AOAAO: The Hybrid algorithm of Arithmetic Optimization algorithm with Aquila Optimizer

Many new algorithms have been proposed to solve the mathematical equations formulated to describe the real-world problems. But there still does not exist one algorithm that could solve the problems all. And most of the proposed algorithms have defects in some aspects, they need to be improved in application. In order to find a more efficient optimization algorithm and inspired by the better performance of the Arithmetic Optimization algorithm (AOA) and Aquila Optimizer (AO), we proposed a hybridization algorithm of them and abbreviated AOAAO in this paper. Considering the better performance of the Harris Hawk optimization (HHO) algorithm, an energy parameter E was also introduced to balance the exploration and exploitation procedures of individuals in AOAAO swarms, and furthermore, piecewise linear map was introduced to decrease the randomness of the energy parameter. Pseudo code of the proposed AOAAO algorithm was presented, Simulation experiments were carried out on the benchmark functions and three classical engineering problems were also involved in optimization. Nine popular well demonstrated algorithms were included for comparison. Results confirmed the AOAAO would be more efficient in optimization with faster convergence rate, and higher convergence accuracy.


I. INTRODUCTION
Along with the development of science and technology, the mechanism of the world is gradually being understood by humans, the mathematical equations formulated to describe the problems are becoming more and more complicated. It becomes more and more difficult to find the solutions, so far as to obtain the analytical solutions. Numerical solutions have been proposed and among all of the numerical simulation methods, the meta-heuristic algorithms [1] have been a research hot spot in optimization [2] due to its simple concept, few parameters, and strong flexibility [3].
Lots of meta-heuristic algorithms have been proposed to solve the complex problems [4]. Literally speaking, the metaheuristic algorithms could be divided into four types: the evolutionary algorithms, physics-based, swarm-based, and human-based algorithms, as shown in Table 1. The arithmetic optimization algorithm (AOA) and Aquila optimizer (AO) [40] are relatively two newly proposed algorithm in physics and swarm-based types respectively. The AO was just proposed and little information in literature have been found while on the contrary, the AOA was proposed months ago, yet lots of improvements have been proposed and a variety of applications have been done in literature. Premkumar et al. proposed a multi-objective arithmetic optimization algorithm (MOAOA) [41] through elite nondominant sorting and distance-based crowding mechanism. Chauhan et al. added a mutation operator by quoting the mutation strategy to provide a balance between the conversion mechanism between the exploration phase and the development phase [42]. Zellagui et al. applied AOA to solve various problems of the optimal installation (location and size) of the distribution static compensator (DSTATCOM) in the power distribution system (EDS) [43]. Wang et al. proposed a new parameter adaptive equation to control the sensitive parameter α [44]. Its function is to balance the capabilities of exploration and development. At the same time, they also proposed a parallel communication strategy and strengthened the communication and information between groups. And accordingly, individuals would avoid falling into the local optimal solution, this improved algorithm has been applied in the robot path planning problem. Abualigah et al. enhanced the local research of AOA by differential technology, and the proposed algorithm was applied to the multi-level threshold problem [45]. A. Ewees et al. combined the traditional arithmetic optimization algorithm (AOA) with the genetic algorithm (GA) operator to enhance their search strategy through the genetic operator [46]. XU et al. introduced a modified version of the extreme learning machine (ELM) model and proposed a developed arithmetic optimization algorithm (dAOA) [47], which was applied to optimize the data of the proton exchange membrane fuel cell (PEMFC). Khatir et al. introduced the Artificial Neural Network Improved Arithmetic Optimization Algorithm (IANN-AOA) and applied it to the study of damage detection, location and quantification in functionally graded material (FGM) plate structures [48]. Agushaka et al. introduced natural logarithm and exponential operators into the AOA algorithm, which can generate high-density values to enhance the exploration ability of AOA, and then be used to solve engineering design problems [49].
In addition, there are many improved methods for the shortcomings of different algorithms. Mosavi et al. [50] proposed an autonomous groups particles swarm optimization (AGPSO) algorithm. He believes that the behavior of groups is usually determined by the individual or representative that they follow. Therefore, different strategies are used to improve c1 and c2, and the experimental results prove that the improvement is successful. Wei et al. [51] introduced a Dynamic Lé vy Flight technology that enables the algorithm to smoothly transit from the exploration phase to the exploitation phase, and improves ability of the algorithm to solve complex problems. Saffari et al. [52] introduced a fuzzy system to adjust the parameters of the grasshopper optimization algorithm. The two stages of exploration and exploitation are balanced, and the convergence of the algorithm is improved. Khishe et al. [53] proposed a weighted improvement. When faced with largescale numerical problems, the chimp optimization algorithm performs very poorly. Therefore, in the position update, a weighted position equation is proposed. The convergence speed is effectively improved by this improved method, and the local optimum is also avoided.
Although the AOA performed better than most of the existed algorithms, individuals in the AOA swarms would also be trapped in local optima easily, and the exploration and exploitation capability is not prominent when optimizing some complex problems. The AOA still needs to be improved and, in this paper, we propose a new improvement hybridizing the AOA with AO. Considering the strong search capability of individuals in AOA and AO swarms, we construct the multiple ways for each individual and every one of them would choose a way with randomness. Furthermore, we also introduce the energy parameter to balance the exploration and exploitation ratio inspired by the giant capability of individuals in the HHO swarm, and the energy parameter would be updated with piecewise linear mapping [54] .
Simulation experiments would be carried out to confirm the better performance. Finally, the results of multiple experiments show that this improved method is effective. The mixing of different algorithms by a good phase transition mechanism may be an improved direction for hybrid algorithm.
The rest of this paper would be arranged as follows: in Section 2, we would give a brief introduction to AOA and AO. And in section 3, shortcomings of the AOA and AO would be talked about and the improvement would be proposed. Simulation experiments would be carried out in Section 4. Discussions would be made and conclusions would be drawn in Section 5.

