Traveling Salesman Problem Solver Research Articles

Application of decision theory and bee-inspired method to railway system route optimization

ABSTRACTRoute planning for multiple destinations via a railway system (RS) is challenging, especially in a complex network with hundreds of stations and interchanges, resulting in a railway traveling salesman problem (RTSP), which is a variant of the traveling salesman problem (TSP). Limited attention has been devoted to solving the RTSP, despite the increase in the number of people using RSs and the added complexity of network expansions. An optimization algorithm and mathematical model based on the foraging behavior of bees and decision theory were used in this paper to identify the optimum RS route to multiple destinations before returning to the first station. Data collected from RSs in Japan and Malaysia were used to create 200 test cases (100 cases with each dataset), and the results were compared to specific solutions and the official optimum route planner to prove that the model is a promising approximation method. The solutions were evaluated and verified by comparing results from another 200 brute-force cases generated by the Wiley TSP Solver. The results obtained from the experiments prove the reliability and capability of the route planning and optimization solutions for RSs with different complexities and in different environments without the assistance of information technology.

A variable neighborhood search for flying sidekick traveling salesman problem

AbstractThe efficiency and dynamism of unmanned aerial vehicles, or drones, have presented substantial application opportunities in several industries in the last years. Notably, logistic companies have given close attention to these vehicles to reduce delivery time and operational cost. A variant of the traveling salesman problem (TSP), called the flying sidekick traveling salesman problem, was introduced involving drone‐assisted parcel delivery. The drone launches from the truck, proceeds to deliver parcels to a customer, and then is recovered by the truck at a third location. While the drone travels through a trip, the truck delivers parcels to other customers as long as the drone has enough battery to hover waiting for the truck. This work proposes a hybrid heuristic where the initial solution is created from the optimal TSP solution reached by a TSP solver. Next, an implementation of the general variable neighborhood search is employed to obtain the delivery routes of truck and drone. Computational experiments show the potential of the algorithm to improve significantly delivery time. Furthermore, we provide a new set of instances based on the well‐known traveling salesman problem library instances.

Construction of optimal run order in design of experiments

It is sometimes favorable to conduct experiments in a systematic run order rather than the conventional random run order. In this article, we propose an algorithm to construct the optimal run order. The algorithm is very flexible: it is applicable to any design and works whenever the optimality criterion can be represented as distance between any two experimental runs. Specifically, the proposed method first formulates the run order problem into a graph, then makes use of the existing traveling salesman problem-solver to obtain the optimal run order. It can always reach the optimal result in an efficient manner.A special case where level change is used as the distance criterion is investigated thoroughly. The optimal run orders for popular two-level designs are obtained and tabulated for practical use. For higher- or mixed-level designs a generic table is not possible, although the proposed algorithm still works well for finding the optimal run order. Some supporting theoretical results are derived.

A note on computational aspects of the Steiner traveling salesman problem

AbstractThe Steiner traveling salesman problem (StTSP) is a variant of the classical traveling salesman problem (TSP). In the StTSP, we are given a graph with edge distances, and a set of terminal nodes, which are a subset of all nodes. The goal is to find a minimum distance closed walk to visit each terminal node at least once. Two recent articles proposed solution approaches to the problem, namely, on the exact side, Letchford et al. proposed compact integer linear programming models, and on the heuristic side, Interian and Ribeiro proposed greedy randomized adaptive search procedure, enhanced with a local search. Using the exact approach, instances with up to 250 nodes could be solved to optimality, with runtimes up to 1400 seconds, and the heuristic approach was used to tackle instances with up to 3353 nodes, with runtimes up to 8500 seconds. In this note, we show that by transforming the problem to the classical TSP, and using a state‐of‐the‐art TSP solver, all instances from the literature can be solved to optimality within 20 seconds, most of them within a second. We provide optimal solution values for 14 instances, where the optimal solution was not known.

Analysis of a High-Performance TSP Solver on the GPU

Graphical Processing Units have been applied to solve NP-hard problems with no known polynomial time solutions. An example of such a problem is the Traveling Salesman Problem (TSP). The TSP is one of the most commonly studied combinatorial optimization problems and has multiple applications in the areas of engineering, transportation, and logistics. This article presents an improved algorithm for approximating the TSP on fully connected, symmetric graphs by utilizing the GPU. Our approach improves an existing 2-opt hill-climbing algorithm with random restarts by considering multiple updates to the current path found in parallel, and it allows k number of updates per iteration, called k-swap . With our k-swap modification, we show a speed-up over the existing algorithm of 4.5× to 22.9× on data sets ranging from 1,400 to 33,810 nodes, respectively.

