Improved whale optimization algorithm based on random hopping update and random control parameter

Abstract

Whale Optimization Algorithm (WOA) is a relatively novel algorithm in the field of meta-heuristic algorithms. WOA can reveal an efficient performance compared with other well-established optimization algorithms, but there is still a problem of premature convergence and easy to fall into local optimal in complex multimodal functions, so this paper presents an improved WOA, and proposes the random hopping update strategy and random control parameter strategy to improve the exploration and exploitation ability of WOA. In this paper, 24 well-known benchmark functions are used to test the algorithm, including 10 unimodal functions and 14 multimodal functions. The experimental results show that the convergence accuracy of the proposed algorithm is better than that of the original algorithm on 21 functions, and better than that of the other 5 algorithms on 23 functions.

Keywords

Whale optimization algorithm Meta-heuristic Function optimization Random hopping update Random control parameter

1 Introduction

In scientific research and industrial applications, many computational problems can be summarized as complex optimization problems, which are often characterized by nonlinearity, non-differentiability, and multimodal. In these complex optimization problems, it is necessary to find the optimal solution of a given problem under highly complex constraints in a reasonable time. In order to solve these problems, various optimization algorithms have been proposed.

These optimization algorithms include traditional optimization algorithms and meta-heuristic algorithms. The traditional optimization algorithms are based on the gradient algorithm, and they have high computational efficiency and strong reliability. They are important and widely used optimization algorithms. However, traditional optimization algorithms search the optimal from a single point and require a lot of time to reach the global optimum. Due to their limited pertinence and complexity of constraints, these algorithms are not very effective for solving complex optimization problems. These shortcomings have led to significant limitations of traditional optimization algorithms in solving a variety of practical problems.

Meta-heuristic algorithm takes inspiration from random phenomena in nature, combines random algorithm with local algorithm, and has a certain probability of jumping out of local optimum to obtain global optimum. These methods are used to solve real-world engineering problems and obtain potential optimal solutions in a given time [1 –4].

Meta-heuristic algorithms mainly have four categories, namely evolutionary algorithm, swarm intelligence algorithm, physics-based algorithm and hybrid algorithm. Evolutionary algorithm is inspired by the evolution of organisms in nature, It generally includes basic operations such as gene coding, population initialization, crossover and mutation operators, operation and reservation mechanism, etc. At present, the famous evolutionary algorithms are Genetic Algorithm (GA) [5], Biogeography-Based Optimization (BBO) [6] and Differential Evolution (DE) Algorithm [7], among which GA is the most popular. It simulates Darwin’s concept of evolution. Each new population is generated by the combination and mutation of the previous generation of individuals. The ergodicity of evolution operators enables GA to search globally in probabilistic sense very effectively. GA provides great flexibility to deal with various special problems, which ensures the effectiveness of the algorithm.

Swarm intelligence algorithms mostly imitate the social behavior of biocenosis in nature. At present, the famous swarm intelligence algorithms are Particle Swarm Optimization (PSO) [8], Artificial Bee Colony (ABC) Algorithm [9], Firefly Algorithm (FA) [10], Grey Wolf Optimizer (GWO) [11], Ant Lion Optimization (ALO) Algorithm [12], Salp Swarm Algorithm (SSA) [13], Butterfly Optimization Algorithm (BOA) [14], Moth-flame Optimization (MFO) Algorithm [15] and Whale Optimization Algorithm (WOA) [16]. The most popular swarm intelligence algorithm is PSO, which introduces the social behavior of bird swarm for the first time, and uses multiple particles to track the location of the best particle and the best position obtained so far. Because of its simple operation and fast convergence, PSO has been widely used in many fields, such as function optimization, image processing, geodesy and so on.

Physics-based algorithms usually imitate physical rules. Some of the most popular algorithms are Sine Cosine Algorithm (SCA) [17], Gravitational Aearch Algorithm (GSA) [18], Atom Search Optimization (ASO) [19], etc. GSA performs the search by using a proxy set with a quality proportional to the fitness function value. In the iterative process, the masses are attracted to each other by the gravity between them. The heavier the mass, the greater the attraction. GSA is simple in principle, simple in operation and high in efficiency in solving various nonlinear functions, which makes GSA applied in many fields.

Hybrid algorithm is usually a new algorithm that combines the advantages of two different optimization algorithms. At present, the common hybrid algorithms are PSO-GA [20], GSA-GA [21], GA-GSA [22], PSO/GSA [23] and so on. GA-GSA combines the characteristics of genetic algorithm and gravity search algorithm, and proposes a hybrid genetic algorithm. By embedding genetic operators in gravity search algorithm, the exploration and exploitation ability of gravity search algorithm are improved.

Whale Optimization Algorithm (WOA) searches for global optimal solutions by surrounding prey, hunting prey and attacking prey. WOA has outstanding properties, such as a small number of parameters controlled, including only two main internal parameters to adjust, easy to implement and high flexibility. WOA smoothly transit between exploration and exploitation depending on only one parameter. Due to the simplicity of WOA in implementation and the low dependence on parameters, the algorithm has been successfully applied in many fields such as image processing [24], prediction of thermal consumption rate of generator sets [25], characteristic selection [26], neural network [27], wind speed prediction [28], fault diagnosis [29] and so on.

At present, the research of WOA can be divided into two categories: (1) Improving the performance of WOA: Kaur and Arora [30] proposed chaotic WOA (CWOA), using chaos theory to adjust the main parameters of WOA to improve its convergence speed; Ling et al. [31] proposed Lévy flight trajectory-based WOA (LWOA). The LWOA employed Lévy flight trajectory to increase the diversity of population; Sun et al. [32] proposed an improved WOA (MWOA) to solve large-scale global optimization problems. MWOA uses Levy flight strategy to improve WOA’s exploration ability, secondary interpolation to improve WOA’s exploitation ability, uses nonlinear control strategy to control the entire search process, and balances WOA’s exploration and exploitation capabilities; Abdel-Basset et al. [33] used Lévy flight and logical chaos mapping to replace and determine the coefficient C in WOA and to switch probability p, improving the exploitation ability of the WOA; Ismai et al. [34] introduced chaos search into the iterative search of WOA, proposed a CWOA that significantly improved WOA’s performance.

(2) Applying WOA to solve some optimization problems: Zhao et al. [35] established the WOA-LSSVM model to predict CO2 emissions by searching for the optimal value of the two parameters of LSSVM; Reddy et al. [36] used WOA to optimize the allocation of renewable resources to reduce waste in distribution systems; Azizd et al. [24] used WOA to determine the optimal threshold problem of image segmentation and solved the time-consuming problem of determining the best threshold at multiple level threshold.

In the present study, an improved WOA is proposed to solve the problem that WOA is not good at exploring space and easily falls into local optimal. On the one hand, the new strategy of random hopping update is introduced to adjust the position update of whales, which makes the algorithm maintain a faster convergence speed in the convergence process and improves the exploitation ability of WOA. On the other hand, by analyzing the influence of control parameter a on the exploration and exploitation ability of WOA, random control parameter is introduced to make WOA jump out of local optimal better, balance the exploration and exploitation ability of WOA and avoid falling into local optimal prematurely. The performance of the improved WOA was tested by 24 typical benchmark function tests and compared with other algorithms, the results show the effectiveness of the proposed algorithm.

2 Whale optimization algorithm

Whale Optimization Algorithm (WOA) is a new meta-heuristic algorithm proposed by Mirjalili in 2016. The algorithm is a simulation of the humpback whale predation behavior. This special hunting mechanism is called bubble net foraging. When humpback whales surround prey, this foraging is done by creating unique bubbles along a circular or “9-shaped” path. The humpback whale can dive underwater for 10-15 meters and then start making spiral bubbles around the prey and then swim to the surface. Its hunting behavior is shown in Fig. 1.

Fig. 1

Bubble-net feeding behavior of humpback whale.[16].

The mathematical model of WOA for surrounding the prey, spiral bubble net foraging action and searching for prey are described as follows:

2.1 Surrounding the prey stage

Humpback whales are able to identify the position of the prey and surround them. Since the position of the optimal solution in the search space is not a priori, the WOA assumes that the current best candidate solution is the target prey or near optimal solution. After the best search agents are defined, other search agents will therefore attempt to update their location to the best search agent. This behavior can be represented by Eq. (1) and Eq. (2): $\vec{D} = | \vec{C} . \vec{X^{*}} (t) - \vec{X} (t) |$ (1) $\vec{X} (t + 1) = \vec{X^{*}} (t) - \vec{A} \cdot \vec{D}$ (2) where t indicates the current number of iterations, $\vec{A}$ and $\vec{C}$ are coefficient vectors, $\vec{X^{*}}$ is the position vector of the best solution obtained, $\vec{X}$ is the individual position vector. The vectors $\vec{A}$ and $\vec{C}$ are calculated as Eq. (3) and Eq. (4): $\vec{A} = 2 \vec{a} \cdot \vec{r} - \vec{a}$ (3) $\vec{C} = 2 \cdot \vec{r}$ (4) where $\vec{a}$ is linearly decreased from 2 to 0 over the course of iterations (in both exploration and exploration phase), and $\vec{r}$ is a random vector in [0,1].

