Salesman Problem Research Articles

Autonomous Exploration Method of Unmanned Ground Vehicles Based on an Incremental B-Spline Probability Roadmap.

Autonomous exploration in unknown environments is a fundamental problem for the practical application of unmanned ground vehicles (UGVs). However, existing exploration methods face difficulties when directly applied to UGVs due to limited sensory coverage, conservative exploration strategies, inappropriate decision frequencies, and the non-holonomic constraints of wheeled vehicles. In this paper, we present IB-PRM, a hierarchical planning method that combines Incremental B-splines with a probabilistic roadmap, which can support rapid exploration by a UGV in complex unknown environments. We define a new frontier structure that includes both information-gain guidance and a B-spline curve segment with different arrival orientations to satisfy the non-holonomic constraint characteristics of UGVs. We construct and maintain local and global graphs to generate and store filtered frontiers. By jointly solving the Traveling Salesman Problem (TSP) using these frontiers, we obtain the optimal global path traversing feasible frontiers. Finally, we optimize the global path based on the Time Elastic Band (TEB) algorithm to obtain a smooth, continuous, and feasible local trajectory. We conducted comparative experiments with existing advanced exploration methods in simulation environments of different scenarios, and the experimental results demonstrate that our method can effectively improve the efficiency of UGV exploration.

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.

Discrete cutting path problems: a general solution framework and industrial applications

The optimal tool routing for cutting machines, also known as cutting path optimisation is an important problem in production research. This problem is relevant in various manufacturing environments such as aeronautic, automotive, garment and semiconductor industries. In this paper, we introduce a general solution framework for the discrete Cutting Path Problem which includes: (i) the universal approach to reduce numerous settings of this problem to the appropriate auxiliary instances of the well-known Precedence Constrained Generalized Traveling Salesman Problem; (ii) the proposition of efficient solution methods for finding (sub-) optimal solutions. We carry out extensive computational experiments in order to evaluate performance of the proposed framework and the obtained results demonstrate its efficiency for real-life industrial instances.

Proposed theoretical value for TSP constant

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points ([Formula: see text]) asymptotically approaches a constant ([Formula: see text] as [Formula: see text] becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size ([Formula: see text] comprise a normal distribution whose mean approaches [Formula: see text] as [Formula: see text] increases. Moreover, as [Formula: see text] approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of [Formula: see text] is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and [Formula: see text], based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data.

Quantifying and reducing the complexity of multi-line charts as a visual aid in multi-criteria decision-making

AbstractMulti-line charts are commonly used in multi-criteria decision-making (MCDM) to represent multiple data series on the same graph. However, the presence of conflicting criteria or divergent viewpoints introduces the challenge of accurately interpreting these charts, necessitating thoughtful design to improve their comprehensibility. In this paper, we model these multi-line charts as connected perfect matching bipartite graphs. We propose a metric called the Coefficient of Complexity (CoC) to quantify the complexity of these multi-line charts. In order to reduce the visual complexity of these charts, we propose to minimize the CoC by modeling it as an integer linear optimization problem (reminiscent of the traveling salesman problem). We demonstrate our techniques through multiple real-life case studies, wherein multi-line charts serve as data visualization across various MCDM software tools. Additionally, multi-line charts with specific requirements have been optimized using our approach, showcasing the adaptability and efficacy of our technique. We also formulate the radar chart as a specialized form of the multi-line chart, and adapt our technique to improve its comprehensibility. The proposed CoC and its optimization are important contributions to the field of analytics, as a number of methods use multi-line charts for visual aid. Consequently, enhancing their comprehensibility can facilitate the decision-making process and help decision-makers gain insights.

On One Implementation of a Natural Approach to Solving the Traveling Salesman Problem

On One Implementation of a Natural Approach to Solving the Traveling Salesman Problem

Fast approximation of the traveling salesman problem shortest route by rectangular cell clustering pattern to parallelize solving

