A Hybrid Grasshopper Optimization Algorithm With Invasive Weed for Global Optimization

The grasshopper optimization algorithm (GOA) is a promising metaheuristic algorithm for optimization. In the current study, a hybrid grasshopper optimization algorithm with invasive weed optimization (IWGOA) is proposed. The invasive weed optimization (IWO) and random walk strategy are helpful for improving the search precision and accelerating the convergence rate. In addition, the exploration and exploitation capability of the IWGOA algorithm are further enhanced by the grouping strategy. The IWGOA algorithm is compared with some typical and latest optimization algorithms including genetic algorithm (GA), moth-flame optimization algorithm (MFO), particle swarm optimization and gravitational search algorithm (PSOGSA), ant lion optimizer (ALO), conventional GOA algorithm, chaotic GOA algorithm (CGOA) and opposition-based learning GOA algorithm (OBLGOA) on 23 benchmark functions and 30 CEC 2014 benchmark functions. The results show that the IWGOA algorithm is able to provide better outcomes than the other algorithms on the majority of the benchmark functions. Additionally, the IWGOA algorithm is applied to multi-level image segmentation, and obtains promising results. All of these findings demonstrate the superiority of the IWGOA algorithm.


I. INTRODUCTION
Optimization exists in many fields such as engineering [1], image processing [2], energy [3], feature selection [4] and industrial applications [5]. Selecting the best solutions from the given alternatives is vital in solving optimization problems. However, the tradition traversal search needs too much computation to find the best solution. Therefore, many optimization algorithms have been proposed and widely used in solving the optimization tasks. Such algorithms employ the genetic algorithm (GA) [6], particle swarm optimization (PSO) [7], black hole algorithm (BH) [8] and moth-flame optimization algorithm (MFO) [9].
The grasshopper optimization algorithm (GOA) is a recent optimization algorithm by emulating the behaviors of The associate editor coordinating the review of this manuscript and approving it for publication was Chao Shen . grasshoppers [10]. In the mathematical model of the GOA algorithm, the movement of the grasshopper is influenced by the social interaction, gravity force and wind advection. In the GOA algorithm, the grasshoppers are guided to move abruptly and locally, which leads the algorithm has an exploration and exploitation capability. It was reported that the GOA was able to obtain superior results than the other algorithms. Further, the GOA has extensive applications. Mirjalili S Z et al. suggested a GOA for multi-objective optimization problems [11]. Pinto H et al. proposed a binary GOA and applied to the knapsack problem [12]. However, the GOA not always performs well in solving optimization tasks, because the algorithm has drawbacks of limited local search capability and easily falling into the local best solutions [13].
In the current study, we will focus on the modification the GOA. In recent years, various versions of the GOA were proposed by the researchers. Saxena A et al. introduced chaos into the GOA. In addition, 10 different variants of the GOA were presented. In the enhanced chaotic GOA, bridging mechanism was improved by chaotic maps [14]. Liang H et al. introduced a novel version GOA, which was promoted by Levy fight algorithm [15]. Wu J et al. proposed an adaptive grasshopper optimization algorithm (AGOA) to solve trajectory optimization problem. In the AGOA, natural selection strategy played an important role in jumping out the local optimal solutions. The advantages were further enhanced by using democratic decision-making mechanism [16]. To overcome the drawbacks of the conventional GOA, Mafarja M et al. used Evolutionary Population Dynamics (EPD) to promote the performance of the algorithm [17]. Jia, Heming, et al. proposed a novel satellite image segmentation technique using improved GOA and minimum cross entropy (MCE). The conventional GOA was enhanced by self-adaptive differential evolution (jDE), namely GOA-jDE [18]. Luo, Jie, et al. introduced a novel GOA, which was improved by three strategies. Population diversity was improved by gaussian mutation. Then Levy-flight strategy was combined to enhance the global exploration capability. Furthermore, the opposition-based learning was incorporated to further promote the search capability [19]. Ewees, Ahmed A. et al. used oppositionbased learning to improve the conventional GOA, called OBLGOA [20].
The invasive weed optimization (IWO) is developed by Mehrabian A R, which mimics the behavior of colonizing weeds [21]. In the IWO algorithm, the plant is able to produce seeds and the number depends on objective function values. Then the produced seeds search the area in a random way and develop to new plant. The best solution is obtained by the competitive exclusion. Further, the superior performance of the IWO algorithm was confirmed by comparative experiments. In recent years, the IWO algorithm has been applied to many fields such as flow-shop scheduling [22], energy consumption [23] and power systems control [24].
In the conventional GOA algorithm, the movement of each grasshopper neglects the objective function value, which leads that the grasshopper with better objective function value may search the best solutions with a big step. Therefore, the exploitation capability of the GOA needs to be improved. Further, all the grasshoppers gather around the target grasshopper, which causes the algorithm may falls into local best solutions. Therefore, a hybrid grasshopper optimization algorithm with invasive weed (IWGOA) for global optimization is proposed. The main contributions of the paper are as follows: (1) The IWO algorithm and random walk strategy are inserted to the GOA algorithm, this mechanism makes the IWGOA algorithm has a perfect exploitation capability. Further, the convergence rate of the GOA algorithm is accelerated.
(2) The grouping strategy is proposed to control the movements of the grasshoppers in a better way. In the IWGOA algorithm, the positions updating types and steps are influenced by the iterative numbers and objective function values.
(3) The proposed IWGOA algorithm is applied to multilevel image segmentation and compare the performance with the GA, MFO, PSOGSA, ALO, GOA, CGOA and OBLGOA based methods.
The structure of rest sections is organized as follows: Section 2 introduces the conventional GOA algorithm. Section 3 explains the proposed IWGOA algorithm. Section 4 shows the experimental results and related discussions. Section 5 presents the application of the IWGOA algorithm in the field of multi-level image segmentation. The last section concludes our work.

