Multiple Traveling Salesman Problem Research Articles

Multi-UAV Path Planning for Inspection of Target Points with Stable Monitoring Frequencies

In this study, we focus on the path-planning problem of unmanned aerial vehicles (UAVs) deployed for inspection missions at target points. The goal is to visit each target point, provide revisits to important target points, and ultimately meet the monitoring requirements with regular and stable monitoring frequencies. Herein, we present MTSP-R, a novel variant of the multiple traveling salesmen problem (MTSP), in which revisits to important target points are allowed. We address the path-planning problem of multi-UAV in two stages. First, we propose a nearest insertion algorithm with revisits (NIA-R) to determine the number of required UAVs and initial inspection paths. We then propose an improved genetic algorithm (IGA) with two-part chromosome encoding to further optimize the inspection paths of the UAVs. The simulation results demonstrate that the IGA can effectively overcome the shortcomings of the original genetic algorithm, providing shorter paths for multiple UAVs and more stable monitoring frequencies for the target points.

Mission-oriented dynamic reconfiguration of airborne photovoltaic array based on multidisciplinary model

The solar-powered stratospheric airship is demonstrated overwhelming superiority because of affordable cost, recyclable material and low carbon emission. Progress has been made in the schemes of energy system consisting of curved photovoltaic array, storage, and energy management devices. However, research on various flight conditions and the resulting presence of non-uniform irradiance on the airborne solar array is rare. Moreover, the methods to reduce the power losses of on-board solar array are not investigated. This paper proposed a dynamic reconfiguration method to improve the output performance of on-board solar array in different mission conditions. The partial shading conditions of airborne and ground-based solar arrays are compared. The multiple traveling salesman problem with solar array is introduced. The Simulated annealing algorithm is adopted. The real-time optimization of solar array is obtained with the dynamic reconfiguration method according to the operating condition consisting of various parameters. The results show the proposed method effectively improves the output power of solar array compared with common total-cross-tied connections. The improvements of daily output energy of the airborne array are more than 13.7 %. The work provides a conducive reference for realizing the utilization prospect of solar energy system on aerial vehicles in a low carbon future.

Visual Reasoning and Multi-Agent Approach in Multimodal Large Language Models (MLLMs): Solving TSP and mTSP Combinatorial Challenges

Multimodal Large Language Models (MLLMs) harness comprehensive knowledge spanning text, images, and audio to adeptly tackle complex problems. This study explores the ability of MLLMs in visually solving the Traveling Salesman Problem (TSP) and Multiple Traveling Salesman Problem (mTSP) using images that portray point distributions on a two-dimensional plane. We introduce a novel approach employing multiple specialized agents within the MLLM framework, each dedicated to optimizing solutions for these combinatorial challenges. We benchmarked our multi-agent model solutions against the Google OR tools, which served as the baseline for comparison. The results demonstrated that both multi-agent models—Multi-Agent 1, which includes the initializer, critic, and scorer agents, and Multi-Agent 2, which comprises only the initializer and critic agents—significantly improved the solution quality for TSP and mTSP problems. Multi-Agent 1 excelled in environments requiring detailed route refinement and evaluation, providing a robust framework for sophisticated optimizations. In contrast, Multi-Agent 2, focusing on iterative refinements by the initializer and critic, proved effective for rapid decision-making scenarios. These experiments yield promising outcomes, showcasing the robust visual reasoning capabilities of MLLMs in addressing diverse combinatorial problems. The findings underscore the potential of MLLMs as powerful tools in computational optimization, offering insights that could inspire further advancements in this promising field.

Cumulative Capacitated Colored Traveling Salesman Problem.