II. Preliminaries
A. AQUILA OPTIMIZER Individuals in AO swarms would catch the prey with four predation strategies.
The first strategy: Flying high in the sky searching prey. In this strategy, Aquila will fly through the hunting area at high altitude and initially search and find the target. Once a prey is found, it will dive vertically towards the prey. This behavior was formulated as follows: Where ( + 1) represents the positions of individuals at the t+1 iteration, ( ) represents the current global best position at t-th iteration. t and T represent the current t-th iteration and the maximum allowed iteration number.
( ) represents the current mean position of individuals at the current iteration. And is a random number in Gauss distribution fallen into an interval of 0 and 1.
The second strategy: contour flight with short glide attack. In this strategy, Aquila will switch from flying at high altitude to hovering on the head of the prey, preparing for Aquila's instinctive predation behavior. This position update can be expressed as: where, ( ) is the random position of Aquila, and D is the size of the dimension. stands for Levy flight function. The y and x represent the shape of the search, which can be expressed as: where, 1 is the number of search cycles from 1 to 20, 1 is a random integer from 1 to dimension D, and is a constant of 0.005.
The third strategy: Low-altitude flight approaching prey and slow attack. In this strategy, Aquila initially discovers and determines the approximate location of the prey, then Aquila will descend vertically for preliminary predation, in case the prey is found to reduce the speed. This initial predation behavior can be expressed as: Where and are adjustment parameters during the development process, fixed at 0.1, and are the upper and lower bounds of the search space, respectively.
The fourth strategy: Walking on land and catching prey. In this strategy, Aquila goes to the land to follow the prey's escape trajectory to chase the prey, and then attack the prey. This predation behavior can be expressed as: Where is the quality function of the average search strategy, 1 is the parameter of random motion in the process of Aquila tracking the prey, and is a random number between [-1,1], and 2 is the flight slope in the process of Aquila tracking the prey, which is represented by 2 Decrease linearly to 0.