II. CONVENTIONAL GRASSHOPPER OPTIMIZATION ALGORITHM
In the GOA algorithm, the position updates can be expressed as follows [10]: where X i denotes the position of the ith grasshopper, S i is the social interaction, G i represents the gravity force, A i is the wind advection. The social interaction S i is presented as follows: where d ij = x j − x i represents the Euclidean distance of ith and jth grasshopper, f and l represent the parameters to adjust the social forces. The gravity force G i can be given by: where g represents the constant, e g is a vector.
Wind advection A i can be expressed as follows: where u refers to a constant and e w denotes a vector.
The mathematical model can be extended as follows: where N represents the number of grasshoppers.
To apply the GOA to solve the optimization problems, a modified mathematical model can be presented as follows: where ub d and lb d denote the upper and lower bound of the dth dimension, T d is the best position, c max denotes the maximum value, c min denotes the minimum value, l is the current iteration, L refers the maximum number of iterations.

III. PROPOSED IWGOA ALGORITHM
In the IWGOA algorithm, three strategies are introduced to improve the conventional GOA. First of all, the local search capability is promoted by integrating with the IWO algorithm and random walk strategy. Further, the grouping strategy is an efficient tool in enhancing the exploration and exploitation capability.

A. IWO ALGORITHM
In the IWO algorithm, the plant with better objective function values will produces more seeds. The number of seeds produced can be calculated as follows [21]: where s max is the maximum number of seeds, s min is the minimum number of seeds, f best is the best objective function value, f worst is the worst objective function value. The produced seeds are normally distributed, which is given in Eq. (10).
where σ init and σ final denote the initial and final standard deviation values, respectively.

B. RANDOM WALK STRATEGY
In the IWGOA algorithm, random walk strategy is proposed to further improve the exploitation capability. New grasshoppers are generated from the best grasshoppers and secondbest grasshoppers in a random way. It can be calculated as follows: where X new is newly generated grasshoppers, X fbest and X sbest are the best and second-best grasshoppers of the current number of iterations, respectively.

