Multiple Traveling Salesman Problem Research Articles

A Cooperative Scheduling Based on Deep Reinforcement Learning for Multi-Agricultural Machines in Emergencies

Effective scheduling of multiple agricultural machines in emergencies can reduce crop losses to a great extent. In this paper, cooperative scheduling based on deep reinforcement learning for multi-agricultural machines with deadlines is designed to minimize makespan. With the asymmetric transfer paths among farmlands, the problem of agricultural machinery scheduling under emergencies is modeled as an asymmetric multiple traveling salesman problem with time windows (AMTSPTW). With the popular encoder-decoder structure, heterogeneous feature fusion attention is designed in the encoder to integrate time windows and asymmetric transfer paths for more comprehensive and better feature extraction. Meanwhile, a path segmentation mask mechanism in the decoder is proposed to divide solutions efficiently by adding virtual depots to assign work to each agricultural machinery. Experimental results show that our proposal outperforms existing modified baselines for the studied problem. Especially, the measurements of computation ratio and makespan are improved by 26.7% and 21.9% on average, respectively. The computation time of our proposed strategy has a significant improvement over these comparisons. Meanwhile, our strategy has a better generalization for larger problems.

BiGNN: Bipartite graph neural network with attention mechanism for solving multiple traveling salesman problems in urban logistics

The multiple traveling salesman problems (MTSP), which arise from real world problems, are essential in urban logistics. Variations such as MinMax-MTSP and Bounded-MTSP aim to distribute workload evenly among salesmen and impose constraints on visited cities, respectively. Branch-and-bound (B&B) provides an exact algorithm solution for these problems. The Learn to Branch (L2B) approach guides branch node selection through deep learning. In this study, we utilize mathematical modeling of Bipartite Graph Neural Network (BiGNN) and an attention mechanism to support B&B in exploring solution spaces through imitation learning. The problems are framed to formulate mixed integer linear programming, which is different from conventional algorithms. It is proposed that a bipartite graph network approach makes a feature representation by setting a structure of constraints and variables. Experimental results showed that our model can generate more accurate solutions than three benchmark models. The BiGNN model can effectively learn the strong branch strategy, which reduces solution time by replacing complex calculations with fast approximations. Additionally, the small-scale instances model can be applied to larger-scale ones.

How to achieve fair and efficient cooperative vehicle routing?

A cooperative is a business entity with the primary objective of providing benefits, services, and goods to its members, who both own and exercise democratic control over it. In the context of a cooperative, a fleet typically consists of vehicles owned by self-concerned individually rational owners who prioritize their own efficiency and the fairness of the system. This fairness refers to how their individual gain aligns with the gain of others. In this paper, we focus on the routing of such cooperative fleets. Considering only the fleet’s efficiency in terms of minimising its overall cost, the studied problem corresponds to the multiple Traveling Salesman Problem (mTSP). However, our interest lies in finding both efficient and fair solutions, so we propose two new variants of this problem that integrate and maximise the fleet’s egalitarian and elitist social welfare. Additionally, to enhance the balance between fleet efficiency and fairness, we propose the systematic elitist and systematic egalitarian social welfare optimisation algorithm. Through simulation results, we observe a wide diversity of routes depending on the approach considered. Therefore, a cooperative may choose a model that best balances its fleet’s efficiency and fairness based on its specific requirements.

Grid Operation and Inspection Resource Scheduling Based on an Adaptive Genetic Algorithm

Grid operation and inspection a key links to ensure the safe operation of the power system, which requires efficient task allocation and resource scheduling. To address this problem, this paper proposes a resource scheduling model for grid operation and inspection based on bi-level programming. Firstly, the O&I process is analyzed and defined as a combined optimization problem of the multiple traveling salesman problem (MTSP) and the job-shop scheduling problem (JSP). Secondly, a bi-level programming model of MTSP and JSP is established according to the characteristics of the problem. Finally, an adaptive genetic algorithm is used to solve the problem. The feasibility of the model and the advancement of the algorithm are verified through the simulation of real scenarios and a large number of tests, which provide strong support for the sustainable development of the power system.

