# Levy Flight-Based Improved Grey Wolf Optimization: A Solution for Various Engineering Problems

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## Abstract

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## 1. Introduction

- Engineering: Metaheuristic algorithms are used in engineering to optimize the design of structures, such as bridges and buildings.
- Computer science: Metaheuristic algorithms are used in computer science to optimize the performance of the algorithms and systems, such as scheduling and routing.
- Operations research: Metaheuristic algorithms are used in operations research to solve problems in areas such as logistics and supply chain management.

## 2. Optimization

- Mathematical programming: These techniques are based on the use of mathematical models to represent the problem and constraints. The solution is then found by solving the mathematical equations.
- Gradient-based methods: These techniques are based on the use of gradients, which are the directions of the steepest ascent or descent. The solution is found by moving in the direction of the gradient until a local or global optimum is reached.
- Metaheuristic algorithm: These techniques are based on the use of metaheuristics, which are rules of thumb, to guide the search for a solution. The solution is found by iteratively improving an initial solution over time.

## 3. Optimization Algorithm and New Proposed Algorithms

#### 3.1. Grey Wolf Optimization

Algorithm 1 GWO Algorithm |

Form the grey wolf population.Evaluate the accuracy of every response.For each iteration:For every single grey wolf in the population:Generate a new solution. Analyze the new solution’s potential. Update a grey wolf’s location in accordance with the updated solution. Sort the grey wolves based on their fitness. Each grey wolf’s location will be updated dependent on where the other grey wolves in the population are. Return the best solution as the result. |

#### 3.2. Improved Grey Wolf Optimization

Algorithm 2 IGWO Algorithms |

- Better convergence: IGWO algorithms have been found to converge faster and produce better solutions than the original GWO algorithm.
- Better handling of constraints: IGWO algorithms have been found to perform better on problems with constraints.
- Better handling of high-dimensional problems: IGWO algorithms have been found to perform better on problems with a vast array of variables.
Set the grey wolf population (solutions).Evaluate the accuracy of every response.Arrange solutions in decreasing fitness order. Make the best response the dominant α wolf. Make the second best response the β wolf. Make the third best response the δ wolf. For each iteration:Generate new solutions for each grey wolf. Examine the appropriateness of each new solution. According to the new answer, adjust each grey wolf’s location. Based on a combination of the α wolf’s answer, the β wolf’s solution, and the δ wolf’s solution, the α wolf’s location should be updated. Using the α wolf’s location and their solutions, adjust the positions of the wolves. Evaluate the fitness of the updated solutions. Arrange the grey wolves according to their most recent fitness. Update the 𝑎, 𝛽, and δ wolves based on the new sorting. Return the best solution (the alpha wolf) as the result. |

#### 3.3. Levy Flight Improved Grey Wolf Optimization

- μ is the parameter for location (the mean of the distribution);
- σ is a scale parameter (the standard deviation of the distribution);
- γ is the tail index parameter (controls the shape of the distribution).

_{i}(t) is the current position of the ith wolf at time t, x

_{i}(t + 1) is the updated position of the ith wolf at time t + 1, A

_{j}is the scaling factor for the jth wolf, and D

_{j}is the distance vector between the ith wolf and the jth wolf. The scaling factors A

_{j}are updated in each iteration as follows:

_{j}is calculated as follows:

_{j}is a random vector in the range [0, 1] that is generated for each wolf in each iteration.

Algorithm 3 Newly Proposed |

Set the grey wolf population (solutions).Evaluate the accuracy of every response.Arrange solutions in decreasing fitness order. Make the best response the dominant α wolf. Make the second best response the β wolf. Make the third best response the δ wolf. Evaluate the new position of the wolves and update it with the initial position.For each iteration:Generate new solutions for each grey wolf using Levy flight.Evaluate the accuracy of every new solution.According to the new answer, adjust each grey wolf’s location. The 𝛼 wolf’s location is updated based on a combination of its solution and the solutions of the 𝛽 and 𝛿 wolves.Based on a combination of the α wolf’s answer, the β wolf’s solution, and the δ wolf’s solution, the α wolf’s location should be updated. Evaluate the fitness of the updated solutions. Arrange the grey wolves according to their most recent fitness. Update the 𝛼, 𝛽, and 𝛿 wolves based on the new sorting.Return the best solution (the 𝛼 wolf) as the result. |

## 4. Results and Discussion

#### 4.1. 31-Level Cascaded H-Bridge MLI

- ${V}_{o\left(t\right)}$ is the output voltage waveform;
- $\omega $ is the resulting waveform’s fundamental frequency;
- $N$ is the inverter’s H-bridge module count;
- ${V}_{dc,k}$ is the DC voltage input of the kth H-bridge module.

