Asymmetric Traveling Salesman Problem Research Articles

Q-learning based scheduling method for continuous pickling process of titanium strips

This article addresses the energy consumption optimization problems of the pickling process for titanium strip manufacturing. The hybrid flow shop scheduling schemes for the pickling process of titanium strips are designed, and a novel shop scheduling method based on reinforcement learning is proposed for the pickling process of titanium strips. In the scheduling scheme, the pickling chemical treatment process of titanium strips are described as an asymmetric hybrid flow shop scheduling problem (AHFSP), and a mathematical model containing a temperature structure is established with the optimization objectives of minimizing pickling time and energy consumption. Based on the proposed scheduling scheme, a novel shop scheduling method based on reinforcement learning for the titanium strip pickling process is proposed. First, a mixed integer linear programing model for the mixed flow shop scheduling problem is established. Second, the flow shop scheduling problem with sequential energy consumption decisions is approximated as an asymmetric traveling sales-man problem (ATSP). Finally, the ATSP is described as a Markov decision processes (MDP), and a Q-learning based scheduling method for titanium strip pickling shops is proposed. Finally, the effectiveness of the proposed method is verified by examples, and the scheduling scheme can reduce the energy consumption by 16.61% on average while maintaining the schedule, which improves the productivity and economic efficiency.

Autonomous Exploration Method for Fast Unknown Environment Mapping by Using UAV Equipped With Limited FOV Sensor

Autonomous exploration is crucial and highly challenging for various intelligent operations of unmanned aerial vehicles (UAVs). However, low-efficiency problems caused by low-quality trajectory or back-and-forth maneuvers (BFM) still exist. To improve the exploration efficiency in unknown environments, a fast autonomous exploration planner (FAEP) is proposed in this paper. We firstly design a novel frontiers exploration sequence generation method to significantly reduce BFM during the exploration by considering multi-level factors in asymmetric traveling salesman problem (ATSP). Then, according to the exploration sequence and the distribution of frontiers, an adaptive yaw planning method is proposed to cover more frontiers by yaw change during an exploration journey. In addition, to increase the fluency of flight, a dynamic replanning strategy is also adopted. The main technical contributions of the proposed approach are to provide a comprehensive global coverage path with less BFM, and use adaptive yaw planning to generate more reasonable yaw trajectory. We present extensive comparison and real-world experiments to validate the effectiveness and correctness of our method. Experimental results show the proposed approach has better performance compared to typical and state-of-the-art methods in these flat scenarios without lots of scattered obstacles. We release our implementation as an open-source package <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{1}$</tex-math></inline-formula> for the community.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.

New strategy for anti-loop formulations

AbstractThis paper presents a strategy based on binary labelling of nodes for the creation of anti-loop formulations from existing strategies. This strategy prevents by default the formation of odd cycles, therefore it can have important role in iterative procedures based on generating subtour elimination constraints. It can also be used to modify the classic strategies used in problems associated to graphs. In this paper we focus on this last application. The behavior of this strategy is analyzed with two problems associated with graphs, the Asymmetric Traveling Salesman Problem (ATSP) and the Steiner Problem, where two configurations that modify the Miller-Tucking-Zemlig proposal to avoid cycles are compared. The experimental analysis shows that this strategy keep a good convergence, highlighting its use for the Steiner problem.

Transfer Reinforcement Learning for Combinatorial Optimization Problems

Reinforcement learning is an important technique in various fields, particularly in automated machine learning for reinforcement learning (AutoRL). The integration of transfer learning (TL) with AutoRL in combinatorial optimization is an area that requires further research. This paper employs both AutoRL and TL to effectively tackle combinatorial optimization challenges, specifically the asymmetric traveling salesman problem (ATSP) and the sequential ordering problem (SOP). A statistical analysis was conducted to assess the impact of TL on the aforementioned problems. Furthermore, the Auto_TL_RL algorithm was introduced as a novel contribution, combining the AutoRL and TL methodologies. Empirical findings strongly support the effectiveness of this integration, resulting in solutions that were significantly more efficient than conventional techniques, with an 85.7% improvement in the preliminary analysis results. Additionally, the computational time was reduced in 13 instances (i.e., in 92.8% of the simulated problems). The TL-integrated model outperformed the optimal benchmarks, demonstrating its superior convergence. The Auto_TL_RL algorithm design allows for smooth transitions between the ATSP and SOP domains. In a comprehensive evaluation, Auto_TL_RL significantly outperformed traditional methodologies in 78% of the instances analyzed.

