**1. Introduction**

As an important branch of computational intelligence, swarm intelligence (SI) [1] provides a competitive solution for dealing with large-scale, nonlinear, and complex problems, and has become an important research direction of artificial intelligence. In the SI model, each individual constitutes an organic whole by simulating the behavior of natural biological groups. Although each individual is very simple, the group shows complex emergen<sup>t</sup> behavior. In particular, it does not require prior knowledge of the problem and has the characteristics of parallelism, so it has significant advantages in dealing with problems that are difficult to solve by traditional optimization algorithms. With the deepening of research, more and more swarm intelligence algorithms have been proposed, such as ant colony optimization algorithm (ACO) [2], particle swarm optimization (PSO) [3], artificial bee colony algorithm (ABC) [4], firefly algorithm (FA) [5], cuckoo algorithm (CA) [6], krill herd algorithm [7], monarch butterfly optimization (MBO) [8], and moth search algorithm [9], etc.

ACO as one of the typical SI is first proposed by Macro Dorigo [2] based on the observation of group behaviors of ants in nature. During the process of food searching, ants will release pheromones in the path when they pass through. Pheromones can be detected by other ants and can affect their further path choices. Generally, the shorter the path is, the more intense the pheromones will be, which means the shortest path will be chosen with the highest probability. The pheromone in other paths will disappear with time. Therefore, given enough time, the optimal path will have the most condensed pheromone. In this way, ants will find the shortest path from their nest to the food source in the end.

ACO has advantages in reasonable robustness, distributed parallel computing, and easy combination with other algorithms. It has been successfully applied in many fields, including traveling salesman problem (TSP) [10,11], satellite control resource scheduling problem [12], knapsack problem [13,14], vehicle routing problem [15,16], and continuous function optimization [17–19]. However, conventional ACO is still far from perfect due to issues like premature convergence and long search time [20].

Many scholars have made substantial contributions to improve ACO, mainly focusing on two perspectives, including model modification and algorithms combination. For example, in the line of model improvement, an ant colony system (ACS) [21] employs a pseudo-random proportional rule, which leads to faster convergence. In ACS, only the pheromone of the optimal path will be increased after each iteration. To prevent premature convergence caused by excessive pheromones concentration in some paths, the max-min ant system (MMAS) modifies AS with three main strategies for pheromone [22], including limitation, maximum initialization, and updating rules. To avoid the early planning of the blind search, an improved ACO algorithm by constructing the unequal allocation initial pheromones is proposed in [23]. Path selecting is based on the pseudo-random rule for state transition. The probability is decided by the number of iterations, and the optimal solution. Introducing a penalty function to the pheromone updating, a novel ACO algorithm is addressed in [24] to improve the solution accuracy.

Considering the other primary kind of modification to the original ACO, algorithm combination, several approaches are proposed as well. A multi-type ant system (MTAS) [25] is proposed combining ACS and MMAS, inheriting advantages from both of these algorithms. Combining particle swarm algorithm (PSO) with ACO, a new ant colony algorithm was proposed in [26] and named PS-ACO. PS-ACO employs pheromones updating rules of ACO and searches mechanisms of PSO simultaneously to keep the trade-off between the exploitation and exploration. A multi-objective evolutionary algorithm via decomposition is combined with ACO, an algorithm, termed MOEA/D-ACO [27], which proposes a series of single-objective optimization problems to solve multi-objective optimization problems. Executing ACO in combination with a genetic algorithm (GA), a new hybrid algorithm is proposed in [28]. Embedding GA into ACO, this method improves ACO in convergence speed and GA in searching ability.

Besides the above primary improvement strategies considering model modification and algorithm combination, approaches based on machine learning are also proposed in recent decades [29]. On the one hand, swarm intelligence can be used to solve the optimization problems in deep learning. In deep neural networks, for example, convolutional neural network (CNN), the optimization of hyperparameters is an NP hard problem. Using the SI method can solve this kind of problem better. PSO, CS, and FA were employed to properly select dropout parameters concerning CNN in [30]. The hybridized algorithm [31] based on original MBO with ABC and FAs was proposed to solve CNN hyperparameters optimization. On the other hand, we can learn from machine learning to improve performance of SI. For example, information feedback models are used to enhance the ability of algorithms [32–34]. In addition, opposition-based learning (OBL) [35], which was first proposed by Tizhoosh, is a famous algorithm. Its main idea is to calculate all the opposite solutions after current iteration, and then optimal solutions are selected among the generated solutions and their opposite solutions for the next round of iteration. OBL has been widely accepted in SI, including ABC [36], differential evolution (DE) [37–39], and PSO [40,41], leading to reasonable performances.

Since opposite solutions to continuous problems are convenient to construct, OBL has been used to solve continuous problems more commonly as above, compared with discrete problems. OBL is combined with ACS and applied to solve the TSP as an example for discrete problems in [42] to acquire the better solution. The solution construction phase and the pheromone updating phase of ACS are the primary foci of this hybrid approach. Besides TSP, the graph coloring problem is also employed as a discrete optimization problem in [43], and an improved DE algorithm based on OBL is proposed, which introduces two different methods of opposition. In [44], a pretreatment step was added in the initial stage when the two-membered evolution strategy was used to solve the total rotation minimization problem. The opposite solutions generated by OBL is compared with the initial solutions randomly generated, and a better solution is selected for the subsequent optimization process.

Inspired by the idea of OBL, in this paper, a series of methods, focusing on the opposite solution construction and the pheromone updating rule, are proposed. Aiming to solve TSP, our proposed methods introduce OBL to ACO and enable ACO no longer limited to the local optimal solutions, avoid premature convergence, and improve its performances.

The rest of this paper is organized as follows. In Section 2, the background knowledge of ACO and OBL are briefly reviewed. In Section 3, the opposition-based extensions to ACO are presented. In Section 4, the effectiveness of the improvement is verified through experiments. Section 5 presents the conclusions of this paper.
