability and the tendency to fall into local optima. To address these shortcomings, this paper proposes augmented multi-strategy beluga optimization (AMBWO). The adaptive population learning strategy is proposed to improve the global exploration capability of BWO. The introduction of the roulette equilibrium selection strategy allows BWO to have more reference points to choose among during the exploitation phase, which enhances the flexibility of the algorithm. In addition, the adaptive avoidance strategy improves the algorithm's ability to escape from local optima and enriches the population quality. In order to validate the performance of the proposed AMBWO, extensive evaluation comparisons with other state-of-the-art improved algorithms were conducted on the CEC2017 and CEC2022 test sets. Statistical tests, convergence analysis, and stability analysis show that the AMBWO exhibits a superior overall performance. Finally, the applicability and superiority of the AMBWO was further verified by several engineering optimization problems.

Enhanced Multi-Strategy Slime Mould Algorithm for Global Optimization Problems

In order to further improve performance of the Slime Mould Algorithm, the Enhanced Multi-Strategy Slime Mould Algorithm (EMSMA) is proposed in this paper. There are three main modifications to SMA. Firstly, a leader covariance learning strategy is proposed to replace the anisotropic search operator in SMA to ensure that the agents can evolve in a better direction during the optimization process. Secondly, the best agent is further modified with an improved non-monopoly search mechanism to boost the algorithm's exploitation and exploration capabilities. Finally, a random differential restart mechanism is developed to assist SMA in escaping from local optimality and increasing population diversity when it is stalled. The impacts of three strategies are discussed, and the performance of EMSMA is evaluated on the CEC2017 suite and CEC2022 test suite. The numerical and statistical results show that EMSMA has excellent performance on both test suites and is superior to the SMA variants such as DTSMA, ISMA, AOSMA, LSMA, ESMA, and MSMA in terms of convergence accuracy, convergence speed, and stability.

Improved multi-strategy artificial rabbits optimization for solving global optimization problems

Artificial rabbits optimization (ARO) is a metaheuristic algorithm based on the survival strategy of rabbits proposed in 2022. ARO has favorable optimization performance, but it still has some shortcomings, such as weak exploitation capacity, easy to fall into local optima, and serious decline of population diversity at the later stage. In order to solve these problems, we propose an improved multi-strategy artificial rabbits optimization, called IMARO, based on ARO algorithm. In this paper, a roulette fitness distance balanced hiding strategy is proposed so that rabbits can find better locations to hide more reasonably. Meanwhile, in order to improve the deficiency of ARO which is easy to fall into local optimum, an improved non-monopoly search strategy based on Gaussian and Cauchy operators is designed to improve the ability of the algorithm to obtain the global optimal solution. Finally, a covariance restart strategy is designed to improve population diversity when the exploitation is stagnant and to improve the convergence accuracy and convergence speed of ARO. The performance of IMARO is verified by comparing original ARO algorithm with six basic algorithms and seven improved algorithms. The results of CEC2014, CEC2017, CEC2022 show that IMARO has a good exploitation and exploration ability and can effectively get rid of local optimum. Moreover, IMARO produces optimal results on six real-world engineering problems, further demonstrating its efficiency in solving real-world optimization challenges.

An Improved Football Team Training Algorithm for Global Optimization

The football team training algorithm (FTTA) is a new metaheuristic algorithm that was proposed in 2024. The FTTA has better performance but faces challenges such as poor convergence accuracy and ease of falling into local optimality due to limitations such as referring too much to the optimal individual for updating and insufficient perturbation of the optimal agent. To address these concerns, this paper presents an improved football team training algorithm called IFTTA. To enhance the exploration ability in the collective training phase, this paper proposes the fitness distance-balanced collective training strategy. This enables the players to train more rationally in the collective training phase and balances the exploration and exploitation capabilities of the algorithm. To further perturb the optimal agent in FTTA, a non-monopoly extra training strategy is designed to enhance the ability to get rid of the local optimum. In addition, a population restart strategy is then designed to boost the convergence accuracy and population diversity of the algorithm. In this paper, we validate the performance of IFTTA and FTTA as well as six comparison algorithms in CEC2017 test suites. The experimental results show that IFTTA has strong optimization performance. Moreover, several engineering-constrained optimization problems confirm the potential of IFTTA to solve real-world optimization problems.

