1. Introduction

Real-world optimization problems have become more challenging, which requires more efficient solution methods. Different scholars have studied various approaches to solve these complex and difficult problems from the real world. A part of researchers solve these optimization problems using traditional methods such as quasi-Newton, conjugate gradient, and sequential quadratic programming methods. However, owing to the nonlinear, nonproductivity characteristics of most real-world optimization problems and the involvement of multiple decision variables and complex constraints, these traditional algorithms are difficult to be solved effectively [1, 2]. The metaheuristic algorithm has the advantages of not relying on the problem model, not requiring gradient information, having strong search capability and wide applicability, and can achieve a good balance between solution quality and computational cost [3]. Therefore, the metaheuristic algorithms have been proposed to solve real-world optimization problems, such as image segmentation [4, 5], feature selection [6, 7], mission planning [8, 9], parameter optimization [10, 11], job shop scheduling [12, 13], etc.

Metaheuristic algorithms are usually classified into three categories [14]: evolution-based algorithms, physical-based algorithms, and swarm-based algorithms. The evolution-based algorithm is inspired by the laws of evolution in nature. Genetic algorithm (GA) [15], inspired by Darwin′s theory of superiority and inferiority, is a well-known evolution-based algorithm. With the popularity of GA, several other widely used evolution-based algorithms have been proposed, including differential evolution (DE) [16], genetic programming (GP) [17], evolutionary strategies (ES) [18], and evolutionary programming (EP) [19]. In addition, several new evolution-based algorithms have been proposed, such as artificial algae algorithm (AAA) [20], biogeography-based optimization (BBO) [21], and monkey king evolutionary (MKE) [22]. The physical-based algorithms are inspired by various laws of physics. One of the most famous algorithms of this category is simulated annealing (SA) [23]. SA is inspired by the law of thermodynamics in which a material is heated up and then cooled slowly. There are other physical-based algorithms proposed, including gravitational search algorithm (GSA) [24], nuclear reaction optimization (NRO) [25], water cycle algorithm (WCA) [26], and sine cosine algorithm (SCA) [27]. The swarm-based algorithms are inspired by the social behavior of different species in natural groups. Particle swarm optimization (PSO) [28] and ant colony optimization (ACO) [29] are two typical swarm-based algorithms. PSO and ACO mimic the aggregation behavior of bird colonies and the foraging behavior of ant colonies, respectively. Some other algorithms of this category include: grey wolf optimizer (GWO) [30], monarch butterfly optimization (MBO) [31], elephant herding optimization (EHO) [32], moth search algorithm (MSA) [33], manta ray foraging optimization (MRFO) [34],earthworm optimization algorithm (EOA) [35], etc. With the development of metaheuristics, a type of human-based metaheuristic algorithm is also emerging. These algorithms are inspired by the characteristics of human activity. Teaching-learning-based optimization (TLBO) [36], inspired by traditional teaching methods, is a typical example of this category among metaheuristic algorithms. Other human-based metaheuristics include: social evolution and learning optimization (SELO) [37], group teaching optimization algorithm (GTOA) [38], heap-based optimizer (HBO) [39], political optimizer (PO) [40], etc.

There is a common feature of all these metaheuristic algorithms that rely on exploration and exploitation in the search space to find the optimal solution [41, 42]. Exploration means that the algorithm searches for promising regions in a wide search space and exploitation is a further search for the best solution in the promising regions. The balance of the two search behaviors affects the quality of the solution. When exploration dominates, exploitation declines, and vice versa. Therefore, it is a big challenge to balance exploration and exploitation for metaheuristics. Although there are constantly new algorithms being developed, the no free lunch (NFL) [43] theory states that no particular algorithm can solve all optimization problems perfectly. The NFL has motivated researchers to develop effective metaheuristic algorithms to solve various fields of optimization problems.

In this paper, a novel swarm-based metaheuristic is presented called tuna swarm optimization (TSO). It is inspired by two types of swarm foraging behavior of tunas. The TSO is evaluated in 23 benchmark functions and 3 engineering design problems. Test results reveal that the method proposed in this paper significantly outperforms those popular and recent metaheuristics. This paper is structured as follows: Section 2 describes the inspiration for TSO and builds the corresponding mathematical model. A benchmark function set and three engineering design problems are employed to check the performance of TSO in Sections 3 and 4, respectively. Section 5 concludes the overall work and provides an outlook for the future.

2. Tuna Swarm Optimization

2.2. Mathematical Model

In this section, the mathematical model of the proposed algorithm is described in detail.

2.2.1. Initialization

Similar to most swarm-based metaheuristics, TSO starts the process of optimization by generating initial populations at random uniformly in the search space,

()

where  is the i^th initial individual, ub and lb are the upper and lower boundaries of the search space, NP is the number of tuna populations, and rand is a uniformly distributed random vector ranging from 0 to 1.

2.2.2. Spiral Foraging

When sardines, herring, and other small schooling fish encounter predators, the entire school of fish forms a dense formation constantly changing the swimming direction, making it difficult for predators to lock a target. At this time, the tuna group chase the prey by forming a tight spiral formation. Although most of the fish in the school have little sense of direction, when a small group of fish swim firmly in a certain direction, the nearby fish will adjust their direction one after another and finally form a large group with the same goal and start to hunt. In addition to spiraling after their prey, schools of tuna also exchange information with each other. Each tuna follows the previous fish, thus enabling information sharing among neighboring tuna. Based on the above principles, the mathematical formula for the spiral foraging strategy is as follows:

()

()

()

()

()

where  is the i^th individual of the t + 1 iteration,  is the current optimal individual (food), α₁ and α₂ are weight coefficients that control the tendency of individuals to move towards the optimal individual and the previous individual, a is a constant used to determine the extent to which the tuna follow the optimal individual and the previous individual in the initial phase, t denotes the number of current iteration, t_max is the maximum iterations, and b is a random number uniformly distributed between 0 and 1.

When all tuna forage spirally around the food, they have good exploitation ability for the search space around the food. However, when the optimal individual fails to find food, blindly following the optimal individual to forage is not conducive to group foraging. Therefore, we consider generating a random coordinate in the search space as a reference point for spiral search. This facilitates each individual to search a wider space and makes TSO with global exploration ability. The specific mathematical model is described as follows:

()

where  is a randomly generated reference point in the search space.

In particular, metaheuristic algorithms usually perform extensive global exploration in the early stage and then gradually transition to precise local exploitation. Therefore, TSO changes the reference points of spiral foraging from random individuals to optimal individuals as the iteration increases. In summary, the final mathematical model of the spiral foraging strategy is as follows:

()

2.2.3. Parabolic Foraging

In addition to feeding by forming a spiral formation, tunas also form a parabolic cooperative feeding. Tuna forms a parabolic formation with food as a reference point. In addition, tuna hunt for food by searching around themselves. These two approaches are performed simultaneously, with the assumption that the selection probability is 50% for both. The specific mathematical model is described as follows:

()

()

where TF is a random number with a value of 1 or −1.

Tuna hunt cooperatively through two foraging strategies and then find their prey. For the optimization process of TSO, the population is first randomly generated in the search space. In each iteration, each individual randomly chooses one of the two foraging strategies to execute, or chooses to regenerate the position in the search space according to probability z. The value of parameter z will be discussed in the parameter setting simulation experiments. During the entire optimization process, all individuals of TSO are continuously updated and calculated until the end condition is met, and then the optimal individual and the corresponding fitness value are returned. The TSO pseudocode is shown in Algorithm 1. The detailed process of TSO is shown in Figure 1.