2.2 Bubble net foraging (development stage)

The bubble whale foraging of the humpback whale is characterized by moving along the spiral path toward the prey in the constricted encirclement. Therefore, the WOA designs the contraction enveloping mechanism and the spiral update position to simulate the unique bubble net predation behavior of the humpback whales.

a) Shrinking encircling mechanism: This behavior is achieved by decreasing the value of a from 2 to 0 in Eq. (3). According to Eq. (3), the value of A decreases as the a decreases, and when a decreases linearly from 2 to 0, A is a random value in the interval [-a, a]; Therefore, when setting random values for A in [-1,1], the new position of a search agent can be defined anywhere between the original position of the agent and the position of the current best agent.

b) Spiral updating position: This approach first calculates the distance between the whale located at (X, Y) and prey located at (X*,Y*). Then create a spiral equation between the position of the whale and the prey to simulate the spiral movement of the humpback whale as Eq. (5): $\vec{X} (t + 1) = \vec{D^{'}} \cdot e^{bl} \cdot cos (2 π l) + \vec{X^{*}} (t)$ (5) where $\vec{D^{'}} = | \vec{X^{*}} (t) - \vec{X} (t) |$ represents the distance from the ith whale to its prey, b is the constant that defines the shape of the logarithmic spiral, and l is the random number in [-1, 1].

As humpback whales swim around their prey in a shrinking circle, they swim along a spiral path. To model this simultaneous behavior, it is assumed that there is a probability of 50% that a choice can be made between a reduced enclosure mechanism and a spiral model to update the position of during optimization. The mathematical model is as Eq. (6): $\vec{X} (t + 1) = {\begin{matrix} \vec{X} (t) - \vec{A} \cdot \vec{D} & , p < 0.5 \\ \vec{D^{'}} \cdot e^{b^{b}} \cdot cos (2 π l) + \vec{X^{*}} (t) & , p \geq 0.5 \end{matrix}$ (6) where p is a random number in [0, 1].

2.3 Search for prey (exploration phase)

In fact, humpback whales search randomly according to the position of each other. Therefore, we use $\vec{A}$ with a random value greater than 1 or less than -1 to force the search agent to move away from the reference whale. In contrast to the exploitation phase, the location of the search agent in the exploration phase is updated based on the randomly selected search agent, rather than the location of the search agent in the exploration phase based on the best search agent currently discovered. This mechanism and $| \vec{A} | > 1$ emphasize exploration allow WOA to perform a global search. The mathematical model is as Eq. (7) and Eq. (8): $\vec{D} = | \vec{C} \cdot \vec{X_{rand}} - \vec{X} |$ (7) $\vec{X} (t + 1) = \vec{X_{rand}} - \vec{A} \cdot \vec{D}$ (8) where $\vec{X_{rand}}$ represents a random position vector (a random whale) chosen from the current population.

The WOA starts with a set of random solutions, search agents update their positions through randomly chosen search agent or current optimal solution during the iteration, the different values of parameter a control the exploration and exploitation of the WOA, and parameter p affects the selective jump execution of the exploration or exploitation. By exploring the process and the parallel iteration of the exploitation of the search agents, the optimal solution of the problem is finally determined.

3 Improved whale optimization algorithm

WOA is briefly introduced in the above section. In spite of having good performance, WOA still cannot perform that better in finding the global optima, which affect the convergence rate of the algorithm. So, to reduce this affect and to improve its efficiency, this paper proposes PAWOA. In order to improve the exploration ability of WOA, in this study random hopping update strategy is integrated into WOA. For better exploration and exploitation balancing, the random control parameter is introduced. The improvement strategies are presented in the next sections.

3.1 Random hopping update strategy (PWOA)

The excellence of the swarm in the iterations of the WOA is dependent on the search agents, which are randomly selected from the previous generation of swarm. From the Eq. (7) and Eq. (8), we can obtain that WOA updates the position of the search agent in the exploration based on a randomly selected search agent, rather than the best search agent so far. In the iterations of the WOA, it is inevitably be relatively small fitness of individuals, if these individuals are selected by randomly selecting mechanisms, the new swarm will certainly affect the convergence speed and accuracy of WOA.

In view of this situation, this paper proposes a random hopping update strategy. It can select search agents more efficiently. For the iterations, the improved strategy is shown in Eq. (9): $\vec{X} (t + 1) = \vec{C} \cdot \vec{D} + \vec{C} \cdot \vec{X} (t)$ (9) where $\vec{C} = 2 \cdot \vec{r}$ .

From Eq. (9), we can conclude that the search agent selected in the iteration process has a better fitness, so the exploration ability of WOA will be improved. Because the coefficient of the equation is a random number, there will be some mutations in the iteration process. Consequently, these mutations can reduce the possibility of the population falling into local optimum and enhance optimization ability. The mutations are generated by random parameters in this paper, the random parameter makes the position of individuals randomly generated from the vicinity of the current optimal agent. This ensures that the individuals within the population will not be infinitely close during the entire process of algorithm iteration. A random distance is kept between the individuals in the population, which ensures that the algorithm will not fall into the local optimal prematurely, thus improving the exploitation ability of the algorithm.

3.2 Random control parameter strategy (AWOA)

In order to obtain better optimization results, the WOA needs to balance exploration and exploitation in the optimization process. when $| \vec{A} | > 1$ , WOA selects random search with a probability of 0.5 to expands the scope of search and improves the ability of global optimization; when $| \vec{A} | < 1$ , WOA performs a local search. Therefore, the parameter A controls the exploration and exploitation ability of WOA. The value of parameter A is changing with the $| \vec{a} |$ , so the exploration and exploitation capabilities of the WOA are controlled by the parameter $| \vec{a} |$ .

In Eq. (3), is decreased from 2 to 0 over the course of iterations. Because of, the position update formula is converted between Eq. (6) and Eq. (8) to making the WOA have better global search capability. However, the global search ability of WOA will decrease rapidly with the increase of iterations, which is making WOA easily to fall into local optimum. Therefore, this paper proposes an improvement method as shown in Eq. (10): $\vec{a} = 2 \cdot \vec{k}$ (10) where $\vec{k}$ is a random vector in [0, 1]. The value of $| \vec{a} |$ changes nonlinearly with the increase of iterations, thus balancing the exploration and exploitation of the WOA.

3.3 Improved WOA based on random hopping update and random control parameter (PAWOA)

In summary, this paper proposed an improved WOA (PAWOA) based on a random hopping update strategy and random control parameter strategy. The flowchart is shown in Fig. 2.

Fig. 2

Flowchart of the PAWOA.

The improved algorithm (PAWOA) optimizes whale predation behavior from two aspects. On the one hand, the position update strategy for whale individuals is changed through random hopping update strategy, so that new individuals are randomly generated near the optimal individuals, and the exploration ability of the algorithm is improved; On the other hand, by introducing random control parameters, the convergence factor a can nonlinearly adjust the ability of global search and local search during the entire iteration of the algorithm, thus ensuring that the algorithm is not easy to fall into local optimal solution.

The performance of the improved algorithm is better than hybrid Particle Swarm Optimization and Gravity Search Algorithm (PSO/GSA), Ant Lion Optimization (ALO) Algorithm, Butterfly Optimization algorithm (BOA), Moth Flame Optimization (MFO) Algorithm and Biogeography-Based Optimization (BBO) on the whole. For comparison of the algorithms, see chapter 4.

4 Experimental results and discussions

To validate the performance of the proposed algorithm, this paper uses 24 well-known benchmark functions for the experiment. These functions are divided into two categories: unimodal functions (ten functions) and multimodal functions (fourteen functions). The experiment consists of two parts: (1) the convergence accuracy test of PWOA, AWOA, PAWOA and WOA with the fixed iterations. (2) the Wilcoxon signed ranks test of PWOA, AWOA, PAWOA and WOA. (3) the comparison experiment of the PAWOA and other five optimization algorithms.

4.1 Benchmark function and parameter settings

In the 24 benchmark functions, f1-f10 are unimodal functions, which have only a global optimum. Thus, they are suitable for evaluating the ability of algorithms in terms of local search and convergence speed. f11-f24 are multimodal functions, which have a massive number of local optima and are suitable to evaluate local optima avoidance and exploration ability of an algorithm. An algorithm should avoid all the local optima to approach and determine the global optimum. Detailed descriptions of these functions are given in Table 1.