APPROXIMATION ALGORITHMS WITH CONSTANT FACTORS FOR A SERIES OF ASYMMETRIC ROUTING PROBLEMS

In this paper, the first fixed-ratio approximation algorithms are proposed for the series of asymmetric settings of the well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all the proposed algorithms rely on original reductions of the considered problems to auxiliary instances of the Asymmetric Traveling Salesman Problem and employ the breakthrough approximation results for this problem obtained recently by O. Svensson, J. Tarnawski, L. Végh, V. Traub and J. Vygen. On the other hand, approximation of the cycle cover problem was proved by more deep extension of their approach.

A comparative analysis of genetic algorithms on a case study of asymmetric traveling salesman problem

A comparative analysis of genetic algorithms on a case study of asymmetric traveling salesman problem

FIXED RATIO POLYNOMIAL TIME APPROXIMATION ALGORITHM FOR THE PRIZE-COLLECTING ASYMMETRIC TRAVELING SALESMAN PROBLEM

We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem, which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph \(G\). Each node of the graph \(G\) can either be visited by the resulting route or skipped, for some penalty, while the arcs of \(G\) are weighted by non-negative transportation costs that fulfill the triangle inequality constraint. The goal is to find a closed walk that minimizes the total transportation costs augmented by the accumulated penalties. We show that an arbitrary \(\alpha\)-approximation algorithm for the Asymmetric Traveling Salesman Problem induces an \((\alpha+1)\)-approximation for the problem in question. In particular, using the recent \((22+\varepsilon)\)-approximation algorithm of V. Traub and J. Vygen that improves the seminal result of O. Svensson, J. Tarnavski, and L. Végh, we obtain \((23+\varepsilon)\)-approximate solutions for the problem.

Optimization of Shipping Routes for Container Ships from Indonesia to the Asia-Pacific Using Heuristic Algorithms

Sea transportation such as that by container ships has an essential role in the economy both locally and internationally. Ships are a major commodity in distributing goods over long distances due to their relatively low price compared to air shipping. The study implemented an optimization method using heuristic algorithms with ship route selection to minimize operational costs based on the parameters of mileage between 12 ports in the Asia-Pacific region. The ship speed, engine power, and fuel prices at each port are processed using asymmetric traveling salesman problem modeling (ATSP). The research uses three different algorithms to compare with the performance of the traveling salesman problem, namely the nearest neighbor algorithm, simulated annealing, and a genetic algorithm, with an objective function of keeping fuel costs that ships will incur to a minimum. The results show that the genetic algorithm provides the route with the lowest fuel cost.

Adjustable Pheromone Reinforcement Strategies for Problems with Efficient Heuristic Information

Ant colony optimization (ACO) is a well-known class of swarm intelligence algorithms suitable for solving many NP-hard problems. An important component of such algorithms is a record of pheromone trails that reflect colonies’ experiences with previously constructed solutions of the problem instance that is being solved. By using pheromones, the algorithm builds a probabilistic model that is exploited for constructing new and, hopefully, better solutions. Traditionally, there are two different strategies for updating pheromone trails. The best-so-far strategy (global best) is rather greedy and can cause a too-fast convergence of the algorithm toward some suboptimal solutions. The other strategy is named iteration best and it promotes exploration and slower convergence, which is sometimes too slow and lacks focus. To allow better adaptability of ant colony optimization algorithms we use κ-best, max-κ-best, and 1/λ-best strategies that form the entire spectrum of strategies between best-so-far and iteration best and go beyond. Selecting a suitable strategy depends on the type of problem, parameters, heuristic information, and conditions in which the ACO is used. In this research, we use two representative combinatorial NP-hard problems, the symmetric traveling salesman problem (TSP) and the asymmetric traveling salesman problem (ATSP), for which very effective heuristic information is widely known, to empirically analyze the influence of strategies on the algorithmic performance. The experiments are carried out on 45 TSP and 47 ATSP instances by using the MAX-MIN ant system variant of ACO with and without local optimizations, with each problem instance repeated 101 times for 24 different pheromone reinforcement strategies. The results show that, by using adjustable pheromone reinforcement strategies, the MMAS outperformed in a large majority of cases the MMAS with classical strategies.

The Solutions to Traveling Salesman Problem

This paper presents solutions to symmetric, asymmetric, and multiple traveling salesman problems. In the symmetric traveling salesman problem, one salesperson must go to five cities, making precisely one stop at each location. A matrix with the distances between each city is provided. Using the branch and bound algorithm and expressing it in Python, the final result is obtained. The formulations of the asymmetric traveling salesman problem and the multiple traveling salesman problem are demonstrated in the paper. The asymmetric problem in the paper is solved by transforming the asymmetric traveling salesman problem into a symmetric traveling salesman problem; then, the branch and bound algorithm has been applied to solve the problem. In the multiple traveling salesman problem, three salesmen are required to visit seven cities in total, each of which is only visited by one salesperson once. Each city is a point in an x-y plane, and the coordinates of all points are given in a graph. By formulating the MTSP problem with an assignment-based double-index integer programming and applying the constraints, the outcome is derived from Python code within a few seconds.

