Research on New Adaptive Whale Algorithm

Bionic algorithms have always played an important role in industrial, agricultural, and scientific research. The optimization of bionics algorithms has always been the focus of scholars in various countries. A whale algorithm based on optimization based on adaptive convergence and Levy features (IWOA) is proposed to overcome the disadvantages, such as low precision, slow convergence speed and tending to involve the local optimum of the whale algorithm. The improved Bernouilli Shift map is used to initialize the population to maintain diversity of the population. Optimizing the adaptive convergence factor is able to balance the local and global optimization ability. The Levy flight mechanism is introduced to optimize foraging behavior and improve global searching ability. In addition, the trigger rule is applied to screen individuals after each iteration to maintain individual vitality and enhance overall performance of the algorithm. In the simulation, IWOA, Ant Colony Optimization, Particle Swarm Optimization, Whale Optimization Algorithm and the optimized whale algorithms CWOA, LWOA are compared using the 20 classical test functions. The simulation results demonstrate that the IWOA algorithm possesses good global and local searching ability, especially in solving multi-peak and high-dimensional functions.

The associate editor coordinating the review of this manuscript and approving it for publication was M. Saif Islam .
These new methods can not only solve many kinds of optimization problems, but also have low algorithm complexity compared with traditional optimization algorithms. Among them, Whale Optimization Algorithm (WOA) is a swarm intelligence algorithm proposed by Mirjalili and Lewis [12], named after the behavior of whales preying in the sea. It exhibits the remarkable advantages of simple principle, simple operation and easy realization. It has been widely used in economic scheduling, photovoltaic MPP system, capacitance location and image segmentation. It has a remarkable effect in dealing with multi-peak and low-dimensional complex functions. In the whale optimization algorithm, humpback whale in search space is a candidate solution in optimization problem, also known as ''search agent''. WOA uses a set of search agents to determine the global optimal solution of the optimization problem. For a given problem, the search process begins with a set of random solutions, and the candidate solution is updated through optimization rules until the end condition is satisfied. However, the WOA has four problems: Problem 1: The population lacks effective initialization. The initialization of the intelligent population algorithm can ensure that the solution generated in the population can be uniformly distributed in the search space to a certain extent, so that a better solution can be obtained in the later iterative calculation for the algorithm, however in Whale optimization VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ algorithm, the spatial position of whale individuals is initialized by random method, which reduces the population diversity of the algorithm to a certain extent, which is not conducive to the generation of the optimal solution of the algorithm. For this problem, the improved Bernouilli Shift mapping is proposed to initialize the population to maintain the diversity of the population.
Problem 2: In the Whale optimization algorithm, parameters is the key to adjusting the local and global search capabilities of the algorithm, and is the important composition of calculating. In the basic WOA algorithm, the convergence factor decreases linearly from 2 to 0 with the number of iterations, yet, the convergence factor's linear decrement strategy makes the algorithm have good global search ability but slow convergence speed in the early stage, speed up the convergence speed in the later stage, but it is easy to fall into the local optimization, and the effect is not good in solving the multi-peak function problem, which shows that the linear decreasing strategy with iterative times of the convergence factor can not fully reflect the actual optimization search process.
For this problem, the improved local and global optimization capability of the adaptive convergence factor balancing algorithm is proposed in this paper.
Problem 3: The individual foraging mode of whales needs to improved. In Whale optimization algorithm, because the reference whale is randomly selected, humpback whales may travel back and forth in a short distance in the process of moving to the position of the reference whale, so it will increase the extra time of the algorithm and lead to the weak global optimization ability of the algorithm.
For this problem, Levy behavior is employed to deal with whale foraging behavior to improve the global search ability.
Problem 4: The individual screening after iteration is lacking. Whale individuals in the Whale optimization algorithm lack the screening of the individual before entering the next iteration through the surrounding, attack, and search steps, which can easily result in a large number of redundant individuals, which affect the performance of the algorithm as a whole.
For this problem, the trigger rules are used to screen the individuals after each iteration, and the individual activity of the algorithm is maintained, and the overall capability of the algorithm is improved.
The main work of this paper is as follows: 1). Improved Whale Optimization Algorithm is proposed, which improves the local and global search ability of the algorithm through priority strategy. The IWOA methods is analyzed and proved theoretically in this paper. 2). In the simulation experiment, the IWOA algorithm is compared with ACO, PSO, WOA, CWOA, LWOA, AGDE [41] and EFADE [42] in 18 classical test functions in detail. The experimental results show that the performance of the IWOA algorithm has been obviously improved, the convergence of the algorithm has improved, and the advantages in solving multi-peak and high-dimensional problems are obvious. Section 2 of this paper describes the research status of the WOA algorithm. Section 3 briefly describes the principle of the WOA algorithm. Because of the current problems of the WOA algorithm, this paper proposes a IWOA algorithm in Section 4 and expounds it from four aspects. In order to illustrate the performance of IWOA algorithm, the IWOA algorithm is compared with other algorithms in section 5 of this paper, and the related simulation experiments are carried out. Finally, the full text is summarized in section 6.