A method of quickly obtaining an approximate solution to the traveling salesman problem (TSP) is suggested, where a dramatic computational speedup is guaranteed. The initial TSP is broken into open-loop TSPs by using a clustering method. The clustering method is based on either imposing a rectangular lattice on the nodes or dividing the dataset iteratively until the open-loop TSPs become sufficiently small. The open-loop TSPs are independent and so they can be solved in parallel without synchronization, whichever the solver is. Then the open-loop subroutes are assembled into an approximately shortest route of the initial TSP via the shortest connections. The assemblage pattern is a symmetric rectangular closed-loop serpentine. The iterative clustering can use the rectangular assembling approach as well. Alternatively, the iterative clustering can use the centroid TSP assembling approach, which requires solving a supplementary closed-loop TSP whose nodes are centroids of the open-loop-TSP clusters. Based on results of numerical simulation, it is ascertained that both the iterative clustering and rectangular cell clustering pattern are roughly equally accurate, but the latter is way more computationally efficient on squarish datasets. Fast approximation of the TSP shortest route serves as an upper bound. In addition, the route can be studied to find its bottlenecks, which subsequently are separated and another bunch of open-loop TSPs is approximately solved. For big-sized TSPs, where the balance of the accuracy loss and computational time is uncertain, the rectangular cell clustering pattern allows obtaining fast solution approximations on multitudinous non-synchronized parallel processor cores.

Travel Salesman Problem Using Dynamic Programming

In the fields of operations research and computer science, the traveling salesman problem (also known as TSP) is a classic algorithmic issue. One method for solving TSP is dynamic programming, which identifies the optimal path at the lowest possible cost. It starts by determining the shortest path between two points, and then it expands to identify routes to other sites. In order to solve the Travel salesman Problem, Dynamic Programming was used in this study as a a solution technique. In addition to a real-world application from AM-EXPRESS NIGERIA LIMITED, illustrative examples of travel salesman difficulties were taken into consideration and handled utilizing the dynamic programming technique. The finding's outcome showed that the least amount of money needed to provide logistical services is ₦6500, and the shortest possible paths or routes were found.

Coordinated last-mile deliveries with trucks and drones: A comparative study of operational modes

Throughout the recent decade, drone delivery technology has been developed to address problems such as last-mile delivery and recurrent traffic jams in metropolitan regions. To resolve the relatively short travel range and limited capacity issues of drones, researchers have proposed to further enhance the utility of drones by deploying them with ground vehicles (e.g., trucks) in tandem. This new concept leads to a new transportation planning problem, i.e., coordinated delivery of trucks and drones (CDTD). In this paper, we conduct a comparative study of three representative operational models: the flying sidekick traveling salesman problem (FSTSP), the traveling salesman problem with drone (TSP-D), and the parallel drone scheduling traveling salesman problem (PDSTSP). Metrics including delivery efficiency and vehicle utilization rate are evaluated for the three models under various scenarios, to extract insights on which operational model is advantageous for which scenario. We also benchmark the algorithms for CDTD by various metrics including result quality, runtime and scalability, and conduct sensitivity analysis to assess the performance of the system under various system parameters, including drone speed, drone range and customer geographic distribution. Finally, a case study is conducted to demonstrate the superior performance of CDTD model as compared to classical TSP model based on real-world data. Experimental results demonstrate that the FSTSP and TSP-D highlight superior performance over PDSTSP and truck-only TSP when the majority of customers are out of the flight range of drone and the customer demand is clustered, as a result of the synchronization and compensation for limitations between the two types of vehicles.

PEGA: A Privacy-Preserving Genetic Algorithm for Combinatorial Optimization.

Evolutionary algorithms (EAs), such as the genetic algorithm (GA), offer an elegant way to handle combinatorial optimization problems (COPs). However, limited by expertise and resources, most users lack the capability to implement EAs for solving COPs. An intuitive and promising solution is to outsource evolutionary operations to a cloud server, however, it poses privacy concerns. To this end, this article proposes a novel computing paradigm called evolutionary computation as a service (ECaaS), where a cloud server renders evolutionary computation services for users while ensuring their privacy. Following the concept of ECaaS, this article presents privacy-preserving genetic algorithm (PEGA), a privacy-preserving GA designed specifically for COPs. PEGA enables users, regardless of their domain expertise or resource availability, to outsource COPs to the cloud server that holds a competitive GA and approximates the optimal solution while safeguarding privacy. Notably, PEGA features the following characteristics. First, PEGA empowers users without domain expertise or sufficient resources to solve COPs effectively. Second, PEGA protects the privacy of users by preventing the leakage of optimization problem details. Third, PEGA performs comparably to the conventional GA when approximating the optimal solution. To realize its functionality, we implement PEGA falling in a twin-server architecture and evaluate it on two widely known COPs: 1) the traveling Salesman problem (TSP) and 2) the 0/1 knapsack problem (KP). Particularly, we utilize encryption cryptography to protect users' privacy and carefully design a suite of secure computing protocols to support evolutionary operators of GA on encrypted chromosomes. Privacy analysis demonstrates that PEGA successfully preserves the confidentiality of COP contents. Experimental evaluation results on several TSP datasets and KP datasets reveal that PEGA performs equivalently to the conventional GA in approximating the optimal solution.

