Optimization algorithms play a pivotal role in enhancing the performance and efficiency of systems across various scientific and engineering disciplines. To enhance the performance and alleviate the limitations of the Northern Goshawk Optimization (NGO) algorithm, particularly its tendency towards premature convergence and entrapment in local optima during function optimization processes, this study introduces an advanced Improved Northern Goshawk Optimization (INGO) algorithm. This algorithm incorporates a multifaceted enhancement strategy to boost operational efficiency. Initially, a tent chaotic map is employed in the initialization phase to generate a diverse initial population, providing high-quality feasible solutions. Subsequently, after the first phase of the NGO’s iterative process, a whale fall strategy is introduced to prevent premature convergence into local optima. This is followed by the integration of T-distribution mutation strategies and the State Transition Algorithm (STA) after the second phase of the NGO, achieving a balanced synergy between the algorithm’s exploitation and exploration. This research evaluates the performance of INGO using 23 benchmark functions alongside the IEEE CEC 2017 benchmark functions, accompanied by a statistical analysis of the results. The experimental outcomes demonstrate INGO’s superior achievements in function optimization tasks. Furthermore, its applicability in solving engineering design problems was verified through simulations on Unmanned Aerial Vehicle (UAV) trajectory planning issues, establishing INGO’s capability in addressing complex optimization challenges.

The growing complexity of optimization problems, combined with the limitations of traditional algorithms, has underscored the need for developing meta-heuristic algorithms in recent years [

Metaheuristic algorithms are categorized into four types: Evolutionary, swarm intelligence-based, physicochemical law-grounded, and social or human behavior-modeled. The Genetic Algorithm (GA) [

Swarm intelligence algorithms offer significant benefits and potential for solving real-world engineering problems. As a result, numerous novel and enhanced swarm intelligence algorithms have been developed to address practical challenges. Introduced in the early 1990s, the Ant Colony Algorithm (ACO) [

Dehghani et al. [

Despite the emergence of numerous NGO variants, they exhibit limitations and excel only in specific problems. This reality persists regardless of the type or extent of modifications researchers apply to any algorithm, as each harbors its inherent constraints. The No Free Lunch (NFL) theorem [

This study synthesizes the high convergence accuracy of physics-based algorithms and the rapid convergence speed of swarm intelligence algorithms by incorporating the STA into the optimization process of NGO, thereby proposing a novel algorithm with superior performance, denoted as INGO.

In the initialization phase, INGO utilizes Tent chaotic mapping to seed the population, thereby augmenting population diversity and boosting the chances of circumventing local optima in pursuit of superior solutions.

The integration of the BWO’s whale fall strategy and the T-distribution variation strategy as mutation strategies significantly augments the global search capability of NGO.

The incorporation of the STA facilitates sampling and comparison within the vicinity of the optimal solution in each iteration of the NGO, thereby enhancing the precision of solution convergence.

The performance of INGO was tested and evaluated using 53 benchmark functions and a path planning scenario of UAV.

The structure of this paper is as follows:

Inspired by the predatory behavior of the northern goshawk, NGO takes each northern goshawk as an individual of a population and abstracts the predatory behavior into two stages. The first stage is to explore and attack prey, which reflects the group behavior by identifying the position of the current best solution and influencing each individual to update their position through the optimal solution, representing the global exploration ability. The second stage is the escape behavior of prey, symbolizing the local exploitation ability.

NGO considers each northern goshawk as an individual of a population, that is, a viable solution, and searches for the best value in the viable solution space by moving. Like most swarm intelligence algorithms, NGO generates initial populations by randomly generating initial populations. In the mathematical model, each individual represents a D-dimensional vector, N individuals constitute the entire population, and the population is an N × D matrix. The mathematical model of the population is given by

Here, the population

The search and attack of prey is the first stage of the NGO in each iteration, which simulates the optimal solution as prey and each individual launches the attack it. In this stage, every individual identifies the location of the current best solution and updates their position based on this optimal solution. The main purpose is to enable the northern goshawk individual to search the feasible solution space more widely. Thus, global search is carried out.

The second stage is the chase and escape of prey, and the local exploitation is carried out by simulating the local activities of prey to affect the individual position of the northern goshawk. The position updating formula of the second stage is shown in

Each iteration within the NGO process encompasses these two phases. The prevailing optimal solution serves as a beacon for other members to gravitate towards improved outcomes. Upon reaching the maximum iteration limit, the algorithm concludes.

The essence of swarm intelligence algorithm lies in the algorithm’s capability to explore and exploit. The improvement of the so-called swarm intelligence algorithm is to help algorithms to strike a balance between the exploration and exploitation. This facilitates the algorithm’s capacity to not only broadly search the feasible solution space but also to refine its approach in achieving a closer approximation to the optimal solution with greater accuracy. Based on previous research, this section presents four effective improvement strategies for function optimization of NGO. By combining all four strategies, the NGO is strengthened in terms of its ability to explore and exploit.

Given the lack of prior knowledge about the global optimal solution for the optimization problem, it is advisable to distribute the population evenly throughout the search space. However, the randomly generated initial population is frequently distributed unevenly, thereby impacting the diversity of the population. Although the intelligent algorithm is insensitive to the initialization method and there are no notable disparities among the methods of population initialization after enough iterations [

Chaotic sequence is used to generate initial population instead of randomly generated initial population in this paper. Tent mapping is widely used in the initial improvement of swarm intelligent algorithm due to its ability to generate more uniform chaotic sequence [

The convergence speed of NGO is quick, which it also makes the local optimal value affecting the algorithm tremendously in the early stage, resulting in the low global search ability of it. In the first stage of NGO search, there is a phenomenon of rapid convergence, and it is quite probable to fall into local optimality. Therefore, improving the global exploratory ability is of great significance to improve NGO performance.

Targeted addition of some strategies is often an effective technique to improve the global exploratory capability of the algorithm. Whale fall refers to the phenomenon wherein a whale dies and its body descends to the bottom of the ocean and it is one of the main reasons life in the deep ocean flourishes. The whale fall phenomenon was introduced into the beluga optimization algorithm. The BWO gives a model of the falling behavior of the whale after death in each iteration. An updated position is established by using the beluga whale’s position and the length of its fall, along with a tiny probability.

Taking into account NGO’s inclination to get stuck in local optimum, this paper incorporated the whale fall phenomenon into the first stage search of it, representing the behavior of the northern goshawk, which has a very small probability of falling or was affected by other random individuals for other reasons after the large-scale exploration behind the first stage search, assists the algorithm in increasing its chances of breaking free from local optimal values.

In the process of each iteration of the NGO, the whale landing operation is carried out after the completion of the prey discovery and attack stage, which can better conduct a global search, so that the algorithm approaches the global optimal solution with a greater probability.

Variation is the most common strategy used for swarm intelligence algorithms to fall into local optimality. Cauchy variation and Gaussian variation are common. According to the characteristics of the NGO, this paper adopts T-distribution mutation strategy [

Here,

Here,

During the initial phase of iteration, the T-distribution resembles the Cauchy distribution, mitigating the impact of local extremities. This alteration increases the likelihood of the algorithm breaking free from local optima, thereby boosting its global exploratory capabilities. In the subsequent phases, the T-distribution approaches the characteristics of a Gaussian distribution, facilitating improved local exploitation and enhanced convergence precision for the algorithm, while simultaneously quickening the rate of convergence.