An asymmetric traveling salesman problem based matheuristic algorithm for flowshop group scheduling problem

The flowshop group scheduling problem (FGSP) has become a hot research problem owing to its practical applications in modern industry in recent years. The FGSP can be regarded as a combination of two coupled sub-problems. One is the group scheduling sub-problem with sequence-dependent setup times. The other is the job scheduling sub-problem within each group. A mixed integer linear programming model is built for the FGSP with the makespan criterion. Based on the problem-specific knowledge, i.e., the sequence-dependent group setup times are greater than the processing time of jobs, and the number of machines is small, the group scheduling sub-problem is approximated into an asymmetric traveling salesman problem (ATSP). Then, a matheuristic algorithm (MA) is proposed by integrating a branch-and-cut algorithm and an iterated greedy (IG) algorithm, where the branch-and-cut algorithm is used to generate the optimal Hamiltonian circuit for sub-group sequences of a group sequence obtained by the IG. On 405 test instances, the proposed MA performs significantly better than several state-of-the-art algorithms in the literature.

Autonomous Recharging and Flight Mission Planning for Battery-Operated Autonomous Drones

Unmanned aerial vehicles (UAVs), commonly known as drones, are being increasingly deployed throughout the globe as a means to streamline monitoring, inspection, mapping, and logistic routines. When dispatched on autonomous missions, drones require an intelligent decision-making system for trajectory planning and tour optimization. Given the limited capacity of their onboard batteries, a key design challenge is to ensure the underlying algorithms can efficiently optimize the mission objectives along with recharging operations during long-haul flights. With this in view, the present work undertakes a comprehensive study on automated tour management systems for an energy-constrained drone: (1) We construct a machine learning model that estimates the energy expenditure of typical multi-rotor drones while accounting for real-world aspects and extrinsic meteorological factors. (2) Leveraging this model, the joint program of flight mission planning and recharging optimization is formulated as a multi-criteria Asymmetric Traveling Salesman Problem (ATSP), wherein a drone seeks for the time-optimal energy-feasible tour that visits all the target sites and refuels whenever necessary. (3) We devise an efficient approximation algorithm with provable worst-case performance guarantees and implement it in a drone management system, which supports real-time flight path tracking and re- computation in dynamic environments. (4) The effectiveness and practicality of the proposed approach are validated through extensive numerical simulations as well as real-world experiments. Note to Practitioners—This study is stimulated by the need for developing pragmatic and provably efficient automated tour management systems for UAVs deployed on energy-constrained, long-distance flight missions. As such, UAVs provide a nifty platform for facilitating environmental monitoring, disaster management, transport of medical supplies, as well as expediting last-mile deliveries. However, existing path planners generally fall short of capturing several crucial aspects, such as detailed power consumption model (e.g., factoring in payload, wind speed and direction) or performance guarantees, potentially leading to underutilized or infeasible routing decisions. To address these issues, the present work proposes a theoretically-backed routing approach with a certifiable degree of optimality and develops an effective, practical power consumption evaluation model for multi-rotor UAVs, verified on multiple drone models.

Discrete Mayfly Algorithm for spherical asymmetric traveling salesman problem

The Asymmetric Traveling Salesman Problem (ATSP), as a branch of the traveling salesman problem, aims to find a group of circuits when a salesman travels all cities at the minimum cost and returns to the starting city under the condition of different round-trip costs between the two cities. Mayfly Algorithm (MA) is a swarm-based metaheuristic algorithm, which has been applied to many fields successfully. In this paper, we arrange all cities on the sphere. A Discrete Mayfly Algorithm (DMA) for solving spherical asymmetric traveling salesman problem is proposed. The algorithm applies inver-over operator, crossover operator and 3-opt operator to simplify the parameters of the algorithm, enrich the diversity of the population and enhance the local search ability. By experimenting, analyzing and comparing the effects of DMA and other metaheuristic algorithms in twelve scales, this study proves that the proposed algorithm is superior to others for the spherical ATSP, especially in high-dimensional cases.

Precedence constrained generalized traveling salesman problem: Polyhedral study, formulations, and branch-and-cut algorithm

The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) is an extension of two well-known combinatorial optimization problems — the Generalized Traveling Salesman Problem (GTSP) and the Precedence Constrained Asymmetric Traveling Salesman Problem (PCATSP), whose path version is known as the Sequential Ordering Problem (SOP). Similarly to the classic GTSP, the goal of the PCGTSP, for a given input digraph and partition of its node set into clusters, is to find a minimum cost cyclic route (tour) visiting each cluster in a single node. In addition, as in the PCATSP, feasible tours are restricted to visit the clusters with respect to the given partial order. Unlike the GTSP and SOP, to the best of our knowledge, the PCGTSP still remain to be weakly studied both in terms of polyhedral theory and algorithms. In this paper, for the first time for the PCGTSP, we propose several families of valid inequalities, establish dimension of the PCGTS polytope and prove sufficient conditions ensuring that the extended Balas’ π- and σ-inequalities become facet-inducing. Relying on these theoretical results and evolving the state-of-the-art algorithmic approaches for the PCATSP and SOP, we introduce a family of MILP-models (formulations) and several variants of the branch-and-cut algorithm for the PCGTSP. We prove their high performance in a competitive numerical evaluation against the public benchmark library PCGTSPLIB, a known adaptation of the classic SOPLIB to the problem in question.