An Improved Multi-Strategy Crayfish Optimization Algorithm for Solving Numerical Optimization Problems

The crayfish optimization algorithm (COA), proposed in 2023, is a metaheuristic optimization algorithm that is based on crayfish's summer escape behavior, competitive behavior, and foraging behavior. COA has a good optimization performance, but it still suffers from the problems of slow convergence speed and sensitivity to the local optimum. To solve these problems, an improved multi-strategy crayfish optimization algorithm for solving numerical optimization problems, called IMCOA, is proposed to address the shortcomings of the original crayfish optimization algorithm for each behavioral strategy. Aiming at the imbalance between local exploitation and global exploration in the summer heat avoidance and competition phases, this paper proposes a cave candidacy strategy and a fitness-distance balanced competition strategy, respectively, so that these two behaviors can better coordinate the global and local optimization capabilities and escape from falling into the local optimum prematurely. The directly foraging formula is modified during the foraging phase. The food covariance learning strategy is utilized to enhance the population diversity and improve the convergence accuracy and convergence speed. Finally, the introduction of an optimal non-monopoly search strategy to perturb the optimal solution for updates improves the algorithm's ability to obtain a global best solution. We evaluated the effectiveness of IMCOA using the CEC2017 and CEC2022 test suites and compared it with eight algorithms. Experiments were conducted using different dimensions of CEC2017 and CEC2022 by performing numerical analyses, convergence analyses, stability analyses, Wilcoxon rank-sum tests and Friedman tests. Experiments on the CEC2017 and CEC2022 test suites show that IMCOA can strike a good balance between exploration and exploitation and outperforms the traditional COA and other optimization algorithms in terms of its convergence speed, optimization accuracy, and ability to avoid premature convergence. Statistical analysis shows that there is a significant difference between the performance of the IMCOA algorithm and other algorithms. Additionally, three engineering design optimization problems confirm the practicality of IMCOA and its potential to solve real-world problems.

Improved Multi-Strategy Sand Cat Swarm Optimization for Solving Global Optimization

The sand cat swarm optimization algorithm (SCSO) is a novel metaheuristic algorithm that has been proposed in recent years. The algorithm optimizes the search ability of individuals by mimicking the hunting behavior of sand cat groups in nature, thereby achieving robust optimization performance. It is characterized by few control parameters and simple operation. However, due to the lack of population diversity, SCSO is less efficient in solving complex problems and is prone to fall into local optimization. To address these shortcomings and refine the algorithm's efficacy, an improved multi-strategy sand cat optimization algorithm (IMSCSO) is proposed in this paper. In IMSCSO, a roulette fitness-distance balancing strategy is used to select codes to replace random agents in the exploration phase and enhance the convergence performance of the algorithm. To bolster population diversity, a novel population perturbation strategy is introduced, aiming to facilitate the algorithm's escape from local optima. Finally, a best-worst perturbation strategy is developed. The approach not only maintains diversity throughout the optimization process but also enhances the algorithm's exploitation capabilities. To evaluate the performance of the proposed IMSCSO, we conducted experiments in the CEC 2017 test suite and compared IMSCSO with seven other algorithms. The results show that the IMSCSO proposed in this paper has better optimization performance.

An improved manta ray foraging optimization algorithm

The Manta Ray Foraging Optimization Algorithm (MRFO) is a metaheuristic algorithm for solving real-world problems. However, MRFO suffers from slow convergence precision and is easily trapped in a local optimal. Hence, to overcome these deficiencies, this paper proposes an Improved MRFO algorithm (IMRFO) that employs Tent chaotic mapping, the bidirectional search strategy, and the Levy flight strategy. Among these strategies, Tent chaotic mapping distributes the manta ray more uniformly and improves the quality of the initial solution, while the bidirectional search strategy expands the search area. The Levy flight strategy strengthens the algorithm's ability to escape from local optimal. To verify IMRFO's performance, the algorithm is compared with 10 other algorithms on 23 benchmark functions, the CEC2017 and CEC2022 benchmark suites, and five engineering problems, with statistical analysis illustrating the superiority and significance of the difference between IMRFO and other algorithms. The results indicate that the IMRFO outperforms the competitor optimization algorithms.

