Grey Wolf Optimization (GWO) is a nature-inspired metaheuristic algorithm that has gained popularity for solving optimization problems. In GWO, the success of the algorithm heavily relies on the efficient updating of the agents’ positions relative to the leader wolves. In this paper, we provide a brief overview of the Grey Wolf Optimization technique and its significance in solving complex optimization problems. Building upon the foundation of GWO, we introduce a novel technique for updating agents’ positions, which aims to enhance the algorithm’s effectiveness and efficiency. To evaluate the performance of our proposed approach, we conduct comprehensive experiments and compare the results with the original Grey Wolf Optimization technique. Our comparative analysis demonstrates that the proposed technique achieves superior optimization outcomes. These findings underscore the potential of our approach in addressing optimization challenges effectively and efficiently, making it a valuable contribution to the field of optimization algorithms.
Nature-Inspired Meta-Heuristic techniques have revolutionized the landscape of computational problem-solving by harnessing the inherent balance and efficiency found in the natural world. These innovative algorithms draw inspiration from the intricate workings of nature’s systems and have emerged as powerful tools across various domains [
The Grey Wolf Optimizer (GWO) has emerged as a versatile and powerful metaheuristic algorithm renowned for its ability to tackle a wide array of optimization problems. Its adaptability and efficiency have earned it a prominent place in the toolbox of researchers and practitioners across diverse domains. In recent years, GWO has found applications in numerous practical scenarios, showcasing its effectiveness in addressing real-world challenges. For instance, it has been extensively applied in the realm of robotics [
In this context, Grey Wolf Optimization (GWO) stands out as an algorithm that closely emulates the social hierarchy and hunting behavior of grey wolves. It manifests an inherent ability to adapt and optimize, making it a valuable tool in the world of optimization. At its core, GWO revolves around the emulation of the pack’s hunting dynamics, with a keen focus on the interplay between leader wolves and their prey.
The key principle that underlies GWO is the precise and efficient updating agents’ positions. This pivotal step dictates its capacity to attain optimal fitness values. As a result, the refinement and enhancement of this crucial aspect have remained at the forefront of research efforts in the field of optimization. In this paper, we embark on a journey to delve deeper into the realm of Grey Wolf Optimization, exploring its theoretical underpinnings and practical implications in solving intricate optimization problems. The motivation behind this endeavor lies in the continuous pursuit of improving optimization algorithms’ performance and applicability in addressing real-world challenges. In the subsequent sections, we provide an extensive overview of the Grey Wolf Optimization technique, elucidating its conceptual framework and mechanisms. While acknowledging its remarkable capabilities, we also recognize the scope for further enhancement. Building upon the foundational principles of GWO, we introduce a novel technique aimed at refining the process of updating agents’ positions. The rationale for this innovation is rooted in the pursuit of maximizing the algorithm’s effectiveness and efficiency, attributes that are indispensable for tackling complex optimization problems efficiently. Our proposed technique is meticulously designed to fine-tune the dynamics of GWO, amplifying its problem-solving capabilities. To substantiate the effectiveness of our proposed approach, we present the results of a comprehensive series of experiments conducted under diverse optimization scenarios.
The crux of this paper lies in a rigorous comparative analysis between our novel technique and the conventional Grey Wolf Optimization method. This analysis is vital for empirically assessing the performance gains and computational efficiency offered by our approach. Preliminary findings indicate that our proposed technique surpasses the performance benchmarks set by traditional GWO. These outcomes underscore the immense potential of our approach in effectively addressing optimization challenges, offering promising avenues for its application across various domains.
As the quest for optimization solutions continues to intensify across industries and research disciplines, our contributions in this paper aim to further the understanding and efficacy of Grey Wolf Optimization. By optimizing the algorithm’s critical agents’ positions updating process, we aim to facilitate more efficient and effective optimization, making a significant stride in the ever-evolving field of optimization algorithms.
The primary issue with the Grey Wolf Optimization (GWO) algorithm lies in its approach to update the position of the agents. GWO traditionally suggests placing the new position at the average point of the three vectors calculated with respect to the leader wolves (alpha, beta, and delta). However, this approach lacks logical coherence, as the average point remains uniform across the three leaders, with no apparent advantage observed for any one of them. It is more reasonable to locate the new position in a manner that maximizes its proximity to the alpha wolf while maintaining relative distances from the beta and delta wolves. Specifically, the distance to the alpha wolf (referred to as “a”) should be less than the distance to the beta wolf (“b”), which in turn should be less than the distance to the delta wolf (“c”). This configuration ensures that the new position is closest to the alpha wolf, relatively close to the beta wolf, and farthest from the delta wolf.