Complexity indices for the traveling salesman problem based on short edge subgraphs

We present the extension of our previous work on complexity indices for the traveling salesman problem (TSP). Since we study the symmetric traveling salesman problem, the instances are represented by complete graphs G with distances between cities as the edge weights. A complexity index is an invariant of an instance I by which the execution time of an exact TSP algorithm for I can be predicted. We consider some subgraphs of G consisting of short edges and define several new invariants related to their connected components. For computational experiments we have used the well-known TSP Solver Concorde. Experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1, 100] show that there exists a notable correlation between the sequences of selected invariants and the sequence of execution times of the TSP Solver Concorde. We provide logical explanations of these phenomena.

A novel two-level optimization approach for clustered vehicle routing problem

Abstract In this paper, we are addressing the clustered vehicle routing problem (CluVRP) which is a variant of the classical capacitated vehicle routing problem (CVRP). The following are the main characteristics of this problem: the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum-cost collection of routes starting and ending at the depot, visiting all the vertices exactly once, except the depot, and with the additional constraint that once a vehicle enters a cluster it visits all the vertices within the cluster before leaving it. We describe a novel two-level optimization approach for CluVRP obtained by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower-level (local) subproblem, and solving them separately. The goal of the first subproblem is to determine the (global) routes visiting the clusters using a genetic algorithm, while the goal of the second subproblem is, to determine for the above mentioned routes, the visiting order within the clusters. The second subproblem is solved by transforming each global route into a traveling salesman problem (TSP) which then is optimally computed using the Concorde TSP solver. Extensive computational results are reported and discussed for an often used set of benchmark instances. The obtained results show an improvement of the quality of the achieved solutions and prove the efficiency of our approach as compared to the existing methods from the literature.

TSP solver: an integrated framework for solving traveling salesman problems consistent with TSPLIB

The Traveling Salesman Problem (TSP) is the subject of study in operational research for more than 30 years. The TSP is considered as NP-complete; consequently, many heuristic and metaheuristic algorithms have been developed to cope with the intractable nature of the problem. Although the problem is well-studied, lack of integrated software that harnesses the new computers' computational power and provides an easy comparison between heuristic algorithms is sensible. TSP solver is the state-of-the-art software that provides a common framework to compare the performance of different algorithms over TSPLIB library. Academicians can focus on developing new methodologies without concerning the availability and correctness of reported algorithms in the literature. Practitioners may also benefit from provided transparency by our software solution and build their own customized packages to flourish their businesses. The proposed software can be a foundation for the future implementations in which users design their algorithm and results would be uploaded automatically on a public server together with the source code.

A Two-Stage Optimized Next-View Planning Framework for 3-D Unknown Environment Exploration, and Structural Reconstruction

In this paper, we present a solution for autonomous exploration and reconstruction in 3-D unknown environments without a priori knowledge of the environments. In our framework, a two-stage heuristic information gain-based next-view planning algorithm is performed to dynamically select and update candidate viewpoints, followed by immediate next-best-view extraction and corresponding motion planning. The two-stage planner consists of a frontier-based boundary coverage planner and a fixed start open travelling salesman problem solver, such that the planner returns an exploration path with the consideration of global optimality in the context of accumulated space information. The effectiveness of our work is evaluated with simulation and experimental tests. The results and comparisons prove our system is able to autonomously explore the 3-D unknown environments and reconstruct the structural model with improved exploration efficiency in terms of path quality and total exploration time.

Non-retraction toolpath generation for irregular compound freeform surfaces with the LKH TSP solver

Toolpath planning for irregular-shaped compound freeform surfaces remains a challenge. This work presents a new approach for compound surface finishing by treating the toolpath planning task as a traveling salesman problem (TSP). The concept of curvature map is proposed. With this concept, the curvatures of the surface patches are recorded to corresponding cells of the map and then the path intervals are determined. The CC points as well as the normal vectors are calculated on the mesh model of the compound surface with a linear algorithm. The obtained CC points are linked as cities in LKH, in which the distance function is redefined to cope with the illegal linking problems. With LKH, tool retractions are no longer necessary. The resulted toolpath is capable of covering the whole compound surface with only one pass. The proposed toolpath generation method is more forgiving on the input. It does not require the surfaces to be perfectly conform on the joining boundaries. Several compound surface examples have been used to verify the feasibility of the proposed method.

De Novo modeling in cryo-EM density maps with Pathwalking.

As electron cryo-microscopy (cryo-EM) can now frequently achieve near atomic resolution, accurate interpretation of these density maps in terms of atomistic detail has become paramount in deciphering macromolecular structure and function. However, there are few software tools for modeling protein structure from cryo-EM density maps in this resolution range. Here, we present an extension of our original Pathwalking protocol, which can automatically trace a protein backbone directly from a near-atomic resolution (3-6Å) density map. The original Pathwalking approach utilized a Traveling Salesman Problem solver for backbone tracing, but manual adjustment was still required during modeling. In the new version, human intervention is minimized and we provide a more robust approach for backbone modeling. This includes iterative secondary structure identification, termini detection and the ability to model multiple subunits without prior segmentation. Overall, the new Pathwalking procedure provides a more complete and robust tool for annotating protein structure function in near-atomic resolution density maps.