A Multiple Mechanism Enhanced Arithmetic Optimization Algorithm for Numerical Problems

The Arithmetic Optimization Algorithm (AOA) is a meta-heuristic algorithm inspired by mathematical operators, which may stagnate in the face of complex optimization issues. Therefore, the convergence and accuracy are reduced. In this paper, an AOA variant called ASFAOA is proposed by integrating a double-opposite learning mechanism, an adaptive spiral search strategy, an offset distribution estimation strategy, and a modified cosine acceleration function formula into the original AOA, aiming to improve the local exploitation and global exploration capability of the original AOA. In the proposed ASFAOA, a dual-opposite learning strategy is utilized to enhance population diversity by searching the problem space a lot better. The spiral search strategy of the tuna swarm optimization is introduced into the addition and subtraction strategy of AOA to enhance the AOA's ability to jump out of the local optimum. An offset distribution estimation strategy is employed to effectively utilize the dominant population information for guiding the correct individual evolution. In addition, an adaptive cosine acceleration function is proposed to perform a better balance between the exploitation and exploration capabilities of the AOA. To demonstrate the superiority of the proposed ASFAOA, two experiments are conducted using existing state-of-the-art algorithms. First, The CEC 2017 benchmark function was applied with the aim of evaluating the performance of ASFAOA on the test function through mean analysis, convergence analysis, stability analysis, Wilcoxon signed rank test, and Friedman's test. The proposed ASFAOA is then utilized to solve the wireless sensor coverage problem and its performance is illustrated by two sets of coverage problems with different dimensions. The results and discussion show that ASFAOA outperforms the original AOA and other comparison algorithms. Therefore, ASFAOA is considered as a useful technique for practical optimization problems.

Recent Developments in Equilibrium Optimizer Algorithm: Its Variants and Applications

There have been many algorithms created and introduced in the literature inspired by various events observable in nature, such as evolutionary phenomena, the actions of social creatures or agents, broad principles based on physical processes, the nature of chemical reactions, human behavior, superiority, and intelligence, intelligent behavior of plants, numerical techniques and mathematics programming procedure and its orientation. Nature-inspired metaheuristic algorithms have dominated the scientific literature and have become a widely used computing paradigm over the past two decades. Equilibrium Optimizer, popularly known as EO, is a population-based, nature-inspired meta-heuristics that belongs to the class of Physics based optimization algorithms, enthused by dynamic source and sink models with a physics foundation that are used to make educated guesses about equilibrium states. EO has achieved massive recognition, and there are quite a few changes made to existing EOs. This article gives a thorough review of EO and its variations. We started with 175 research articles published by several major publishers. Additionally, we discuss the strengths and weaknesses of the algorithms to help researchers find the variant that best suits their needs. The core optimization problems from numerous application areas using EO are also covered in the study, including image classification, scheduling problems, and many others. Lastly, this work recommends a few potential areas for EO research in the future.

Research on Parameter Optimization Method of Sliding Mode Controller for the Grid-Connected Composite Device Based on IMFO Algorithm

In order to make the grid-connected composite device (GCCD) controller meet the requirements of different operating modes and complex working conditions of power grid, this paper proposes to introduce sliding mode control (SMC) into GCCD controller. Firstly, the mathematical model of MMC converter is established, and the sliding mode controller is designed based on the SMC principle. Then, aiming at the problems of complex controller structure and difficult parameter tuning in multiple modes of the GCCD, this paper proposes a controller parameter optimization method based on improved Month Flame optimization (IMFO) algorithm. This method improves the MFO algorithm by introducing good point set (GPS) initialization and Levy flight strategy, which accelerates the convergence speed of the algorithm while avoiding falling into local optimization, and realizes the optimization of converter controller parameters. Under a variety of standard test functions, the advantages of the proposed IMFO algorithm are verified by comparing it with the traditional algorithm. Finally, in order to realize the automatic tuning of control parameters, the Python-PSCAD joint simulation method is studied and implemented. Taking the comprehensive integral of time and absolute error (CITAE) index as the objective function, the parameters of the sliding mode controller are optimized. The simulation results show that the controller parameters optimized by the IMFO algorithm can make the GCCD have better dynamic performance.