ASP: Learn a Universal Neural Solver!

Applying machine learning to combinatorial optimization problems has the potential to improve both efficiency and accuracy. However, existing learning-based solvers often struggle with generalization when faced with changes in problem distributions and scales. In this paper, we propose a new approach called ASP: Adaptive Staircase Policy Space Response Oracle to address these generalization issues and learn a universal neural solver. ASP consists of two components: Distributional Exploration, which enhances the solver's ability to handle unknown distributions using Policy Space Response Oracles, and Persistent Scale Adaption, which improves scalability through curriculum learning. We have tested ASP on several challenging COPs, including the traveling salesman problem, the vehicle routing problem, and the prize collecting TSP, as well as the real-world instances from TSPLib and CVRPLib. Our results show that even with the same model size and weak training signal, ASP can help neural solvers explore and adapt to unseen distributions and varying scales, achieving superior performance. In particular, compared with the same neural solvers under a standard training pipeline, ASP produces a remarkable decrease in terms of the optimality gap with 90.9% and 47.43% on generated instances and real-world instances for TSP, and a decrease of 19% and 45.57% for CVRP.

Optimizing UAV Routes: An Implementation and Evaluation of Ant-Colony Optimization Algorithm on Crazyflie Quadcopter for Solving the Traveling Salesman Problem

Optimizing UAV Routes: An Implementation and Evaluation of Ant-Colony Optimization Algorithm on Crazyflie Quadcopter for Solving the Traveling Salesman Problem

A Hierarchical Framework for Solving the Constrained Multiple Depot Traveling Salesman Problem

A Hierarchical Framework for Solving the Constrained Multiple Depot Traveling Salesman Problem

Circulant TSP special cases: Easily-solvable cases and improved approximations

Circulant TSP is an intriguing special case of the Traveling Salesman Problem, whose complexity remains an often-cited open problem. In this note, we present three results: We show that circulant TSP can be efficiently solved whenever the input number of vertices is a prime-squared; we show that the {1,2}-TSP can be easily and efficiently solved when specialized to circulant instances; and we present a substantially-improved approximation factor for finding a minimum-cost Eulerian connected sub-(multi)graph on two-stripe circulant instances.

A Full-Coverage Path Planning Method for an Orchard Mower Based on the Dung Beetle Optimization Algorithm

In order to optimize the operating path of orchard mowers and improve their efficiency, we propose an MI-DBO (multi-strategy improved dung beetle optimization algorithm) to solve the problem of full-coverage path planning for mowers in standardized quadrilateral orchard environments. First, we analyzed the operation scenario of lawn mowers in standardized orchards, transformed the full-coverage path planning problem into a TSP (traveling salesman problem), and mathematically modeled the U-turn and T-turn strategies based on the characteristics of lawn mowers in orchards. Furthermore, in order to overcome the issue of uneven distribution of individual positions in the DBO (dung beetle optimization) algorithm and the tendency to fall into local optimal solutions, we incorporated Bernoulli mapping and the convex lens reverse-learning strategy in the initialization stage of DBO to ensure a uniform distribution of the initial population. During the algorithm iteration stage, we incorporated the Levy flight strategy into the position update formulas of breeding beetles, foraging beetles, and stealing beetles in the DBO algorithm, allowing them to escape from local optimal solutions. Simulation experiments show that for 18 types of orchards with different parameters, MI-DBO can find the mowing machine’s operation paths. Compared with other common swarm intelligence algorithms, MI-DBO has the shortest average path length of 456.36 m and can ensure faster optimization efficiency. Field experiments indicate that the algorithm-optimized paths do not effectively reduce the mowing machine’s missed mowing rate, but the overall missed mowing rate is controlled below 0.8%, allowing for the completion of mowing operations effectively. Compared with other algorithms, MI-DBO has the least time and fuel consumption for operations. Compared to the row-by-row operation method, using paths generated by MI-DBO reduces the operation time by an average of 1193.67 s and the fuel consumption rate by an average of 9.99%. Compared to paths generated by DBO, the operation time is reduced by an average of 314.33 s and the fuel consumption rate by an average of 2.79%.