Table 1
Test functions

Name Function Dim Range f _min

f1 Bohachevsky $f (x) = x_{1}^{2} + 2 x_{2}^{2} - 0.3 cos (3 π x_{1}) - 0.4 cos (4 π x_{2}) + 0.7$ 2 [-100,100] 0

f2 Perm Function 0, d, β $f (x) = \sum_{i = 1}^{d} {(\sum_{j = 1}^{d} (j + β) (x_{j}^{i} - \frac{1}{j^{i}}))}^{2}$ 30 [-30,30] 0

f3 Sum of Different Powers $f (x) = \sum_{i = 1}^{d} {| x_{i} |}^{i + 1}$ 30 [-1,1] 0

f4 Sum Squares $f (x) = \sum_{i = 1}^{d} i x_{i}^{2}$ 30 [-10, 10] 0

f5 Booth f (x) = (x₁ + 2x₂ - 7) ² + (2x₁ + x₂ - 5) ² 2 [-10, 10] 0

f6 Matyas $f (x) = 0.26 (x_{1}^{2} + x_{2}^{2}) - 0.48 x_{1} x_{2}$ 2 [-10, 10] 0

f7 Zakharov $f (x) = \sum_{i = 1}^{d} x_{i}^{2} + {(\sum_{i = 1}^{d} 0.5 i x_{i})}^{2} + {(\sum_{i = 1}^{d} 0.5 i x_{i})}^{4}$ 30 [-5, 10] 0

f8 Three-Hump Camel $f (x) = 2 x_{1}^{2} - 1.05 x_{1}^{4} + \frac{x_{1}^{6}}{6} + x_{1} x_{2} + x_{2}^{2}$ 2 [-5, 5] 0

f9 Dixon-Price $f (x) = {(x_{1} - 1)}^{2} + \sum_{i = 2}^{d} i {(2 x_{i}^{2} - x_{i - 1})}^{2}$ 30 [-10,10] 0

f10 Rosenbrock $f (x) = \sum_{i = 1}^{d - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$ 30 [-5, 10] 0

f11 Shubert $f (x) = (\sum_{i = 1}^{5} i cos ((i + 1) x_{1} + i)) (\sum_{i = 1}^{5} i cos ((i + 1) x_{2} + i))$ 2 [-5.12,5.12] -186.7309

f12 De Joung $f (x) = {(0.002 + \sum_{i = 1}^{25} \frac{1}{i + {(x_{1} - a_{1 i})}^{6} + {(x_{2} - a_{2 i})}^{6}})}^{- 1}$ 3 [-65.365,65.365] 0

f13 Shekel(5) $f (x) = \sum_{i = 1}^{5} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$ 4 [0,10] -10.1532

f14 Shekel(7) $f (x) = \sum_{i = 1}^{7} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$ 4 [0,10] -10.4029

f15 Shekel(10) $f (x) = \sum_{i = 1}^{10} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$ 4 [0,10] -10.53641

f16 Rastrigin(20) $f (x) = 10 d + \sum_{i = 1}^{d} [x_{i}^{2} - 10 cos (2 π x_{i})]$ 20 [-5.12,5.12] 0

f17 Griewank(20) $f (x) = \sum_{i = 1}^{d} \frac{x_{i}^{2}}{4000} - \prod_{i = 1}^{d} cos (\frac{x_{i}}{\sqrt{i}}) + 1$ 20 [-600,600] 0

f18 Powell(24) $f (x) = \sum_{i = 1}^{d / 4} [{(x_{4 i - 3} + 10 x_{4 i - 2})}^{2} + 5 {(x_{4 i - 1} - x_{4 i})}^{2}$ + (x_4i-2 - 2x_4i-1) ⁴ + 10 (x_4i-3 - x_4i) ⁴] 24 [-4,5] 0

f19 Levy(30) $f (x) = {sin}^{2} (π ω_{1}) + \sum_{i = 1}^{d - 1} {(ω_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π ω_{i} + 1)]$ + (ω_d - 1) ² [1 + sin ² (2πω_d)] 30 [-10,10] 0

f20 Ackley(30) $f (x) = - a exp (- b \sqrt{\frac{1}{d} \sum_{i = 1}^{d} x_{i}^{2}}) - exp (\frac{1}{d} \sum_{i = 1}^{d} cos ({cx}_{i}))$ +a + exp(1) 30 [-32.768,32.768] 0

f21 Bukin06 $f (x) = 100 \sqrt{| x_{2} - 0.01 x_{1}^{2} |} + 0.01 | x_{1} + 10 |$ 2 [-15,3] 0

f22 EggCrate $f (x) = x_{1}^{2} + x_{2}^{2} + 25 ({sin}^{2} x_{1} + {sin}^{2} x_{2})$ 2 [-5,5] 0

f23 Price02 $f (x) = 1 + {sin}^{2} (x_{1}) + {sin}^{2} (x_{2}) - 0.1 exp (- x_{1}^{2} - x_{2}^{2})$ 2 [-10,10] 0.9

f24 XinSheYang02 $f (x) = \frac{\sum_{i = 1}^{n} | x_{i} |}{exp (\sum_{i = 1}^{n} sin (x_{i}^{2}))}$ 2 [-2pi,2pi] 0

Name	Function	Dim	Range	f _min
f1	Bohachevsky	$f (x) = x_{1}^{2} + 2 x_{2}^{2} - 0.3 cos (3 π x_{1}) - 0.4 cos (4 π x_{2}) + 0.7$	2	[-100,100]	0
f2	Perm Function 0, d, β	$f (x) = \sum_{i = 1}^{d} {(\sum_{j = 1}^{d} (j + β) (x_{j}^{i} - \frac{1}{j^{i}}))}^{2}$	30	[-30,30]	0
f3	Sum of Different Powers	$f (x) = \sum_{i = 1}^{d} {\| x_{i} \|}^{i + 1}$	30	[-1,1]	0
f4	Sum Squares	$f (x) = \sum_{i = 1}^{d} i x_{i}^{2}$	30	[-10, 10]	0
f5	Booth	f (x) = (x₁ + 2x₂ - 7) ² + (2x₁ + x₂ - 5) ²	2	[-10, 10]	0
f6	Matyas	$f (x) = 0.26 (x_{1}^{2} + x_{2}^{2}) - 0.48 x_{1} x_{2}$	2	[-10, 10]	0
f7	Zakharov	$f (x) = \sum_{i = 1}^{d} x_{i}^{2} + {(\sum_{i = 1}^{d} 0.5 i x_{i})}^{2} + {(\sum_{i = 1}^{d} 0.5 i x_{i})}^{4}$	30	[-5, 10]	0
f8	Three-Hump Camel	$f (x) = 2 x_{1}^{2} - 1.05 x_{1}^{4} + \frac{x_{1}^{6}}{6} + x_{1} x_{2} + x_{2}^{2}$	2	[-5, 5]	0
f9	Dixon-Price	$f (x) = {(x_{1} - 1)}^{2} + \sum_{i = 2}^{d} i {(2 x_{i}^{2} - x_{i - 1})}^{2}$	30	[-10,10]	0
f10	Rosenbrock	$f (x) = \sum_{i = 1}^{d - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$	30	[-5, 10]	0
f11	Shubert	$f (x) = (\sum_{i = 1}^{5} i cos ((i + 1) x_{1} + i)) (\sum_{i = 1}^{5} i cos ((i + 1) x_{2} + i))$	2	[-5.12,5.12]	-186.7309
f12	De Joung	$f (x) = {(0.002 + \sum_{i = 1}^{25} \frac{1}{i + {(x_{1} - a_{1 i})}^{6} + {(x_{2} - a_{2 i})}^{6}})}^{- 1}$	3	[-65.365,65.365]	0
f13	Shekel(5)	$f (x) = \sum_{i = 1}^{5} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$	4	[0,10]	-10.1532
f14	Shekel(7)	$f (x) = \sum_{i = 1}^{7} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$	4	[0,10]	-10.4029
f15	Shekel(10)	$f (x) = \sum_{i = 1}^{10} \frac{1}{c_{i} + \sum_{j = 1}^{n} {(x_{j} - a_{ij})}^{2}}$	4	[0,10]	-10.53641
f16	Rastrigin(20)	$f (x) = 10 d + \sum_{i = 1}^{d} [x_{i}^{2} - 10 cos (2 π x_{i})]$	20	[-5.12,5.12]	0
f17	Griewank(20)	$f (x) = \sum_{i = 1}^{d} \frac{x_{i}^{2}}{4000} - \prod_{i = 1}^{d} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	20	[-600,600]	0
f18	Powell(24)	$f (x) = \sum_{i = 1}^{d / 4} [{(x_{4 i - 3} + 10 x_{4 i - 2})}^{2} + 5 {(x_{4 i - 1} - x_{4 i})}^{2}$ + (x_4i-2 - 2x_4i-1) ⁴ + 10 (x_4i-3 - x_4i) ⁴]	24	[-4,5]	0
f19	Levy(30)	$f (x) = {sin}^{2} (π ω_{1}) + \sum_{i = 1}^{d - 1} {(ω_{i} - 1)}^{2} [1 + 10 {sin}^{2} (π ω_{i} + 1)]$ + (ω_d - 1) ² [1 + sin ² (2πω_d)]	30	[-10,10]	0
f20	Ackley(30)	$f (x) = - a exp (- b \sqrt{\frac{1}{d} \sum_{i = 1}^{d} x_{i}^{2}}) - exp (\frac{1}{d} \sum_{i = 1}^{d} cos ({cx}_{i}))$ +a + exp(1)	30	[-32.768,32.768]	0
f21	Bukin06	$f (x) = 100 \sqrt{\| x_{2} - 0.01 x_{1}^{2} \|} + 0.01 \| x_{1} + 10 \|$	2	[-15,3]	0
f22	EggCrate	$f (x) = x_{1}^{2} + x_{2}^{2} + 25 ({sin}^{2} x_{1} + {sin}^{2} x_{2})$	2	[-5,5]	0
f23	Price02	$f (x) = 1 + {sin}^{2} (x_{1}) + {sin}^{2} (x_{2}) - 0.1 exp (- x_{1}^{2} - x_{2}^{2})$	2	[-10,10]	0.9
f24	XinSheYang02	$f (x) = \frac{\sum_{i = 1}^{n} \| x_{i} \|}{exp (\sum_{i = 1}^{n} sin (x_{i}^{2}))}$	2	[-2pi,2pi]	0