Algorithm fusion is the trend of swarm intelligence algorithm improvement recently. The main purpose is to combine the advantages of various algorithms so that new algorithms have stronger exploration and exploitation ability. STA is a competitive intelligent algorithm proposed by Zhou in 2012. It uses the state space expression as the framework to design four transformation operators, considering the process of the feasible solution iteration as a process of state transition. Based on the current optimal solution, some state transition operator is run, and the candidate solution set is generated by generating neighborhood and sampling in the neighborhood, and then compared with the current best solution, a new optimal solution is obtained.

STA utilizes the state space expression as a framework to simulate the iterative updating process of the solution, as shown in

Here,

The expression of the rotation transformation operator is shown in

Here,

The expression of translation transformation operator is shown in

Here,

The expression of expansion transformation operator is shown in

Here,

The expression of axesion transformation operator is shown in

Here,

For the existing optimal solution, a neighborhood is constructed through a candidate solution set generated by a specific transformation operator. Sampling within this neighborhood is executed via a defined sampling method, from which the superior solution is identified. Subsequently, optimal solutions undergo updates through greedy mechanisms; namely, comparing the derived solutions with the current optimal and selecting the most advantageous one. The greedy mechanism is shown in

Aimed at the characteristics of NGO, which has high search speed but low search accuracy, STA is introduced into NGO to enhance its convergence accuracy, so that it could get better optimization results when solving function optimization problems. STA possesses strong local exploitation capability and is capable of thoroughly exploring the surrounding space of the current optimal solution. Upon discovering the best outcome during each NGO iteration, STA employs its four transformation operators to explore the vicinity of the optimal solution, thereby guaranteeing enhanced local exploitation of the algorithm.

Algorithm 1 outlines INGO’s pseudo-code. At line 3, the Tent chaotic map initializes the population. Line 11 updates positions through the whale fall strategy, and line 18 applies the T-distribution for variations. STA’s utilization for neighborhood searches of the optimal solution is marked at line 20.

This section mainly analyzes the performance of INGO. Firstly, the effectiveness of the strategy is analyzed by gradually adding the strategy, secondly, the scalability of the algorithm is verified by testing on different dimensions, and finally, the algorithm is compared with other algorithms on 23 benchmark functions [

Algorithm | Parameter |
---|---|

AO | |

HPO | C: [1,0.02], |

SCSO | Sensitivity range: [2,0] |

HHO | Escaping probability: 0.5 |

Escaping energy: 0.5 | |

WOA | |

GWO | ^{*}rand (0, 1) |

SCA | N/A |

DA | |

SSA | p: 0.5 |

BWO | N/A |

INGOfu |

A set of experiments is set up in this section to analyze the effect of the four strategies introduced in INGO algorithm on the algorithm. Four strategies are added to NGO successively, and different combinations are shown in