II. RELATED RESEARCH
At present, the research on the WOA algorithm is mainly from three aspects: the improvement of its algorithm performance, the fusion of WOA algorithm and other intelligent algorithms, and the solution of practical problems by the WOA algorithm. Jangir et al. [13] introduced adaptive strategy into the whale optimization algorithm. Through the optimization of 10 classical function problems, the simulation results show that the proposed optimization Whale optimization algorithm is superior to the basic Whale optimization algorithm both in convergence speed and accuracy. Kaveh and Ghazaan [14] proposed an enhanced whale optimization algorithm for position updating in the Whale optimization algorithm. Simulation results show that the performance of the algorithm is better than that of the basic Whale optimization algorithm; Kaur and Arora [15] proposed a whale optimization algorithm-CWOA based on chaotic mapping. The simulation experiment is improved; Abdel-Basset et al. [16] proposed a hybrid Whale optimization algorithm-LWOA based on local search optimization, the simulation experiment shows that the performance of the algorithm is better than that of the basic Whale optimization algorithm; Sayed et al. [17] proposed a chaotic mapping whale optimization algorithm based on feature selection. The simulation results show that the performance of the chaotic mapping whale optimization algorithm is obviously improved compared with Whale optimization algorithm; Ling et al. [18] proposed an improved WOA algorithm (LWOA) based on Levy behavior. The performance of this algorithm is much better than that of the Whale optimization algorithm.
Mirjalili [19] introduced the idea of simulated annealing into the whale optimization algorithm and proposed a hybrid whale optimization algorithm (WOA-SA). The simulation results show that the algorithm has good performance. Trivedi et al. [20] proposed a new hybrid PSO-WOA algorithm for global numerical function optimization. Simulation experiments show that the hybrid algorithm has better convergence. Kaveh and Moghaddam [21] proposed a hybrid algorithm of CBO-WOA, which was applied to the layout of construction engineering and achieved good results. Masadeh et al. [22] proposed a hybrid algorithm based on GWO-WOA, which is applied to task priority scheduling in software engineering with good results.
Touma [23] apply the whale optimization algorithm to the economic scheduling problems. The effectiveness of WOA algorithm is verified by using the IEEE-30 bus system, which shows that WOA algorithm has a good effect on economic scheduling. Cherukuri and Rayapudi [24] used the whale optimization algorithm in the tracking system of the global MPP photovoltaic system to achieve the use of energy. The effectiveness of the WOA algorithm in solving this problem is verified by photovoltaic arrays under different conditions. Prakash and Lakshminarayana [25] applied the whale optimization algorithm to capacitance location in the network. Experiments show that the WOA algorithm is superior to other comparative algorithms in maintaining voltage stability and optimization cost. Aljarah et al. [26] apply the Whale optimization algorithm to the weight optimization of neural networks. Compared with the basic swarm intelligence algorithm, the optimization effect of Whale optimization algorithm is better. Mostafa et al. [27] used the whale optimization algorithm to medical nuclear magnetic resonance image segmentation. Experiments show that Whale optimization algorithm has achieved good segmentation results. Reddy et al. [28] used Whale optimization algorithm to optimize distributed renewable resources. Experiments were carried out through different distributed systems. The experiments show that the Whale optimization algorithm has good performance. El Aziz et al. [29] used the Whale optimization algorithm in the multi-threshold image segmentation problem. The experimental results show that the performance of the Whale optimization algorithm is better than that of other contrast algorithms. Hassan and Hassanien [30] applied the whale optimization algorithm to retinal image segmentation to improve the accuracy of image segmentation. Oliva and El Aziz [31] used the whale optimization algorithm to predict panel parameters of solar cells and photovoltaic modules. The experimental results show that the improved algorithm can solve the prediction accuracy of the problem. Mafarja and Mirjalili [32] proposed that the Whale optimization algorithm has a better selection effect in feature selection of data sets, especially in searching for optimal feature subsets. Mehne and Mirjalili [33] proposed that Whale optimization algorithm be used for parallel processing in the optimal control problem. The experimental results show that the processing effect of using the Whale optimization algorithm is better. Nasiri and Khiyabani [34] proposed to solve the problem of using Whale optimization algorithm for clustering analysis. It is found in the simulation experiment that compared with ACO, PSO, ABC and other algorithms, it can improve the clustering effect.
From the above research results, the application of the Whale optimization algorithm to solve practical problems has achieved good results, which shows that the performance of Whale optimization algorithm is worthy of affirmation. Thus, it is of great significance to further improve the Whale optimization algorithm. Based on this consideration, an improved Whale optimization algorithm is proposed in this paper to improve the performance of the algorithm.