Sustainable multiple travelling salesman problem solved by reference point aspiration level based multi objective quasi oppositional Jaya algorithm under uncertain environment

Sustainable multiple travelling salesman problem solved by reference point aspiration level based multi objective quasi oppositional Jaya algorithm under uncertain environment

Solving the single depot open close multiple travelling salesman problem through a multi-chromosome based genetic algorithm

The multiple travelling salesman problem (MTSP) extends the classical travelling salesman problem (TSP) by involving multiple salesman in the solution. MTSP has found widespread applications in various domains, such as transportation, robotics, and networking. Despite extensive research on MTSP and its variants, there has been limited attention given to the open close multiple travelling salesman problem (OCMTSP) and its variants in the literature. To the best of the author's knowledge, only one study has addressed OCMTSP, introducing an exact algorithm designed for optimal solutions. However, the efficiency of this existing algorithm diminishes for larger instances due to computational complexity. Therefore, there is a crucial need for a high-level metaheuristic to provide optimal/best solutions within a reasonable timeframe. Addressing this gap, this study proposes a first meta-heuristic called multi-chromosome-based Genetic Algorithm (GA) for solving OCMTSP. The effectiveness of the developed algorithm is demonstrated through a comparative study on distinct asymmetric benchmark instances sourced from the TSPLIB dataset. Additionally, results from comprehensive experiments conducted on 90 OCMTSP symmetric instances, generated from the renowned TSPLIB benchmark dataset, highlight the efficiency of the proposed GA in addressing the OCMTSP. Notably, the proposed multi-chromosome-based GA stands out as the top-performing approach in terms of overall performance. Further, solutions to symmetric TSPLIB benchmark instances are also reported, which will be used as a basis for future studies.

Single Depot Multiple Travelling Salesman Problem Solved With Preference-Based Stepping Ahead Firefly Algorithm

Single Depot Multiple Travelling Salesman Problem Solved With Preference-Based Stepping Ahead Firefly Algorithm

A comparative study between SA and GA in solving MTSP

The multiple traveling salesmen problems (MTSP) is a combinatorial optimization and np-hard problem. In practice, the computational resource required to solve such problems is usually prohibitive, and, in most cases, using heuristic algorithms is the only practical option. This paper implements genetic algorithms (GA) and simulated annealing (SA) to solve the MTSP and does an experimental study based on a benchmark from the TSPLIB instance to compare the performance of two algorithms in reality. The results show that GA can achieve an acceptable solution in a shorter time for any of the MTSP cases and is more accurate when the data size is small. Meanwhile, SA is more robust and achieves a better solution than GA for complex MTSP cases, but it takes more time to converge. Therefore, the result indicates that it is hard to identify which algorithm is comprehensively superior to the other one. However, It also provides an essential reference to developers who want to choose algorithms to solve MTSP in real life, facilitating them to balance the algorithms performance on different metrics they value.

K-DSA for the Multiple Traveling Salesman Problem

Aimed at a multiple traveling salesman problem (MTSP) with multiple depots and closed paths, this paper proposes a k-means clustering donkey and a smuggler algorithm (K-DSA). The algorithm first uses the k-means clustering method to divide all cities into several categories based on the center of various samples; the large-scale MTSP is divided into multiple separate traveling salesman problems (TSPs), and the TSP is solved through the DSA. The proposed algorithm adopts a solution strategy of clustering first and then carrying out, which can not only greatly reduce the search space of the algorithm but also make the search space more fully explored so that the optimal solution of the problem can be more quickly obtained. The experimental results from solving several test cases in the TSPLIB database show that compared with other related intelligent algorithms, the K-DSA has good solving performance and computational efficiency in MTSPs of different scales, especially with large-scale MTSP and when the convergence speed is faster; thus, the advantages of this algorithm are more obvious compared to other algorithms.