Addressing this problem involves two key tasks: Localizing the Critical Point: The first task is to determine the precise location of the critical point where the agents in the coming iteration should be positioned within the GWO algorithm. This critical point, ideally situated to maximize the algorithm’s efficiency, needs to be identified through a systematic and precise method.
Optimizing the Distance Ratios (a, b, c): The second task involves determining the optimal ratio between the three distances (a, b, and c). Finding the right balance between these distances is crucial for the algorithm’s performance. This optimization seeks to ensure that the new position adheres to the logical hierarchy of distances from the leader wolves. By addressing these tasks, we aim to refine and enhance the updating step within the GWO algorithm. This approach holds the potential to significantly improve the algorithm’s efficiency and effectiveness in solving optimization problems, offering a more coherent and biologically inspired solution.
In this section, we introduce an innovative approach designed to enhance the efficiency and effectiveness of the Grey Wolf Optimization (GWO) algorithm by revisiting the pivotal step of updating agents’ positions.
Regarding the coordinates of best three agent fitness value x a l p h a , x b e t a , and x d e l t a , respectively.
The primary objective of our proposed technique is to replace the conventional practice of positioning wolves at the average point among the three coordinates x i a l p h a x i b e t a and x i d e l t a where x i a l p h a x i b e t a and x i d e l t a are the relative coordinates of the wolf current position with respect to the three leaders wolves (alpha, beta, and delta) and the index “I” refers to a certain wolf.
x i a l p h a = x a l p h a − A 1 D a l p h a (1)
D a l p h a = | C 1 x a l p h a − x i k | (2)
C 1 = 2 ∗ r n d (3)
A 1 = 2 ∗ a ∗ r n d − a (4)
where x i k refers to the position of a certain wolf “i” in the iteration number “k”
a = (number of iterations −1) for the first iteration, and 0 ≤ r n d ≤ 1 .
Applying the same steps to get x i b e t a , and x i d e l t a .
In conventional GWO algorithm the new position of a certain wolf is calculated as following:
x i k + 1 = x i a l p h a + x i b e t a + x i d e l t a 3 (5)
where x i k + 1 is the new position of the same wolf “i” with a more refined and biologically inspired approach. This new positioning strategy prioritizes the alpha wolf with the highest emphasis, followed by a slightly reduced priority on the beta wolf, and lastly the delta wolf. The fundamental aim is to align the new position within the hierarchy of the wolf pack, thus optimizing its proximity to the alpha wolf while maintaining appropriate relative distances from the beta and delta wolves.
To achieve this objective, we employ a rigorous mathematical analysis. Our approach entails calculating the optimal coordinates for the wolf new position, instead of having the average of x i a l p h a x i b e t a and x i d e l t a , giving priority to x i a l p h a then x i b e t a .
The calculation of the agent’s new position involves determining a point that fulfills the following criteria:
Distance “a” (to the alpha wolf) is minimized to position the new location of the agents closest to the alpha wolf.
Distance “b” (to the beta wolf) is adjusted to provide a slightly lower priority than “a”.
Distance “c” (to the delta wolf) is maximized to maintain the relative distance hierarchy.
Additionally, we delve into the optimization of the ratios between these distances “a”, “b”, and “c”. Seeking the optimal balance between these ratios further enhances the precision of our proposed technique. By systematically solving these mathematical equations and optimizing the parameters involved, our proposed technique ensures a more biologically plausible and efficient approach for updating the agents’ positions within the GWO algorithm. In the subsequent sections, we present the mathematical formulations and detailed algorithms utilized in our approach, along with empirical results that highlight the advantages of this novel technique in terms of optimization outcomes and computational efficiency.
In GWO, in case of minimization problem the fitness values F of the three leader wolves are as following: F a l p h a w o l f < F b e t a w o l f < F d e l t a w o l f .
To ensure that, the distance a < distance b < distance c, we can make them relative to the fitness values of the leader wolves as following:
a : b : c = F a l p h a w o l f F d e l t a w o l f : F b e t a w o l f F d e l t a w o l f : 1 (6)
The three vectors ( x 1 , y 1 ) , ( x 2 , y 2 ) , and ( x 3 , y 3 ) represents x i a l p h a x i b e t a and x i d e l t a which are related to the three leaders alpha wolf, beta wolf, and delta wolf respectively and their values differ from one agent to another. The new position x i k + 1 is (x, y) for a certain agent can be calculated mathematically as follows:
( x 1 − x ) 2 + ( y 1 − y ) 2 = a 2 (7)
( x 2 − x ) 2 + ( y 2 − y ) 2 = b 2 (8)
( x 3 − x ) 2 + ( y 3 − y ) 2 = c 2 (9)
Each equation represents a circle equation, and we solve for the value of x and y. Actually, the solution of this system is not always real numbers, but the previous analysis gives us a hint for an alternative approach.