#### 4.2. Harmonics

- Increased current levels: Harmonics can cause an increase in the RMS current levels, which can result in the overloading of conductors, transformers, and other electrical equipment.
- Decreased power factor: A reduction in the power factor, a measurement of an electrical system’s efficiency, can be brought on by the presence of harmonics. This can result in increased energy costs and decreased system efficiency.
- Increased heating: The additional current levels caused by harmonics can result in increased heating in conductors and transformers, which can reduce their life expectancy and cause safety issues.
- Interference with communication systems: Harmonics can interfere with communication systems and cause problems such as data corruption and interference with radio and television signals.

#### 4.3. Total Harmonics Distortion

## 5. Engineering Problems

#### 5.1. Tension Compression Spring

#### 5.2. Pressure Vessel Design

#### 5.3. Welded Beam Design

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**(

**a**) Search space for multi model function F08 (

**b**) Comparison of convergence curve for F08.

**Figure 11.**(

**a**) Search space for multi model function F09 (

**b**) Comparison of convergence curve for F09.

**Figure 12.**(

**a**) Search space for multi model function F10 (

**b**) Comparison of convergence curve for F10.

**Figure 13.**(

**a**) Search space for multi model function F11 (

**b**) Comparison of convergence curve for F11.

**Figure 14.**(

**a**) Search space for multi model function F12 (

**b**) Comparison of convergence curve for F12.

**Figure 15.**(

**a**) Search space for multi model function F13 (

**b**) Comparison of convergence curve for F13.

**Figure 16.**(

**a**) Search space for fixed dimensional function F14 (

**b**) Comparison of convergence curve for F14.

**Figure 17.**(

**a**) Search space for fixed dimensional function F15 (

**b**) Comparison of convergence curve for F15.

**Figure 18.**(

**a**) Search space for fixed dimensional function F16 (

**b**) Comparison of convergence curve for F16.

**Figure 19.**(

**a**) Search space for fixed dimensional function F17 (

**b**) Comparison of convergence curve for F17.

**Figure 20.**(

**a**) Search space for fixed dimensional function F18 (

**b**) Comparison of convergence curve for F18.

**Figure 21.**(

**a**) Search space for fixed dimensional function F19 (

**b**) Comparison of convergence curve for F19.

**Figure 22.**(

**a**) Search space for fixed dimensional function F20 (

**b**) Comparison of convergence curve for F20.

**Figure 23.**(

**a**) Search space for fixed dimensional function F21 (

**b**) Comparison of convergence curve for F21.

**Figure 24.**(

**a**) Search space for fixed dimensional function F22 (

**b**) Comparison of convergence curve for F22.

**Figure 25.**(

**a**) Search space for fixed dimensional function F23 (

**b**) Comparison of convergence curve for F23.

**Figure 28.**In this figure, (

**a**) represents LF-IGWO waveform and (

**b**) represents the THD spectrum of LF-IGWO.

**Figure 29.**In this figure, (

**a**) represents IGWO waveform and (

**b**) represents the THD spectrum of IGWO.

Optimization Methods | Description |
---|---|

Simulated Annealing | A probabilistic technique is used to identify a function’s global optimum by iteratively perturbing a candidate solution and accepting it with a probability based on the temperature parameter. At each iteration, the algorithm compares the energy of the new approach and the existing solution and accepts the new solution if it is better than the current solution or with a probability that decreases with time [20]. |

Gravitational Search | The law of gravity and the interaction of masses served as inspiration for this population-based optimization system. The algorithm uses a set of masses, which represent candidate solutions that are attracted or repelled by each other based on their positions and masses [21]. |

Artificial Electric Field | The electrostatic force in physics served as the inspiration for this metaheuristic optimization approach. The algorithm represents each potential solution as a charged particle that interacts with other particles via the electrostatic force [22]. |

Sine Cosine Algorithm | The sine and cosine functions in mathematics served as inspiration for this metaheuristic optimization approach. Each potential solution is represented by a position vector in the SCA algorithm’s high-dimensional search space. The technique creates random vectors that represent the search directions using the sine and cosine functions [23]. |

Equilibrium Optimizer | It is a metaheuristic optimization method that draws its motivation from physics’ equilibrium concept. Each potential solution is modeled by the algorithm as a particle that interacts with other particles via the forces of gravity and elastic deformation [24]. |

Artificial Bee Colony | It is a metaheuristic optimization method that draws inspiration from honey bee feeding habits. Each bee in the ABC algorithm represents a potential solution to the optimization problem and represents a population of candidate solutions. Three different types of bees are used in the algorithm: working bees, observers, and scout bees [25]. |

Particle Swarm Optimization | It is a metaheuristic optimization system that draws on social behavior cues from flocks of birds or schools of fish. Each potential solution is represented by the algorithm as a particle in a multidimensional search space. The particles move across the search space, modifying their positions and velocities in response to their own experiences as well as those of their nearby neighbors [7]. |