C. GROUPING STRATEGY
In recent years, various inertia weights are proposed to improve the performance of optimization algorithms [25]- [29]. In the conventional GOA algorithm, the coefficient c is the key to balance exploration and exploitation, which decreases with the increasing number of iterations with a linear way. By this way, all the grasshopper positions update in the same way, and there is room for improvement. Therefore, the grouping strategy is proposed to optimize the search process. In the grouping strategy, the grasshoppers are divided into three groups based on objective function values named elite grasshoppers (EG), onlooker grasshoppers (OG)  and scout grasshoppers (SG). Elite grasshoppers with better objective function values are near the theory best solution, so they need move with small steps. The coefficient c for elite  grasshopper is calculated as follows: where ζ is the adjust constant. The onlooker grasshoppers with median objective function values update the positions keep the same way with the conventional GOA algorithm. The scout grasshoppers play important roles in helping the IWGOA algorithm jumps out of the local best trap. At the first three-quarters of iterations, the positions of scout grasshoppers with worse objective functions are randomly generated. For the rest quarter of iterations, the scout grasshoppers move with big steps, which further enhance the exploration capability. The coefficient c for scout grasshopper is given by: The improvement mathematical model can be presented as follows: The pseudo codes of the proposed IWGOA algorithm is showed in Fig.1. Additionally, the flowchart of the proposed IWGOA algorithm is illustrated in Fig.2.

IV. EXPERIMENTS
To demonstrate the superiority of the proposed IWGOA algorithm, various benchmark functions and algorithms are employed. Firstly, the comparative experiments are carried on 23 well-known benchmark functions. Three families of benchmark functions called unimodal test functions, multimodal test functions and fixed-dimension multimodal test functions are introduced to evaluate the performance of the IWGOA [30]- [33]. The description of the benchmark functions is listed in Tables 1-3. The unimodal test functions have only one extreme point. They are suitable for assessing the convergence rate and the exploitation of the algorithm. Multimodal benchmark functions are used to assess the local avoidance and exploration capability of the algorithm, because they have multiple solutions. Fixed-dimension multimodal benchmark functions have consistent functions with multimodal test functions. The differences are the multimodal benchmark functions have higher dimensionality. Thirty test functions provided by CEC 2014 special session are listed in Table 4, to further verify the IWGOA algorithm [34].
Some representative and latest meta-heuristic algorithms such as GA [35], MFO [9], PSOGSA [36], ALO [37], GOA [10] CGOA [13] and OBLGOA [20] are used to make comparisons. The parameters setting of the algorithms are consistent with the literatures, which are showed in Table 5. In addition, search agents N = 30 and the number of  iterations Iter max = 300 are used for all the optimization algorithms.
Each algorithm runs 30 times, and the best, worst, mean and standard deviation (Std) of objective function values are recorded. Moreover, Wilcoxon statistical test at 5% significance level is used to compare the results in a quantitative way [38], [39]. Then the experiments are visually displayed by the Box-and-whisker images. Finally, the convergence rate of the eight algorithms is studied to make the comparisons more comprehensive.  As Table 6 shows, the proposed IWGOA algorithm is able to obtain competitive outcomes on the unimodal test functions. The IWGOA outperforms the GA, MFO, PSOGSA, ALO, GOA, CGOA and OBLGOA for f 1 , f 2 , f 3 , f 4 , f 5 and f 6 . Further, the IWGOA obtains the second-best results in terms of the mean objective function values for f 7 . As we can see from the Table 7, p-values are much smaller than 0.05, which confirms the superiority of the IWGOA is significant. Figs 3-9 show the box-and-whiskers of the unimodal test functions for the eight algorithms. It can be seen from Table 6 and Figs 3-9, the standard deviation values of the IWGOA are less than the alternatives on the majority of the VOLUME 8, 2020  cases, which illustrates the IWGOA obtains consistent results by each run. It indicates that the IWGOA is more stable than the other algorithms. From the experiment results, it is obvious that the IWGOA has an excellent exploitation capability. The reason why the IWGOA obtains outperform results is due to the integrated IWO algorithm, random walk strategy and grouping strategy. Comparisons of the convergence rate for the eight algorithms are showed in Figs. 10-16. It should be noted all the convergence curves are obtained by the averaged best objective function values with 30 runs. From  Figs 10-16, the IWGOA has a faster convergence rate than the alternatives on the majority of the cases. By inserting the three strategies, the convergence rate of the GOA is significantly improved. The IWO algorithm makes the search agents with better objective function values proceed further local search. This mechanism helps the IWGOA has a better target    position by each iteration. Moreover, random walk strategy is able to search the candidate space with an efficient way. Furthermore, the grouping strategy makes the elite grasshoppers move with smaller steps. Therefore, we can conclude the IWGOA has a higher exploitation capability and faster convergence rate than the other algorithms. Table 8 shows that the IWGOA is able to obtain better results than the other alternatives on the multimodal test functions for the most cases. The IWGOA exhibits the best                   From all these findings, it is further proved that our modifications are able to enhance the exploration capability of the GOA. In the IWGOA, the grasshopper with worse objective function value keeps a strong global search capability because