III. THE BASIC IDEA OF WOA ALGORITHM
WOA is a meta heuristic optimization algorithm. The main difference between the current work and other swarm optimization algorithms is that they use random or optimal search agents to simulate hunting behavior, and use spiral to simulate the bubble net attack mechanism of humpback whales. The most interesting thing about humpback whales is their special hunting methods. This foraging behavior is called bubble net foraging. Humpback whales like to hunt krill or small fish near the sea. It is worth mentioning that the bubble net predation is a unique behavior, which can only be observed in humpback whales.
The Whale optimization algorithm mainly consists of three stages: encircle predation, bubble attack, and prey search. In the WOA algorithm, the whale population size is set to N , the search space is d dimensions, and the position of the i whale in the d dimensional space is represented as The position of prey corresponds to the global optimal solution of the problem.

A. ENCIRCLE PREDATION
At the beginning of the algorithm, whales can identify the location of their prey and surround it. Because there is no priority in the global optimal position of the algorithm, it is assumed that the optimal position in the current population is the prey, and the optimal individual is surrounded by other whale individuals in the population. Use formula (1) to update the location: where, t is the current number of iterations, X p (t) = (X 1 p , X 2 p , · · · X D p )is the local optimal solution, A×|C ×X p (t)− X (t)| is the step of surrounding, The expressions of Aand C are as follows: where, rand 1 and rand 2 represent the random number between (0, 1), a is a convergence factor, and as the number of iterations increases, it decreases linearly from 2 to 0. The expression is as follows where, t max is the maximum number of iterations.

B. BUBBLE ATTACK
In the process of whale predation, bubbles are used to attack, and the behavior of whale predation bubbles is simulated by shrinking encirclement and spiral renewal position, to achieve the purpose of whale local optimization. VOLUME 8, 2020 According to formula (1), get the whale population to shrink and surround. When |A| < 1, the whale individual gets close to the whale individual in the current optimal position, the larger the value of |A|, the greater the pace of whale swimming, on the contrary, the smaller the pace of whale swimming.

2) SPIRAL UPDATE POSITION
Whale individuals first calculate the distance from their current prey and then search the prey in a spiral mode. The mathematical model of the spiral walk mode is as follows: where D = |X p (t) − X (t)|indicates the distance between the i whale and its prey, b is a constant used to define the shape of a logarithmic spiral, l is a random number between-1 and 1.In the optimization process, the probability of selecting the shrinkage encirclement mechanism and spiral position update is the same, which is 0.5 [12].