It is required to obtain a relation for the proposed point “q” to obtain gradual increase of the three vectors, which makes sense and gives priority to the alpha wolf. It’s a weighted average of the point of the three leader wolves. First, we discuss the power of the weighted average in shaping the distances and then we delve to the appropriate values of this weights.
“p” represents the new position in case of traditional GWO, however “q” represents the new position in case of our new proposed technique.
As shown in
p = ( x 1 , y 1 ) + ( x 2 , y 2 ) + ( x 3 , y 3 ) 3 (10)
Distancetoalphawolf = ( x 1 − x ) 2 + ( y 1 − y ) 2 (11)
Distancetobetawolf = ( x 2 − x ) 2 + ( y 2 − y ) 2 (12)
Distancetodeltawolf = ( x 3 − x ) 2 + ( y 3 − y ) 2 (13)
| α wolf | Beta wolf | Delta wolf | p | Distance to alpha | Distance to beta | Distance to delta | q | Distance to alpha | Distance to beta | Distance to delta |
|---|---|---|---|---|---|---|---|---|---|---|
| (1, 3) | (3, 5) | (7, 2) | (3.66, 3.33) | 2.6 | 1.7 | 3.5 | (2.11, 3.33) | 1.16 | 1.88 | 5.06 |
| (−2, 5) | (3, −4) | (2, 0) | (1, 0.33) | 5.54 | 4.77 | 1.05 | (−0.44, 2.44) | 2.99 | 7.33 | 3.45 |
| (−7, −3) | (0, 0) | (4, −2) | (−1, −1.66) | 6.146 | 1.94 | 5.011 | (−4.22, −2.22) | 2.88 | 4.77 | 8.22 |
| (2, 7) | (0, −3) | (−1, −1) | (−0.33, 1) | 6.22 | 4.01 | 2.4 | (1.22, 3.88) | 3.2 | 6.99 | 5.37 |
| (5, 3) | (2, −1) | (0, 3) | (2.33, 1.66) | 2.98 | 2.68 | 2.68 | (3.77, 2.11) | 1.51 | 3.58 | 3.88 |
Point “q” which is the new proposed position of the agents can be calculated as following:
q = 1 r 1 ∗ ( x 1 , y 1 ) + 1 r 2 ∗ ( x 2 , y 2 ) + 1 r 3 ∗ ( x 3 , y 3 ) 1 r 1 + 1 r 2 + 1 r 3 (14)
where r1, r2, and r3 are values satisfying r 1 < r 2 < r 3 . In this table, r 1 = 0.1 , r 2 = 0.3 , and r 3 = 0.6 .
As shown in
On the other hand, in our new proposed technique, there is a gradual increase from alpha wolf and beta wolf, ending by delta wolf.
The values of, r1, r2, and r3 are very critical and have to be optimized for best performance, so different cases of r1, r2, and r3 are set to compare between then regarding the fitness value. One of a suggested values of the three ratios r1, r2, and r3 is the ratio between the fitness value of the three leader wolves.
r 1 : r 2 : r 3 = F a l p h a w o l f F d e l t a w o l f : F b e t a w o l f F d e l t a w o l f : 1 (15)
But there is no guarantee that the previous ratio is the best ratio to obtain the best final value of the model so, a general formula in Equation (16) was formulated to cover all possibilities and ensure best ratio.
q = 1 k ∗ ( x 1 , y 1 ) + 1 m ∗ ( x 2 , y 2 ) + 1 n ∗ ( x 3 , y 3 ) 1 k + 1 m + 1 n (16)
where k , m , n have values from 0.1 to 1 step size 0.1.
Using three nested loops to iterate over all possible vector of ( k , m , n ) where k < m < n to ensure that 1 k > 1 m > 1 n .