Tool path generation for multi-axis freeform surface finishing with the LKH TSP solver

In freeform surface finishing, there are three major types of tool path topologies: the direction-parallel type, the contour–parallel type and the space-filling curve (SFC) type. The SFC topology is capable of covering the whole surface with only one path. In this paper, we present a new way of planning the SFC type tool path by formulating the planning task as a traveling salesman problem (TSP). The optimal path is generated in two steps. Firstly, a set of regular cutter contact (CC) points is generated on the input surface. A cutting simulation method is developed to evaluate the scallop error and determine the position of the next CC point in cross-feed direction. This method is free of local surface curvature assumptions and is therefore accurate for big cutters. Secondly, the obtained CC points are input into an efficient TSP solver LHK for the optimal CC point linking sequences. To stop the CC points from diagonal linking or penetrating linking, the Euclidean distance evaluation function for two CC points is redefined in LHK. The proposed tool path generation method is verified with several freeform surface examples; the results show that the method can automatically find the optimal feed direction and it can generate shorter tool path than the traditional SFC method. The feasibility of the proposed method is also verified by a cutting experiment.

Solving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun Algorithm

The equality generalized traveling salesman problem (E-GTSP) is an extension of the traveling salesman problem (TSP) where the set of cities is partitioned into clusters, and the salesman has to visit every cluster exactly once. It is well known that any instance of E-GTSP can be transformed into a standard asymmetric instance of the TSP, and therefore solved with a TSP solver. This paper evaluates the performance of the state-of-the art TSP solver Lin–Kernighan–Helsgaun (LKH) on transformed E-GTSP instances. Although LKH is used without any modifications, the computational evaluation shows that all instances in a well-known library of benchmark instances, GTSPLIB, could be solved to optimality in a reasonable time. In addition, it was possible to solve a series of new very-large-scale instances with up to 17,180 clusters and 85,900 vertices. Optima for these instances are not known but it is conjectured that LKH has been able to find solutions of a very high quality. The program’s performance has also been evaluated on a large number of instances generated by transforming arc routing problem instances into E-GTSP instances. The program is free of charge for academic and non-commercial use and can be downloaded in source code.

Benchmarking Optimization Algorithms: An Open Source Framework for the Traveling Salesman Problem

We introduce an experimentation procedure for evaluating and comparing optimization algorithms based on the Traveling Salesman Problem (TSP). We argue that end-of-run results alone do not give sufficient information about an algorithm's performance, so our approach analyzes the algorithm's progress over time. Comparisons of performance curves in diagrams can be formalized by comparing the areas under them. Algorithms can be ranked according to a performance metric. Rankings based on different metrics can then be aggregated into a global ranking, which provides a quick overview of the quality of algorithms in comparison. An open source software framework, the TSP Suite, applies this experimental procedure to the TSP. The framework can support researchers in implementing TSP solvers, unit testing them, and running experiments in a parallel and distributed fashion. It also has an evaluator component, which implements the proposed evaluation process and produces detailed reports. We test the approach by using the TSP Suite to benchmark several local search and evolutionary computation methods. This results in a large set of baseline data, which will be made available to the research community. Our experiments show that the tested pure global optimization algorithms are outperformed by local search, but the best results come from hybrid algorithms.

Path Optimization for Line Scanning on Flat Panel Displays Using a Self Organizing Map

This paper presents a methodology for optimization of line scan path on fixed panel displays (FPD) using a modified self organizing map (SOM). The existing scanning method of defective thin film transistor (TFT) lines on FPDs by a non-contact moving sensor is top down and unidirectional which is time consuming. First, the problem is approximated to a well known asymmetric traveling salesman problem (TSP) with precedence constraints. Then the SOM based TSP solver is proposed together with modifications for neurons before mapping them onto input space. Finally, simulation results are presented and shown the feasibility and superiority of proposed method over the existing method.