Name | A | B | C | D |
---|---|---|---|---|

NGO | 0 | 0 | 0 | 0 |

NGO-A | 1 | 0 | 0 | 0 |

NGO-B | 1 | 1 | 0 | 0 |

NGO-C | 1 | 1 | 1 | 0 |

NGO-D | 1 | 1 | 1 | 1 |

Index | FUN | F1 | F2 | F3 | F4 | F5 | F6 |
---|---|---|---|---|---|---|---|

NGO | MEAN | 1.27E−88 | 7.12E−46 | 1.87E−20 | 1.26E−37 | 2.53E+01 | 1.94E−05 |

STD | 1.86E−88 | 4.71E−46 | 1.02E−19 | 1.56E−37 | 4.20E−01 | 1.02E−04 | |

NGO-A | MEAN | 1.33E−88 | 9.78E−46 | 5.52E−22 | 1.25E−37 | 2.52E+01 | 1.41E−05 |

STD | 2.08E−88 | 7.57E−46 | 1.54E−21 | 1.17E−37 | 4.05E−01 | 5.76E−06 | |

NGO-B | MEAN | 5.82E−89 | 2.84E−46 | 8.74E−23 | 3.25E−38 | 1.38E−06 | 3.09E−06 |

STD | 1.49E−88 | 2.78E−46 | 4.25E−22 | 2.05E−38 | 4.52E−06 | 2.14E−06 | |

NGO-C | MEAN | 4.23E−07 | 7.13E−07 | ||||

STD | 5.09E−07 | 3.75E−07 | |||||

NGO-D | MEAN | ||||||

STD | |||||||

Index | FUN | F8 | F9 | F10 | F11 | F12 | F13 |

NGO | MEAN | −7.69E+03 | 5.98E−15 | 1.04E−06 | 6.36E−02 | ||

STD | 4.87E+02 | 1.79E−15 | 4.41E−06 | 6.82E−02 | |||

NGO-A | MEAN | −7.64E+03 | 7.21E+00 | 8.82E−07 | 7.01E−02 | ||

STD | 1.36E+03 | 7.56E+00 | 4.44E−07 | 5.86E−02 | |||

NGO-B | MEAN | −3.17E+41 | 6.10E−15 | 7.63E−08 | 1.81E−10 | ||

STD | 1.58E+42 | 1.80E−15 | 1.35E−07 | 2.76E−10 | |||

NGO-C | MEAN | 8.88E−16 | 2.11E−08 | 1.52E−10 | |||

STD | 1.00E−31 | 1.29E−08 | 2.23E−10 | ||||

NGO-D | MEAN | −1.70E+23 | |||||

STD | 7.68E+23 |

Name | NGO | NGO-A | NGO-B | NGO-C | NGO-D |
---|---|---|---|---|---|

F1 | 2.636E+06 | 4.271E+06 | 1.621E+05 | 1.319E+05 | |

F2 | 5.091E+16 | 5.719E+18 | 1.003E+16 | 1.466E+16 | |

F3 | 6.222E+04 | 6.157E+04 | 4.998E+04 | 4.104E+04 | |

F4 | 5.184E+02 | 5.117E+02 | 5.124E+02 | 4.973E+02 | |

F5 | 6.836E+02 | 6.676E+02 | 6.770E+02 | 6.362E+02 | |

F6 | 6.256E+02 | 6.266E+02 | 6.058E+02 | 6.080E+02 | |

F7 | 9.855E+02 | 9.833E+02 | 9.377E+02 | 8.950E+02 | |

F8 | 9.528E+02 | 9.463E+02 | 9.420E+02 | 9.351E+02 | |

F9 | 3.379E+03 | 3.464E+03 | 1.647E+03 | 1.621E+03 | |

F10 | 5.431E+03 | 5.647E+03 | 6.135E+03 | 6.220E+03 | |

F11 | 1.235E+03 | 1.246E+03 | 1.243E+03 | 1.234E+03 | |

F12 | 1.201E+06 | 1.336E+06 | 1.936E+06 | 1.654E+06 | |

F13 | 1.951E+04 | 2.350E+04 | 7.213E+03 | 8.851E+03 | |

F14 | 1.276E+04 | 2.550E+04 | 1.696E+04 | 1.577E+04 | |

F15 | 4.986E+03 | 4.726E+03 | 3.147E+03 | 3.773E+03 | |

F16 | 2.750E+03 | 2.724E+03 | 3.138E+03 | 2.628E+03 | |

F17 | 2.038E+03 | 1.999E+03 | 2.024E+03 | 1.946E+03 | |

F18 | 1.532E+05 | 1.834E+05 | 7.345E+05 | 3.544E+05 | |

F19 | 5.196E+03 | 6.539E+03 | 5.851E+03 | 5.953E+03 | |

F20 | 2.439E+03 | 2.412E+03 | 2.589E+03 | 2.440E+03 | |

F21 | 2.441E+03 | 2.429E+03 | 2.387E+03 | 2.381E+03 | |

F22 | 2.313E+03 | 2.653E+03 | 2.317E+03 | ||

F23 | 2.797E+03 | 2.797E+03 | 2.752E+03 | 2.750E+03 | |

F24 | 2.933E+03 | 2.935E+03 | 2.948E+03 | ||

F25 | 2.932E+03 | 2.928E+03 | 2.897E+03 | 2.896E+03 | |

F26 | 3.899E+03 | 5.179E+03 | 4.256E+03 | 4.349E+03 | |

F27 | 3.227E+03 | 3.230E+03 | |||

F28 | |||||

F29 | 4.040E+03 | 3.906E+03 | 3.833E+03 | 3.730E+03 | |

F30 | 2.205E+04 | 2.276E+04 | 1.856E+04 | 1.690E+04 |

The dimensionality of optimization problems exerts a discernible influence on the operational efficiency of algorithms. The intricacy of high-dimensional optimization problems serves as a more effective measure for assessing an algorithm’s performance. In this section, the first 13 functions of 23 benchmark functions (F1–F13) are used to conduct dimensional tests on INGO. The performance of NGO and INGO was tested on four dimensions: 30, 50, 100 and 500.

F/dim | 30 | 50 | 100 | 500 | ||||
---|---|---|---|---|---|---|---|---|

NGO | INGO | NGO | INGO | NGO | INGO | NGO | INGO | |

F1 | 2.35E−88 | 1.50E−88 | 2.11E−88 | 2.33E−88 | ||||

F2 | 6.56E−46 | 8.13E−46 | 8.37E−46 | 8.70E−46 | ||||

F3 | 1.92E−21 | 3.15E−22 | 4.51E−22 | 5.05E−22 | ||||

F4 | 7.99E−38 | 1.27E−37 | 9.21E−38 | 1.02E−37 | ||||

F5 | 2.55E+01 | 2.54E+01 | 2.54E+01 | 2.54E+01 | ||||

F6 | 9.66E−07 | 3.27E−06 | 1.63E−06 | 9.13E−07 | ||||

F7 | 1.46E−02 | 1.42E−02 | 1.77E−02 | 1.70E−02 | ||||

F8 | −7.75E+03 | −7.74E+03 | −7.67E+03 | −7.69E+03 | ||||

F9 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | ||||

F10 | 5.74E−15 | 5.39E−15 | 5.63E−15 | 6.10E−15 | ||||

F11 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | ||||

F12 | 2.10E−7 | 1.88E−07 | 2.29E−07 | 4.18E−07 | ||||

F13 | 7.55E−02 | 7.75E−02 | 7.75E−02 | 6.07E−02 |

INGO’s function optimization performance is tested using 23 classic benchmark functions. There are 7 single-peak functions (F1–F7), 6 multi-peak functions (F8–F13) and 10 fixed-dimension multi-peak test functions (F14–F23). The algorithm’s exploitation ability is verified using unimodal functions. Multimodal function is used to reveal the algorithm’s exploration ability. INGO is compared with the NGO, AO, HPO, SCSO, HHO, WOA, GWO, SCA and DA.

Function | Index | INGO | NGO | AO | HPO | SCSO | HHO | WOA | GWO | DA | SCA |
---|---|---|---|---|---|---|---|---|---|---|---|

F1 | MEAN | 1.02E−177 | 5.18E−200 | 0.00E+00 | 1.10E−231 | 1.01E−183 | 5.36E−147 | 6.18E−59 | 1.29E+03 | 3.21E−02 | |

STD | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 2.92E−146 | 2.16E−58 | 6.18E+02 | 6.63E−02 | ||

F2 | MEAN | 1.37E−92 | 6.60E−100 | 2.61E−186 | 4.33E−123 | 5.46E−95 | 2.02E−101 | 1.09E−34 | 1.55E+01 | 2.03E−05 | |

STD | 2.11E−92 | 3.28E−99 | 0.00E+00 | 1.35E−122 | 2.98E−94 | 1.10E−100 | 1.60E−34 | 5.87E+00 | 4.60E−05 | ||

F3 | MEAN | 1.65E−49 | 7.13E−199 | 1.14E−299 | 1.15E−197 | 4.38E−144 | 1.96E+04 | 1.07E−13 | 9.77E+03 | 4.49E+03 | |

STD | 5.47E−49 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 2.40E−143 | 1.14E+04 | 3.27E−13 | 5.57E+03 | 4.56E+03 | ||

F4 | MEAN | 2.38E−76 | 1.30E−103 | 1.74E−153 | 1.32E−103 | 6.03E−94 | 4.93E+01 | 1.51E−14 | 2.04E+01 | 2.09E+01 | |

STD | 4.66E−76 | 6.92E−103 | 9.51E−153 | 4.33E−103 | 2.42E−93 | 3.09E+01 | 1.52E−14 | 7.45E+00 | 1.19E+01 | ||

F5 | MEAN | 2.37E+01 | 1.08E−03 | 2.17E+01 | 2.77E+01 | 2.62E−03 | 2.71E+01 | 2.69E+01 | 1.54E+05 | 9.46E+02 | |

STD | 5.25E−01 | 1.24E−03 | 5.03E−01 | 8.33E−01 | 3.69E−03 | 3.84E−01 | 7.52E−01 | 1.57E+05 | 2.21E+03 | ||

F6 | MEAN | 6.59E−08 | 1.74E−05 | 4.00E−11 | 1.96E+00 | 4.44E−05 | 4.25E−02 | 5.54E−01 | 1.35E+03 | 4.68E+00 | |

STD | 6.37E−08 | 4.72E−05 | 2.11E−10 | 5.77E−01 | 5.82E−05 | 5.83E−02 | 3.35E−01 | 7.14E+02 | 3.74E−01 | ||

F7 | MEAN | 3.28E−02 | 2.77E−04 | 9.67E−05 | 1.00E−04 | 7.76E−05 | 2.08E−03 | 8.55E−04 | 3.47E−01 | 3.30E−02 | |

STD | 2.69E−02 | 9.17E−05 | 9.83E−05 | 1.11E−04 | 6.09E−05 | 2.24E−03 | 6.16E−04 | 1.73E−01 | 1.96E−02 |

The test results for the multimodal functions F8–F13 are displayed in

Function | Index | INGO | NGO | AO | HPO | SCSO | HHO | WOA | GWO | DA | SCA |
---|---|---|---|---|---|---|---|---|---|---|---|