In order to ensure the fairness and rationality of the experiment and avoid the influence of the randomness of the algorithm on the experiment, the population size and maximum iterations of all algorithms are set to 30 and 500. All results are recorded and compared based on 30 independent runs. The experimentation and algorithms are implemented in MATLAB R2014a and simulations are tested on Core i5 processor with 2.4 GHz and 8 GB main memory.

4.2 Convergence accuracy test of fixed iterations

This paper compares the performance of PWOA, AWOA, PAWOA and WOA. To statistically asses the proposed algorithm, this paper evaluates the performance of the algorithm by four evaluation criteria: average (mean), standard deviation (std), worst values (worst) between and best values (best). “best” and “worst” respectively denote the maximum and minimum fitness values of the optimal solution obtained by the algorithm, “mean” denotes the average of the result fitness values, and “std” denotes standard deviation. “std” indicates the stability of the algorithm, the smaller the standard deviation,the greater the stability of the algorithm. The results of the experiment are shown in the Table 2.

Table 2
Experimental results of fixed number of iterations

Fun WOA PWOA AWOA PAWOA

f1 mean 0 0 0 0

std 0 0 0 0

best 0 0 0 0

worst 0 0 0 0

f2 mean 1082.8932 474.4647 844.7097 658.9085

std 970.7269 408.5380 774.3744 575.2776

best 179.3717 54.3772 71.8856 7.4499

worst 3.3119e+03 1.8160e+03 3.3023e+03 1.8340e+03

f3 mean 2.0574e-100 2.7864e-147 1.0576e-147 0

std 1.1269e-99 1.3409e-146 5.7925e-147 0

best 4.6287e-142 4.7770e-180 1.2727e-183 0

worst 6.1723e-99 7.3358e-146 3.1727e-146 0

f4 mean 1.716e-71 1.2207e-99 2.0443e-86 1.9094e-197

std 8.5199e-71 6.6863e-99 9.5938e-86 0

best 1.1238e-87 8.7494e-125 2.3367e-105 1.1820e-236

worst 4.6577e-70 3.6622e-98 5.2425e-85 5.5989e-196

f5 mean 0.0012394 0.00066298 0.00052644 0.00033209

std 0.0022 8.3271e-04 5.6824e-04 4.6627e-04

best 2.7766e-06 9.1405e-06 3.7789e-06 2.8783e-08

worst 0.0112 0.0031 0.0028 0.0018

f6 mean 2.725e-196 1.7926e-211 1.57e-280 0

std 0 0 0 0

best 3.7227e-225 8.8805e-250 2.4725e-304 0

worst 5.4628e-195 5.3777e-210 4.4591e-279 0

f7 mean 5.8426e-79 7.1121e-117 2.335e-126 1.2035e-263

std 2.6413e-78 4.3384e-147 8.9471e-126 0

best 4.8545e-104 3.8576e-116 1.6281e-158 1.5732e-311

worst 1.4350e-77 1.518e-117 3.9147e-125 3.6101e-262

f8 mean 5.0787e-51 6.8723e-107 0.049773 2.3326e-267

std 2.7817e-50 3.5105e-106 0.1132 0

best 2.7478e-97 4.3646e-139 6.2303e-158 3.7917e-316

worst 1.5236e-49 1.9226e-105 0.2986 6.9355e-266

f9 mean 0.66708 0.6668 0.66669 0.6667

std 2.8267e-04 1.2505e-04 1.6479e-05 1.3825e-05

best 0.6667 0.6667 0.6667 0.6667

worst 0.6679 0.6672 0.6667 0.6668

f10 mean 27.6882 26.508 27.4071 27.2505

std 0.4108 4.9535 0.6931 0.5541

best 27.0261 0.3494 26.7062 26.1975

worst 28.7488 28.4633 28.7472 28.7271

f11 mean -24.1465 -24.1465 -24.1465 -24.1465

std 3.6134e-15 3.6134e-15 3.6134e-15 3.6134e-15

best -24.1465 -24.1465 -24.1465 -24.1465

worst -24.1465 -24.1465 -24.1465 -24.1465

f12 mean 3.7375e-93 3.8659e-124 1.4181e-128 5.0638e-244

std 1.8771e-92 2.1092e-123 7.7672e-128 0

best 7.9376e-116 1.3863e-153 1.0639e-153 8.8464e-306

worst 1.0272e-91 1.1554e-122 4.2543e-127 1.5191e-242

f13 mean -8.7789 -9.0614 -7.7722 -9.6159

std 2.2844 2.0444 2.5850 1.6522

best -10.1522 -10.1520 -10.1531 -10.1530

worst -5.0512 -5.0537 -5.0551 -2.6293

f14 mean -7.5356 -9.649 -6.9749 -9.2823

std 3.1412 1.8237 3.3686 2.5899

best -10.4002 -10.4028 -10.4028 -10.4025

worst -2.7640 -5.0872 -1.8366 -1.8371

f15 mean -7.0577 -9.5092 -6.3841 -9.5252

std 3.6278 2.0082 3.1327 2.5835

best -10.5360 -10.5358 -10.5361 -10.5362

worst -1.8594 -3.8326 -2.4216 -1.8594

f16 mean 3.7896e-15 0 0 0

std 1.2340e-14 0 0 0

best 0 0 0 0

worst 5.6843e-14 0 0 0

f17 mean 0.023102 0 0 0

std 0.0537 0 0 0

best 0 0 0 0

worst 0.2011 0 0 0

f18 mean 2.7797e-06 1.9125e-53 3.9046e-08 8.9994e-184

std 6.0019e-06 1.0475e-52 2.0787e-07 0

best 2.5709e-60 1.5628e-122 8.5096e-97 1.7751e-236

worst 2.4856e-05 5.7374e-52 1.1392e-06 2.6998e-182

f19 mean 0.43975 0.17069 0.98123 0.52745

std 0.2887 0.1748 0.3719 0.3606

best 0.1477 0.0247 0.3399 0.0234

worst 1.3954 0.7402 1.8759 1.2220

f20 mean 5.033e-15 8.8818e-16 4.5593e-15 8.8818e-16

std 2.6526e-15 0 2.3756e-15 0

best 8.8818e-16 8.8818e-16 8.8818e-16 8.8818e-16

worst 7.9936e-15 8.8818e-16 7.9936e-15 8.8818e-16

f21 mean 0.10492 0.1177 0.11232 0.12111

std 0.0788 0.0559 0.0714 0.0742

best 0.0145 0.0167 0.0244 0.0048

worst 0.2551 0.2647 0.2476 0.2441

f22 mean 5.847e-109 4.2991e-136 7.6118e-187 1.2841e-288

std 3.2025e-108 2.1201e-135 0 0

best 3.2926e-134 1.6325e-174 5.4377e-223 0

worst 1.7541e-107 1.1583e-134 2.0998e-185 3.8520e-287

f23 mean 0.92667 0.9 0.93667 0.9

std 0.0450 4.5168e-16 0.0490 4.5168e-16

best 0.9000 0.9000 0.9000 0.9000

worst 1.0000 0.9000 1.0000 0.9000

f24 mean 0.054948 5.3911e-88 0.021979 1.0803e-154

std 0.1250 2.6607e-87 0.0836 4.1309e-154

best 1.1958e-84 5.8885e-102 1.3701e-125 4.4046e-176

worst 0.3297 1.4555e-86 0.3297 1.7740e-153

Fun		WOA	PWOA	AWOA	PAWOA
f1	mean	0	0	0	0
	std	0	0	0	0
	best	0	0	0	0
	worst	0	0	0	0
f2	mean	1082.8932	474.4647	844.7097	658.9085
	std	970.7269	408.5380	774.3744	575.2776
	best	179.3717	54.3772	71.8856	7.4499
	worst	3.3119e+03	1.8160e+03	3.3023e+03	1.8340e+03
f3	mean	2.0574e-100	2.7864e-147	1.0576e-147	0
	std	1.1269e-99	1.3409e-146	5.7925e-147	0
	best	4.6287e-142	4.7770e-180	1.2727e-183	0
	worst	6.1723e-99	7.3358e-146	3.1727e-146	0
f4	mean	1.716e-71	1.2207e-99	2.0443e-86	1.9094e-197
	std	8.5199e-71	6.6863e-99	9.5938e-86	0
	best	1.1238e-87	8.7494e-125	2.3367e-105	1.1820e-236
	worst	4.6577e-70	3.6622e-98	5.2425e-85	5.5989e-196
f5	mean	0.0012394	0.00066298	0.00052644	0.00033209
	std	0.0022	8.3271e-04	5.6824e-04	4.6627e-04
	best	2.7766e-06	9.1405e-06	3.7789e-06	2.8783e-08
	worst	0.0112	0.0031	0.0028	0.0018
f6	mean	2.725e-196	1.7926e-211	1.57e-280	0
	std	0	0	0	0
	best	3.7227e-225	8.8805e-250	2.4725e-304	0
	worst	5.4628e-195	5.3777e-210	4.4591e-279	0