B. ARITHMETIC OPTIMIZATION ALGORITHM
In the AOA algorithm, four basic arithmetic operators are introduced to optimize the exploration phase and development phase of the algorithm respectively.
Firstly, the exploration phase. According to arithmetic operators, the mathematical calculations carried out by the division operator and the multiplication operator can obtain highly distributed values. It is precise because of its highly distributed nature that it is impossible to approach the target easily. Obviously, the characteristics of these two operators are very suitable to be used in the search phase. It can be expressed as: Where is a small integer and is the control parameter during the search process, which is fixed at 0.5. Math Optimizer probability (MOP) is a coefficient, is a sensitive coefficient, used to define the development accuracy, fixed at 5. Math Optimizer Accelerated (MOA) is used to select the search phase, Max and Min are the maximum and minimum values of the acceleration function.
Secondly, the exploitation phase. According to arithmetic operators, subtraction operators and addition operators can obtain high-density results through mathematical calculations, because these two operators have low dispersion and are easy to approach the target, which is undoubtedly in line with the characteristics of the exploitation stage. Therefore, these two operators are introduced into the exploitation stage.

III. The defects and improved algorithm
Individuals in the AO swarms perform Aquila's fast flight and hunting in the search space during the exploration procedure. The direct addition of the global best position in updating the positions of individuals leads to faster convergence and strong search capability, but meanwhile, the individuals would also lack the capability to escape from local optima, and consequently, the individuals would be trapped in local optima easily. While for the individuals in AOA swarms, the experimental results showed that the convergence speed of the division and multiplication operators in the exploration is quite low, the population diversity is insufficient, and the volatility of the search agents is insufficient, too. In addition, the mechanism switching from the exploration to the exploitation procedure is also not perfect. Therefore, improvements should be made to overcome these shortcomings.

A. HYBRIDIZATION OF AO WITH AOA ALGORITHM
Considering the difference between equations (1), (2), (9) and (11), individuals in AO swarms would perform more random search than individuals in AOA swarms. However, in the exploitation procedure as formulated in equations (5), (6), (9) and (11), individuals in the AO swarms would perform worse than individuals in the AOA swarms. Although both of these two algorithms perform well in optimization, The exploitation capability of individuals in AO swarms would be not satisfactory and meanwhile, the exploration capability of individuals in AOA swarms would be less capable than the individuals in AO swarms. Therefore, there might be better if the exploration procedure of individuals in AO swarms is combined with the exploitation procedure of individuals in AOA swarms.

B. ENERGY PARAMETER BALANCING EXPLORATION AND EXPLOITATION RATIO
VOLUME XX, 2017 1 To combine the exploration procedure of individuals in AO swarms with the exploitation procedure of individuals in AOA swarms, another parameter should be involved to show the way to the individuals in the new combined algorithm. Random numbers might be a choice, however, considering the better performance of individuals in the Harris Hawk optimization (HHO) algorithm in the early days, the energy parameter E is introduced to do so.
Where 0 is the initial energy of the prey, which is a random value of [-1, 1]. According to the HHO algorithm, when | | ≥1, the Harris Hawk starts to look for the position of the prey, and when | | <1, the Harris Hawk starts to capture the prey.

C. PIECEWISE LINEAR MAP
Literally speaking, the chaos is a good replacement of randomness. Therefore, we introduce the chaotic mappings to this algorithm. In this paper, the piecewise linear map is introduced and its formulation is shown as follows: And then, a chaotic map enabled energy parameter would be introduced to the new hybridized algorithm: where, the value of is 0.6, The values of and are 0.9 and 0.5 respectively, and represents the current solution. The improved parameter will increase volatility, enabling more individuals to stay in the exploration at the end of the iteration, as shown in Figure 1.

D. THE PROPOSED AOAAO ALGORITHM
The proposed piecewise linear map enabled hybridization algorithm of AOA and AO could be abbreviated as AOAAO.
Individuals in AOAAO swarms would perform a rather complicated choice of ways in updating their positions, as shown in Figure 2, and its pseudo code is listed in Table 2..  With the pseudo code of the improved AOAAO algorithm in Table 2, we can find that although more parameters are involved, the total time complexity is not increased too much, O( ⋅ ⋅ + ).