Ant Colony Optimization | It is a metaheuristic optimization method that takes its cues from how ants forage. Each ant in the algorithm’s representation of a population of potential solutions as an ant colony stands for a potential solution to the optimization issue. The algorithm mimics the actions of ants as they look for food, with the food serving as the ideal answer to the issue [8]. |

Artificial Fish Swarm | It is a metaheuristic optimization technique that draws its inspiration from how fish forage. Each fish in the algorithm’s representation of a population of potential solutions is a potential solution to the optimization issue. The algorithm mimics the actions of fish as they swim and search for food, with the food serving as the ideal answer to the issue [26]. |

Bacterial Foraging Optimization | It is a metaheuristic optimization algorithm that draws inspiration from how bacteria forage. Each bacterium in the algorithm’s model of a population of candidate solutions serves as a potential solution to the optimization problem. The algorithm mimics how bacteria scavenge for nutrition, with the nutrients serving as the ideal solution to the issue [27]. |

Harmony Search Optimization | It is a metaheuristic algorithm that draws inspiration from the process of musical improvisation. The method searches for a function’s global optimum using a population-based approach. The algorithm selects elements from the already-existing solutions and randomly adds some randomness to them in order to produce a new harmony at each iteration. A memory-based method is also incorporated into the algorithm to speed up the search’s convergence [13]. |

Teaching Learning-based Optimization | It is a metaheuristic optimization method that draws its inspiration from classroom teaching and learning procedures. Each student in the algorithm’s representation of a population of candidate solutions serves as a potential solution to the optimization problem. The algorithm mimics how students act as they interact with the teacher and one another and learn new things [28]. |

Imperialist Competition Algorithm | The social rivalry and hierarchical organization principles serve as the foundation for this metaheuristic optimization method. A population of potential solutions is modeled by the algorithm as a collection of empires, where each empire consists of one imperialist and one or more colonies. The algorithm mimics how empires act as they compete and work together to increase their influence and power [29]. |

Brain Storm Optimization | It is a metaheuristic optimization algorithm that draws inspiration from how human brain neurons behave. The programmer simulates the brainstorming process, in which a group of people come up with and assess solutions to a problem. Each person in BSO represents a potential solution to the optimization problem, and the algorithm adjusts each person’s position based on how they interact with other people [30]. |

Political Optimizer | It is a metaheuristic optimization method that draws inspiration from how politicians act in a given political system. The algorithm mimics the process of political rivalry, in which politicians face off against one another and work together to accomplish their objectives. The algorithm in PO adjusts the positions of the politicians depending on their interactions with other politicians in the population. Each possible solution to the optimization issue is represented in PO as a politician [31]. |

Differential Evolution | For the purpose of resolving optimization issues, it is a stochastic optimization algorithm. DE is a population-based method that uses natural selection to gradually weed out suboptimal solutions from a population of candidate solutions. The fundamental strategy is to generate a population of potential solutions, referred to as individuals, and then develop them utilizing the three crucial operators of mutation, crossover, and selection [9]. |

Genetic Algorithm | It is a metaheuristic optimization technique that draws inspiration from the evolution and natural selection processes. A population of potential solutions, or people, is created, and genetic operators such as crossover and mutation are used to gradually evolve them over generations. Every member of the population stands for a potential answer to the optimization issue [6]. |

Evolutionary Strategy | It draws inspiration from the course of evolution and natural selection. However, there are some significant ways in which ES is different from GA. The fundamental tenet of ES is to generate a population of potential solutions, referred to as individuals, and evolve them through mutation and selection across generations. Every member of the population is a potential answer to the optimization problem. |

Evolutionary Programming | Similar to evolutionary strategy (ES) and the genetic algorithm (GA), it is a family of optimization algorithms that draws its inspiration from the processes of natural selection and evolution. The fundamental tenet of EP is to generate a population of potential solutions, or individuals, and to use mutation and selection to gradually evolve them over generations. Every member of the population is a potential answer to the optimization problem. |

Genetic Programming | It is a machine learning technique that uses a form of evolutionary computation to automatically discover computer programs that solve a problem. GP is a variant of the genetic algorithm (GA) and evolutionary programming (EP), but instead of evolving vectors or individuals, it evolves computer programs represented as trees. |

Optimization of Meat and Poultry Farm Inventory Stock Using Data Analytics for Green Supply Chain Network | Optimizing inventory stock in meat and poultry farms is important for maintaining a sustainable and efficient supply chain network. Data analytics can be employed to analyze various factors that affect inventory levels, such as demand, production capacity, and supply chain lead time, to optimize inventory stock levels. The optimization process involves creating a model that considers various factors such as demand patterns, production schedules, and storage capacity. The model is trained using historical data on inventory levels, sales, and other relevant metrics. The trained model can then be used to predict the optimal inventory levels for each item in the meat and poultry farm [32]. |