C. HUNTING STAGE
Whales can also look for food at random. In fact, individual whales search randomly according to each other's position, and the expression is as follows: where, X rand (t) is the randomly selected individual position of whales in the current population.

IV. IMPROVED WHALE OPTIMIZATION
LGORITHM-IWOA Given the shortcomings of the WOA algorithm, it is easy to fall into local optimization and thus slow convergence speed. This paper proposed an Improved Whale Optimization Algorithm (IWOA), It has been improved in the following four areas. First, chaotic mapping is used to initialize the population, so that the diversity of the population can kept and the algorithm can be avoided from falling into ''precocious''. Secondly, the local and global optimization ability of the optimization adaptive convergence factor balance algorithm is used. Third, the Levy flight mechanism is used to optimize foraging behavior and improve the global searching ability. Fourth, the trigger rule is used to update the individual after each iteration to maintain the development ability of the algorithm.

A. POPULATION INITIALIZTTION
The diversity of the initialization population will affect the convergence speed and accuracy of the swarm intelligence algorithm to a great extent, but the basic Whale optimization algorithm can't guarantee the population diversity by initializing the population randomly. Chaotic maps are widely used in the optimization of intelligent algorithms because of their randomness, ergodicity and regularity. In order to make better use of the space of solution, in this paper, an improved Bernouilli Shift map with the best chaotic effect in onedimensional space [35] is introduced to initialize the population x n+1 = 2(x n + 0.1 × rand(0, 1)) mod 1 The concrete steps of generating the initial population using improved Bernouilli Shift chaos are shown in algorithm 1.

Algorithm 1 Population Initialization Method Based on
Bernouilli Shift Chaos Set population size N, dimension D and the maximum number of chaotic iterative step K for i = 1 to N do for j = 1 to D do for k = 1 to K do Where, x k−1,j represents j dimensional individual in the k − 1 iteration, x k,j represents the j dimensional individual in the k iteration after Bernouilli Shif chaotic mapping is used, x i,j represents the i individual in dimension j, x max,j and x min,j respectively represent the upper and lower bounds in j dimensional space.
According to the above principles, the Bernouilli Shift mapping of the population initialization can make better using of the space of solution through the equation of x k,j = 2 x k−1,j + 0.1 × rand (0, 1) mod 1.

B. ADAPTIVE CONVERGENCE FACTOR
In the basic WOA algorithm, A is used to adjust the local and global search ability of the algorithm. When A is larger than 1, the algorithm will expand the search range to find a better candidate solution, whereas the algorithm will narrow the range and carry out a fine search in the local range. From formula (2), it can be seen that the value of A is affected by a to a great extent. When the value of a is large, the algorithm has better global search ability and is not easy to fall into local optimization. On the contrary, the algorithm has strong local searching ability and fast convergence speed. Thus, the dynamic adjustment of a and the balance of the search ability of the algorithm are helpful to improve the optimization performance of the algorithm as a whole, which is expressed as follows: where, t max is the maximum number of iterations, t is the current number of iterations, f max obj (x t i ) and f min obj (x i ) epresents the maximum and minimum values of the fitness value of the current individual i under the current number of iterations, respectively, ζ is a random number between [1], [2], Figure 1 shows the fluctuation range of the convergence factor. Through a large number of experiments, it is found that when a 1 and a 2 are 0.6 and 0.4, respectively, the effect is better. On the whole, in the early stage of the algorithm, the a value is larger, which makes the algorithm pay more attention to the global optimal search, but with the iterative operation of the algorithm, the a value decreases gradually, which is convenient for fine search in the later stage, and improves the convergence accuracy of the algorithm. In the later iteration stage of the algorithm, the a value increases gradually in order to jump out of the local optimal. According to the above principles, the adaptive convergence factor balancing algorithm based on the equation 8 can solve the problem of local optimization.