f7	mean	5.8426e-79	7.1121e-117	2.335e-126	1.2035e-263
	std	2.6413e-78	4.3384e-147	8.9471e-126	0
	best	4.8545e-104	3.8576e-116	1.6281e-158	1.5732e-311
	worst	1.4350e-77	1.518e-117	3.9147e-125	3.6101e-262
f8	mean	5.0787e-51	6.8723e-107	0.049773	2.3326e-267
	std	2.7817e-50	3.5105e-106	0.1132	0
	best	2.7478e-97	4.3646e-139	6.2303e-158	3.7917e-316
	worst	1.5236e-49	1.9226e-105	0.2986	6.9355e-266
f9	mean	0.66708	0.6668	0.66669	0.6667
	std	2.8267e-04	1.2505e-04	1.6479e-05	1.3825e-05
	best	0.6667	0.6667	0.6667	0.6667
	worst	0.6679	0.6672	0.6667	0.6668
f10	mean	27.6882	26.508	27.4071	27.2505
	std	0.4108	4.9535	0.6931	0.5541
	best	27.0261	0.3494	26.7062	26.1975
	worst	28.7488	28.4633	28.7472	28.7271
f11	mean	-24.1465	-24.1465	-24.1465	-24.1465
	std	3.6134e-15	3.6134e-15	3.6134e-15	3.6134e-15
	best	-24.1465	-24.1465	-24.1465	-24.1465
	worst	-24.1465	-24.1465	-24.1465	-24.1465
f12	mean	3.7375e-93	3.8659e-124	1.4181e-128	5.0638e-244
	std	1.8771e-92	2.1092e-123	7.7672e-128	0
	best	7.9376e-116	1.3863e-153	1.0639e-153	8.8464e-306
	worst	1.0272e-91	1.1554e-122	4.2543e-127	1.5191e-242
f13	mean	-8.7789	-9.0614	-7.7722	-9.6159
	std	2.2844	2.0444	2.5850	1.6522
	best	-10.1522	-10.1520	-10.1531	-10.1530
	worst	-5.0512	-5.0537	-5.0551	-2.6293
f14	mean	-7.5356	-9.649	-6.9749	-9.2823
	std	3.1412	1.8237	3.3686	2.5899
	best	-10.4002	-10.4028	-10.4028	-10.4025
	worst	-2.7640	-5.0872	-1.8366	-1.8371
f15	mean	-7.0577	-9.5092	-6.3841	-9.5252
	std	3.6278	2.0082	3.1327	2.5835
	best	-10.5360	-10.5358	-10.5361	-10.5362
	worst	-1.8594	-3.8326	-2.4216	-1.8594
f16	mean	3.7896e-15	0	0	0
	std	1.2340e-14	0	0	0
	best	0	0	0	0
	worst	5.6843e-14	0	0	0
f17	mean	0.023102	0	0	0
	std	0.0537	0	0	0
	best	0	0	0	0
	worst	0.2011	0	0	0
f18	mean	2.7797e-06	1.9125e-53	3.9046e-08	8.9994e-184
	std	6.0019e-06	1.0475e-52	2.0787e-07	0
	best	2.5709e-60	1.5628e-122	8.5096e-97	1.7751e-236
	worst	2.4856e-05	5.7374e-52	1.1392e-06	2.6998e-182
f19	mean	0.43975	0.17069	0.98123	0.52745
	std	0.2887	0.1748	0.3719	0.3606
	best	0.1477	0.0247	0.3399	0.0234
	worst	1.3954	0.7402	1.8759	1.2220
f20	mean	5.033e-15	8.8818e-16	4.5593e-15	8.8818e-16
	std	2.6526e-15	0	2.3756e-15	0
	best	8.8818e-16	8.8818e-16	8.8818e-16	8.8818e-16
	worst	7.9936e-15	8.8818e-16	7.9936e-15	8.8818e-16
f21	mean	0.10492	0.1177	0.11232	0.12111
	std	0.0788	0.0559	0.0714	0.0742
	best	0.0145	0.0167	0.0244	0.0048
	worst	0.2551	0.2647	0.2476	0.2441
f22	mean	5.847e-109	4.2991e-136	7.6118e-187	1.2841e-288
	std	3.2025e-108	2.1201e-135	0	0
	best	3.2926e-134	1.6325e-174	5.4377e-223	0
	worst	1.7541e-107	1.1583e-134	2.0998e-185	3.8520e-287
f23	mean	0.92667	0.9	0.93667	0.9
	std	0.0450	4.5168e-16	0.0490	4.5168e-16
	best	0.9000	0.9000	0.9000	0.9000
	worst	1.0000	0.9000	1.0000	0.9000
f24	mean	0.054948	5.3911e-88	0.021979	1.0803e-154
	std	0.1250	2.6607e-87	0.0836	4.1309e-154
	best	1.1958e-84	5.8885e-102	1.3701e-125	4.4046e-176
	worst	0.3297	1.4555e-86	0.3297	1.7740e-153

Table 2 shows that PAWOA is very competitive as compared with other competitors. The PAWOA can obtain the best results on f1, f3, f6, f16, f17, and PAWOA are the best algorithm in terms of standard deviation. For f7, f8, f12, f13, f18, f22,f24, although PAWOA do not reach the minimum accuracy, but also significantly better than PWOA, AWOA and WOA. For f9, f10, f11, f12, f21 there is no significant difference between these algorithms. For f2, f10, f14, PWOA is the best algorithms in terms of average and standard deviation, and PAWOA is the second. For most of the cases, PAWOA gives good results for the standard deviation. This means that the proposed algorithm has higher stability for finding the optimum values. Based on overall performance, we can sort these algorithms as follows: PAWOA>PWOA>AWOA>WOA.

In order to visually observe the performance difference between the improved algorithm and the original algorithm, the convergence curves of 24 benchmark functions are given in this paper, as shown in Fig. 3. The red curve in the figure represents the convergence curve of WOA, and the pink curve, the blue curve and the green curve represent the convergence curve of PWOA, AWOA and PAWOA respectively.

Fig. 3

Convergence curves of the test functions.

It can be clearly seen that in most functions, the convergence speed and accuracy of the curve of PWOA have been significantly improved compared with that of WOA, indicating that random hopping has a strong effectiveness in improving the performance of the algorithm. For most of the convergence curves, such as f3 f4 f12 f18, PWOA is easy to fall into the local optimal, but when AWOA is added to the global search equation, the algorithm converges quickly.

From the figures we observed that for most of the cases both the convergence speeds and the final results of our approach PAWOA are better than PWOA, AWOA and WOA. These convergence curves show that the convergence of PAWOA is more rapid and more accurate, and the convergences of PWOA and AWOA are better than WOA, but for but for the functions f8, f14, f15, f19, f23, AWOA is easy to fall into the local optimal.

4.3 P-value test of Wilcoxon rank

In order to test the statistical significance, this paper uses Wilcoxon signed rank sum test to analyze the performance of the algorithm and sets the significance level to 0.05. The Wilcoxon signed rank test results of the optimal values of 24 test functions optimized by the algorithm after 30 independent runs are given in Table 3. p-value is the significance probability of whether the population of the two comparative samples is the same, h returns the result of the hypothesis test. If p-value is less than the set significance level, the overall difference between the two comparison samples is significant, and the value of h is 1; If p-value is exceeds the set significance level, the overall difference between the two comparison samples is not significant, and the value of h is 0; If the result of p-value is NaN, there is no difference between the two samples. Wilcoxon rank sum test further shows that the improved algorithm has significant advantages over the WOA.