A colored traveling salesman problem (CTSP) is a generalization of the well-known multiple traveling salesman problem, which introduces colors to distinguish the accessibility of its cities to salesmen. This work proposes a city/customer-centric model called cumulative capacitated CTSP (C2-CTSP) to tackle some practical problems with fast response requirements. Its hypergraph and mathematical programming formulations are developed for the first time. A general variable neighborhood search (GVNS) metaheuristic is designed to solve it. Specifically, greedy backtracking is proposed to initialize a solution taking into account the cumulative cost and two constraints including colors and capacities. Next, 2-swap, reinsertion, and double-bridge operations are randomly selected and carried out to execute the perturbation. Moreover, neighborhood-list-2-opt, relocation move, and generalized partition crossover are organized as variable neighborhood descent to constitute the local search for better solutions. Extensive experiments are conducted to compare the proposed GVNS with four genetic algorithms, two hybrid ant colony systems, two variable neighborhood search methods, and a perturb-based local search in 20 regular and random cases. The statistical results demonstrate that GVNS is superior to all competitors tuned by irace package in terms of both search ability and convergence rate. In addition, the study of six GVNS variants lacking different operators validates the significant role of each corresponding operator in GVNS's outstanding performance.

Enhancing the multiple traveling salesman problem solutions through Harris Hawks optimization and machine learning techniques

Enhancing the multiple traveling salesman problem solutions through Harris Hawks optimization and machine learning techniques

An ALNS metaheuristic for the family multiple traveling salesman problem

This paper presents a novel variant of the Family Traveling Salesman Problem (FTSP), a well-known NP-hard problem introduced by Morán-Mirabal et al. (2014). In the FTSP, the set of nodes is partitioned into several subsets called families, and the aim is to determine the minimum-cost cycle that begins and ends at the depot and visits a predefined number of nodes in each family. We extend the FTSP by requiring that m tours be generated, each having a number of nodes between a minimum and a maximum quantity, thus yielding the family multiple traveling salesman problem (FmTSP). We present a two-index MIP formulation and develop an ALNS metaheuristic as a solution method for large-sized instances. Our proposed ALNS was initially employed to solve the FTSP benchmark instances and allowed us to improve some of the best-known solutions. The results for new derived instances requiring multiple tours show that our ALNS also achieves optimality for all instances with proven-optimal solutions.

Online task allocation and scheduling in multi-manipulator system considering collision constraints and unknown tasks

Compared to a single robot, multi-robot systems (MRS) offer several advantages in complex multi-task scenarios. The overall efficiency of MRS relies heavily on an efficient task allocation and scheduling process. Multi-robot task allocation (MRTA) is often formulated as a multiple traveling salesman problem, which is NP-hard and typically addressed offline. This paper specifically addresses the online allocation problem in multi-manipulator systems within multi-task scenarios. The tasks are initially pre-allocated to alleviate the computational burden of online allocation. Subsequently, considering collision constraints, we search for the current feasible set of manipulators and employ greedy algorithms to achieve local optima as the online allocation result within this set. Our method can handle the online addition of new, unknown tasks to the task list. Moreover, we demonstrate the feasibility of our approach through simulations and on a realistic platform, where multiple manipulators are tasked with polishing the white body of automobile parts. The results demonstrate that our method is effective and efficient for online allocation and scheduling scenarios.

Optimization of offshore wind farm inspection paths based on K-means-GA.

As global demand for offshore wind energy continues to rise, the imperative to enhance the profitability of wind power projects and reduce their operational costs becomes increasingly urgent. This study proposes an innovative approach to optimize the inspection routes of offshore wind farms, which integrates the K-means clustering algorithm and genetic algorithm (GA). In this paper, the inspection route planning problem is formulated as a multiple traveling salesman problem (mTSP), and the advantages of the K-means clustering algorithm in distance similarity are utilized to effectively group the positions of wind turbines, thereby optimizing the inspection schedule for vessels. Subsequently, by harnessing the powerful optimization capability and robustness of genetic algorithms, further refinement is conducted to search for the optimal inspection routes, aiming to achieve cost reduction objectives. The results of simulation experiments demonstrate the effectiveness of this integrated approach. Compared to traditional genetic algorithms, the inspection route length has been significantly reduced, from 93 kilometers to 79.36 kilometers. Simultaneously, operational costs have also experienced a notable decrease, dropping from 141,500 Chinese Yuan to 125,600 Chinese Yuan.

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.