V. MULTI-LEVEL IMAGE SEGMENTATION
In multi-level image segmentation, finding the optimal thresholds using optimization algorithms has been a popular     method in the literature [40]- [42]. However, it is still challenging to set the optimal thresholds especially the number of thresholds is high. One of the reasons is that the optimization VOLUME 8, 2020     algorithm has limited exploration and exploitation capability. Therefore, the proposed IWGOA algorithm is used to choose the optimal thresholds.     Kapur's entropy is a prevalent technique that used as a criterion for multi-level image segmentation [43]. Therefore, Kapur's entropy is investigated as objective function.     In Kapur's entropy method, a higher objective function value implies better segmented image quality, and it can be expressed as follows: where p i denotes the probability of the gray level i. VOLUME 8, 2020           and 10-level thresholds are investigated in the experiments. Further, two well-known performance indicator peak-signalto-noise ratio (PSNR) [44] and mean structural similarity (MSSIM) [45] are selected to assess the segmented image     quality. Mainly, the segmented images with better PSNR and MSSIM values denote better image quality. The PSNR is given by: PSNR (x,y) = 10 log where x and y represent the original and segmented images with sizes a × b.     The MSSIM is formulated as: where u x denotes the average of the x, u y is the average of the y, σ 2 x indicates the standard deviation of x, σ 2 y expresses     the standard deviation of y, σ xy defines the covariance of x and y. C 1 = (K 1 L) 2 , C 2 = (K 2 L) 2 , K 1 1, K 2 1, L is the range of the gray level, M is the number of local windows.      The segmented images with different levels using the GA, MFO, PSOGSA, ALO, GOA, CGOA, OBLGOA and IWGOA based on Kapur's entropy are showed in Fig. 110.     The segmented images using optimal thresholds obtained by the IWGOA are able to extract the objects from the background. As an example, the segmented images with 8-level      thresholds of Bee using the proposed IWGOA-based method seems better than the other methods, because it reserves more details.     The optimal thresholds with best objective function values are recorded in Table 14. Table 15 reports the best, mean and standard deviation of the objective function values. VOLUME 8, 2020   The results of Table 15 suggest that the proposed IWGOA algorithm is able to obtain promising performance for Lena, Building, Bee, Baboon on 7-level, 8-level, 9-                     established that the IWGOA based method is able to offer much better image quality than the other methods. These findings demonstrate the proposed IWGOA algorithm is VOLUME 8, 2020     an efficient tool for high number of thresholds image segmentation.
Figs. 127-142 show the convergence rate of the eight methods, which demonstrates that the IWGOA based method need          also has an excellent capability of jumping out of the local best solutions.     As a summary, the proposed IWGOA obtains promising results on the majority of the 23 and 30 CEC VOLUME 8, 2020    2014 benchmark functions. Then the IWGOA algorithm is applied in multi-level image segmentation, and provides better results than the other methods.   It is obvious that the performance of the conventional GOA is significantly enhanced by combing the three strategies.

VI. CONCLUSION
In the current work, an improved grasshopper optimization algorithm named IWGOA is proposed. The IWO algorithm and random walk strategy are integrated into the GOA algorithm to enhance the exploitation capability. Moreover, the grouping strategy is employed to update the positions of grasshoppers in a better way. The performance of the proposed IWGOA algorithm is assessed on the unimodal test functions, multimodal test functions, fixed-dimension multimodal test functions and CEC 2014 benchmark functions. The proposed IWGOA algorithm obtains outperform results, which illustrates that the IWGOA has a high exploration and exploitation capability. In addition, the IWGOA also performs well in convergence rate. The results also prove that our modifications are effective.
We also apply the proposed IWGOA algorithm to multilevel image segmentation. The results indicate that the IWGOA based method is able to obtain better objective function, PSNR and MSSIM values. It demonstrates that the IWGOA algorithm is an efficient tool for the multi-level image segmentation.
For further studies, we will focus on developing more versions of the GOA algorithm. Furthermore, the proposed IWGOA algorithm will be applied to solve more practical optimization problems such as infinite impulse response (IIR) model identification, truss design and image enhancement.