**5. Conclusions**

The performances of swarm optimization algorithms based on OBL present advantages when handling problems of continuous optimization. However, there are only a few approaches proposed to solve problems of discrete optimization. The difficulty in opposite solution construction is considered as one top reason. To solve this problem, two different strategies, direction and indirection, of constructing opposite paths are presented individually in this paper. For indirection strategy, other than using the order of cities from the current solution directly, it studies the positions, noted as indices, of the cities rearranged in a circle, and then calculates the opposite indices. While for direction strategy, opposite operations are carried out directly to the cities in each path.

To use the information of the opposite path, three different frameworks of opposite-based ACO, called ACO-Index, ACO-MaxIt, and ACO-Rand, are also proposed. All ants need to ge<sup>t</sup> the increment of pheromone in three improved frameworks. Among three proposed algorithms, ACO-Index employs the strategy of indirection to construct the opposite path and introduces it to pheromone updating. ACO-MaxIt also employs direction strategy to obtain opposite path but only adopts it in the early updating period. Similar to ACO-MaxIt in opposite path construction, ACO-Rand employs this opposite path throughout the stage of pheromone updating. In order to verify the effectiveness of the improvement strategy, AS and PS-ACO are used in three frameworks, respectively. Experiments demonstrate that all three methods, As-Index, As-MaxIt, and AS-Rand, outperform original AS in the cases of small-scale and medium-scale cities while AS-Index performs best when facing large-scale cities. The three improved PS-ACO also showed good performance.

Constructing the opposite path mentioned in this paper is only suitable for symmetric TSP. This is mainly because the path (solution) of the problem is an arrangemen<sup>t</sup> without considering the direction. However, if it is replaced by the asymmetric TSP, this method needs to be modified. In addition, if it is replaced by a more general combinatorial optimization problem, it is necessary to restudy how to construct the opposite solution according to the characteristics of the problem. Therefore, our current method of constructing opposite solution is not universal. This is one of the limitations of this study. At the same time, the improved algorithm requires all ants to participate in pheromone updating in order to use the information of opposite path. However, now many algorithms use the best ant to update pheromone, so the method in this paper will have some limitations when it is extended to more ant colony algorithm. However, we also find that it is effective to apply reverse learning to combinatorial optimization problems. Therefore, we will carry out our future research work from two aspects. On the one hand, we plan to continue to study the construction method of more general opposite solution for combinatorial optimization problems, so as to improve its generality. In addition, it will be applied to practical problems such as path optimization to further expand the scope of application. Meanwhile, applying OBL to more widely used algorithms is also one interesting and promising topic. Therefore, on the other hand, we plan to study more effective use of the reverse solution and extend it to the more wildly used ACO, such as MMAS and ACS, and even some other optimization algorithms such as PSO and ABC, to solve more combinatorial optimization problems more effectively.

**Author Contributions:** Conceptualization, Z.Z.; methodology, Z.Z.and Z.X.; software, Z.X. and X.L.; formal analysis, Z.Z.and Z.X.; resources, Z.Z.; writing—original draft preparation, Z.X.; writing—review and editing, Z.Z.and S.L.; supervision, Z.Z.and Y.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No. 61703256, 61801197), Jiangsu Natural Science Foundation (Grant No. BK20181004), Natural Science Basic Research Plan In Shaanxi Province of China (Program No. 2017JQ6070), and the Fundamental Research Funds for the Central Universities (Grant No. GK201603014, GK201803020).

**Acknowledgments:** The authors are grateful to the anonymous reviewers and the editor for the constructive comments and valuable suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