Table 3
Experimental results of fixed number of iterations

Fun PWOA vs WOA AWOA vs WOA PAWOA vs WOA

p-value h p-value h p-value h

f1 NaN 0 NaN 0 NaN 0

f2 0.0436 1 0.0150 1 0.0087 1

f3 3.0199e-11 1 3.0199e-11 1 1.7203e-12 1

f4 3.0199e-11 1 3.0199e-11 1 3.0199e-11 1

f5 0.0364 1 3.9881 e-04 1 1.1674e-05 1

f6 1.0937e-10 1 3.0199e-11 1 1.2118e-12 1

f7 3.0199e-11 1 3.0199e-11 1 3.0199e-11 1

f8 3.0199e-11 1 1.4298e-05 1 3.0199e-11 1

f9 1.3367-05 1 4.9752e-11 1 1.3289e-10 1

f10 0.0012 1 2.1327e-05 1 3.8349e-06 1

f11 NaN 0 NaN 0 NaN 0

f12 3.0199e-11 1 3.0199e-11 1 3.0199e-11 1

f13 0.0061 1 0.0095 1 2.6806e-04 1

f14 0.0468 1 0.1154 0 0.0112 11

f15 0.0378 1 0.5394 0 0.0133 1

f16 NaN 0 NaN 0 NaN 0

f17 0.0419 1 0.0419 1 0.0419 1

f18 3.0199e-11 1 8.8411e-07 1 3.0199e-11 1

f19 1.0407e-04 1 1.7479e-05 1 0.5011 0

f20 9.8284e-08 1 0.9029 0 2.7194e-10 1

f21 0.6100 0 0.6414 0 0.2973 0

f22 4.0772e-11 1 3.0199e-11 1 2.8646e-11 1

f23 0.0216 1 0.2314 0 0.0216 1

f24 4.0772e-11 1 0.0017 1 3.0199e-11 1

4.4 Algorithm comparison

In this experiment, to evaluate the performance of the proposed algorithm, five well-known metaheuristics are chosen for comparison. These algorithms are hybrid particle swarm and gravity search algorithm (PSO/GSA), ant lion optimization algorithm (ALO), butterfly optimization algorithm (BOA), moth flame optimization algorithm (MFO) and biogeography-based optimization (BBO). These algorithms are evaluated on the conventional benchmark functions, and the parameter values of these algorithms are set as they are recommended in their original papers. The experimental results of the compared algorithms for the functions are shown in Table 4, in which the bold values show the best solutions.