Optimum Design for the Magnification Mechanisms Employing Fuzzy Logic–ANFIS | To optimize the design of a centrifugal pump, fuzzy logic and ANFIS (Adaptive Neuro-Fuzzy Inference System) can be employed. Fuzzy logic is a mathematical technique that deals with uncertainty and imprecision in data and is commonly used in control systems. ANFIS is a type of fuzzy inference system that uses neural networks to model the fuzzy logic. The design process involves various parameters such as impeller diameter, number of blades, blade angle, and outlet diameter, which need to be optimized to achieve the desired performance [33]. |

Minimizing Warpage for Macro-sized Fused Deposition Modeling Parts | There are several methods to minimize warpage in macro-sized FDM parts. The first method is to optimize the design of the part. The design should be modified to avoid features that are susceptible to warping, such as sharp corners, thin walls, and unsupported overhangs. Additionally, the part should be designed with proper wall thickness and infill density to ensure structural integrity and dimensional stability. Minimizing warpage in macro-sized FDM parts involves optimizing the part design, printing process parameters, and support structures, as well as using a heated build platform. By implementing these methods, high-quality parts with minimal warpage can be achieved [34]. |

Optimal Switching Angle Scheme for a Cascaded H-Bridge Inverter using Pigeon-Inspired Optimization | A cascaded H-bridge inverter is a type of multilevel inverter that is widely used in high-power applications such as electric vehicles, renewable energy systems, and industrial motor drives. It is made up of several H-bridge modules connected in series to produce a stepped waveform output. Each H-bridge module is made up of four power switches (IGBTs or MOSFETs) and a DC voltage source. The switches are controlled with the help of firing angles to create a sinusoidal waveform output [35]. |

Function Name | Equation | Range | Dim | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

${F}_{1}\left(x\right)$ | ${\displaystyle \sum}_{t=1}^{n}}{x}_{i}^{2$ | [−100, 100] | 30 | 0 |

${F}_{2}\left(x\right)$ | ${\displaystyle \sum}_{t=1}^{n}}\left|{x}_{i}\right|+{{\displaystyle \prod}}_{t=1}^{n}{x}_{i$ | [−10, 10] | 30 | 0 |

${F}_{3}\left(x\right)$ | ${\displaystyle \sum}_{i=1}^{n}}{\left({{\displaystyle \sum}}_{j-1}^{i}{x}_{j}\right)}^{2$ | [−100, 100] | 30 | 0 |

${F}_{4}\left(x\right)$ | $ma{x}_{i}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | [−100, 100] | 30 | 0 |

${F}_{5}\left(x\right)$ | ${\displaystyle \sum}_{t=1}^{n-1}}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | [−30, 30] | 30 | 0 |

${F}_{6}\left(x\right)$ | ${\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}+0.5\right)}^{2$ | [−100, 100] | 30 | 0 |

${F}_{7}\left(x\right)$ | ${\displaystyle \sum}_{i=1}^{n}}i{x}_{i}^{4}+random\left[0,1\right]$ | [−1.28, 1.28] | 30 | 0 |

Function Name | Equation | Range | Dim | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

${F}_{8}\left(x\right)$ | ${{\displaystyle \sum}}_{i=1}^{n}-{x}_{i}\mathrm{sin}(\surd {x}_{i})$ | [−500, 500] | 30 | 418.9829 ×D |

${F}_{9}\left(x\right)$ | ${\displaystyle \sum}_{i=1}^{n}}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | [−5.12, 5.12] | 30 | 0 |

${F}_{10}\left(x\right)$ | $-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | [−32, 32] | 30 | 0 |

${F}_{11}\left(x\right)$ | $\frac{1}{4000}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | [−600, 600] | 30 | 0 |

${F}_{12}\left(x\right)$ | $\frac{\pi}{n}\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}{\left({y}_{i}-1\right)}^{2}\left[1+10si{n}^{2}\left(\pi {y}_{i}+z\right)\right]+{\left({y}_{n-1}\right)}^{2}\}+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}u\left({x}_{i},10,100,4\right)$ ${y}_{i}=1+\frac{{x}_{i}+1}{4},u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}a\\ 0,-a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}a\end{array}\right.$ | [−50, 50] | 30 | 0 |

${F}_{13}\left(x\right)$ | $0.1\left\{si{n}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}-1\right)}^{2}\left[1+si{n}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+si{n}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}u\left({x}_{i},5,100,4\right)$ | [−50, 50] | 30 | 0 |

Function Name | Equation | Range | Dim | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|---|

${F}_{14}\left(x\right)$ | ${\left(\frac{1}{500}+{\displaystyle {\displaystyle \sum}_{j=1}^{25}}\frac{1}{j+{{\displaystyle \sum}}_{i=1}^{2}{\left({x}_{i}-{a}_{ij}\right)}^{6}}\right)}^{-1}$ | [−65, 65] | 2 | 1 |