Path planning optimization in unmanned aerial vehicles using meta-heuristic algorithms: a systematic review

Unmanned aerial vehicles (UAVs) have recently been increasingly popular in various areas, fields, and applications. Military, disaster management, rescue operations, public services, agriculture, and various other areas are examples. As a result, UAV path planning is concerned with determining the optimal path from the source to the destination while avoiding collisions with lowering the cost of time, energy, and other resources. This review aims to assort academic studies on the path planning optimization in UAV using meta-heuristic algorithms, summarize the results of each optimization algorithm, and extend the understanding of the current state of the path planning in UAV in the meta-heuristic optimization field. For this purpose, we implemented a broad, automated search using Boolean and snowballing searching methods to find academic works on path planning in UAVs. Studies and papers have been distinguished, and the following information was obtained and aggregated from each article: authors, publication's year, the journal name or the conference name, proposed algorithms, the aim of the study, the outcome, and the quality of each study. According to the findings, the meta-heuristic algorithm is a standard optimization method for tackling single and multi-objective problems. Besides, the findings show that meta-heuristic algorithms have a great compact on the path planning optimization in UAVs, and there is good progress in this field. However, the problem still exists mainly in complex and dynamic environments, on battlefields, in rescue missions, mobile obstacles, and with multiple UAVs.

A Modified Reptile Search Algorithm for Numerical Optimization Problems

The reptile search algorithm (RSA) is a swarm-based metaheuristic algorithm inspired by the encirclement and hunt mechanisms of crocodiles. Compared with other algorithms, RSA is competitive but still suffers from low population diversity, unbalanced exploitation and exploration, and the tendency to fall into local optima. To overcome these shortcomings, a modified variant of RSA, named MRSA, is proposed in this paper. First, an adaptive chaotic reverse learning strategy is employed to enhance the population diversity. Second, an elite alternative pooling strategy is proposed to balance exploitation and exploration. Finally, a shifted distribution estimation strategy is used to correct the evolutionary direction and improve the algorithm performance. Subsequently, the superiority of MRSA is verified using 23 benchmark functions, IEEE CEC2017 benchmark functions, and robot path planning problems. The Friedman test, the Wilcoxon signed-rank test, and simulation results show that the proposed MRSA outperforms other comparative algorithms in terms of convergence accuracy, convergence speed, and stability.

A novel equilibrium optimized double-loop control scheme for unstable and integrating chemical processes involving dead time

Integrating and unstable chemical processes showcase instability in open-loop configuration due to the existence of poles at the origin and right-half of the s-plane. They present challenging control requirements due to their non-self-regulating nature. The presence of dead time demands more sophisticated control requirements for the above-mentioned processes. So double-loop control strategies are preferred over PID controllers in single-loop configuration. In this work, a novel IMC-PD double-loop control strategy is proposed for unstable and integrating plants with dead time. The inner-loop consists of PD controller whose initial settings are derived using Routh–Hurwitz stability conditions. The outer-loop consists of an IMC controller whose parameter along with that of the PD controller is optimized using the metaheuristic algorithm called equilibrium optimizer algorithm (EOA). EOA utilizes the range of controller settings from RH criteria for stable operation and provides the optimal settings by minimizing the integral square error (ISE). Merits of the suggested strategy is illustrated with the help of benchmark plant models of unstable/integrating chemical processes and that of a bioreactor. By computing quantitative performance measures, the dynamic responses resulting from the proposed control scheme is found to be more effective than the reported works.