F8 | MEAN | −7.93E+03 | −1.01E+04 | −9.29E+03 | −6.73E+03 | −1.26E+04 | −1.06E+04 | −5.99E+03 | −5.81E+03 | 3.87E+03 | |

STD | 5.26E+02 | 3.62E+03 | 6.69E+02 | 6.97E+02 | 2.84E−01 | 1.88E+03 | 8.00E+02 | 8.19E+02 | 2.55E+02 | ||

F9 | MEAN | 3.79E−15 | 4.50E−01 | 1.67E+02 | 1.56E+01 | ||||||

STD | 1.44E−14 | 1.51E+00 | 3.37E+01 | 2.41E+01 | |||||||

F10 | MEAN | 5.86E−15 | 4.09E−15 | 1.69E−14 | 8.99E+00 | 9.90E+00 | |||||

STD | 1.77E−15 | 1.71E−15 | 2.91E−15 | 1.47E+00 | 9.49E+00 | ||||||

F11 | MEAN | 1.86E−03 | 2.34E−03 | 1.13E+01 | 2.91E−01 | ||||||

STD | 1.02E−02 | 6.46E−03 | 4.84E+00 | 2.70E−01 | |||||||

F12 | MEAN | 1.56E−12 | 5.33E−09 | 1.11E−06 | 1.02E−01 | 1.44E−06 | 7.71E−03 | 3.91E−02 | 1.60E+02 | 4.05E+01 | |

STD | 1.19E−12 | 4.51E−09 | 1.82E−06 | 5.00E−02 | 1.57E−06 | 9.67E−03 | 2.04E−02 | 4.15E+02 | 1.99E+02 | ||

F13 | MEAN | 7.88E−02 | 5.21E−06 | 9.54E−02 | 2.38E+00 | 1.69E−05 | 2.83E−01 | 4.79E−01 | 7.00E+04 | 6.34E+00 | |

STD | 8.99E−02 | 6.59E−06 | 1.36E−01 | 3.60E−01 | 2.09E−05 | 2.52E−01 | 2.05E−01 | 1.27E+05 | 8.19E+00 |

The experimental outcomes for functions F14–F23 are detailed in

Function | Index | INGO | NGO | AO | HPO | SCSO | HHO | WOA | GWO | SCA | DA |
---|---|---|---|---|---|---|---|---|---|---|---|

F14 | MEAN | 1.98E+00 | 2.50E+00 | 3.48E+00 | 1.03E+00 | 2.02E+00 | 4.45E+00 | 1.03E+00 | 1.26E+00 | ||

STD | 2.72E+00 | 2.93E+00 | 3.79E+00 | 1.81E−01 | 2.00E+00 | 4.71E+00 | 1.81E−01 | 6.86E−01 | |||

F15 | MEAN | 4.28E−04 | 3.06E−03 | 3.78E−04 | 3.61E−04 | 6.55E−04 | 3.71E−03 | 4.38E−03 | 1.01E−03 | ||

STD | 6.87E−08 | 6.11E−05 | 6.91E−03 | 2.48E−04 | 1.84E−04 | 3.56E−04 | 7.58E−03 | 7.14E−03 | 3.76E−04 | ||

F16 | MEAN | ||||||||||

STD | 1.84E−04 | 6.60E−11 | 1.17E−11 | 3.92E−11 | 5.26E−09 | 4.54E−06 | 3.19E−05 | ||||

F17 | MEAN | 3.99E−01 | |||||||||

STD | 8.14E−05 | 2.08E−08 | 7.31E−07 | 3.30E−07 | 8.21E−06 | 6.07E−06 | 2.05E−03 | ||||

F18 | MEAN | 3.01E+00 | 5.70E+00 | ||||||||

STD | 1.48E−02 | 1.48E+01 | 2.92E−06 | 5.40E−08 | 2.64E−05 | 9.17E−06 | 3.25E−05 | 1.94E−05 | |||

F19 | MEAN | ||||||||||

STD | 2.61E−03 | 2.00E−03 | 3.26E−03 | 2.82E−03 | 4.35E−03 | 2.32E−03 | 3.67E−05 | 2.09E−03 | |||

F20 | MEAN | −3.19E+00 | −3.26E+00 | −3.18E+00 | −3.14E+00 | −3.27E+00 | −3.25E+00 | −3.26E+00 | −2.88E+00 | ||

STD | 8.88E−02 | 8.33E−02 | 2.83E−01 | 8.15E−02 | 8.36E−02 | 7.84E−02 | 7.13E−02 | 3.89E−01 | |||

F21 | MEAN | −1.01E+01 | −7.21E+00 | −6.08E+00 | −5.05E+00 | −7.85E+00 | −9.31E+00 | −7.53E+00 | −2.48E+00 | ||

STD | 7.64E−07 | 5.25E−03 | 2.92E+00 | 2.07E+00 | 1.00E−03 | 2.71E+00 | 1.92E+00 | 2.69E+00 | 2.46E+00 | ||

F22 | MEAN | −7.12E+00 | −6.33E+00 | −5.43E+00 | −8.71E+00 | −7.19E+00 | −3.57E+00 | ||||

STD | 4.68E−07 | 9.98E−03 | 2.74E+00 | 2.29E+00 | 1.29E+00 | 2.64E+00 | 2.78E−04 | 3.17E+00 | 2.04E+00 | ||

F23 | MEAN | −7.21E+00 | −7.69E+00 | −5.47E+00 | −8.43E+00 | −7.74E+00 | −4.04E+00 | ||||

STD | 7.87E−03 | 2.78E+00 | 2.99E+00 | 1.31E+00 | 2.85E+00 | 2.78E−04 | 3.36E+00 | 1.65E+00 |

Convergence curves offer a more intuitive reflection of an algorithm’s optimization process. The partial convergence curves for unimodal and multimodal functions are depicted in

To mitigate the impact of randomness on the evaluation of algorithm performance, we employed two statistical tests: The Wilcoxon signed-rank sum and the Friedman tests.