C. INTRODUCING LEVY TO OPTIMIZE FORAGING BEHAVIRO
Edwards et al. [36] studies the activity characteristics of specific animals and draws the conclusion that it accords with the flight characteristics of Levy. That is to say, this flight feature can not only satisfy the local search in a small range, but also satisfy the global search in a large range, and effectively balance the relationship between the local and the global. The distribution density function of the Levy flight step size change can be approximately expressed as follows: Formula (9) shows that s is the random motion step of the Levy flight behavior. According to reference [37], the expression of s is as follows: Parameter µ, v obey normal distribution. where Whale optimization algorithm has similar Levy flight characteristics like other population intelligent algorithms. In the foraging stage, because the whale position as the reference object is random, it is easy for whales to fall into local optimization when they are getting close to their position. Therefore, Levy -based flight mechanism is introduced into foraging individual renewal behavior. The formula is as following: (12) where rand is the random number between [−1, 1]. sign(rand) is the Levy flight direction as shown in formula (13), a(t) is the scale coefficient, as shown in formula (14).
where, f obj (x t i ) and f best obj (x t i ) represent the current individual fitness value and the optimal adaptation value, respectively. t is the current iterations number, a init is the initial scale coefficient.
Compared with the formula (6), formula (12) for foraging behavior has the possibility of random large step search after small step search, so that Whale optimization algorithm can search in different ranges and jump out of local optimality. At the same time, the construction of scale coefficient can ensure that in the early stage of the Whale optimization algorithm, the Levy range is large, the search range is expanded, and the approximate global optimal solution is found. When the Levy flight search range tends to be stable in the later stage, the Levy flight search range can be reduced, the algorithm can be prevented from oscillating near the optimal value, and the optimal solution can be approximated as soon as possible.
According to the above principles, Levy behavior can improve the global search ability through the equation 12.

D. INTRODUCTION OF TRIGGER RULES FOR INDIVIDUAL SCREENING
In each iteration, the basic Whale optimization algorithm lacks the screening of effective individuals in the existing population before moving directly to the next iteration through the steps of encirclement, attack, and search. Therefore, in this paper, the whale individuals are re-screened after each local update, and the trigger rules as described perviously [38], [39] are used to update the individual. By adding the operator and deleting the operator to update the individual, the overall performance of the algorithm can be improved. VOLUME 8, 2020 Rule 1: If the optimal individual is continuously updated in the 2 GP generation and ps > PS min , the delete operator is executed, delete n dec individual; Rule 2: If the optimal individual is not updated continuously in the GP generation and ps = PS max , the delete operator is executed, delete n dec individual; Rule 3: If the optimal individual continuous GP generation is not updated and ps < PS max , the addition operator is executed, add n inc individual.
Where, PS min and PS max is the maximum and minimum of population size, ps represents the number of individuals in the current population, GP is the rising period.

1) ADDITION OPERATOR DESIGN
The function of addition operator is that after each local optimal solution is obtained, the individual of the whole population can be updated again, the information of the excellent individual can be shared, and the lack of diversity of the population can be avoided. The steps are as follows: a: Determine the increase i the number of individuals First, a set S of individuals of size n inc is generated, And then randomly select two individuals x 1 and x 2 from S. According to formula (16) [40], a new individual is crossgenerated, of which α is the random number between 0 and 1.
The ways to generate individual S are as follows: a) Generates a random number n 1 between [1, n inc ], the current population is randomly divided into n 1 group, and the optimal individuals in each group are made up of S 1 b) n 2 ← n inc − n 1 , Randomly generate n 2 individuals, to compose S 2 c) S ← S 1 ∪ S 2 The new individuals generated according to the above methods have the following three possibilities: If x 1 , x 2 ∈ S 1 , then the new individual focuses on the learning of the current population. If x 1 , x 2 ∈ S 2 , then the focus of the new individual will enhance the diversity of the population. If x 1 , x 2 belong to S 1 and S 2 , then the new individual may be located in other areas that have not yet been explored. Even if the current whale population falls into local optimization, it can bring new information to the population by adding operators, thus improving the efficiency of the effective operation of the algorithm and the ability to explore other extreme regions. Grouping optimal rather than directly selecting the optimal n1 individual to compose S 1 is beneficial to inhibit the premature convergence of the Whale optimization algorithm.