IV. Simulation experiment
In this section, we would carry on some simulation experiments to verify the capability of the proposed AOAAO algorithm.

A. EXPERIMENTAL SETUP
For simplicity, benchmark functions would be involved in the experiments and 9 unimodal, 9 multimodal non-scalable, and 9 multimodal scalable benchmark functions are introduced, as shown in Table 3, Table 4, and Table 5. These benchmark functions are commonly introduced to evaluate the capability of optimization algorithms in literature.  To compare the capability of the proposed AOAAO algorithm to the original, same simulation circumstances should be maintained. Considering the better performance of modern algorithms and shortening the computer complexity, a minimum 100 iterations of run would be fixed to all benchmark functions during optimization, all of the individuals in each swarm would be fixed to be 1000 in population and the dimensionality, if not fixed, would be chosen to be 30. To reduce the influence of random numbers involved in every algorithm, 30 Monte Carlo simulation experiments would be carried out and the final results, if averaged, would be the overall averaged values. Detailed setup could be seen in Table 6. Notes: ChOA represents the chimp optimization algorithm, WOA represents the whale optimization algorithm, SSA represents the sparrow search algorithm, STOA represents the sooty tern optimization algorithm, and WOAGWO represents the hybridization algorithm of whale optimization algorithm and grey wolf optimizer algorithm respectively.

B. QUALITATIVE ANALYSIS
To have a glance at the capability in optimization, qualitative analysis would be usually carried out. In this experiment, the proposed AOAAO algorithm would be compared to the original AO and AOA, results were shown in Figure 3.

Figure 3 Qualitative analysis results
Apparently, the proposed AOAAO would perform better than all of its originality.

C. INTENSIFICATION CAPABILITY ANALYSIS
For the unimodal benchmark functions, there would always be one global optimum, therefore, individuals in swarms would all approach the global optima if they are allowed. Literally speaking, the better the algorithm performed, the faster individuals converged. Simulation experiments on unimodal benchmark functions could also be treated as the intensification capability experiment. For a better understanding of the capabilities, eight algorithms would be involved in intensification capability analysis and the best, worst, median, standard derivation would be calculated over 1000 Monte Carlo simulation results, as shown in Table 7. We can see from Table 7 that both the AOA and AO algorithm perform quite better in optimizing unimodal benchmark functions, and the proposed AOAAO algorithm would perform more better than the originality.

D. DIVERSIFICATION CAPABILITY ANALYSIS
Unlike the unimodal benchmark functions, the multimodal benchmark functions have many local optima and one global optimum. During optimization, individuals in swarms would approach to the nearest local optima if they are allowed to do so. And consequently, they could be trapped around the local optima. If the algorithm is better enough, individuals would diverse themselves and leave the local optima and approach the global optimum. Therefore, experiments on multimodal benchmark functions would be usually involved to verify the diversification capability. Considering the scalable characteristics, results on multimodal benchmark functions were shown in Table 8 and Table 9.  Results confirmed the better performance of the AO, AOA and AOAAO algorithms, and the proposed AOAAO algorithm would perform better than the originality.

E. ACCELERATION CONVERGENCE ALAYSIS
The convergence of the hybrid function is one of the goals of algorithm performance research. Figure 3 depicts the number of iterations of the hybrid algorithm in each benchmark function and the optimal solution obtained. It can be seen from Figure 4 that under the unimodal function (F1-F9), the development ability of the mixed function is extremely strong compared with other algorithms, and it converges faster. In the multimodal function (F10-F27), the convergence curve shows the transition between exploration and development phases. Compared with other algorithms, the hybrid algorithm in F10,  F12, F13, F14, F15, F16, F17, F18, 19, F20, F22, F23, F24,  F25, F26 can quickly obtain the optimal solution. In F21, the hybrid algorithm gradually obtains the optimal solution, and updates the optimal solution in iterations. In F11 and F27, the balance conversion between the exploration phase and the development phase can be seen. VOLUME XX, 2017 1