DAO2: Overcoming Overall Storage Overflow in Intermittently Connected Sensor Networks

Many emerging sensor network applications operate in challenging environments wherein the base station is unavailable. Data generated from such intermittently connected sensor networks (ICSNs) must be stored inside the network for some unpredictable time before uploading opportunities become available. Consequently, sensory data could overflow the limited storage capacity available in the entire network, making discarding valuable data inevitable. To overcome such overall storage overflow in ICSNs, we propose and study a new algorithmic framework called data aggregation for overall storage overflow ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{DAO}^2$</tex-math> </inline-formula> ). Utilizing spatial data correlation that commonly exists among sensory data, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{DAO}^2$</tex-math> </inline-formula> employs data aggregation techniques to reduce the overflow data size while minimizing the total energy consumption in data aggregation. At the core of our framework are two new graph theoretical problems that have not been studied. We refer to them as traveling salesmen placement problem ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{TSP}^2$</tex-math> </inline-formula> ) and quota traveling salesmen placement problem (Q- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{TSP}^2$</tex-math> </inline-formula> ). Different from the well-known multiple traveling salesman problem (mTSP) and its variants, which mainly focus on the routing of multiple salesmen initially located at fixed locations, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{TSP}^2$</tex-math> </inline-formula> and Q- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{TSP}^2$</tex-math> </inline-formula> must decide the placement as well as the routing of the traveling salesmen. We prove that both problems are NP-hard and design approximation, heuristic, and distributed algorithms. Our algorithms outperform the state-of-the-art data aggregation work with base stations by up to 71.8% in energy consumption.

A hybrid genetic algorithm for the min–max Multiple Traveling Salesman Problem

This paper proposes a hybrid genetic algorithm for solving the Multiple Traveling Salesman Problem (mTSP) to minimize the length of the longest tour. The genetic algorithm utilizes a TSP sequence as the representation of each individual, and a dynamic programming algorithm is employed to evaluate the individual and find the optimal mTSP solution for the given sequence of cities. A novel crossover operator is designed to combine similar tours from two parents and offers great diversity for the population. For some of the generated offspring, we detect and remove intersections between tours to obtain a solution with no intersections. This is particularly useful for the min–max mTSP. The generated offspring are also improved by a self-adaptive random local search and a thorough neighborhood search. Our algorithm outperforms all existing algorithms on average, with similar cutoff time thresholds, when tested against multiple benchmark sets found in the literature. Additionally, we improve the best-known solutions for 21 out of 89 instances on four benchmark sets.

Multi-UAV Data Collection and Path Planning Method for Large-Scale Terminal Access.

In the context of the relentless evolution of network and communication technologies, the need for enhanced communication content and quality continues to escalate. Addressing the demands of data collection from the abundance of terminals within Internet of Things (IoT) scenarios, this paper presents an advanced approach to multi-Unmanned Aerial Vehicle (UAV) data collection and path planning tailored for extensive terminal accessibility. This paper focuses on optimizing the complex interplay between task completion time and task volume equilibrium. To this end, a novel strategy is devised that integrates sensor area partitioning and flight trajectory planning for multiple UAVs, forming an optimization framework geared towards minimizing task completion duration. The core idea of this work involves designing an innovative k-means algorithm capable of balancing data quantities within each cluster, thereby achieving balanced sensor node partitioning based on data volume. Then, the UAV flight trajectory paths are discretely modeled, and a grouped, improved genetic algorithm is used to solve the Multiple Traveling Salesman Problem (MTSP). The algorithm introduces a 2-opt optimization operator to improve the computational efficiency of the genetic algorithm. Empirical validation through comprehensive simulations clearly underscores the efficacy of the proposed approach. In particular, the method demonstrates a remarkable capacity to rectify the historical issue of diverse task volumes among multiple UAVs, all the while significantly reducing task completion times. Moreover, its convergence rate substantially outperforms that of the conventional genetic algorithm, attesting to its computational efficiency. This paper contributes an innovative and efficient paradigm to improve the problem of data collection from IoT terminals through the use of multiple UAVs. As a result, it not only augments the efficiency and balance of task distribution but also showcases the potential of tailored algorithm solutions for realizing optimal outcomes in complex engineering scenarios.

Three-Dimensional Multi-UAV Trajectory Design for Cooperative Video Inspection and Uploading

This paper studies the multiple unmanned aerial vehicles (UAVs) trajectory design for post-disaster cooperative video inspection and information uploading. The locations to be inspected are modeled as points of interest (PoIs) with heterogeneous resolution requirements. The three-dimensional (3D) trajectories for multiple cooperative UAVs are designed to minimize the time for successfully inspecting all the PoIs and maximize the communication time for the video uploading. An UAV inspection model is built to clarify the criteria for successful inspection, which is related to the UAV's 3D positions and velocity. The proposed UAV trajectory design involves two steps. First, the PoIs in proximity are grouped using convex optimization. Then, the multiple UAV trajectories are jointly designed by solving a multiple traveling salesman problem (MTSP). Numerical results show that the proposed design can significantly reduce the mission completion time.