2) DELETION OPERATOR DESIGN
The population inevitably produces useless individuals in the process of evolution, if these individuals cannot be removed. It will certainly reduce the efficiency of the algorithm, so using the delete operator to remove redundant individuals, the design steps are as follows: a: Determine the number of deleted individuals n dec = ps × (PS max − ps) PS max (17) b: According to Algorithm 2, divide into n dec class, the worst individuals in these classes are deleted so that the remaining individuals can be evenly distributed in the population, which is beneficial to preserve the diversity of the population.

Algorithm 2
Step 1: Generate a reference point R within the search range Step 2: Select a point P closest to R from the current population individual X Step 3: Find the nearest point of P\{X } and M − 1 from X to form a subpopulation Step 4: Delete these P individuals in the M Step 5: Repeat Step 2-Step 4 until the population is divided into N p \M classes.
In the process of iteration, it is possible that the algorithm has reached the upper limit of scale when the algorithm has not yet found the optimal solution, so it is necessary to improve the population by adding new individuals. Therefore, the deletion operator is used to remove the worst fitness individual n dec , to save space for resulting new individuals.
According to the above principles, the trigger rules are used to screen the individuals after each iteration. The individuals can maintain activity through the equation 18.

V. SIMULATION EXPERIMENT A. BASIC SETTING OF ALGORITHM
In order to further illustrate the advantages of the algorithm in this paper, the I7 CPU processor, 16GHz memory is, Win7 64-bit operating system, and the simulation software of Mat-lab2013b were selected. The proposed algorithm (IWOA) is compared with the basic ant colony algorithm (ACO), particle swarm optimization algorithm (PSO), Whale optimization algorithm (WOA) and two improved Whale optimization algorithm (CWOA algorithm in reference [15], LWOA algorithm in reference [16], AGDE algorithm in reference [41], EFADE algorithm in reference [42], EBLSHADE algorithm in reference [50] and EAGDE algorithm in reference [51]). The optimal solutions for each test problem and the obtained best, median, mean, worst values and the standard deviations of error from optimum solution of the proposed algorithms over 50 runs for all 20 benchmark functions. The main parameters required for the algorithm are shown in Table 1.

B. CLASSICAL TEST FUNCTION
In this paper, 10 representative classical test functions and the all basic Functions from CEC2017 [52] ( Table 2 shows Basic function of CEC2017) are selected to evaluate the performance of the proposed algorithm. These test   functions have both high dimensions (30,50,100) and low dimensions (2,5,10), which can be compared with the other five algorithms in all aspects.
The average value, the minimum value, the maximum value and the standard deviation are selected as the evaluation indexes, in which the maximum and the minimum value reflect the quality of the solution, the average value reflects the accuracy that the algorithm can achieve under a given number of iterations, and the standard deviation reflects the convergence speed of the algorithm.

