A comprehensive survey on the Multiple Traveling Salesman Problem: Applications, approaches and taxonomy

doi:10.1016/j.cosrev.2021.100369

Computer Science Review

Volume 40, May 2021, 100369

https://doi.org/10.1016/j.cosrev.2021.100369 Get rights and content

Abstract

The Multiple Traveling Salesman Problem (MTSP) is among the most interesting combinatorial optimization problems because it is widely adopted in real-life applications, including robotics, transportation, networking, etc. Although the importance of this optimization problem, there is no survey dedicated to reviewing recent MTSP contributions. In this paper, we aim to fill this gap by providing a comprehensive review of existing studies on MTSP. In this survey, we focus on MTSP’s recent contributions to both classical vehicles/robots and unmanned aerial vehicles. We highlight the approaches applied to solve the MTSP as well as its application domains. We analyze the MTSP variants and propose a taxonomy and a classification of recent studies.

Section snippets

Introduction and motivation

The Multiple Traveling Salesman Problem (MTSP) is a generalization of the well-known Traveling Salesman Problem (TSP), where multiple salesmen are involved to visit a given number of cities exactly once and return to the initial position with the minimum traveling cost. MTSP is highly related to other optimization problems such as Vehicle Routing Problem (VRP) [1] and Task Assignment problem [2]. Indeed, MTSP is a relaxation of VRP with neither considering the vehicle capacity or customer

MTSP application fields

For years, mobile robots, vehicles, and UAVs, which are aircraft operating without a human pilot on board, have been considered as emerging technologies that have made many complex missions safer and easier. In order to achieve their missions, it is important to determine a path for each vehicle that optimizes a given objective while considering some constraints. MTSP has been adopted in different real-life applications to obtain optimized multiple vehicle routes. The main applications of MTSP

MTSP definition and variants

MTSP is widely studied and was originally defined as which, given a set of cities, one depot, $m$ salesmen and a cost function (e.g. time or distance), MTSP aims to determine a set of routes for $m$ salesmen minimizing the total cost of the $m$ routes, such that, each route starts and ends at the depot and each city is visited exactly once by one salesman.

MTSP has been applied to different application domains, which gave rise to new variants of this optimization problem. In this section, we analyze

MTSP approaches

As its name suggests, MTSP was originally designed to optimize traveling salesman problems. It was then generalized to handle optimization tasks of ground vehicles or robots. However, over recent decades, UAVs, the new flying vehicle, have emerged. UAVs were first used in dangerous military missions to ensure the safety of pilots. Subsequently, these flying vehicles have attracted a great deal of interest in various civilian applications, and their use continues to grow. In the literature,

Taxonomy, classification and analysis

In this section, we first provide an extended taxonomy for MTSP, which is based on the MTSP variants, the applied optimization approaches, and the application domains for which the solution was proposed. After that, the previously reviewed solutions are classified according to this proposed taxonomy. This classification presents an overview of the existing MTSP studies which can help the readers to select the suitable MTSP variant for a given application, as well as the approach used to solve

Discussion and future directions

The previous two sections have been devoted to reviewing contributions proposed for MTSP. Indeed, we have provided an overview of the different approaches proposed in the literature to address MTSP while highlighting the application’s areas.

Even though MTSP is very relevant for real-life applications, we pointed out that several studies have solved the general context MTSP without considering a specific application. However, when the study is within a given context such as parcel delivery, data

Conclusion

The multiple traveling salesman problem is one of the most interesting combinatorial optimization problems due to its ability to describe and formulate real-life applications. Indeed, this survey showed that MTSP is used to formulate optimization problems in several fields, including transportation and delivery, data collection, search and rescue, multi-robot task allocation and scheduling, etc. Although the MTSP importance, there is a lack of a survey that describes existing solutions. This

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the reviewers for their comments. Dr. Omar Cheikhrouhou thanks Taif university for its support under the project Taif University Researchers Supporting Project number (TURSP-2020/55), Taif university, Taif, Saudi arabia .

