# Ant-Balanced Multiple Traveling Salesmen: ACO-BmTSP
## 1 Introduction  
In recent years, intelligent computation technologies have received increasing attention from researchers. Most of these technologies are inspired by natural phenomena and attempt to use the virtues of elements of behavior. A technique for problem solving inspired by the behavior of ants when finding paths from their nest to food sources (ant colony optimization--ACO) gained special importance due to its successful results when applied to optimization problems [1; 2]. The original ACO underwent several modifications in order to adapt it to different real-world problems, such as assignment problems, graph coloring, scheduling, circuit design, communication networks, bioinformatics, vehicle routing, etc. Several ACO extensions have been created since then, such as elitist AS [3], Ant-Q [4], Ant Colony System [5], rank-based AS [6], population-based ACO [7], Beam-ACO [8], etc. Later, the ACO algorithm was combined with other algorithms to create denominated hybrid methods and to obtain better results. Some examples of hybrid methods using ACO are found in [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. When the traveling salesman problem (TSP) is extended to consider multiple salesmen (mTSP), other kinds of algorithms can use this technique to solve more general problems. Some works focused on solving the mTSP are reviewed in the following. [PERSON] et al. [19] proposed a hybrid algorithm, AC2 OptGA, for solving the mTSP. The algorithm used genetic (GA), ant colony, and two-opt algorithms. The ant colony algorithm was used to generate solutions, and the two-opt algorithm and GA were used to enhance the solutions that were obtained. These algorithms used instances for \(n=\{51,100,150\}\) and \(m=\{3,5,10\}\). [PERSON] et al. [20] implemented a GA in a vehicle-routing problem (VRP). The goal was to optimize daily routes for workers assigned to distribution for a company located in a city(Covilha, Portugal). They claimed to reduce the total distance traveled by 618 km per week in comparison with the company runs. They considered \(n=270\) nodes, three zones, and a five-day window. The primary goal was to minimize the cost and distance instead of balancing the tours. The cost-balanced TSP is one of the recent variants of the TSP. Thus, [PERSON] and [PERSON] [21] proposed a solution based on the variable neighborhood search algorithm to solve the cost-balanced TSP. They used twenty-two datasets, which included small-, medium-, and large-scale instances, to obtain computational results. [PERSON] and [PERSON] [22] focused their research on fitness landscape analysis (FLA) for the multiple traveling salesman problem (mTSP) and capacitated VRP (CVRP). They proposed a new FLA measure that provided valuable information for characterizing the fitness landscape's structure under specific scenarios and obtained several relationships between the fitness landscape's structure and algorithmic performance. [PERSON] [23] developed a balanced dynamic closest vehicle heuristic (BD-CVH) and a balanced dynamic assignment vehicle heuristic (BD-AVH) to solve a dynamic routing problem for the balanced mTSP. The obtained distances indicated that, in general, the BD-AVH algorithm led to slightly better solutions. [PERSON] and [PERSON] [24] implemented a hybrid algorithm for the balanced mTSP based on genetic algorithms and the two-opt algorithm. [PERSON] et al. [25] proposed a review of the existing literature devoted to UAV (unmanned aerial vehicle) path optimization problems related to two existing general classic problems, the traveling salesman problem and the vehicle-routing problem. They provided a synthetic overview of the resolution techniques and performance metrics, and they obtained numerical results. [PERSON] et al. [18] proposed a hybrid method, GACO2, for solving the mTSP based on the genetic algorithm, ant colony optimization, and the two-opt algorithm to improve the solution. They applied this method in two instances--mTSP51 and mTSP100--with 3, 5, and 10 travelers.