In this section, 10 benchmark functions used are presented in
The condition of the code running is as following: the maximum iterations are 500, and the number of search agents is 30. The initial random positions, as well as the random parameters in Equations (3), and (4) are fixed to be 0.33 for the traditional GWO and the new proposed technique to ensure that the only parameter affect the model efficiency is the way of positioning the wolves in each iteration.
| Objective Function | Range | Dimension | Fminmun | |
|---|---|---|---|---|
| F1 | F 1 ( x ) = ∑ i = 1 m x i 2 | [−100, 100] | 10 | 0 |
| F2 | F 2 ( x ) = ∑ i = 1 m | x i | + ∏ i = 1 m | x i | | [−10, 10] | 10 | 0 |
| F3 | F 3 ( x ) = ∑ i = 1 m ( ∑ j = 1 i x i ) 2 | [−100, 100] | 10 | 0 |
| F4 | F 4 ( x ) = max { | x i | , 1 ≤ i ≤ m } | [−100, 100] | 10 | 0 |
| F5 | F 5 ( x ) = ∑ i = 1 n − 1 ( x i x i + 1 + 1 + x i + 1 x i + 1 ) | [−1, 4] | 10 | 0 |
| F6 | F 6 ( x ) = − 20 exp ( − 0.2 1 D ∑ i = 1 D x i 2 ) − exp ( 1 D ∑ i = 1 D cos ( 2 π x i ) ) + 20 + e | [−100, 100] | 10 | 0 |
| F7 | F 7 ( x ) = ∑ i = 1 D − 1 ( x i 2 ) x i + 1 2 + 1 + ( x i + 1 2 ) x i 2 + 1 | [−1.28, 1.28] | 10 | 0 |
| F8 | F 8 ( x ) = ∑ i = 1 D | x i | i + 1 | [−5, 5] | 10 | 0 |
| F9 | F 9 ( x ) = ∑ i = 1 D x i 2 4000 − ∏ i = 1 D cos ( x i i ) + 1 | [−5.12, 5.12] | 10 | 0 |
| F10 | F 10 ( x ) = 1 − cos ( 2 π ∑ i = 1 D x i 2 ) + 0.1 ∑ i = 1 D x i 2 | [−32, 32] | 10 | 0 |
| Traditional | Best enhancement | |||||
|---|---|---|---|---|---|---|
| Value e−6 | k | m | n | Value e−6 | Reduced by % | |
| F1 | 5.48 | 0.5 | 0.7 | 1 | 1.89 | 65.5 |
| F2 | 7.46 | 0.4 | 0.9 | 1 | 5.57 | 25.3 |
| F3 | 7.03 | 0.3 | 0.4 | 0.9 | 2.32 | 66.99 |
| F4 | 9.52 | 0.1 | 0.4 | 1 | 5.39 | 43.38 |
| F5 | 6.38 | 0.1 | 0.5 | 1 | 1.64 | 74.29 |
| F6 | 8.80 | 0.1 | 0.4 | 0.8 | 2.34 | 73.4 |
| F7 | 769.9 | 0.1 | 0.8 | 1 | 19.2 | 97.50 |
| F8 | 31.6 | 0.3 | 0.8 | 1 | 5.3 | 83.22 |
| F9 | 4.262 | 0.6 | 0.7 | 0.9 | 2.94 | 31.01 |
| F10 | 8 | 0.2 | 0.6 | 1 | 5.92 | 26 |
As shown in
In conclusion, Grey Wolf Optimization (GWO) has established itself as a robust and effective optimization algorithm, drawing inspiration from the coordinated hunting behavior of grey wolves. Over the years, it has proven its mettle in solving complex optimization problems, showcasing adaptability and efficiency. In this paper, we embarked on a comprehensive exploration of GWO, shedding light on its theoretical foundations, mechanisms, and practical applications. While recognizing the substantial achievements of GWO, we introduced a novel technique aimed at enhancing its performance by refining the process of updating the position of the agents. Our experimental results have demonstrated the superiority of our proposed approach over the traditional GWO method.
In the future, we can refine our proposed technique by optimizing its parameters and mechanisms for various problem domains. Exploring hybrid approaches with other optimization algorithms offers potential for improved results and increased robustness. Adapting GWO for multi-objective tasks addresses real-world complexity. Investigating parallel computing enhances scalability, enabling larger challenges. Applying our technique to practical problems in fields like engineering and finance validates its effectiveness. A thorough theoretical analysis will provide deeper insights, and user-friendly software tools can broaden its adoption, advancing optimization algorithms.
The authors declare no conflicts of interest regarding the publication of this paper.
Khatab, M., El-Gamel, M., Saleh, A.I., Rabie, A.H. and El-Shenawy, A. (2024) Optimizing Grey Wolf Optimization: A Novel Agents’ Positions Updating Technique for Enhanced Efficiency and Performance. Open Journal of Optimization, 13, 21-30. https://doi.org/10.4236/ojop.2024.131002