${F}_{15}\left(x\right)$ | ${\displaystyle \sum}_{i=1}^{11}}{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2$ | [−5, 5] | 4 | 0.00030 |

${F}_{16}\left(x\right)$ | $4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | [−5, 5] | 2 | −1.0316 |

${F}_{17}\left(x\right)$ | ${\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi}\mathrm{cos}{x}_{1}+10\right)$ | [−5, 5] | 2 | 0.398 |

${F}_{18}\left(x\right)$ | $\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]$$\left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)\right]$ | [−2, 2] | 2 | 3 |

${F}_{19}\left(x\right)$ | $-{{\displaystyle \sum}}_{i=1}^{4}{c}_{i}exp\left(-{{\displaystyle \sum}}_{j=1}^{3}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)$ | [1, 3] | 3 | −3.86 |

${F}_{20}\left(x\right)$ | $-{{\displaystyle \sum}}_{i=1}^{4}{c}_{i}exp\left(-{{\displaystyle \sum}}_{j=1}^{6}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)$ | [0, 1] | 6 | −3.32 |

${F}_{21}\left(x\right)$ | $-{{\displaystyle \sum}}_{i=1}^{5}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | [0, 10] | 4 | −10.1532 |

${F}_{22}\left(x\right)$ | $-{{\displaystyle \sum}}_{i=1}^{7}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | [0, 10] | 4 | −10.4028 |

${F}_{23}\left(x\right)$ | $-{{\displaystyle \sum}}_{i=1}^{10}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | [0, 10] | 4 | −10.5363 |

Algorithms | Parameters |
---|---|

LF-IGWO | a linearly declines from 2 to 0 |

IGWO | a linearly declines from 2 to 0 |

GWO | a linearly declines from 2 to 0 |

PSO | w decreases linearly from 0.9 to 0.2, and c1 = c2 = 2. |

GA | 0.3 is the crossover probability, while 0.1 is the mutation probability |

SSR | R_{f} = 15 and M = 5. |

Function Number | PSO | GA | GWO | IGWO | SSR | LF-IGWO |
---|---|---|---|---|---|---|