Table 4
Experimental results of algorithm comparison

Fun algorithm mean std best worst

f1 PAWOA 0 0 0 0

PSO/GSA 0 0 0 0

ALO 8.7536e-11 1.4376e-10 3.7221e-12 7.7689e-10

BOA 2.5649e-12 4.7202e-12 0 1.8635e-11

MFO 0 0 0 0

BBO 0.00035779 0.0012998 2.4399e-09 0.0070825

f2 PAWOA 654.8563 601.5312 4.1431 1800.6152

PSO/GSA 1.4076e+58 4.4873e+58 5.9353e+40 1.9156e+59

ALO 4.3873e+52 2.1942e+53 3.1281e+23 1.2004e+54

BOA 9.5624e+88 1.9146e+89 3.7366e+86 9.9562e+89

MFO 1.2186e+76 6.6747e+76 8.3026e+48 3.6559e+77

BBO 1.2495e+41 4.7645e+41 1.7936e+27 1.9980e+42

f3 PAWOA 0 0 0 0

PSO/GSA 7.4017e-09 6.1113e-09 3.31e-11 2.0173e-08

ALO 1.0965e-06 5.6688e-07 3.1099e-07 2.4425e-06

BOA 5.7174e-13 4.4318e-13 1.268e-13 1.6622e-12

MFO 2.0897e-09 4.6223e-09 6.9887e-13 2.1762e-08

BBO 1.4935e+11 5.7574e+11 1.6536e+07 3.0676e+12

f4 PAWOA 1.6336e-206 0 1.3731e-231 4.8965e-205

PSO/GSA 1.6112e-05 8.744e-05 1.6092e-17 0.00047906

ALO 3.9549 2.8125 0.14426 12.2863

BOA 2.1153e-07 7.0632e-08 8.9096e-08 3.8558e-07

MFO 594.0017 987.3133 0.059048 4900.0194

BBO 198.6948 64.2561 93.9324 338.6846

f5 PAWOA 0.00050278 0.00078171 8.6867e-06 0.0034012

PSO/GSA 3.9733e-23 4.7519e-23 5.0914e-26 2.0222e-22

ALO 1.8836e-13 1.7447e-13 1.7229e-14 6.153e-13

BOA 0.00012763 0.00013823 4.4052e-07 0.00044633

MFO 0 0 0 0

BBO 0.023916 0.043653 3.8519e-09 0.21445

f6 PAWOA 2.1734e-320 0 0 6.5199e-319

PSO/GSA 2.2109e-24 2.5632e-24 8.0044e-27 1.1195e-23

ALO 4.159e-15 3.5469e-15 2.181e-16 1.5378e-14

BOA 8.1404e-18 3.0708e-17 5.0582e-23 1.6552e-16

MFO 2.0008e-25 1.0922e-24 7.3732e-89 5.983e-24

BBO 0.011191 0.021742 8.8673e-06 0.10739

f7 PAWOA 5.2386e-266 0 1.3827e-309 1.5692e-264

PSO/GSA 3.1701e-23 3.0414e-23 6.048e-25 1.1485e-22

ALO 2.6176e-14 2.2524e-14 2.3437e-16 7.8077e-14

BOA 1.4547e-18 6.347e-18 1.0748e-21 3.4644e-17

MFO 1.9547e-105 5.795e-105 5.6981e-120 2.6934e-104

BBO 0.00011584 0.00020281 1.9967e-11 0.00061101

f8 PAWOA 5.1948e-259 0 6.8198e-314 1.5584e-257

PSO/GSA 2.3904e-23 1.6038e-23 4.2839e-25 5.0087e-23

ALO 1.0607e-14 1.1763e-14 2.409e-17 4.0211e-14

BOA 1.1393e-13 3.608e-13 3.1057e-18 1.8801e-12

MFO 6.2038e-103 3.3967e-102 4.4121e-117 1.8605e-101

BBO 0.04993 70.11313 2.5534e-10 0.29868

f9 PAWOA 0.6667 2.4303e-05 0.66667 0.66678

PSO/GSA 11.6781 58.4279 0.66667 321

ALO 6.5336 6.1715 0.74421 27.887

BOA 0.88154 0.031998 0.81871 0.9495

MFO 10515.1719 40843.1799 4.5572 195136.5612

BBO 3553.0913 2716.9992 733.8688 13428.8555

f10 PAWOA 27.2788 0.52099 26.4571 28.7264

PSO/GSA 1903.3413 10143.1898 23.6287 55607.589

ALO 76.9407 67.993 25.6132 249.1785

BOA 28.8054 0.030134 28.7342 28.8499

MFO 67457.8063 58895.9642 34.7379 175160.4493

BBO 6978.8944 3741.8006 1854.6201 13496.0947

f11 PAWOA -24.1465 0 -24.1465 -24.1465

PSO/GSA -24.1465 0 -24.1465 -24.1465

ALO -24.1465 0 -24.1465 -24.1465

BOA -24.1465 0 -24.1465 -24.1465

MFO -24.1465 0 -24.1465 -24.1465

BBO -19.2739 5.5335 -24.1465 6.6429

f12 PAWOA 3.5611e-258 0 8.3063e-299 3.1116e-257

PSO/GSA 2.2144e-22 1.537e-22 9.0593e-24 7.1955e-22

ALO 3.3966e-14 3.1835e-14 3.1851e-15 1.527e-13

BOA 1.1287e-14 1.9771e-14 4.2662e-18 9.3908e-14

MFO 9.89e-70 3.3319e-69 9.6009e-78 1.3988e-68

BBO 3.4883e-05 7.0119e-05 3.3549e-09 0.00034784

f13 PAWOA -9.4693 1.8424 -10.1531 -2.6301

PSO/GSA -6.4694 3.4009 -10.1532 -2.6305

ALO -6.1145 2.8422 -10.1532 -2.6305

BOA -5.5012 0.88921 -8.2174 -4.5784

MFO -7.4707 3.222 -10.1532 -2.6305

BBO -6.3103 3.4107 -10.1474 -2.6062

f14 PAWOA -9.1017 2.6859 -10.4027 -1.8368

PSO/GSA -5.896 3.5658 -10.4029 -1.8376

ALO -6.3463 3.047 -10.4029 -1.8376

BOA -5.5042 1.0043 7.3465 -4.1576

MFO 6.8027 3.7166 -10.4029 1.8376

BBO -5.3986 3.3888 -10.388 -1.8373

f15 PAWOA -9.6053 3.2619 -10.5361 -2.4212

PSO/GSA 4.2771 3.2454 -10.5364 -1.8595

ALO -5.6401 3.5986 -10.5364 -1.8595

BOA -5.2229 1.084 -7.912 -4.2051

MFO -7.9818 3.6871 -10.5364 -2.4217

BBO -6.1608 3.4964 -10.5334 -2.4212

PAWOA 0 0 0 0

PSO/GSA 277.773 257.3542 110.4398 1538.7659

ALO 162.0447 53.4057 87.5566 298.4854

BOA 87.9537 45.8245 1.5844e-07 125.5535

MFO 200.5102 404.4439 25.8698 1438.2774

BBO 71.4208 14.1315 50.5744 109.8521

f17 PAWOA 0 0 0 0

PSO/GSA 1.6538 5.7273 4.1408e-06 22.773

ALO 0.024812 0.025595 0.00022144 0.08639

BOA 1.7743e-08 1.6829e-08 3.015e-09 7.2311e-08

MFO 0.025635 0.024133 0.00013243 0.081141

BBO 0.29599 0.14716 0.074719 0.69897

f18 PAWOA 4.795e-178 0 5.5703e-232 1.4385e-176

PSO/GSA 0.3391 0.57046 0.0034407 2.5835

ALO 2.4825 1.4578 0.16516 6.9804

BOA 3.7482e-09 2.9497e-09 3.3288e-10 1.1949e-08

MFO 391.6595 687.3749 0.074706 3022.3662

BBO 2391.9813 1771.7683 407.3664 8202.6666

f19 PAWOA 0.35548 0.36866 0.025106 1.556

PSO/GSA 31.0549 11.8551 16.4223 65.3857

ALO 10.0546 4.436 3.355 23.1558

BOA 2.3191 0.26323 1.7431 2.7774

MFO 32.8351 10.5067 13.011 53.1446

BBO 7.1994 2.7355 2.4908 12.4964

f20 PAWOA 8.8818e-16 0 8.8818e-16 8.8818e-16

PSO/GSA 13.4173 2.4858 6.799 17.1351

ALO 4.384 1.7901 1.1557 7.7013

BOA 3.6401e-05 7.9504e-06 2.0037e-05 5.1676e-05

MFO 11.4716 4.0523 0.62805 16.1006

BBO 4.1126 0.518 2.9982 5.2576

f21 PAWOA 0.12101 0.06546 0.018022 0.30643

PSO/GSA 0.12797 0.072133 0.01803 0.25004

ALO 0.12132 0.072405 0.0034315 0.26261

BOA 0.17334 0.087778 0.066888 0.40089

MFO 0.13967 0.089277 0.016758 0.25

BBO 0.11296 0.068466 0.024861 0.37133

f22 PAWOA 1.333e-296 0 0 3.7244e-295

PSO/GSA 0.31627 1.7323 2.0494e-23 9.4882

ALO 4.6432e-13 5.2266e-13 8.4307e-15 1.8579e-12

BOA 3.3768e-13 1.0683e-12 1.288e-18 4.2515e-12

MFO 4.1158e-102 2.1565e-101 8.5859e-119 1.1818e-100

BBO 0.63356 2.407 2.1992e-13 9.4883

f23 PAWOA 0.9 0 0.9 0.9

PSO/GSA 0.99 0.030513 0.9 1

ALO 0.94 0.049827 0.9 1

BOA 0.92538 0.042697 0.9 1.0005

MFO 0.97 0.046609 0.9 1

BBO 0.98667 0.034576 0.9 1.0001

f24 PAWOA 1.7211e-154 9.4042e-154 2.5819e-171 5.1513e-153

PSO/GSA 0.054948 0.12497 1.5755e-13 0.32969

ALO 1.6519e-07 8.4251e-08 3.9666e-08 3.3832e-07

BOA 0.0008023 0.0039063 1.2046e-07 0.021438

MFO 0.021979 0.083645 8.2629e-62 0.32969

BBO 0.18721 0.16572 3.3364e-07 0.32972

Fun	algorithm	mean	std	best	worst
f1	PAWOA	0	0	0	0
	PSO/GSA	0	0	0	0
	ALO	8.7536e-11	1.4376e-10	3.7221e-12	7.7689e-10
	BOA	2.5649e-12	4.7202e-12	0	1.8635e-11
	MFO	0	0	0	0
	BBO	0.00035779	0.0012998	2.4399e-09	0.0070825
f2	PAWOA	654.8563	601.5312	4.1431	1800.6152
	PSO/GSA	1.4076e+58	4.4873e+58	5.9353e+40	1.9156e+59
	ALO	4.3873e+52	2.1942e+53	3.1281e+23	1.2004e+54
	BOA	9.5624e+88	1.9146e+89	3.7366e+86	9.9562e+89
	MFO	1.2186e+76	6.6747e+76	8.3026e+48	3.6559e+77
	BBO	1.2495e+41	4.7645e+41	1.7936e+27	1.9980e+42
f3	PAWOA	0	0	0	0
	PSO/GSA	7.4017e-09	6.1113e-09	3.31e-11	2.0173e-08
	ALO	1.0965e-06	5.6688e-07	3.1099e-07	2.4425e-06
	BOA	5.7174e-13	4.4318e-13	1.268e-13	1.6622e-12
	MFO	2.0897e-09	4.6223e-09	6.9887e-13	2.1762e-08
	BBO	1.4935e+11	5.7574e+11	1.6536e+07	3.0676e+12
f4	PAWOA	1.6336e-206	0	1.3731e-231	4.8965e-205
	PSO/GSA	1.6112e-05	8.744e-05	1.6092e-17	0.00047906
	ALO	3.9549	2.8125	0.14426	12.2863
	BOA	2.1153e-07	7.0632e-08	8.9096e-08	3.8558e-07
	MFO	594.0017	987.3133	0.059048	4900.0194
	BBO	198.6948	64.2561	93.9324	338.6846
f5	PAWOA	0.00050278	0.00078171	8.6867e-06	0.0034012
	PSO/GSA	3.9733e-23	4.7519e-23	5.0914e-26	2.0222e-22
	ALO	1.8836e-13	1.7447e-13	1.7229e-14	6.153e-13
	BOA	0.00012763	0.00013823	4.4052e-07	0.00044633
	MFO	0	0	0	0
	BBO	0.023916	0.043653	3.8519e-09	0.21445
f6	PAWOA	2.1734e-320	0	0	6.5199e-319
	PSO/GSA	2.2109e-24	2.5632e-24	8.0044e-27	1.1195e-23
	ALO	4.159e-15	3.5469e-15	2.181e-16	1.5378e-14
	BOA	8.1404e-18	3.0708e-17	5.0582e-23	1.6552e-16