A Novel Multi-Robot Task Assignment Scheme Based on a Multi-Angle K-Means Clustering Algorithm and a Two-Stage Load-Balancing Strategy

A practical task assignment is one of the core issues of a multi-robot system. In this paper, a multi-robot task assignment strategy based on load balancing is proposed to effectively balance and plan out the execution cost of each robot when it has a large number of working task points. Considering the variability of the execution task cost in practical situations with different task point categories, the multi-robot task assignment (MRTA) problem is transformed into a multiple traveling salesman problem (MTSP) using a multi-angle K-means clustering algorithm. To solve the problem of unbalanced cost consumed by each robot after clustering assignment, which leads to low efficiency of system completion, a two-stage load-balancing strategy is presented. The first stage of this strategy makes a large adjustment to the unbalanced task set, and the second stage achieves a small fine-tuning to the unbalanced task set. The experimental results show that the standard deviation of the cost ratio of each set decreases when four robots perform the task between 100 and 550 work points using the load-balancing strategy. The reduction in standard deviation is between 3.53% and 83.44%. The maximum cost of individual robots decreases between 0.18% and 14.27%. The proposed method can effectively solve the uneven execution cost of each robot in the task assignment process and improve the efficiency of the system in completing tasks.

Spherical vector-based artificial gorilla troops optimization for spherical asymmetric multiple traveling salesman problem

Spherical vector-based artificial gorilla troops optimization for spherical asymmetric multiple traveling salesman problem

Heuristics and Learning Models for Dubins MinMax Traveling Salesman Problem.

This paper addresses a MinMax variant of the Dubins multiple traveling salesman problem (mTSP). This routing problem arises naturally in mission planning applications involving fixed-wing unmanned vehicles and ground robots. We first formulate the routing problem, referred to as the one-in-a-set Dubins mTSP problem (MD-GmTSP), as a mixed-integer linear program (MILP). We then develop heuristic-based search methods for the MD-GmTSP using tour construction algorithms to generate initial feasible solutions relatively fast and then improve on these solutions using variants of the variable neighborhood search (VNS) metaheuristic. Finally, we also explore a graph neural network to implicitly learn policies for the MD-GmTSP using a learning-based approach; specifically, we employ an S-sample batch reinforcement learning method on a shared graph neural network architecture and distributed policy networks to solve the MD-GMTSP. All the proposed algorithms are implemented on modified TSPLIB instances, and the performance of all the proposed algorithms is corroborated. The results show that learning based approaches work well for smaller sized instances, while the VNS based heuristics find the best solutions for larger instances.

Planning of UAV Flight Routes in the Problems of Group Patrolling of the Extended Territories

Currently one of the promising areas of joint use of unmanned aerial vehicles (UAVs) is group air patrolling of large territories. Here the organization of patrolling assumes the solving the planning problem of routes flight of UAV group. The paper considers the problem of optimal planning of flight routes of the same type of UAVs during group patrolling of large territories. The territorial waters or narrow border areas of any State may serve as an example of such territories. It is suggested that the patrolled area has an elongated shape and is divided into a chain of adjacent patrol zones prescribed by a separate UAV. The drone’s flight route passes through adjacent zones. The flight task performed periodically by each drone consists in moving it to a given flight zone, collecting operational data and transmitting this data to a control point (center, station). The optimization aspect of UAV flight route planning is to minimize the maximum time required to complete flight tasks. The considered problem of group patrolling reduced to the multiple traveling salesman problem — one of the classic intractable combinatorial optimization problems. A brief analysis of modern methods for solving the multiple traveling salesman problem is given. Due to the lack of effective exact methods for solving this problem, it is natural to use approximate heuristic and metaheuristic methods focused on solving NP-hard optimization problems, reducing the full search and giving a solution close to the exact one. The multiple traveling salesman problem considered in this paper is reduced to the problem of integer linear programming, for the solution of which a genetic algorithm implemented in MATLAB based on the mathematical package Global Optimization Toolbox is proposed. An illustrative example of patrolling by three UAVs of an extended territory with 11 adjacent zones is considered. Computational experiments confirm the effectiveness of the algorithmic solutions proposed in the work.