F1 | 3.80 × 10^{−8} | 2.31 × 10^{1} | 4.96 × 10^{−14} | 2.67 × 10^{−60} | 6.35 × 10^{−9} | 7.37 × 10^{−61} |

F2 | 4.80 × 10^{−8} | 1.07 × 10^{0} | 2.40 × 10^{−11} | 2.25 × 10^{−37} | 3.52 × 10^{−5} | 7.72 × 10^{−38} |

F3 | 1.53 × 10^{1} | 5.60 × 10^{3} | 1.86 × 10 | 1.08 × 10^{−10} | 1.82 × 10^{−1} | 9.07 × 10^{−15} |

F4 | 6.05 × 10^{−1} | 1.58 × 10^{−1} | 1.08 × 10 | 2.08 × 10^{−11} | 3.03 × 10^{−5} | 5.56 × 10^{−13} |

F5 | 6.03 × 10^{1} | 1.18 × 10^{1} | 28.93 | 23.36 | 1.02 × 10^{2} | 22.65 |

F6 | 3.69 × 10^{−8} | 1.10 × 10^{3} | 4.29 | 6.98 × 10^{−6} | 6.29 × 10^{−9} | 3.65 × 10^{−6} |

F7 | 7.07 × 10^{−2} | 1.01 × 10^{−2} | 7.90 × 10^{−4} | 1.92 × 10^{−3} | 1.01 × 10^{−2} | 6.61 × 10^{−4} |

F8 | −6.06 × 10^{2} | −2.09 × 10^{3} | −6.07 × 10^{3} | −5.58 × 10^{3} | −6.84 × 10^{3} | −1.02 × 10^{4} |

F9 | 4.67 × 10^{3} | 6.59 × 10^{−1} | 23.30 × 10 | 2.54 × 10^{1} | 4.77 × 10^{1} | 7.96 × 10^{0} |

F10 | 7.33 × 10^{−2} | 9.56 × 10^{−1} | 1.67 × 10^{−8} | 1.51 × 10^{−14} | 1.86 × 10^{−5} | 7.99 × 10^{−15} |

F11 | 9.28 × 10^{−3} | 4.88 × 10^{−1} | 6.89 × 10^{−13} | 0 | 1.38 × 10^{−3} | 0 |

F12 | 7.26 × 10^{−3} | 1.11 × 10^{−1} | 6.42 × 10^{−1} | 2.67 × 10^{−7} | 1.45 × 10^{−11} | 1.96 × 10^{−7} |

F13 | 2.53 × 10^{−3} | 1.29 × 10^{−1} | 12.67 × 10 | 9.72 × 10^{−2} | 2.24 × 10^{−10} | 8.98 × 10^{−6} |

F14 | 3.46 × 10^{0} | 1.26 × 10^{0} | 7.87 × 10^{0} | 9.98 × 10^{−1} | 1.16 × 10^{0} | 9.98 × 10^{−1} |

F15 | 8.94 × 10^{−4} | 4.00 × 10^{−3} | 4.54 × 10^{−4} | 3.07 × 10^{−4} | 1.48 × 10^{−4} | 3.07 × 10^{−4} |

F16 | −1.03 × 10^{0} | −1.03 × 10^{0} | −1.03 × 10^{0} | −1.03 × 10^{0} | −1.03 × 10^{0} | −1.03 × 10^{0} |

F17 | 3.99 × 10^{−1} | 4.00 × 10^{−1} | 3.98 × 10^{−1} | 3.97 × 10^{−1} | 3.98 × 10^{−1} | 3.097 × 10^{−1} |

F18 | 3.00 × 10^{0} | 5.70 × 10^{0} | 3.00 × 10^{0} | 3.00 × 10^{0} | 3.00 × 10^{0} | 3.00 × 10^{0} |

F19 | −3.86 × 10^{0} | −3.86 × 10^{0} | −3.86 × 10^{0} | −3.86 × 10^{0} | −3.86 × 10^{0} | −3.86 × 10^{0} |

F20 | −3.27 × 10^{0} | −3.31 × 10^{0} | −3.24 × 10^{0} | −3.32 × 10^{0} | −3.32 × 10^{0} | −3.32 × 10^{0} |

F21 | −8.60 × 10^{0} | −5.66 × 10^{0} | −5.05 × 10^{0} | −10.15 × 10^{0} | −8.60 × 10^{0} | −10.15 × 10^{0} |

F22 | −9.07 × 10^{0} | −7.34 × 10^{0} | −10.39 × 10^{0} | −10.40 × 10^{0} | −1.04 × 10^{1} | −10.40 × 10^{0} |

F23 | −9.20 × 10^{0} | −6.25 × 10^{0} | −1.04 × 10^{1} | −1.05 × 10^{1} | −1.05 × 10^{1} | −1.05 × 10^{1} |

Function Number | PSO | SSR | GWO | I-GWO | LF-IGWO |
---|---|---|---|---|---|

F1 | 1.50 × 10^{11} | 1.11 × 10^{11} | 3.00 × 10^{10} | 1.59 × 10^{10} | 5.04 × 10^{7} |

F2 | 2.64 × 10^{8} | 1.00 × 10^{9} | 2.84 × 10^{7} | 3.49 × 10^{6} | 4.68 × 10^{5} |

F3 | 4.12 × 10^{6} | 1.51 × 10^{8} | 5.19 × 10^{4} | 8.59 × 10^{4} | 2.92 × 10^{4} |

F4 | 4.53 × 10^{4} | 2.38 × 10^{4} | 5.30 × 10^{3} | 1.082 × 10^{3} | 491.677 |

F5 | 1.13 × 10^{3} | 1.00 × 10^{3} | 778.48 | 787.30 | 564.402 |

F6 | 751.76 | 699.66 | 660.77 | 646.87 | 601.758 |

F7 | 3.37 × 10^{3} | 1.76 × 10^{3} | 1.22 × 10^{3} | 1.45 × 10^{3} | 805.982 |

F8 | 1.37 × 10^{3} | 1.28 × 10^{3} | 1.05 × 10^{3} | 1.068 × 10^{3} | 849.817 |

F9 | 4.47 × 10^{4} | 2.51 × 10^{4} | 5.48 × 10^{3} | 6.911 × 10^{3} | 1.07 × 10^{3} |