	MFO	2.0008e-25	1.0922e-24	7.3732e-89	5.983e-24
	BBO	0.011191	0.021742	8.8673e-06	0.10739
f7	PAWOA	5.2386e-266	0	1.3827e-309	1.5692e-264
	PSO/GSA	3.1701e-23	3.0414e-23	6.048e-25	1.1485e-22
	ALO	2.6176e-14	2.2524e-14	2.3437e-16	7.8077e-14
	BOA	1.4547e-18	6.347e-18	1.0748e-21	3.4644e-17
	MFO	1.9547e-105	5.795e-105	5.6981e-120	2.6934e-104
	BBO	0.00011584	0.00020281	1.9967e-11	0.00061101
f8	PAWOA	5.1948e-259	0	6.8198e-314	1.5584e-257
	PSO/GSA	2.3904e-23	1.6038e-23	4.2839e-25	5.0087e-23
	ALO	1.0607e-14	1.1763e-14	2.409e-17	4.0211e-14
	BOA	1.1393e-13	3.608e-13	3.1057e-18	1.8801e-12
	MFO	6.2038e-103	3.3967e-102	4.4121e-117	1.8605e-101
	BBO	0.04993	70.11313	2.5534e-10	0.29868
f9	PAWOA	0.6667	2.4303e-05	0.66667	0.66678
	PSO/GSA	11.6781	58.4279	0.66667	321
	ALO	6.5336	6.1715	0.74421	27.887
	BOA	0.88154	0.031998	0.81871	0.9495
	MFO	10515.1719	40843.1799	4.5572	195136.5612
	BBO	3553.0913	2716.9992	733.8688	13428.8555
f10	PAWOA	27.2788	0.52099	26.4571	28.7264
	PSO/GSA	1903.3413	10143.1898	23.6287	55607.589
	ALO	76.9407	67.993	25.6132	249.1785
	BOA	28.8054	0.030134	28.7342	28.8499
	MFO	67457.8063	58895.9642	34.7379	175160.4493
	BBO	6978.8944	3741.8006	1854.6201	13496.0947
f11	PAWOA	-24.1465	0	-24.1465	-24.1465
	PSO/GSA	-24.1465	0	-24.1465	-24.1465
	ALO	-24.1465	0	-24.1465	-24.1465
	BOA	-24.1465	0	-24.1465	-24.1465
	MFO	-24.1465	0	-24.1465	-24.1465
	BBO	-19.2739	5.5335	-24.1465	6.6429
f12	PAWOA	3.5611e-258	0	8.3063e-299	3.1116e-257
	PSO/GSA	2.2144e-22	1.537e-22	9.0593e-24	7.1955e-22
	ALO	3.3966e-14	3.1835e-14	3.1851e-15	1.527e-13
	BOA	1.1287e-14	1.9771e-14	4.2662e-18	9.3908e-14
	MFO	9.89e-70	3.3319e-69	9.6009e-78	1.3988e-68
	BBO	3.4883e-05	7.0119e-05	3.3549e-09	0.00034784
f13	PAWOA	-9.4693	1.8424	-10.1531	-2.6301
	PSO/GSA	-6.4694	3.4009	-10.1532	-2.6305
	ALO	-6.1145	2.8422	-10.1532	-2.6305
	BOA	-5.5012	0.88921	-8.2174	-4.5784
	MFO	-7.4707	3.222	-10.1532	-2.6305
	BBO	-6.3103	3.4107	-10.1474	-2.6062
f14	PAWOA	-9.1017	2.6859	-10.4027	-1.8368
	PSO/GSA	-5.896	3.5658	-10.4029	-1.8376
	ALO	-6.3463	3.047	-10.4029	-1.8376
	BOA	-5.5042	1.0043	7.3465	-4.1576
	MFO	6.8027	3.7166	-10.4029	1.8376
	BBO	-5.3986	3.3888	-10.388	-1.8373
f15	PAWOA	-9.6053	3.2619	-10.5361	-2.4212
	PSO/GSA	4.2771	3.2454	-10.5364	-1.8595
	ALO	-5.6401	3.5986	-10.5364	-1.8595
	BOA	-5.2229	1.084	-7.912	-4.2051
	MFO	-7.9818	3.6871	-10.5364	-2.4217
	BBO	-6.1608	3.4964	-10.5334	-2.4212
	PAWOA	0	0	0	0
	PSO/GSA	277.773	257.3542	110.4398	1538.7659
	ALO	162.0447	53.4057	87.5566	298.4854
	BOA	87.9537	45.8245	1.5844e-07	125.5535
	MFO	200.5102	404.4439	25.8698	1438.2774
	BBO	71.4208	14.1315	50.5744	109.8521
f17	PAWOA	0	0	0	0
	PSO/GSA	1.6538	5.7273	4.1408e-06	22.773
	ALO	0.024812	0.025595	0.00022144	0.08639
	BOA	1.7743e-08	1.6829e-08	3.015e-09	7.2311e-08
	MFO	0.025635	0.024133	0.00013243	0.081141
	BBO	0.29599	0.14716	0.074719	0.69897
f18	PAWOA	4.795e-178	0	5.5703e-232	1.4385e-176
	PSO/GSA	0.3391	0.57046	0.0034407	2.5835
	ALO	2.4825	1.4578	0.16516	6.9804
	BOA	3.7482e-09	2.9497e-09	3.3288e-10	1.1949e-08
	MFO	391.6595	687.3749	0.074706	3022.3662
	BBO	2391.9813	1771.7683	407.3664	8202.6666
f19	PAWOA	0.35548	0.36866	0.025106	1.556
	PSO/GSA	31.0549	11.8551	16.4223	65.3857
	ALO	10.0546	4.436	3.355	23.1558
	BOA	2.3191	0.26323	1.7431	2.7774
	MFO	32.8351	10.5067	13.011	53.1446
	BBO	7.1994	2.7355	2.4908	12.4964
f20	PAWOA	8.8818e-16	0	8.8818e-16	8.8818e-16
	PSO/GSA	13.4173	2.4858	6.799	17.1351
	ALO	4.384	1.7901	1.1557	7.7013
	BOA	3.6401e-05	7.9504e-06	2.0037e-05	5.1676e-05
	MFO	11.4716	4.0523	0.62805	16.1006
	BBO	4.1126	0.518	2.9982	5.2576
f21	PAWOA	0.12101	0.06546	0.018022	0.30643
	PSO/GSA	0.12797	0.072133	0.01803	0.25004
	ALO	0.12132	0.072405	0.0034315	0.26261
	BOA	0.17334	0.087778	0.066888	0.40089
	MFO	0.13967	0.089277	0.016758	0.25
	BBO	0.11296	0.068466	0.024861	0.37133
f22	PAWOA	1.333e-296	0	0	3.7244e-295
	PSO/GSA	0.31627	1.7323	2.0494e-23	9.4882
	ALO	4.6432e-13	5.2266e-13	8.4307e-15	1.8579e-12
	BOA	3.3768e-13	1.0683e-12	1.288e-18	4.2515e-12
	MFO	4.1158e-102	2.1565e-101	8.5859e-119	1.1818e-100
	BBO	0.63356	2.407	2.1992e-13	9.4883
f23	PAWOA	0.9	0	0.9	0.9
	PSO/GSA	0.99	0.030513	0.9	1
	ALO	0.94	0.049827	0.9	1
	BOA	0.92538	0.042697	0.9	1.0005
	MFO	0.97	0.046609	0.9	1
	BBO	0.98667	0.034576	0.9	1.0001
f24	PAWOA	1.7211e-154	9.4042e-154	2.5819e-171	5.1513e-153
	PSO/GSA	0.054948	0.12497	1.5755e-13	0.32969
	ALO	1.6519e-07	8.4251e-08	3.9666e-08	3.3832e-07
	BOA	0.0008023	0.0039063	1.2046e-07	0.021438
	MFO	0.021979	0.083645	8.2629e-62	0.32969
	BBO	0.18721	0.16572	3.3364e-07	0.32972

In this paper, functions f1, f3, f16, f17 are selected with representative and optimal values of 0, and the convergence curve of the six algorithms is given, as shown in Fig.4. From the convergence curve, we can see that PAWOA has the characteristics of fast convergence speed and high convergence precision, strong robustness, and the overall performance is obviously better than the other five algorithms.

Fig. 4

Convergence curve of algorithm comparison.

It can be seen from Table 4 that PAWOA has obvious advantages in convergence speed and accuracy. In comparison with five algorithms, PAWOA has significantly better performance than PSO/GSA on 20 problems, better than ALO, BOA and MFO on 21 problems, better than BBO on 23 problems. For f11 and f21, there is no significant difference among these algorithms. Therefore, it can be concluded that the results of PAWOA are significantly effective than competitive approaches. Overall, the results reveal that PAWOA outperforms other algorithms.

5 Conclusions and future scope

This paper improved on the basis of the WOA. In order to improve the exploitation ability of the algorithm, a random hopping strategy is proposed. In order to improve the exploration ability of the algorithm, a random parameter strategy is proposed. PAWOA combines the two strategies to better balance the exploration and exploitation ability of the algorithm. According to comparative studies, it can be found that the improved WOA algorithm has a higher success rate in finding global optimal for different benchmark functions. In the future, with the deepening of research, many potential directions can be explored, which are mainly divided into two categories: the mechanism of swarm intelligence and specific practical applications.

Footnotes

Acknowledgments

This work is supported by National Nature Science Foundation of China [61401307]; Tianjin Research Program of Science and Technology Commissioner of China [18JCTPJC57500]; Hebei Province Research Project of Higher Education Science and Technology of China [ZD2018045].

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Fun	PWOA vs WOA		AWOA vs WOA		PAWOA vs WOA
	p-value	h	p-value	h	p-value	h
f1	NaN	0	NaN	0	NaN	0
f2	0.0436	1	0.0150	1	0.0087	1
f3	3.0199e-11	1	3.0199e-11	1	1.7203e-12	1
f4	3.0199e-11	1	3.0199e-11	1	3.0199e-11	1
f5	0.0364	1	3.9881 e-04	1	1.1674e-05	1
f6	1.0937e-10	1	3.0199e-11	1	1.2118e-12	1
f7	3.0199e-11	1	3.0199e-11	1	3.0199e-11	1
f8	3.0199e-11	1	1.4298e-05	1	3.0199e-11	1
f9	1.3367-05	1	4.9752e-11	1	1.3289e-10	1
f10	0.0012	1	2.1327e-05	1	3.8349e-06	1
f11	NaN	0	NaN	0	NaN	0
f12	3.0199e-11	1	3.0199e-11	1	3.0199e-11	1
f13	0.0061	1	0.0095	1	2.6806e-04	1
f14	0.0468	1	0.1154	0	0.0112	11
f15	0.0378	1	0.5394	0	0.0133	1
f16	NaN	0	NaN	0	NaN	0
f17	0.0419	1	0.0419	1	0.0419	1
f18	3.0199e-11	1	8.8411e-07	1	3.0199e-11	1
f19	1.0407e-04	1	1.7479e-05	1	0.5011	0
f20	9.8284e-08	1	0.9029	0	2.7194e-10	1
f21	0.6100	0	0.6414	0	0.2973	0
f22	4.0772e-11	1	3.0199e-11	1	2.8646e-11	1
f23	0.0216	1	0.2314	0	0.0216	1
f24	4.0772e-11	1	0.0017	1	3.0199e-11	1