Experimental analysis of heuristic solutions for the moving target traveling salesman problem applied to a moving targets monitoring system

Highlights

• Crowd monitoring mapped as a Moving Target Traveling Salesman Problem instance.

• Design of a solution to address the analyzed problem.

• Experimental evaluation of different heuristics approaches to the analyzed problem.

• Discussions about the pros and cons of each approach.

• Conclusion in favor of Genetic Algorithm compared to the other heuristics.

Abstract

The Traveling Salesman Problem (TSP) is an important problem in computer science which consists in finding a path linking a set of cities so that each of then can be visited once, before the traveler comes back to the starting point. This is highly relevant because several real world problems can be mapped to it. A special case of TSP is the one in which the cities (the points to be visited) are not static as the cities, but mobile, changing their positions as the time passes. This variation is known as Moving Target TSP (MT-TSP). Emerging systems for crowd monitoring and control based on unmanned aerial vehicles (UAVs) can be mapped to this variation of the TSP problem, as a number of persons (targets) in the crowd can be assigned to be monitored by a given number of UAVs, which by their turn divide the targets among them. These target persons have to be visited from time to time, in a similar way to the cities in the traditional TSP. Aiming at finding a suitable solution for this type of crowd monitoring application, and considering the fact that exact solutions are too complex to perform in a reasonable time, this work explores and compares different heuristic methods for the intended solution. The performed experiments showed that the Genetic Algorithms present the best performance in finding acceptable solutions for the problem in restricted time and processing power situations, performing better compared to Ant Colony Optimization and Simulated Annealing Algorithms.

Introduction

The well-known Traveling Salesman Problem (TSP) consists in finding the shortest path to visit a set of cities exactly once and return to the initial point (first visited city). Its specializations and generalizations have been a common and important topic of study for both mathematicians and computer scientists along the years, as many problems can be reduced to the TSP (Rajesh Matai & Mittal, 2010). With its first mentions in literature dating back to the nineteenth century, this routing problem has evolved with the technology around it. Nowadays TSP’s are found as sub-problems of several other more general applications such as transportation infrastructure planning, logistics, PCI (Printed Circuit Board) projects, DNA sequencing, communication routing and even military intervention planning (Rajesh Matai & Mittal, 2010). These are just a small enumeration of the problems that are mapped to and need to deal with TSP in their solutions.

A great number of different approaches for solving the TSP has been proposed during recent decades (Osaba, Yang, Diaz, Lopez-Garcia, Carballedo, 2016, Saenphon, Phimoltares, Lursinsap, 2014, Xing, Chen, Yang, Hou, Shen, Cai, 2008). Most of these methods can be divided into heuristic and exact methods. Exact methods search for the optimal solution and can only optimally solve small problems since their approaches result in exponential computational complexities for large problems. Heuristic methods, however, trade optimality for speed, being satisfied by local optimal solutions presenting good results in solving large problems.

Recently, the usage of new technological assets such as Unmanned Aerial Vehicles (UAVs) and autonomous agents such as robots or autonomous vehicles (including autonomous UAVs) has exponentially increased since cheaper and more complete autonomous platforms have appeared on the market (Keller, 2015). This trend, allied to other technological developments in both software and hardware fields has originated a great number of new applications, some of which need to deal with TSP-like problems. Autonomous cars, for example, need to calculate the best route to drive to their destination given traffic conditions, a common example of a TSP usage in this new scenario.

Among other usages, autonomous agents have become an important tool to law enforcement forces, allowing them to track and follow targets or survey areas or buildings quietly, safely and almost undetectably. Amidst these uses, crowd control and monitoring applications for law enforcement present a set of interesting and complex characteristics that represents a special case of the TSP, Moving Target Traveling Salesman Problem (MT-TSP) (Helvig, Robins, & Zelikovsky, 2003) or Time Dependant Traveling Salesman Problem (TD-TSP).

The MT-TSP is a generalization of the Traveling Salesman Problem (TSP) in which the cost to travel from each city to the next depends on the time spent since the tour has started, or the position of these cities on the tour. An even more special case, and the one that appears constantly in law enforcement applications is the one in which not only the cost to travel between cities changes with the time, but the cities, or targets, in this case, move with a given speed as time passes. An example of such a scenario would be the interception of multiple evading targets on a crowd. Imagining that an electrical autonomous law-enforcement UAV has to intercept each evading target to identify it and do that as to minimize battery consumption, it is easy to recognize that this is a hard task as it is to find the best route to accomplish this multi-target interception mission. This special case of both TSP and TD-TSP is especially challenging because the position of each city (in this case a target) must be recalculated each time a new target is reached on a tour considered as a potential solution for the problem. Moreover, it must be done to all the tours considered as potential solutions, considerably increasing the computational cost to find such solutions.

Considering the presented scenario, this work investigates different heuristic and artificial intelligence methods’ performances on solving the MT-TSP. Moreover, a second contribution presented here is the development of a movement prediction module to be associated with the investigated heuristics, to compose a complete solution. The work reported here is part of a research on law-enforcement usage of autonomous vehicles for crowd control scenarios, and thus focuses its efforts on solving the MT-TSP for a small number of targets that move considerably slower than the traveler or interceptor.

The work is organized as follows: Section 2 presents a deeper discussion on Time Dependant and Moving Target Traveling Salesman Problems and the related works that served as inspiration for this research; Section 3 presents an overview of the meta-heuristic models implemented to solve the MT-TSP; Section 4 explains how the solution and the algorithms were implemented to solve the MT-TSP problem; in Section 5, the results of this proposed solution is presented, the different methods are compared and evaluated; Section 5 provides a detailed discussion on the results presented in the previous section and their possible application; Section 7, in turn, closes the paper presenting the conclusions and future work directions.