F10 | 1.19 × 10^{4} | 1.08 × 10^{4} | 6.32 × 10^{3} | 7.529 × 10^{3} | 3.01 × 10^{3} |

F11 | 5.02 × 10^{4} | 1.18 × 10^{4} | 4.44 × 10^{3} | 2.379 × 10^{3} | 1.31 × 10^{3} |

F12 | 3.44 × 10^{10} | 1.66 × 10^{10} | 7.21 × 10^{9} | 3.58 × 10^{8} | 4.38 × 10^{6} |

F13 | 3.82 × 10^{10} | 2.06 × 10^{10} | 2.89 × 10^{9} | 6.27 × 10^{7} | 2.68 × 10^{4} |

F14 | 1.59 × 10^{8} | 2.59 × 10^{7} | 1.71 × 10^{4} | 5.87 × 10^{4} | 3.92 × 10^{3} |

F15 | 8.30 × 10^{9} | 4.22 × 10^{9} | 1.94 × 10^{4} | 1.03 × 10^{7} | 1.11 × 10^{4} |

F16 | 9.62 × 10^{3} | 1.30 × 10^{4} | 3.566 × 10^{3} | 3.184 × 10^{3} | 2.10 × 10^{3} |

F17 | 1.56 × 10^{5} | 7.15 × 10^{3} | 2.269 × 10^{3} | 2.458 × 10^{3} | 1.78 × 10^{3} |

F18 | 9.70 × 10^{8} | 2.56 × 10^{8} | 7.17 × 10^{5} | 1.05 × 10^{5} | 5.98 × 10^{4} |

F19 | 7.79 × 10^{9} | 1.00 × 10^{9} | 1.77 × 10^{6} | 2.70 × 10^{7} | 2.13 × 10^{4} |

F20 | 4.09 × 10^{3} | 4.11 × 10^{3} | 2.240 × 10^{3} | 2.484 × 10^{3} | 2.20 × 10^{3} |

F21 | 2.98 × 10^{3} | 2.79 × 10^{3} | 2.577 × 10^{3} | 2.552 × 10^{3} | 2.36 × 10^{3} |

F22 | 1.22 × 10^{4} | 1.29 × 10^{4} | 6.644 × 10^{3} | 4.186 × 10^{3} | 2.44 × 10^{3} |

F23 | 4.27 × 10^{3} | 4.01 × 10^{3} | 3.207 × 10^{3} | 2.902 × 10^{3} | 2.70 × 10^{3} |

F24 | 4.49 × 10^{3} | 4.41 × 10^{3} | 3.377 × 10^{3} | 3.071 × 10^{3} | 2.87 × 10^{3} |

F25 | 2.35 × 10^{4} | 6.47 × 10^{3} | 4.228 × 10^{3} | 3.669 × 10^{3} | 2.94 × 10^{3} |

F26 | 2.02 × 10^{4} | 1.32 × 10^{4} | 9.251 × 10^{3} | 5.105 × 10^{3} | 3.40 × 10^{3} |

F27 | 4.90 × 10^{3} | 5.86 × 10^{3} | 3.798 × 10^{3} | 3.298 × 10^{3} | 3.20 × 10^{3} |

F28 | 1.56 × 10^{4} | 1.04 × 10^{4} | 5.270 × 10^{3} | 3.686 × 10^{3} | 3.33 × 10^{3} |

F29 | 6.16 × 10^{4} | 5.22 × 10^{4} | 5.208 × 10^{3} | 4.106 × 10^{3} | 3.56 × 10^{3} |

F30 | 8.49 × 10^{9} | 3.25 × 10^{9} | 2.1 × 10^{8} | 1.86 × 10^{7} | 4.53 × 10^{5} |

Angles | LF-IGWO | IGWO | GWO | PSO |
---|---|---|---|---|

A1 | 0.036788 | −0.25222 | 0.2074 | 0.2675 |

A2 | 0.84503 | 0.73168 | 0 | −0.6905 |

A3 | 0.24646 | 0.13577 | 0.6342 | −0.3538 |

A4 | 0.94454 | −0.87421 | 0.0864 | 0.8315 |

A5 | 0.17788 | 0.18032 | 0.5053 | −0.3885 |

A6 | 0.69653 | −0.26365 | 0.0428 | 1.1988 |

A7 | 0.47166 | 0.33732 | 0.0575 | −0.7247 |

A8 | 0.29501 | 0.42087 | 0.1335 | −1.0523 |

A9 | 0.4211 | −0.4421 | 0.74 | 0.8072 |

A10 | 0.77193 | 0.5118 | 0.3837 | 0.9868 |

A11 | 1.2306 | −0.086737 | 0.3183 | 0.2315 |

A12 | 0.56625 | −0.48201 | 0.5527 | −0.4606 |

A13 | 0.11742 | −0.010185 | 0.8574 | 0.0215 |

A14 | 0.63186 | 0.071222 | 0.2855 | −0.1755 |

A15 | 0.36631 | 0.67509 | 0.4286 | −0.3821 |

Algorithm | LF-IGWO | IGWO | GWO | PSO |
---|---|---|---|---|

THD (%) | 4.54 | 13.81 | 14.68 | 5.92 |

Algorithms | d | D2 | N | Op. Value |
---|---|---|---|---|

LF-IGWO | 0.0527485 | 0.367865 | 10.665 | 0.01267 |

IGWO | 0.0517029 | 0.356964 | 11.2893 | 0.012681 |

GWO | 0.05169 | 0.3567 | 11.2888 | 0.012666 |

PSO | 0.051728 | 0.357644 | 11.24454 | 0.012674 |

GA | 0.05148 | 0.35166 | 11.6322 | 0.012704 |

Algorithms | ${\mathit{T}}_{\mathit{s}}$ | ${\mathit{T}}_{\mathit{h}}$ | $\mathit{R}$ | $\mathit{L}$ | Op. Value |
---|---|---|---|---|---|