INGO(VS) | INGO | NGO | AO | HPO | SCSO | ||||
---|---|---|---|---|---|---|---|---|---|

P | S | P | S | P | S | P | S | ||

F1 | N/A | 1.21E−12 | + | 1.21E−12 | + | NaN | 1.21E−12 | + | |

F2 | N/A | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

F3 | N/A | 1.21E−12 | + | 1.21E−12 | + | 4.57E−12 | + | 1.21E−12 | + |

F4 | N/A | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

F5 | N/A | 2.96E−11 | + | 8.99E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F6 | N/A | 3.02E−11 | + | 3.02E−11 | + | 2.20E−07 | + | 3.02E−11 | + |

F7 | N/A | 3.00E−11 | – | 3.02E−11 | – | 3.02E−11 | – | 3.02E−11 | – |

F8 | N/A | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F9 | N/A | NaN | NaN | NaN | NaN | ||||

F10 | N/A | 4.17E−13 | + | NaN | NaN | NaN | |||

F11 | N/A | NaN | = | NaN | NaN | NaN | |||

F12 | N/A | 3.02E−11 | + | 3.02E−11 | + | 3.80E−08 | + | 3.02E−11 | + |

F13 | N/A | 3.00E−11 | + | 3.02E−11 | + | 3.28E−03 | + | 3.02E−11 | + |

F14 | N/A | NaN | 1.04E−12 | + | 4.27E−13 | + | 6.87E−13 | + | |

F15 | N/A | 1.21E−12 | + | 3.20E−13 | + | 1.20E−12 | + | ||

F16 | N/A | NaN | 1.21E−12 | + | 1.69E−14 | + | 1.69E−14 | + | |

F17 | N/A | NaN | 1.54E−04 | + | 1.69E−14 | + | 1.16E−12 | + | |

F18 | N/A | NaN | 1.21E−12 | + | 1.21E−12 | + | |||

F19 | N/A | NaN | 5.86E−11 | + | 1.53E−04 | + | |||

F20 | N/A | NaN | 1.21E−12 | + | |||||

F21 | N/A | NaN | 1.21E−12 | + | 5.64E−13 | + | 1.21E−12 | + | |

F22 | N/A | NaN | 9.30E−04 | + | 1.54E−04 | + | |||

F23 | N/A | NaN | 1.21E−12 | + | |||||

+/−/= | 10/1/2 | 18/1/1 | 14/1/4 | 17/1/2 | |||||

ARV | 2.61 | 4.20 | 4.04 | 4.61 | 5.72 | ||||

RANK | 1 | 3 | 2 | 4 | 6 | ||||

HHO | WOA | GWO | SCA | DA | |||||

P | S | P | S | P | S | P | S | P | S |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

3.69E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.02E−11 | – | 4.20E−10 | – | 5.49E−11 | – | 4.08E−11 | + | ||

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

NaN | 6.56E−04 | + | 1.21E−12 | + | 1.21E−12 | + | |||

NaN | 1.42E−10 | + | 3.12E−13 | + | 1.21E−12 | + | 1.21E−12 | + | |

NaN | 4.19E−02 | + | 1.21E−12 | + | 1.21E−12 | + | |||

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

4.16E−14 | + | 7.32E−13 | + | 6.53E−13 | + | 1.20E−13 | + | 1.21E−12 | + |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

1.69E−14 | + | 1.69E−14 | + | 1.15E−12 | + | 1.10E−12 | + | 1.21E−12 | + |

7.60E−13 | + | 1.06E−12 | + | 1.21E−12 | + | 1.06E−12 | + | 7.47E−10 | + |

1.25E−05 | + | 1.21E−12 | + | 1.21E−12 | + | 2.21E−06 | + | 1.21E−12 | + |

2.05E−05 | + | 1.91E−07 | + | 1.21E−12 | + | 1.33E−08 | + | ||

1.21E−12 | + | 1.21E−12 | + | ||||||

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + |

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | ||||

1.21E−12 | + | 1.21E−12 | + | 1.21E−12 | + | ||||

+/−/= | 19/1/0 | 16/1/6 | 21/1/1 | 20/0/3 | 22/0/1 | ||||

ARV | 4.96 | 5.98 | 6.50 | 8.50 | 7.89 | ||||

RANK | 5 | 7 | 8 | 10 | 9 |

INGO’s performance was tested and evaluated using the IEEE CEC2017 test set from the Conference on Evolutionary Computation 2017, which contains 30 test functions. CEC2017 has greater computational complexity than the common 23 reference functions and CEC2014. Moreover, the wide range of function types demonstrates that CEC2017 effectively assesses algorithm performance. In this test set, NGO, AO, HPO, SCSO, SSA, BWO, HHO, DA and INGOfu are compared.

Function | Index | AO | HPO | SCSO | SSA | BWO | HHO | DA | INGOfu | NGO | INGO |
---|---|---|---|---|---|---|---|---|---|---|---|

F1 | MEAN | 1.010E+08 | 1.133E+04 | 3.912E+09 | 6.503E+03 | 1.874E+07 | 4.769E+08 | 1.927E+09 | 3.599E+09 | 6.239E+03 | |

STD | 3.902E+07 | 8.564E+03 | 2.521E+09 | 6.308E+03 | 3.348E+06 | 3.167E+08 | 1.501E+09 | 1.448E+09 | 6.065E+03 | ||

F2 | MEAN | 3.850E+30 | 9.984E+07 | 1.723E+33 | 2.037E+16 | 4.557E+22 | 4.501E+29 | 6.095E+30 | 2.347E+38 | 2.644E+13 | |

STD | 1.833E+31 | 3.147E+08 | 4.633E+33 | 8.361E+16 | 2.482E+23 | 1.290E+30 | 2.445E+31 | 9.507E+38 | 8.524E+13 | ||

F3 | MEAN | 4.516E+04 | 3.839E+03 | 4.593E+04 | 1.702E+04 | 2.706E+04 | 2.425E+05 | 4.436E+04 | 1.038E+05 | 4.475E+04 | |

STD | 4.902E+03 | 3.125E+03 | 9.292E+03 | 1.013E+04 | 5.425E+03 | 6.869E+04 | 1.079E+04 | 2.970E+04 | 5.751E+03 | ||

F4 | MEAN | 5.965E+02 | 4.847E+02 | 6.866E+02 | 5.028E+02 | 5.420E+02 | 6.954E+02 | 5.753E+02 | 9.552E+02 | 4.961E+02 | |

STD | 4.266E+01 | 1.986E+01 | 1.340E+02 | 2.340E+01 | 2.656E+01 | 9.835E+01 | 4.627E+01 | 2.973E+02 | 2.559E+01 | ||

F5 | MEAN | 6.840E+02 | 6.610E+02 | 7.190E+02 | 6.544E+02 | 7.381E+02 | 8.170E+02 | 6.023E+02 | 8.455E+02 | 6.443E+02 | |

STD | 3.235E+01 | 3.217E+01 | 4.224E+01 | 5.569E+01 | 3.147E+01 | 4.912E+01 | 2.907E+01 | 6.304E+01 | 1.907E+01 | ||

F6 | MEAN | 6.524E+02 | 6.327E+02 | 6.608E+02 | 6.466E+02 | 6.626E+02 | 6.730E+02 | 6.083E+02 | 6.828E+02 | 6.226E+02 | |

STD | 5.514E+00 | 9.664E+00 | 1.120E+01 | 1.542E+01 | 5.486E+00 | 9.743E+00 | 3.159E+00 | 1.345E+01 | 1.381E+01 | ||

F7 | MEAN | 1.071E+03 | 9.925E+02 | 1.121E+03 | 9.064E+02 | 1.284E+03 | 1.254E+03 | 8.627E+02 | 1.125E+03 | 9.738E+02 | |

STD | 6.536E+01 | 7.163E+01 | 8.817E+01 | 5.268E+01 | 7.367E+01 | 1.065E+02 | 4.135E+01 | 1.154E+02 | 5.619E+01 | ||

F8 | MEAN | 9.509E+02 | 9.618E+02 | 9.945E+02 | 9.390E+02 | 9.779E+02 | 1.026E+03 | 1.088E+03 | 1.107E+03 | 9.211E+02 | |