Combining reinforcement learning algorithm and genetic algorithm to solve the traveling salesman problem

AbstractWith the growing recognition of the unique advantages of reinforcement learning and genetic algorithms in addressing combinatorial optimization problems, this study aims to integrate these two methods to collectively tackle the classic combinatorial optimization challenge of the travelling salesman problem (TSP). The TSP stands as a quintessential combinatorial optimization challenge, tasked with determining the shortest path among designated cities. This paper introduces an innovative approach by amalgamating reinforcement learning's path selection prowess with genetic algorithms' global search strategy, aiming to uncover superior solutions in TSP. Specifically, the experiment employs a dual Q‐learning algorithm within reinforcement learning to identify multiple optimal paths, serving as progenitors for the genetic algorithm to further enhance performance. The paper meticulously outlines the problem modelling process, elucidating TSP instance definitions, descriptions, and precise objective function definitions. Experimental findings underscore the substantial enhancements achievable in TSP optimization through this comprehensive approach, offering a fresh perspective and methodology for tackling combinatorial optimization challenges.

New techniques to improve neighborhood exploration in pareto local search

Pareto Local Search (PLS) is an extension of local search to multiobjective combinatorial optimization problems (MCOPs). It is widely used as a building block in many multiobjective metaheuristics. However, it suffers from poor anytime behavior and long convergence time. In this paper, we focus on improving the neighborhood exploration mechanism to accelerate PLS. In total, we propose three new search strategies, namely, Promising List, Don’t Look Bits and Continuing Search and implement them on the multiobjective Traveling Salesman Problem (mTSP) and the multiobjective Unconstrained Binary Quadratic Programming (mUBQP) problem. In Promising List, features of problem structure are utilized to guide the local search. In Don’t Look Bits, neighborhood moves that are less promising to produce improving solutions are skipped. In Continuing Search, the agent will start from where previous neighborhood exploration stops to avoid repetitive neighborhood moves. The experimental results verify the effectiveness of the three techniques both individually and collaboratively on most of the test instances. Besides, one proposed PLS variant outperforms two state-of-the-art PLS-based algorithms.

Route optimization of battery electric vehicles using dynamic charging on electrified roads

The Electric Road System (ERS) is an innovative technology enabling electric vehicles (EVs) to charge while in motion, offering increased EV range and cost-effective battery solutions. This advancement is particularly valuable for logistics companies seeking to enhance their EV adoption. To provide long-distance delivery routes for heavy-duty EVs through ERSs, we present a first-time attempt to solve the Electric Vehicle Travelling Salesman Problem with Dynamic Charging, where the EV can be charged while driving on the electrified road segments. We formulate this problem by considering path choices between consecutive customers and propose a network reduction heuristic (NRH) to solve large-size instances. We test our model and NRH on grid-like synthetic instances and real-world data developed from the UK road network. Our results demonstrate the effectiveness of the NRH and highlight the potential of ERSs in improving logistics activities and decarbonising road transport. Our experiments revealed that although the typical driving range of most EVs falls short of 400 kilometres, with the widespread deployment of ERS, EVs can traverse distances exceeding 2000 kilometres without requiring recharging stops. Our experiments also revealed that establishing infrastructure on approximately 30 % to 40 % of all roadways is adequate to ensure reliable access for EVs.

Geometric Approaches to Solving the Traveling Salesman Problem

Geometric Approaches to Solving the Traveling Salesman Problem

A hybrid genetic algorithm with type-aware chromosomes for Traveling Salesman Problems with Drone

There are emerging transportation problems known as the Traveling Salesman Problem with Drone (TSPD) and the Flying Sidekick Traveling Salesman Problem (FSTSP) that involve using a drone in conjunction with a truck for package delivery. This study presents a hybrid genetic algorithm for solving TSPD and FSTSP by incorporating local search and dynamic programming. Similar algorithms exist in the literature. Our algorithm, however, considers more sophisticated chromosomes and less computationally complex dynamic programming to enable broader exploration by the genetic algorithm and efficient exploitation through dynamic programming and local search. The key contribution of this paper is the discovery of how decision-making processes for solving TSPD and FSTSP should be divided among the layers of genetic algorithm, dynamic programming, and local search. In particular, our genetic algorithm generates the truck and the drone sequences separately and encodes them in a type-aware chromosome, wherein each customer is assigned to either the truck or the drone. We apply local search to each chromosome, which is decoded by dynamic programming for fitness evaluation. Our new algorithm is shown to outperform existing algorithms on most benchmark instances in both quality and time. Our algorithms found the new best solutions for 538 TSPD instances out of 920 and 74 FSTSP instances out of 132.