References (89)

EksiogluB. et al.
The vehicle routing problem: A taxonomic review
Comput. Ind. Eng.
(2009)
LahyaniR. et al.
Rich vehicle routing problems: From a taxonomy to a definition
European J. Oper. Res.
(2015)
BraekersK. et al.
The vehicle routing problem : State of the art classification and review
Comput. Ind. Eng.
(2016)
BektasT.
The multiple traveling salesman problem: an overview of formulations and solution procedures
Omega
(2006)
CoutinhoW.P. et al.
The unmanned aerial vehicle routing and trajectory optimisation problem, a taxonomic review
Comput. Ind. Eng.
(2018)
MurrayC.C. et al.
The multiple flying sidekicks traveling salesman problem: Parcel delivery with multiple drones
Transp. Res. C
(2020)
KitjacharoenchaiP. et al.
Multiple traveling salesman problem with drones: Mathematical model and heuristic approach
Comput. Ind. Eng.
(2019)
HamA.M.
Integrated scheduling of m-truck, m-drone, and m-depot constrained by time-window, drop-pickup, and m-visit using constraint programming
Transp. Res. C
(2018)
WichmannA. et al.
Smooth path construction and adjustment for multiple mobile sinks in wireless sensor networks
Comput. Commun.
(2015)
HuangH. et al.
Viable path planning for data collection robots in a sensing field with obstacles
Comput. Commun.
(2017)

K. Braekers, K. Ramaekers, I. Nieuwenhuyse, The vehicle routing problem: State of the art classification and review,...

OncanT.

A survey of the generalized assignment problem and its applications

Infor

(2007)

MorA. et al.

Vehicle routing problems over time: a survey

4OR

(2020)

MacHaretD.G. et al.

A survey on routing problems and robotic systems

Robotica

(2018)

QamarN. et al.

Comparative analysis of evolutionary algorithms for multi-objective travelling salesman problem

Int. J. Adv. Comput. Sci. Appl.

(2018)

KhoufiI. et al.

A survey of recent extended variants of the traveling salesman and vehicle routing problems for unmanned aerial vehicles

Drones

(2019)

OttoA. et al.

Optimization approaches for civil applications of unmanned aerial vehicles (uavs) or aerial drones: A survey

Networks

(2018)

Amazon primeair, https://www.amazon.com/Amazon-Prime-Air/b?ie=UTF8&node=8037720011, accessed:...

Wing: Transforming the way goods are transported, https://x.company/projects/wing/, accessed:...

LiuX. et al.

Latency-aware path planning for disconnected sensor networks with mobile sinks

IEEE Trans. Ind. Inf.

(2019)

Z. Wei, C. Xia, X. Yuan, R. Sun, Z. Lyu, L. Shi, J. Ji, The path planning scheme for joint charging and data collection...

BarrocaC. et al.

Improving message delivery in uav-based delay tolerant networks

HarounabadiM. et al.

Evolutionary path planning for multiple uavs in message ferry networks applying genetic algorithm

ZhaoW. et al.

A heuristic distributed task allocation method for multivehicle multitask problems and its application to search and rescue scenario

IEEE Trans. Cybern.

(2015)

S. Hayat, E. Yanmaz, T.X. Brown, C. Bettstetter, Multi-objective UAV path planning for search and rescue, in:...

LiX. et al.

A cloud-assisted region monitoring strategy of mobile robot in smart greenhouse

Mob. Inf. Syst.

(2019)

CheikhrouhouO. et al.

A cloud based disaster management system

J. Sensor Actuator Netw.

(2020)

TriguiS. et al.

An analytical hierarchy process-based approach to solve the multi-objective multiple traveling salesman problem

Intell. Serv. Robot.

(2018)

GuX. et al.

Cooperative trajectory planning for multi-ucav using multiple traveling salesman problem

Cited by (158)