F. SENSITIVITY ANALYSIS
The sensitivity [58] of the algorithm is one of the indicators [59] to measure the performance of the algorithm. It can reflect the efficiency of the algorithm in dealing with problems. It is expressed by the following formula: where, _ represents the number of iterations when the best value is obtained, and _ represents the total number of iterations. The experiment was run 10 times, and the average value was taken. The experimental results of this section are shown in Table 10. where, NaN represents that the algorithm has fallen into a local optimum from the beginning to the end. Experimental results prove that the performance of the proposed AOAAO algorithm is the best in most cases. Because the proposed AOAAO algorithm obtains the best value with very few iterations.

G. SCALABILITY ANALYSIS
The larger number of parameters involved, the more difficult to optimize. Therefore, almost all of the algorithms would be verified their scalability capability. In this experiment, 18 functions, F1-F9 and F19-F27 would be involved and the dimensionality would be varied with 60, 100, 300, 500 respectively. Results were shown in Table 11 ~Table 14 .

V. Engineering design problem
In order to verify the practicality of the algorithm, three engineering design problems were selected including the three-bar truss design problem, the speed reducer design problem and the welded beam design problem. For three experiments, each algorithm was run 15 times.

A. THREE-BAR TRUSS DESIGN PROBLEM
The three-bar truss design problem is a well-known engineering optimization problem. This problem is mainly to minimize the weight of the three-bar truss through two parameters (A1 A2). There are three constraints to this problem. Consider:

B. SPEED REDUCER DESIGN PROBLEM
The purpose of this question is to optimize 7 variables to minimize the weight of the reducer. The speed reducer design problem has 11 constraints. Consider:  5.0 ≤ 7 ≤ 5.5 The experimental results of the speed reducer design problem are shown in Table 16. The best value is obtained by the proposed AOAAO algorithm. By observing the P value, the results of the AOAAO algorithm are different from other algorithms.

C. WELDED BEAM DESIGN PROBLEM
The purpose of the welded beam design problem is to reduce the manufacturing cost of the design. This question involves four variables: weld thickness (h), the length (l), height (t), and weld thickness (h) of the bar. This question contains 7 constraints. Consider:

VI. Discussions and conclusions
Although the meta-heuristic algorithms have been the hot spot in literature for decades of years, and lots of algorithms have been proposed, however, the problems we human facing become more and more complicated and lots of the problems remain unsolved. There still does not exist an algorithm that could solve the problems all. And consequently, new algorithms are still under demand and all of the improvements are in need to find a better way to solve the current problems. Recently, the AOA and AO was proposed and they were proved to be capable in optimization. However, detailed studies proved that they still have defects. Individuals in AOA swarms perform capable exploration and exploitation procedure in optimization, but the overall performance was nor satisfactory, while on the contrary, individuals in AO swarms would perform quite better in optimization, but the exploitation capability was so bad as to the results would remain unchanged in quantity. Therefore, we proposed a hybridization of the two algorithms and proposed AOAAO algorithm henceforth. To ensure a balanced transition between the exploration and exploitation phases, a phase transition mechanism inspired by the HHO algorithm is introduced into the hybrid algorithm, while this transition mechanism is improved again and chaotic mapping is introduced. Simulation experiments were carried out and almost all of the results proved that new improved algorithm could perform better than all of the compared algorithms, which were popular algorithms applied in solving real-world engineering problems, or optimization.
Simulation experiments proved that the combination of AOA and AO algorithms could perform well in optimization, either on unimodal or multimodal, fixed dimensionality or scalability in characteristics. The improvement would perform better than all of the compared algorithms in most cases, and sometimes, if the problems were easy to optimize, it would not perform worse than most of the compared algorithms.
We have carried out several kinds of simulation experiments, and results confirmed the better performance of the proposed AOAAO algorithm. AOAAO algorithm is ready to be applied in real complicated problems. The multiobjective version of the AOAAO algorithm may be created to further solve complex multi-objective problems.