STD | 2.795E+01 | 3.580E+01 | 2.837E+01 | 3.227E+01 | 2.698E+01 | 5.510E+01 | 3.277E+01 | 5.275E+01 | 1.808E+01 | ||

F9 | MEAN | 6.113E+03 | 4.699E+03 | 5.744E+03 | 4.396E+03 | 7.605E+03 | 9.709E+03 | 1.935E+03 | 1.184E+04 | 3.275E+03 | |

STD | 9.379E+02 | 1.120E+03 | 7.999E+02 | 1.456E+03 | 9.218E+02 | 3.586E+03 | 6.568E+02 | 3.328E+03 | 5.496E+02 | ||

F10 | MEAN | 5.617E+03 | 4.867E+03 | 5.832E+03 | 5.004E+03 | 5.769E+03 | 6.962E+03 | 4.662E+03 | 7.506E+03 | 4.991E+03 | |

STD | 6.331E+02 | 5.910E+02 | 6.399E+02 | 8.813E+02 | 6.189E+02 | 8.728E+02 | 1.163E+03 | 9.058E+02 | 2.824E+02 | ||

F11 | MEAN | 1.715E+03 | 1.332E+03 | 2.187E+03 | 1.319E+03 | 1.273E+03 | 5.021E+03 | 1.702E+03 | 2.679E+03 | 1.206E+03 | |

STD | 2.496E+02 | 9.272E+01 | 9.252E+02 | 5.662E+01 | 4.743E+01 | 2.646E+03 | 5.609E+02 | 1.056E+03 | 2.665E+01 | ||

F12 | MEAN | 4.042E+07 | 1.967E+06 | 1.290E+08 | 1.521E+07 | 2.088E+07 | 1.789E+08 | 8.350E+07 | 4.262E+08 | 4.402E+05 | |

STD | 2.575E+07 | 1.476E+06 | 1.652E+08 | 1.700E+07 | 1.585E+07 | 1.453E+08 | 9.517E+07 | 3.581E+08 | 3.580E+05 | ||

F13 | MEAN | 4.988E+05 | 4.859E+04 | 8.658E+06 | 1.033E+05 | 5.288E+05 | 8.276E+05 | 4.032E+07 | 3.327E+07 | 1.662E+04 | |

STD | 3.207E+05 | 4.050E+04 | 3.021E+07 | 7.226E+04 | 2.134E+05 | 1.014E+06 | 1.073E+08 | 3.372E+07 | 1.312E+04 | ||

F14 | MEAN | 4.984E+05 | 1.599E+04 | 4.518E+05 | 5.175E+04 | 3.596E+05 | 1.896E+06 | 6.574E+05 | 8.276E+05 | 6.250E+03 | |

STD | 5.448E+05 | 1.153E+04 | 7.153E+05 | 4.545E+04 | 3.386E+05 | 1.934E+06 | 1.114E+06 | 9.337E+05 | 5.011E+03 | ||

F15 | MEAN | 9.090E+04 | 1.872E+04 | 6.949E+05 | 8.819E+04 | 5.990E+04 | 3.780E+05 | 1.250E+06 | 3.174E+05 | 3.299E+03 | |

STD | 5.337E+04 | 1.570E+04 | 1.371E+06 | 6.250E+04 | 3.181E+04 | 4.021E+05 | 4.525E+06 | 2.080E+05 | 2.299E+03 | ||

F16 | MEAN | 3.256E+03 | 2.881E+03 | 3.152E+03 | 2.858E+03 | 3.398E+03 | 3.815E+03 | 2.500E+03 | 3.891E+03 | 2.527E+03 | |

STD | 4.271E+02 | 3.471E+02 | 4.293E+02 | 2.985E+02 | 3.895E+02 | 4.923E+02 | 3.278E+02 | 4.721E+02 | 1.816E+02 | ||

F17 | MEAN | 2.292E+03 | 2.323E+03 | 2.311E+03 | 2.231E+03 | 2.582E+03 | 2.723E+03 | 1.950E+03 | 2.646E+03 | 1.922E+03 | |

STD | 2.894E+02 | 3.042E+02 | 2.027E+02 | 2.137E+02 | 2.807E+02 | 2.614E+02 | 1.250E+02 | 2.282E+02 | 6.039E+01 | ||

F18 | MEAN | 2.573E+06 | 1.750E+05 | 2.239E+06 | 7.801E+05 | 1.870E+06 | 6.585E+06 | 1.210E+06 | 5.221E+06 | 1.224E+05 | |

STD | 2.104E+06 | 1.459E+05 | 2.238E+06 | 8.972E+05 | 2.640E+06 | 5.928E+06 | 1.809E+06 | 5.739E+06 | 7.009E+04 | ||

F19 | MEAN | 1.548E+06 | 1.253E+04 | 6.131E+06 | 2.328E+06 | 6.954E+05 | 1.170E+07 | 7.106E+05 | 3.159E+07 | 5.024E+03 | |

STD | 1.010E+06 | 1.715E+04 | 2.479E+07 | 1.279E+06 | 5.597E+05 | 1.027E+07 | 1.118E+06 | 2.563E+07 | 2.666E+03 | ||

F20 | MEAN | 2.583E+03 | 2.674E+03 | 2.690E+03 | 2.558E+03 | 2.802E+03 | 2.822E+03 | 2.434E+03 | 2.841E+03 | 2.360E+03 | |

STD | 1.930E+02 | 2.618E+02 | 1.528E+02 | 1.984E+02 | 1.980E+02 | 2.124E+02 | 1.774E+02 | 1.946E+02 | 8.031E+01 | ||

F21 | MEAN | 2.469E+03 | 2.485E+03 | 2.515E+03 | 2.438E+03 | 2.565E+03 | 2.614E+03 | 2.386E+03 | 2.650E+03 | 2.417E+03 | |

STD | 2.666E+01 | 3.546E+01 | 3.803E+01 | 3.539E+01 | 4.495E+01 | 5.413E+01 | 1.960E+01 | 6.081E+01 | 1.658E+01 | ||

F22 | MEAN | 2.414E+03 | 4.017E+03 | 3.466E+03 | 4.044E+03 | 6.522E+03 | 7.759E+03 | 4.985E+03 | 7.824E+03 | 2.301E+03 | |

STD | 8.520E+01 | 2.211E+03 | 1.171E+03 | 2.226E+03 | 1.875E+03 | 1.612E+03 | 2.046E+03 | 2.427E+03 | 2.055E+00 | ||

F23 | MEAN | 2.917E+03 | 2.853E+03 | 2.924E+03 | 2.772E+03 | 3.197E+03 | 3.084E+03 | 2.769E+03 | 3.240E+03 | 2.764E+03 | |

STD | 5.389E+01 | 4.188E+01 | 6.594E+01 | 3.624E+01 | 1.274E+02 | 1.064E+02 | 5.021E+01 | 1.444E+02 | 2.353E+01 | ||

F24 | MEAN | 3.059E+03 | 3.006E+03 | 3.064E+03 | 2.935E+03 | 3.442E+03 | 3.221E+03 | 2.916E+03 | 3.443E+03 | 2.927E+03 | |