Air cargo load and route planning in pickup and delivery operations
2024, Expert Systems with Applications
In the aerial pickup and delivery of goods in a distribution network, transport aviation faces risks of load imbalance due to the urgency required for loading, immediate take-off, and mission accomplishment. Transport planners deal with trip itineraries, prioritization of items, building up pallets, and balanced loading, but there are no commercially available systems that can integrally assist in all these requirements. This enables other risks, such as improper delivery, excessive fuel burn, and possible safety issues due to cargo imbalance, as well as a longer than necessary turn-around time. This NP-hard problem, named Air Cargo Load Planning with Routing, Pickup, and Delivery Problem (ACLP+RPDP), is mathematically modelled using standardized pallets in fixed positions. We developed a strategy to solve this problem, considering historical transport data from some Brazilian hub networks, and performed several experiments with a commercial solver, five known meta-heuristics, and a new heuristic designed specifically for this problem. By using a portable computer, our strategy quickly found practical solutions to a wide range of real problems in much less than operationally acceptable time.
CGKOA: An enhanced Kepler optimization algorithm for multi-domain optimization problems
2024, Computer Methods in Applied Mechanics and Engineering
Kepler Optimization Algorithm (KOA) is a physically based meta-heuristic algorithm inspired by Kepler's laws to simulate planetary motions, KOA shows strong performance on different test sets as well as various optimization problems. However, it also suffers from imbalanced exploration and exploitation, delayed convergence, and insufficient convergence accuracy in dealing with high-dimensional and complex applications. To address these shortcomings, this paper proposes an enhanced Kepler optimization algorithm called CGKOA with stronger performance by combining adaptive function, sinusoidal chaotic gravity, lateral crossover, and elite gold rush strategies. Firstly, the adaptive function and sinusoidal chaotic gravity are adjustments to the internal structure of KOA algorithm, which successfully balances the exploration and exploitation, and increases the population diversity. Secondly, the lateral crossover strategy strengthens the spatial exploration ability of the algorithm, eliminates the poor quality individuals and accelerate the output of high-quality population, and finally, the proposed elite gold rush strategy provides an in-depth and rational exploration of elite groups from multiple perspectives, improves solving accuracy and accelerates the convergence speed. Experimental comparisons of CGKOA with a variety of state-of-the-art and high-performance algorithms on different dimensions of the 2017 and 2020 test sets are conducted, and the experimental results show the superiority and robustness of CGKOA algorithm. In addition, the effectiveness and practicability of CGKOA for real problems are verified by solving 50 complex engineering applications. Last, the algorithm is applied to the difficult problems in the fields of path planning, job-shop scheduling, variant travelers, robot machining trajectory planning, and complex truss topology optimization, and the excellent results obtained by CGKOA demonstrate its applicability and development potential for optimization tasks in various fields. Therefore, CGKOA is an efficient and competitive algorithm for solving complex optimization problems with different dimensions.
Multi-agent collaborative path planning algorithm with reinforcement learning and combined prioritized experience replay in Internet of Things
2024, Computers and Electrical Engineering
As one of the most promising research areas in the field of reinforcement learning, multi-agent collaborative decision-making faces the challenge of real-time control in dynamic environments. This paper focuses on the path planning problem of multi-mobile sink collaborative data collection in the Internet of Things. Based on the excellent collaborative decision-making performance of multi-agent deep deterministic policy gradients (MADDPG), a novel path planning algorithm is proposed in the constructed wireless sensor network model. Furthermore, a combined prioritized experience replay strategy is designed to increase the utilization of important experiences in MADDPG. Extensive experiments conducted in various conditions show that compared with traditional MADDPG and MADDPG with prioritized experience replay, the proposed algorithm demonstrates the best performance. The accelerated convergence speed, enhanced training effectiveness and shortened paths indicate the capability of the proposed algorithm in multi-agent collaborative path planning problem.
A branch-and-price algorithm for integrating urban rail crew scheduling and rostering problems
2024, Transportation Research Part B: Methodological
Crew planning, involving how to best schedule crew members during a given period, is a significant problem for urban rail transit companies. This paper proposes a new integer linear programming (ILP) model that can simultaneously optimize urban rail crew scheduling and rostering problems. The proposed ILP model is a set partitioning-based model with only one type of important duty selection variable that connects the two-level problem and circumvents the drawbacks of conventional approaches that usually formulate the crew scheduling and rostering problems separately and couple these two problems through linking constraints. This study demonstrates that the structure of the underlying network used to model the problem enables the development of an effective, heuristic branch-and-price procedure. The study compares the proposed approach with two other decomposition methods, namely Lagrangian relaxation and alternating direction method of multipliers (ADMM), on problems of different sizes and shows that the method provides lower bounds that are on average 16.4% better than Lagrangian relaxation and 5.03% better than ADMM, respectively. Furthermore, the study shows that, with an average optimality gap of 3.28%, the proposed approach obtains high-quality integer solutions to the integrated problem.
A comprehensive survey on the generalized traveling salesman problem
2024, European Journal of Operational Research
The generalized traveling salesman problem (GTSP) is an extension of the classical traveling salesman problem (TSP) and it is among the most researched combinatorial optimization problems due to its theoretical properties, complexity aspects and real-life applications in various areas: location-routing problems, material flow design problem, distribution of medical supplies, urban waste collection management, airport selection and routing the courier airplanes, image retrieval and ranking, digital garment manufacturing, etc. Even though the importance of this combinatorial optimization problem was highlighted in several publications and there were developed several methods for solving it, there is no survey dedicated to the GTSP. The scope of this paper is to close this gap by providing a comprehensive survey on mathematical formulations, solution approaches and latest advances regarding the GTSP. The paper is organized around the following issues: problem definition, variations and related problems, real-life applications of the GTSP, mathematical formulations, solution approaches designed for solving the investigated problem, datasets, computational results and comparative analysis of the performance of the existing state-of-the-art algorithms. Additionally, we discuss certain open problems and potential research directions.
The AddACO: A bio-inspired modified version of the ant colony optimization algorithm to solve travel salesman problems
2024, Mathematics and Computers in Simulation
The Travel Salesman Problem (TSP) consists in finding the minimal-length closed tour that connects the entire group of nodes of a given graph. We propose to solve such a combinatorial optimization problem with the AddACO algorithm: it is a version of the Ant Colony Optimization method that is characterized by a modified probabilistic law at the basis of the exploratory movement of the artificial insects. In particular, the ant decisional rule is here set to amount in a linear convex combination of competing behavioral stimuli and has therefore an additive form (hence the name of our algorithm), rather than the canonical multiplicative one. The AddACO intends to address two conceptual shortcomings that characterize classical ACO methods: (i) the population of artificial insects is in principle allowed to simultaneously minimize/maximize all migratory guidance cues (which is in implausible from a biological/ecological point of view) and (ii) a given edge of the graph has a null probability to be explored if at least one of the movement trait is therein equal to zero, i.e., regardless the intensity of the others (this in principle reduces the exploratory potential of the ant colony). Three possible variants of our method are then specified: the AddACO-V1, which includes pheromone trail and visibility as insect decisional variables, and the AddACO-V2 and the AddACO-V3, which in turn add random effects and inertia, respectively, to the two classical migratory stimuli. The three versions of our algorithm are tested on benchmark middle-scale TPS instances, in order to assess their performance and to find their optimal parameter setting. The best performing variant is finally applied to large-scale TSPs, compared to the naive Ant-Cycle Ant System, proposed by Dorigo and colleagues, and evaluated in terms of quality of the solutions, computational time, and convergence speed. The aim is in fact to show that the proposed transition probability, as long as its conceptual advantages, is competitive from a performance perspective, i.e., if it does not reduce the exploratory capacity of the ant population w.r.t. the canonical one (at least in the case of selected TSPs). A theoretical study of the asymptotic behavior of the AddACO is given in the appendix of the work, whose conclusive section contains some hints for further improvements of our algorithm, also in the perspective of its application to other optimization problems.