Multi-UAV cooperative reconnaissance mission planning novel method under multi-radar detection

Past swarm intelligence algorithms for solving UAV path planning problems have suffered from slow convergence, lack of complex constraints and guidance for local optimisation. It no longer meets the requirements of the Multi-UAV Cooperative Reconnaissance Mission Planning (MUCRMP) problem in the context of multi-radar detection. In this paper, a global optimisation model with the objective of a shorter distance within radar detection range of the UAV is proposed at first, including the planning of reconnaissance sequence between and within target groups, relative position to targets. More importantly, the imaging characteristics of the UAV and its minimum turning radius have been considered in depth in this study. Then an improved synthetic heuristic algorithm is proposed to solve the model, which obtains valuable reconnaissance mission plan. Finally, an example solution for a problem with 68 target point sizes is carried out, and the validity and feasibility of the model and algorithm are illustrated through the analysis given. Compared with the existing algorithms, the improved synthetic heuristic algorithm can give better anti-radar attributes to the UAV and efficiently improved the convergence speed in the specific reconnaissance mission.

A Tent Marine Predators Algorithm with Estimation Distribution Algorithm and Gaussian Random Walk for Continuous Optimization Problems

The marine predators algorithm (MPA) is a novel population-based optimization method that has been widely used in real-world optimization applications. However, MPA can easily fall into a local optimum because of the lack of population diversity in the late stage of optimization. To overcome this shortcoming, this paper proposes an MPA variant with a hybrid estimation distribution algorithm (EDA) and a Gaussian random walk strategy, namely, HEGMPA. The initial population is constructed using cubic mapping to enhance the diversity of individuals in the population. Then, EDA is adapted into MPA to modify the evolutionary direction using the population distribution information, thus improving the convergence performance of the algorithm. In addition, a Gaussian random walk strategy with medium solution is used to help the algorithm get rid of stagnation. The proposed algorithm is verified by simulation using the CEC2014 test suite. Simulation results show that the performance of HEGMPA is more competitive than other comparative algorithms, with significant improvements in terms of convergence accuracy and convergence speed.

A Hybrid Butterfly Optimization Algorithm for Numerical Optimization Problems

The butterfly optimization algorithm (BOA) is a swarm-based metaheuristic algorithm inspired by the foraging behaviour and information sharing of butterflies. BOA has been applied to various fields of optimization problems due to its performance. However, BOA also suffers from drawbacks such as diminished population diversity and the tendency to get trapped in local optimum. In this paper, a hybrid butterfly optimization algorithm based on a Gaussian distribution estimation strategy, called GDEBOA, is proposed. A Gaussian distribution estimation strategy is used to sample dominant population information and thus modify the evolutionary direction of butterfly populations, improving the exploitation and exploration capabilities of the algorithm. To evaluate the superiority of the proposed algorithm, GDEBOA was compared with six state-of-the-art algorithms in CEC2017. In addition, GDEBOA was employed to solve the UAV path planning problem. The simulation results show that GDEBOA is highly competitive.

A Modified Slime Mould Algorithm for Global Optimization

Slime mould algorithm (SMA) is a population-based metaheuristic algorithm inspired by the phenomenon of slime mould oscillation. The SMA is competitive compared to other algorithms but still suffers from the disadvantages of unbalanced exploitation and exploration and is easy to fall into local optima. To address these shortcomings, an improved variant of SMA named MSMA is proposed in this paper. Firstly, a chaotic opposition-based learning strategy is used to enhance population diversity. Secondly, two adaptive parameter control strategies are proposed to balance exploitation and exploration. Finally, a spiral search strategy is used to help SMA get rid of local optimum. The superiority of MSMA is verified in 13 multidimensional test functions and 10 fixed-dimensional test functions. In addition, two engineering optimization problems are used to verify the potential of MSMA to solve real-world optimization problems. The simulation results show that the proposed MSMA outperforms other comparative algorithms in terms of convergence accuracy, convergence speed, and stability.

Tuna Swarm Optimization: A Novel Swarm-Based Metaheuristic Algorithm for Global Optimization

In this paper, a novel swarm-based metaheuristic algorithm is proposed, which is called tuna swarm optimization (TSO). The main inspiration for TSO is based on the cooperative foraging behavior of tuna swarm. The work mimics two foraging behaviors of tuna swarm, including spiral foraging and parabolic foraging, for developing an effective metaheuristic algorithm. The performance of TSO is evaluated by comparison with other metaheuristics on a set of benchmark functions and several real engineering problems. Sensitivity, scalability, robustness, and convergence analyses were used and combined with the Wilcoxon rank-sum test and Friedman test. The simulation results show that TSO performs better compared to other comparative algorithms.