STD | 5.831E+01 | 5.664E+01 | 6.381E+01 | 3.177E+01 | 1.394E+02 | 8.048E+01 | 4.362E+01 | 1.526E+02 | 2.270E+01 | ||

F25 | MEAN | 2.953E+03 | 2.902E+03 | 3.070E+03 | 2.919E+03 | 2.929E+03 | 3.053E+03 | 2.984E+03 | 3.228E+03 | 2.906E+03 | |

STD | 2.940E+01 | 2.132E+01 | 8.566E+01 | 2.148E+01 | 2.145E+01 | 5.299E+01 | 5.647E+01 | 1.566E+02 | 1.608E+01 | ||

F26 | MEAN | 5.164E+03 | 5.855E+03 | 6.052E+03 | 4.861E+03 | 7.108E+03 | 8.039E+03 | 4.632E+03 | 7.997E+03 | 4.150E+03 | |

STD | 1.675E+03 | 8.682E+02 | 1.099E+03 | 9.589E+02 | 1.863E+03 | 9.025E+02 | 2.541E+02 | 1.833E+03 | 7.974E+02 | ||

F27 | MEAN | 3.344E+03 | 3.245E+03 | 3.339E+03 | 3.260E+03 | 3.413E+03 | 3.402E+03 | 3.247E+03 | 3.477E+03 | 3.221E+03 | |

STD | 6.648E+01 | 2.218E+01 | 5.332E+01 | 3.630E+01 | 9.864E+01 | 1.016E+02 | 2.211E+01 | 1.372E+02 | 1.145E+01 | ||

F28 | MEAN | 3.369E+03 | 3.543E+03 | 3.258E+03 | 3.299E+03 | 3.461E+03 | 3.402E+03 | 3.759E+03 | 3.248E+03 | 3.300E+03 | |

STD | 5.107E+01 | 1.527E+02 | 4.097E+01 | 3.717E+01 | 9.048E+01 | 5.758E+01 | 2.857E+02 | 2.811E+01 | 5.118E−09 | ||

F29 | MEAN | 4.543E+03 | 4.079E+03 | 4.549E+03 | 4.186E+03 | 4.530E+03 | 5.099E+03 | 3.774E+03 | 5.126E+03 | 3.867E+03 | |

STD | 2.712E+02 | 3.034E+02 | 3.112E+02 | 2.273E+02 | 3.047E+02 | 4.559E+02 | 1.837E+02 | 4.247E+02 | 1.462E+02 | ||

F30 | MEAN | 1.340E+07 | 3.413E+04 | 1.301E+07 | 6.554E+06 | 3.543E+06 | 2.764E+07 | 1.153E+07 | 4.237E+07 | 9.997E+03 | |

STD | 1.294E+07 | 2.582E+04 | 1.209E+07 | 6.005E+06 | 2.142E+06 | 2.169E+07 | 1.049E+07 | 3.184E+07 | 5.169E+03 |

INGO(VS) | INGO | NGO | AO | HPO | SCSO | ||||
---|---|---|---|---|---|---|---|---|---|

P | S | P | S | P | S | P | S | ||

F1 | N/A | 2.07E−02 | + | 3.02E−11 | + | 4.98E−04 | + | 3.02E−11 | + |

F2 | N/A | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F3 | N/A | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | ||

F4 | N/A | 6.97E−03 | + | 3.02E−11 | + | 4.50E−11 | + | ||

F5 | N/A | 6.72E−10 | + | 5.49E−11 | + | 2.15E−10 | + | 3.34E−11 | + |

F6 | N/A | 8.99E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F7 | N/A | 5.07E−10 | + | 3.02E−11 | + | 1.69E−09 | + | 3.02E−11 | + |

F8 | N/A | 1.17E−03 | + | 1.20E−08 | + | 2.19E−08 | + | 8.15E−11 | + |

F9 | N/A | 3.34E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F10 | N/A | 1.19E−06 | + | 7.12E−09 | + | 1.12E−02 | + | 5.07E−10 | + |

F11 | N/A | 3.02E−11 | + | 4.18E−09 | + | 3.02E−11 | + | ||

F12 | N/A | 3.02E−11 | + | 1.25E−07 | + | 3.02E−11 | + | ||

F13 | N/A | 2.62E−03 | + | 3.02E−11 | + | 2.67E−09 | + | 3.69E−11 | + |

F14 | N/A | 4.08E−11 | + | 7.20E−05 | + | 8.99E−11 | + | ||

F15 | N/A | 3.02E−11 | + | 2.19E−08 | + | 3.02E−11 | + | ||

F16 | N/A | 1.43E−08 | + | 2.43E−05 | + | 7.09E−08 | + | ||

F17 | N/A | 7.70E−04 | + | 6.52E−09 | + | 9.26E−09 | + | 7.39E−11 | + |

F18 | N/A | 6.07E−11 | + | 2.92E−02 | + | 1.09E−10 | + | ||

F19 | N/A | 3.02E−11 | + | 3.02E−11 | + | ||||

F20 | N/A | 7.20E−05 | + | 4.57E−09 | + | 1.69E−09 | + | 4.50E−11 | + |

F21 | N/A | 8.15E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F22 | N/A | 2.54E−08 | + | 2.92E−11 | + | 2.97E−04 | + | 2.92E−11 | + |

F23 | N/A | 7.69E−08 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

F24 | N/A | 1.10E−08 | + | 3.02E−11 | + | 3.34E−11 | + | 3.02E−11 | + |

F25 | N/A | 1.64E−05 | + | 1.09E−10 | + | 1.06E−03 | + | 3.02E−11 | + |

F26 | N/A | = | 3.82E−10 | + | 1.16E−07 | + | |||

F27 | N/A | 5.19E−10 | + | 2.79E−11 | + | 2.79E−11 | + | 2.79E−11 | + |

F28 | N/A | 1.21E−12 | – | 7.47E−10 | + | 1.21E−12 | – | 3.36E−11 | + |

F29 | N/A | 8.89E−10 | + | 3.02E−11 | + | 5.07E−10 | + | 3.02E−11 | + |

F30 | N/A | 3.02E−11 | + | 1.09E−10 | + | 3.02E−11 | + | ||

21/0/9 | 29/0/1 | 26/1/3 | 30/0/0 | ||||||

1.20 | 2.53 | 6.12 | 3.93 | 7.35 | |||||

1 | 2 | 6 | 3 | 8 | |||||

SSA | BWO | HHO | DA | INGOfu | |||||

P | S | P | S | P | S | P | S | P | S |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | ||

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

9.92E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.01E−04 | + | 1.41E−09 | + | 3.02E−11 | + | 4.08E−11 | + | 3.02E−11 | + |

5.53E−08 | + | 3.02E−11 | + | 3.02E−11 | + | 2.62E−03 | + | 3.02E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 7.39E−11 | + | 3.02E−11 | + |

1.32E−04 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | ||

9.51E−06 | + | 8.99E−11 | + | 3.34E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