View all citing articles on Scopus

View full text

Review articleA comprehensive survey on the Multiple Traveling Salesman Problem: Applications, approaches and taxonomy

Abstract

Section snippets

Introduction and motivation

MTSP application fields

MTSP definition and variants

MTSP approaches

Taxonomy, classification and analysis

Discussion and future directions

Conclusion

Declaration of Competing Interest

Acknowledgments

Comput. Ind. Eng.

European J. Oper. Res.

Comput. Ind. Eng.

Omega

Comput. Ind. Eng.

Transp. Res. C

Comput. Ind. Eng.

Transp. Res. C

Comput. Commun.

Comput. Commun.

Comput. Netw.

Expert Syst. Appl.

Comput. Electron. Agric.

Procedia Comput. Sci.

Appl. Soft Comput.

Procedia Comput. Sci.

Appl. Soft Comput.

Appl. Soft Comput.

Expert Syst. Appl.

Swarm Evol. Comput.

Expert Syst. Appl.

Appl. Soft Comput.

Math. Model.

Appl. Soft Comput.

Transp. Res. C

A survey of the generalized assignment problem and its applications

Infor

Vehicle routing problems over time: a survey

4OR

A survey on routing problems and robotic systems

Robotica

Comparative analysis of evolutionary algorithms for multi-objective travelling salesman problem

Int. J. Adv. Comput. Sci. Appl.

A survey of recent extended variants of the traveling salesman and vehicle routing problems for unmanned aerial vehicles

Drones

Optimization approaches for civil applications of unmanned aerial vehicles (uavs) or aerial drones: A survey

Networks

Latency-aware path planning for disconnected sensor networks with mobile sinks

IEEE Trans. Ind. Inf.

Improving message delivery in uav-based delay tolerant networks

Evolutionary path planning for multiple uavs in message ferry networks applying genetic algorithm

A heuristic distributed task allocation method for multivehicle multitask problems and its application to search and rescue scenario

IEEE Trans. Cybern.

A cloud-assisted region monitoring strategy of mobile robot in smart greenhouse

Mob. Inf. Syst.

A cloud based disaster management system

J. Sensor Actuator Netw.

An analytical hierarchy process-based approach to solve the multi-objective multiple traveling salesman problem

Intell. Serv. Robot.

Cooperative trajectory planning for multi-ucav using multiple traveling salesman problem

Review article
A comprehensive survey on the Multiple Traveling Salesman Problem: Applications, approaches and taxonomy