Salp swarm algorithm with crossover scheme and Lévy flight for global optimization

1 Introduction

Meta-heuristic algorithm has been successfully applied to many optimization problems, such as image processing [1], feature selection [2], parameter extraction of solar photovoltaic models [3], path planning [4], etc. Due to the gradient-free characteristic, it shows better performance than traditional method on certain specific problems, and has received extensive attention from scholars [5, 6].

Generally speaking, the process of meta-heuristic algorithm to solve optimization problems can be divided into two stages: exploration and exploitation, in which the global and local search operations are carried out respectively. It is worth mentioning that the balance between these two stages is significant, which is related to the convergence performance of the algorithm. More exploration processes will lead to non-optimal solutions, while more exploitation processes will lead to local optimality. In fact, various algorithms have been proposed in succession because of their performance differences in different optimization problems. For example, an algorithm shows excellent performance in the field of multilevel image thresholding, but its ability in path planning is not necessarily outstanding. Algorithms proposed in recent years include but are not limited to: salp swarm algorithm (SSA) [7], grasshopper optimization algorithm (GOA) [8], sailfish optimizer (SFO) [9], seagull optimization algorithm (SOA) [10], emperor penguin optimizer (EPO) [11], moth search algorithm (MSA) [12].

Salp swarm algorithm (SSA) is a novel optimization algorithm proposed by Mirjalili in 2017, which is inspired from the swarming behavior of salps [13, 14]. There are mainly two types of salps in the population, namely leader and followers. The leader decides the direction of the population and the followers move along with each other. Although the SSA algorithm has been successfully applied into many engineering problems such as feature selection [15], and parameter identification of photovoltaic cell model [16], the performance of algorithm itself still needs to be improved to better suit complex problems. Faris et al. [17] proposed to use the crossover operator to replace average operator and enhance the exploratory behavior of SSA algorithm. The experiments result of 22 well-known UCI datasets and 5 feature selection comparing methods fully demonstrated the superiority of the proposed method on feature selection task. Zhang et al. [18] proposed an improved salp swarm algorithm based on Levy flight and sine cosine operator (called LSC-SSA) to solve high-dimensional largescale optimization problems, exhibiting remarkable performance on avoiding failing into the local optimum and obtaining ideal accuracy. These successful applications greatly stimulated our exploration for the improvement of this simple yet efficient algorithm.

In this paper, an improved version of SSA algorithm is proposed, which is known as SSACL. Crossover scheme and Lévy flight strategy are introduced to enhance the optimization capability. To be more specific, the crossover mechanism provides an alternative way to update the position of salp leader, which significantly improves the diversity and randomness of population. Lévy flight is added to the update formula of salp followers to guide the movement. With the occasional long step of Lévy distribution, the exploration ability of the SSACL algorithm has been improved to prevent falling into local optimum. Furthermore, the new updating formula refers to the position of food, which can improve the convergence effect of the algorithm to some extent. Compared to the previous approaches, our work pays more attention to the utilization of information about differences among population in order to guide individual updates. The crossover strategy scales the differences between two groups of random salps, not just the hard selection on the dimensions [17]. As for the Levy flight adopted, the information about food (i.e., approximate optimal solution) is referenced with the purpose of improving convergence capability, rather than only the location information of individual itself [18].

In order to evaluate the performance of proposed technique, various functions including unimodal, multimodal, and composite have been selected. Six state-of-the-art meta-heuristic algorithms are chosen for the experimental comparison. A non-parametric Wilcoxon’s rank sum test is also conducted to investigate the difference between algorithms. From the experimental results, it can be found that SSACL based method performs higher accuracy and better quality of solutions in general, which can provide technical support for solving practical engineering problems.

The structure of this paper is described as follows: Section 2 gives a brief review of traditional SSA algorithm and the pseudo code is included. In Section 3, the proposed method namely SSACL algorithm is presented in details and the flow chart is also given. A series of experiments and analysis are conducted in Section 4. Finally, the relevant conclusion and future work direction are presented in Section 5.

2 Salp swarm algorithm

Salp swarm algorithm is inspired by the swarming behavior of salps in nature. It is firstly proposed by Mirjalili to solve the problem of marine vehicles propeller optimization [7]. Just like other swarm-based techniques, the position vector of salp is defined in a d-dimensional space, where d denotes the number of decision variables of an optimization problem [19, 20]. Thus, the population X of n salp individuals can be represented by a n × d-dimensional matrix, which is shown in Equation (1). X = [ x 1 , 1 , x 1 , 2 , … , x 1 , d x 2 , 1 , x 2 , 2 , … , x 2 , d ⋮ ⋮ ⋯ ⋮ x n , 1 , x n , 2 , … , x n , d ](1)

There are two types of salps in the population, namely leader and followers. The leader decides the direction of the population and the followers move along with each other. And they form a chain structure called salp chain for better movement and foraging. The shape of salp individual and salp chain is presented in Fig. 1 [7]. Besides, the mathematical model is given as follows:

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Fig. 1

The shape of individual and chain of salps.

The position of the leader can be updated by: x L , j = { F j + c 1 × ( ( ub j - lb j ) × c 2 + lb j ) c 3 ⩾ 0 F j - c 1 × ( ( ub j - lb j ) × c 2 + lb j ) c 3 < 0(2) where the x_L,j is the position of leader salp, F _j represents the position of food source. ub _j and lb _j denote the upper and lower bound of jth-dimension, respectively. c₂ and c₃ represent random values which are in the range [0, 1]. c₁ is a significant parameter of SSA that can balance the exploration and exploitation of optimization. It decreases with the number of iterations and can be calculated by: c 1 = 2 e - ( 4 t / T ) 2(3) where t is the current iteration and T indicates the maximum number of iterations.

The position of follower salp can be updated by: x i , j t + 1 = 1 2 × ( x i , j t + x i - 1 , j t )(4) where i ⩾ 2 and x i , j t shows the ith salp at jth dimension and tth iteration.

Pseudo code of traditional salp swarm algorithm for optimization problem has been given in Algorithm 1.

In this section, a detailed introduction of SSACL algorithm is given. The optimization capability of the proposed algorithm is improved through two strategies. To be more specific, an efficient crossover scheme is used to increase the diversity of population. This part of SSACL can be regarded as a DE-like algorithm [21]. Besides, Lévy flight strategy is also introduced in the form of attempted motion, reducing the risk of falling into local optimum [22]. The flowchart of SSACL algorithm for finding the optimal solution is presented in Fig. 2.

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Fig. 2

The overall framework of SSACL algorithm.

A modified version of SSA algorithm is proposed in this paper, which is called SSACL. Crossover scheme and Lévy flight strategy are utilized to improve the movement patterns of leader and followers in the salp population, respectively.

In order to evaluate the performance of SSACL, the authors conduct a series of experiments on nineteen different mathematical functions, including unimodal, multimodal, and composite functions. Six state-of-the-art meta-heuristic algorithms are used for comparison. In addition, the non-parametric Wilcoxon’s rank sum test is adopted to evaluate the superior performance of proposed technique in a statistical and meaningful way. From the experimental results, it can be found that the SSACL algorithm can obtain the solutions more accurately, which is suitable for practical engineering problems.

In the future, we hope to introduce the crossover scheme and Lévy flight strategy into more meta-heuristic algorithms, and investigate their performance in the field of image processing. Besides, other improvement strategies will be used to improve the optimization capabilities of meta-heuristic algorithms, such as chaos initialization, opposition-based learning, Gaussian disturbance, etc.