6.70E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 1.70E−08 | + | 3.02E−11 | + |

5.32E−03 | + | 2.87E−10 | + | 3.02E−11 | + | 4.08E−11 | + | ||

2.15E−10 | + | 3.96E−08 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.02E−11 | + | 3.02E−1 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

8.15E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 4.98E−11 | + | 3.02E−11 | + |

5.00E−09 | + | 6.07E−11 | + | 3.02E−11 | + | 2.15E−10 | + | 3.34E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

3.83E−05 | + | 3.82E−10 | + | 8.99E−11 | + | 3.69E−11 | + | ||

1.86E−09 | + | 3.34E−11 | + | 3.02E−11 | + | 1.24E−03 | + | 3.02E−11 | + |

3.35E−08 | + | 9.76E−10 | + | 3.69E−11 | + | 2.23E−09 | + | 4.98E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 1.33E−10 | + | 3.02E−11 | + |

2.03E−07 | + | 3.34E−11 | + | 3.69E−11 | + | 1.17E−04 | + | 3.34E−11 | + |

1.21E−10 | + | 3.02E−11 | + | 3.02E−11 | + | 2.25E−04 | + | 3.02E−11 | + |

3.72E−09 | + | 2.92E−11 | + | 2.92E−11 | + | 2.92E−11 | + | 2.92E−11 | + |

3.81E−07 | + | 3.02E−11 | + | 3.02E−11 | + | 5.86E−06 | + | 3.02E−11 | + |

1.29E−09 | + | 3.02E−11 | + | 3.02E−11 | + | 2.39E−04 | + | 3.02E−11 | + |

1.60E−07 | + | 2.87E−10 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

8.66E−05 | + | 5.18E−07 | + | 3.02E−11 | + | 3.39E−02 | + | 4.68E−08 | + |

2.79E−11 | + | 2.79E−11 | + | 2.79E−11 | + | 2.79E−11 | + | 2.79E−11 | + |

1.33E−08 | – | 1.21E−12 | + | 1.21E−12 | – | 1.21E−12 | + | ||

8.15E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 2.78E−07 | + | 3.02E−11 | + |

3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + | 3.02E−11 | + |

28/1/1 | 29/0/1 | 30/0/0 | 26/1/3 | 30/0/0 | |||||

4.17 | 6.47 | 8.73 | 4.95 | 9.55 | |||||

4 | 7 | 9 | 5 | 10 |

To verify INGO’s performance in solving real-world problems, the algorithm is tested on a UAV flight path planning problem. UAV path planning represents a constrained optimization challenge where Swarm Intelligence (SI) finds widespread application [

Constrained optimization problems are typically addressed using the penalty function approach and the multi-objective approach [

UAV path planning is actually a multi-constraint combinatorial optimization problem [

Here,

To plan UAV flight path using Swarm Intelligent Optimization algorithm, the objective function should be determined first, and the total flight path evaluation function composed of the flight path length function, flight height function and angle cost function should be optimized. The total cost function is taken as the objective function of the algorithm. The indexes affecting UAV performance mainly include track length, flight altitude, minimum step length, corner cost, maximum climb angle, etc. [

Here,

In the route planning problem processed by swarm intelligent algorithm,

The mathematical model of the standard deviation of flight altitude

The mathematical model of turning angle

To fully test the performance of the algorithm, the flight space is set to

No. | |||||
---|---|---|---|---|---|

1 | 20 | 35 | 30 | 20 | 18 |

2 | 16 | 130 | 35 | 14 | 17 |

3 | 17 | 40 | 80 | 19 | 19 |

4 | 19 | 160 | 85 | 20 | 20 |

5 | 15 | 100 | 100 | 10 | 13 |

6 | 20 | 40 | 100 | 20 | 20 |

7 | 23 | 170 | 100 | 20 | 20 |

8 | 18 | 90 | 125 | 15 | 15 |

9 | 24 | 18 | 71 | 18 | 19 |

10 | 25 | 155 | 170 | 17 | 18 |

11 | 19 | 90 | 60 | 18 | 20 |

12 | 21 | 117 | 78 | 20 | 20 |

13 | 16 | 179 | 146 | 15 | 15 |

Under identical conditions, the INGO’s outcomes are compared to those of NGO, SSA, BWO, DA and ChOA through 30 repetitions of the experiment.

No. | Indicators | INGO | NGO | SSA | BWO | DA | ChOA |
---|---|---|---|---|---|---|---|

1 | Optimal | 217.2375 | 231.1058 | 220.5907 | 248.8895 | 216.9916 | |

2 | Worst | 257.9022 | 246.7527 | 368.2322 | 286.4377 | 319.5354 | |

3 | Mean | 241.8038 | 242.7427 | 285.0163 | 234.9963 | 246.0827 | |

4 | STD | 7.4185 | 32.6120 | 34.4144 | 15.7019 | 24.1947 | |

5 | Time (s) | 12.8619 | 85.0672 | 7.8375 | 3.0306 | 4.5729 |

Within the realm of function optimization, the NGO algorithm exhibits diminished convergence accuracy and a propensity for entrapment in local optima. This paper proposes four strategic enhancements aimed at harmonizing the exploratory and exploitative abilities of NGO. First, introducing tent chaotic mapping aims to uniformly distribute the initial population, increasing diversity. Next, incorporating whale fall phenomena and T-distribution variations increases the algorithm’s chance of escaping local optima, reducing the risk of falling into local minima. Lastly, integrating the state transition algorithm’s neighborhood search capability strengthens INGO’s local utilization. INGO’s performance, assessed on public test sets and UAV trajectory planning, was compared with classic and new algorithms. Statistical methods analyzed the experimental results, summarizing the algorithm’s strengths and areas for improvement. The findings show that INGO has higher convergence accuracy and a better ability to escape local extremes in function optimization.

Moving forward, our efforts will focus on three main avenues: Firstly, we aim to integrate various optimization strategies into advanced intelligent optimization algorithms, creating competitive solutions for real-world problems. Secondly, considering INGO’s limitations, we will explore the inclusion of adaptive adjustment strategies to enhance algorithmic performance. Lastly, we will extend the application of INGO to tackle more sophisticated engineering scenarios, such as the complex task of UAV trajectory planning that involves additional constraints, broadening our approach to encompass a wider spectrum of intricate engineering problems.

The authors wish to acknowledge the editor and anonymous reviewers for their insightful comments, which have improved the quality of this publication.

This work was in part supported by the Key Research and Development Project of Hubei Province (No. 2023BAB094), the Key Project of Science and Technology Research Program of Hubei Educational Committee (No. D20211402), and the Open Foundation of Hubei Key Laboratory for High-Efficiency Utilization of Solar Energy and Operation Control of Energy Storage System (No. HBSEES202309).

Study conception and design: Mai Hu, Liang Zeng; Data collection: Chenning Zhang, Quan Yuan and Shanshan Wang; Analysis and interpretation of results: All authors; Draft manuscript preparation: Mai Hu. All authors reviewed the results and approved the final version of the manuscript.

Data and materials are available.

The authors declare that they have no conflicts of interest to report regarding the present study.