The drone-assisted variable speed asymmetric traveling salesman problem

We introduce and solve the Drone-Assisted Variable Speed Asymmetric Traveling Salesman Problem, which considers different flight times for the drone depending on the selected speed level and weather conditions, as a result of the natural asymmetry intrinsically involved in the delivery operations. Moreover, we provide an extension that introduces an additional decision regarding the speed of the drone influencing energy consumption. We formulate mixed-integer linear programming models (MILP) and develop two metaheuristic approaches: a variable neighborhood descent (VND) and a multi-neighborhood tabu search (MTS) algorithm. To assess their performance, a set of instances based on existing benchmarks are proposed. Results point out that delivery systems are strongly sensitive to the features considered in this research, showing that the symmetric approach is not able to find feasible solutions when incorporating operational aspects, such as wind conditions and energy consumption. Furthermore, we demonstrate that the option to fly at lower speeds results in a decreasing makespan due to the increased use of the drone. Finally, we show that for all larger instances, the heuristics VND and MTS outperform the MILP solutions by at least 48% and 53%, respectively.

Energy Efficient Path Planning for 3D Aerial Inspections

The use of Unmanned Aerial Vehicles (UAVs) in different inspection tasks is increasing. This technology reduces inspection costs and collects high quality data of distinct structures, including areas that are not easily accessible by human operators. However, the reduced energy available on the UAVs limits their flight endurance. To increase the autonomy of a single flight, it is important to optimize the path to be performed by the UAV, in terms of energy loss. Therefore, this work presents a novel formulation of the Travelling Salesman Problem (TSP) and a path planning algorithm that uses a UAV energy model to solve this optimization problem. The novel TSP formulation is defined as Asymmetric Travelling Salesman Problem with Precedence Loss (ATSP-PL), where the cost of moving the UAV depends on the previous position. The energy model relates each UAV movement with its energy consumption, while the path planning algorithm is focused on minimizing the energy loss of the UAV, ensuring that the structure is fully covered. The developed algorithm was tested in both simulated and real scenarios. The simulated experiments were performed with realistic models of wind turbines and a UAV, whereas the real experiments were performed with a real UAV and an illumination tower. The inspection paths generated presented improvements over 24% and 8%, when compared with other methods, for the simulated and real experiments, respectively, optimizing the energy consumption of the UAV.

Hyper-heuristics with Path Relinking applied to the Generalised Time-Dependent ATSP in air travel

In this work we propose the use of path relinking within a hyper-heuristic framework to solve the Generalised Time-Dependent Asymmetric Traveling Salesman problem applied to Air Travel. We implemented several heuristic selection methods, such as Simple Random, Random Descent, Random Permutation and Reinforcement Learning. We were able to achieve very good solutions for 13 of the 14 instances from a well-known benchmark set, evidencing that the adequate use of a hyper-heuristic framework with path relinking can be very efficient in improving the final results.

Learned Upper Bounds for the Time-Dependent Travelling Salesman Problem

Given a graph whose arc traversal times vary over time, the Time-Dependent Travelling Salesman Problem consists in finding a Hamiltonian tour of least total duration covering the vertices of the graph. The main goal of this work is to define tight upper bounds for this problem by reusing the information gained when solving instances with similar features. This is customary in distribution management, where vehicle routes have to be generated over and over again with similar input data. To this aim, we devise an upper bounding technique based on the solution of a classical (and simpler) time-independent Asymmetric Travelling Salesman Problem, where the constant arc costs are suitably defined by the combined use of a Linear Program and a mix of unsupervised and supervised Machine Learning techniques. The effectiveness of this approach has been assessed through a computational campaign on the real travel time functions of two European cities: Paris and London. The overall average gap between our heuristic and the best-known solutions is about 0.001\%. For 31 instances, new best solutions have been obtained.

Constant-Factor Approximation Algorithms for a Series of Combinatorial Routing Problems Based on the Reduction to the Asymmetric Traveling Salesman Problem

Constant-Factor Approximation Algorithms for a Series of Combinatorial Routing Problems Based on the Reduction to the Asymmetric Traveling Salesman Problem