Single Objective Mayfly Algorithm with Balancing Parameter for Multiple Traveling Salesman Problem

The Multiple Travelling Salesman Problem (MTSP) is a challenging combinatorial problem that involves multiple salesman visiting a set of cities, each exactly once, starting and ending at the same depot. The aim is to determine the optimal route with minimal cost and node cuts for each salesman while ensuring that at least one salesman visits each city. As the problem is NP-Hard, a single-objective metaheuristic algorithm, called the Mayfly Algorithm, inspired by the collective behavior of mayflies, is employed to solve the problem using the TSPlib95 test data. Since the Mayfly Algorithm employs a single fitness function, a balancing parameter is added to perform multiobjective optimization. Three balancing parameters in the optimization process: SumRoute represents the total cost of all salesmen travelling, StdRoute balances each salesman cost, and StdNodes balances the number of nodes for each salesman. The values of these parameters are determined based on the results of various tests, as they significantly impact the MTSP optimization process. With the appropriate parameter values, the single-objective Mayfly Algorithm can produce optimal solutions and avoid premature convergence. Overall, the Mayfly Algorithm shows promise as a practical approach to solving the MTSP problem. Using multiobjective optimization with balancing parameters enables the algorithm to achieve optimal results and avoid convergence issues. The TSPlib95 dataset provides a robust testing ground for evaluating the algorithm’s effectiveness, demonstrating its ability to solve MTSP effectively with multiple salesman.

Global path planning and waypoint following for heterogeneous unmanned surface vehicles assisting inland water monitoring

The idea of dispatching multiple unmanned surface vehicles (USVs) to undertake marine missions has ignited a burgeoning enthusiasm on a global scale. Embarking on a quest to facilitate inland water monitoring, this paper presents a systematical approach concerning global path planning and path following for heterogeneous USVs. Specifically, by capturing the heterogeneous nature, an extended multiple travelling salesman problem (EMTSP) model, which seamlessly bridges the gap between various disparate constraints and optimization objectives, is formulated for the first time. Then, a novel Greedy Partheno Genetic Algorithm (GPGA) is devised to consistently address the problem from two aspects: (1) Incorporating the greedy randomized initialization and local exploration strategy, GPGA merits strong global and local searching ability, providing high-quality solutions for EMTSP. (2) A novel mutation strategy which not only inherits all advantages of PGA but also maintains the best individual in the offspring is devised, contributing to the local escaping efficiently. Finally, to track the waypoint permutations generated by GPGA, control input is generated by the nonlinear model predictive controller (NMPC), ensuring the USV corresponds with the reference path and smoothen the motion under constrained dynamics. Simulations and comparisons in various scenarios demonstrated the effectiveness and superiority of the proposed scheme.

A genetic algorithm improvement by tour constraint violation penalty discount for maritime cargo delivery

The problem of minimizing the cost of maritime cargo delivery is considered. The cost is equivalent to the sum of the tour lengths of feeders used for the delivery. The problem is formulated as a multiple traveling salesman problem. In order to find its solution as the shortest route of the tours of feeders, a genetic algorithm is used where we present two inequalities constraining the tour length of every feeder to lie between the shortest and longest lengths. Apart from the constant tour constraint violation penalty in the genetic algorithm, we suggest a changeable penalty as an exponential function of the algorithm iteration, where we maintain the possibility of the penalty rate to be either increasing or decreasing, whose steepness is controlled by a positive parameter. Our tests show that the changeable penalty algorithm may return shorter routes, although the constant penalty algorithms cannot be neglected. As the longest possible tour of the feeder is shortened, the changeable penalty becomes more useful owing to a penalty discount required either at the beginning or at the end of the algorithm run to improve the selectivity of the best feeder tours. In optimizing maritime cargo delivery, we propose to run the genetic algorithm by the low and constant penalties along with the increasing and decreasing penalties. The solution is the minimal value of the four route lengths. In addition, we recommend that four algorithm versions be initialized by four different pseudorandom number generator states. The expected gain is a few percent, by which the route length is shortened, but it substantially reduces expenses for maritime cargo delivery.