LF-IGWO | 0.8021493 | 0.5113717 | 41.54997 | 183.573 | 6282.2292 |

GWO | 0.8125 | 0.4345 | 42.089181 | 176.758731 | 6051.5639 |

GA | 0.8125 | 0.4345 | 40.3239 | 200 | 6288.7445 |

PSO | 0.883044 | 0.533053 | 45.38829 | 190.0616 | 7865.233 |

IGWO | 0.9035907 | 0.5319827 | 44.20703 | 154.0763 | 6793.5848 |

Algorithms | ${\mathit{T}}_{\mathit{s}}$ | ${\mathit{T}}_{\mathit{h}}$ | $\mathit{R}$ | $\mathit{L}$ | Op. Value |
---|---|---|---|---|---|

LF-IGWO | 0.8021493 | 0.5113717 | 41.54997 | 183.573 | 6282.2292 |

GWO | 0.8125 | 0.4345 | 42.089181 | 176.758731 | 6051.5639 |

GA | 0.8125 | 0.4345 | 40.3239 | 200 | 6288.7445 |

PSO | 0.883044 | 0.533053 | 45.38829 | 190.0616 | 7865.233 |

IGWO | 0.9035907 | 0.5319827 | 44.20703 | 154.0763 | 6793.5848 |

Function Number | PSO | GA | GWO | IGWO | SSR |
---|---|---|---|---|---|

F1 | 0.001588 | 0.000156 | 0.003177 | 0.027086 | 0.10499 |

F2 | 0.004522 | 0.000145 | 0.003177 | 0.18577 | 0.10499 |

F3 | 0.000145 | 0.000145 | 0.000145 | 1 | 0.10499 |

F4 | 0.009524 | 0.000145 | 0.000145 | 0.37904 | 0.10499 |

F5 | 0.000145 | 0.28378 | 0.000145 | 0.077272 | 0.10499 |

F6 | 0.000145 | 0.000145 | 0.000145 | 0.27071 | 0.10499 |

F7 | 0.0001268 | 0.10499 | 0.000145 | 0.48252 | 0.10499 |

F8 | 0.30815 | 0.37904 | 0.31815 | 0.28378 | 0.10499 |

F9 | 0.62527 | 0.69913 | 0.10221 | 0.98231 | 0.10499 |

F10 | 0.000294 | 0.000156 | 0.000145 | 0.23985 | 0.10499 |

F11 | 0.000156 | 0.000156 | 0.000145 | 0.46427 | 0.10499 |

F12 | 0.351889 | 0.000145 | 0.000145 | 0.063533 | 0.10499 |

F13 | 0.168452 | 0.000156 | 0.000145 | 0.59969 | 0.10499 |

F14 | 0.061837 | 0.01857 | 0.071429 | 0.66667 | 0.66667 |

F15 | 0.62483 | 0.35684 | 0.47619 | 0.88571 | 0.4 |

F16 | 0.07467 | 0.071429 | 0.071429 | 0.66667 | 0.66667 |

F17 | 0.071498 | 0.071429 | 0.071429 | 0.66667 | 0.66667 |

F18 | 0.072795 | 0.66667 | 0.071429 | 1 | 0.66667 |

F19 | 0.069426 | 0.061584 | 0.071429 | 0.7 | 0.5 |

F20 | 0.49509 | 0.51455 | 0.48485 | 0.69913 | 0.28571 |

F21 | 0.0064732 | 0.0008616 | 0.009524 | 0.68571 | 0.4 |

F22 | 0.018654 | 0.02666 | 0.009524 | 0.88571 | 0.4 |

F23 | 0.72364 | 0.5426 | 0.009524 | 1 | 0.4 |

Tension Compression Spring | 0.7569 | 0.6 | 0.71429 | 1 | 0.5 |

Welded Beam | 0.09852 | 0.854554 | 0.47619 | 0.68571 | 1 |

Pressure Vessel | 0.09852 | 0.854554 | 0.25714 | 0.68571 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Bhatt, B.; Sharma, H.; Arora, K.; Joshi, G.P.; Shrestha, B.
Levy Flight-Based Improved Grey Wolf Optimization: A Solution for Various Engineering Problems. *Mathematics* **2023**, *11*, 1745.
https://doi.org/10.3390/math11071745

**AMA Style**

Bhatt B, Sharma H, Arora K, Joshi GP, Shrestha B.
Levy Flight-Based Improved Grey Wolf Optimization: A Solution for Various Engineering Problems. *Mathematics*. 2023; 11(7):1745.
https://doi.org/10.3390/math11071745

**Chicago/Turabian Style**

Bhatt, Bhargav, Himanshu Sharma, Krishan Arora, Gyanendra Prasad Joshi, and Bhanu Shrestha.
2023. "Levy Flight-Based Improved Grey Wolf Optimization: A Solution for Various Engineering Problems" *Mathematics* 11, no. 7: 1745.
https://doi.org/10.3390/math11071745