C. PERFORMANCE COMPARISON OF FOUR IMPROVEMENTS
First of all, we take the Sphere function as an example to compare the improvement of traditional WOA optimization       From the results shown in Table 3, we can see that the IWOA provides higher R+ values and R-values than other 4 single improvements.              and EBLSHADE algorithm will fall into local optimization earlier and difficult to jump out, and can't find the theoretical optimal value and the optimization  This shows that the improvement of the Whale optimization algorithm in this paper has a certain effect, which can effectively improve the convergence accuracy of the algorithm. (c) From the overall effect, the algorithm in this paper has a good effect on the optimization of F4, F6-F8, F11, F13 both in convergence rate and accuracy. These test functions show that the algorithm has a flat trend in the second half of its value range, which shows that the algorithm has a certain effect in improving the accuracy.
In summary, the proposed algorithm outperforms the other 9 algorithms in 20 classical test functions with faster convergence speed and convergence accuracy.   Table 23 shows the comparison of statistical results, which includes minimum value, maximum value, average value and standard deviation test index of the 10 algorithms in different dimensions of 20 test functions from CEC 2017.
Let' take the first test function for examples, from low dimension to high dimension, the statistical minimum value of IWOA is from 8.73E-07 to 4.64E-45. The statistical maximum value of IWOA is from 7.16Ee-73 to 6.88E-36. The statistical average value of IWOA is from 1.50E-74 to 3.34E-37. The statistical standard deviation of IWOA is from 1.86E-73 to 1.07E-36. The performance of these four indexes is better than other 9 algorithm. On the other hand, according to the performance of test functions in six different dimensions, it can be concluded that the performance of the IWOA algorithm slightly diminishes and it is still more robust against the curse of dimensionality.
Other test functions can also be made a similar conclusion analysis. Therefore, according to the results of tables 4∼23, it can be clearly seen that IWOA succeeded at solving most of the problems. In F1,F2, F4,F6-F14,F18-20 test functions, under the condition that the dimensions are 2, 5, 10, 30,50,100 the results of the proposed algorithm is optimal and the advantages are obvious, especially when the dimension is 2, the minimum value is 0, which shows that the proposed algorithm has a good quality of solution. While in F3, F5, F15, F17 test functions, the results of the EFADE algorithm is the best. Therefore, from the results of the above 20 test functions, the proposed algorithm has some advantages over the other 9 algorithms, especially the convergence speed is obviously improved compared with the basic WOA algorithm, and the quality effect of the solution is further enhanced.

F. WILCOXON'S TEST
According to the result of table 4∼23, the performance of the 10 algorithms can be sorted into the following order: IWOA, EBLSHADE, EAGDE,EFADE, CWOA, LWOA, AGDE, WOA, PSO, ACO. Additionally, due to the importance of the multiple-problem statistical analysis, Table 24 also gives the statistical analysis resultsthroug Wilcoxon's test between IWOA and other 7 compared algorithms. The parameters of Wilcoxon's test are α = 0.01 and 0.05.
From the results shown in Table 24, we can see that IWOA provides higher R+ values than R-in all the cases. Therefore, we can obtain the conclusions: IWOA is better than EBLSHADE, EAGDE,EFADE, CWOA, LWOA, AGDE, WOA, PSO, ACO significantly. Table 25 shows the comparison of the usage time of 10 algorithms in different dimensions under 20 test functions. It is found that the usage time of this algorithm is higher than that of the other 2 algorithms includes CWOA and LWOA in all dimensions. While the usage time of the proposed algorithm is less than that of the algorithms of AGDE, EFADE, EAGDE and EBLSHADE in most dimensions.

VI. DISCUSSION
In this paper, the author presented a new swarm-based optimization algorithm based on the Whale Optimization Algorithm. According to the disadvantages of the WOA, the paper proposed four improvements, which includes population initialization by Bernouilli Shift mapping, adaptive convergence factor, levy optimization and new trigger rules.    In order to prove the performance of the proposed IWOA algorithm, the statistical test index is simulated in different dimensions of 20 test functions from CEC 2017.
The simulation results show that the proposed algorithm has the advantages over the other 9 algorithms both in the convergence speed and convergence accuracy. However, there are still some defects in the algorithm proposed in this paper. Due to the addition of four improvement points, the complexity of the algorithm is far greater than other traditional optimization algorithms. On the other hand, the improved algorithm proposed in this paper has not reached the optimal performance in some test functions, such as F3, F5, F15, F17 test functions.

VII. CONCLUSION
In views of the shortcomings of WOA algorithm, which is slow in convergence speed and easy to fall into local optimization, this paper proposes an improved Whale optimization algorithm-IWOA, to improve the performance of the algorithm as a whole through population initialization, optimizing adaptive convergence factor, adopting Levy behavior and using trigger rules. In the simulation experiment, IWOA and ACO, PSO, WOA, CWOA,LWOA, AGDE,EFADE,EAGDE and EBLSHADE are compared with the minimum, maximum, average, standard deviation and time of different dimension dimensions of 10 test functions. The experimental results show that the IWO algorithm has a better effect on the aspects of the quality of the solution, the accuracy of the solution, the speed of convergence, etc. The accuracy of the convergence of the algorithm is obviously improved.