On the empirical scaling of run-time for finding optimal solutions to the travelling salesman problem

The travelling salesman problem (TSP) is one of the most prominent NP-hard combinatorial optimisation problems. After over fifty years of intense study, the TSP continues to be of broad theoretical and practical interest. Using a novel approach to empirical scaling analysis, which in principle is applicable to solvers for many other problems, we demonstrate that some of the most widely studied types of TSP instances tend to be much easier than expected from previous theoretical and empirical results. In particular, we show that the empirical median run-time required for finding optimal solutions to so-called random uniform Euclidean (RUE) instances – one of the most widely studied classes of TSP instances – scales substantially better than Θ(2n) with the number n of cities to be visited. The Concorde solver, for which we achieved this result, is the best-performing exact TSP solver we are aware of, and has been applied to a broad range of real-world problems. Furthermore, we show that even when applied to a broad range of instances from the prominent TSPLIB benchmark collection for the TSP, Concorde exhibits run-times that are surprisingly consistent with our empirical model of Concorde’s scaling behaviour on RUE instances. This result suggests that the behaviour observed for the simple random structure underlying RUE is very similar to that obtained on the structured instances arising in various applications.

Study of Some Recent Crossovers Effects on Speed and Accuracy of Genetic Algorithm, using Symmetric Travelling Salesman Problem

The Travelling Salesman Problem (TSP) is one of the most famous optimization problems. The Genetic Algorithm (GA) is one of metaheuristics that have been applied to TSP. The Crossover and mutation operators are two important elements of GA. There are many TSP solver crossover operators. In this paper, we state implementation of some recent TSP solver crossovers at first and then we use each of them in GA to solve some Symmetric TSP (STSP) instances and finally compare their effects on speed and accuracy of presented GA.

Computation of the travelling salesman problem by a shrinking blob

The travelling salesman problem (TSP) is a well known and challenging combinatorial optimisation problem. Its computational intractability has attracted a number of heuristic approaches to generate satisfactory, if not optimal, candidate solutions. Some methods take their inspiration from natural systems, extracting the salient features of such systems for use in classical computer algorithms. In this paper we demonstrate a simple unconventional computation method to approximate the Euclidean TSP using a virtual material approach. The morphological adaptation behaviour of the material emerges from the low-level interactions of a population of particles moving within a diffusive lattice. A `blob' of this material is placed over a set of data points projected into the lattice, representing TSP city locations, and the blob is reduced in size over time. As the blob shrinks it morphologically adapts to the configuration of the cities. The shrinkage process automatically stops when the blob no longer completely covers all cities. By manually tracing the perimeter of the blob a path between cities is elicited corresponding to a TSP tour. Over 10 runs on 20 randomly generated datasets consisting of 20 cities this simple and unguided method found tours with a mean average tour length of 6.41 % longer than the minimum tours computed by a TSP solver (mean best performance was 4.27 % longer and mean worst performance was 9.22 % longer). We examine the insertion mechanism by which the blob constructs a tour, note some properties and limitations of its performance, and discuss the relationship between the blob TSP and proximity graphs which group points on the plane. The method is notable for its simplicity, novelty and the spatially represented mechanical mode of its operation. We discuss similarities between this method and previously suggested models of human performance on the TSP and suggest possibilities for further improvement.

Placement Optimization of Distributed-Sensing Fiber Optic Sensors Using Genetic Algorithms

In this paper, we present methodology for the placement optimization of both discretely and continuously distributed fiber-optic sensors in which many point or quasi-point sensors can be multiplexed along a single optical fiber. A unified sensor performance metric is defined for vibration monitoring and fault detection as the integration/ summation of a weighted functional of some strain measure over the optical fiber length for both discretely and continuously distributed fiber-optic sensors. The optical fiber is represented by a nonuniform rational B-splines curve. The design variables include the control point coordinates of the nonuniform rational B-splines curve and the arclength coordinates of the point sensing element positions along the fiber. The constraints common to all kinds of optical sensors include the maximum fiber length and maximum allowed initial curvature. The constraints are treated as the exact penalty functions in the fitness function for any genetic algorithm. For discretely distributed fiber-optic sensors, we briefly show by an example how to use a genetic algorithm-based traveling salesman problem solver to optimally connect those optimally distributed sensors using one single optical fiber by minimizing the total fiber length. For continuously distributed fiber-optic sensors, two sets of examples are given as applications to fluttering or vibrating panels in aerospace engineering and a clinical smart bed for patient vital signal monitoring in biomedical engineering. In both cases, the integrating bend-sensing fiber-optic sensors are used.

Assigning Micro UAVs to Task Tours in an Urban Terrain

An optimization algorithm for assigning in realtime multiple unmanned aerial vehicles (UAVs) to task tours is presented and tested as part of a flight demonstration program. The scenario of interest is one where multiple microaerial vehicles are launched from a small UAV in order to investigate selected targets in an urban terrain. For path planning, we use the Dubin's car model so that the vehicles' dynamic constraint of minimum turning radius is taken into account. Due to the prohibitive computational complexity of the coupled path optimization and assignment problem, we solve the problem by ordering a set of tasks based on the Euclidean distance, utilizing a traveling salesman problem solver. We apply upper and lower bounding procedures iteratively on active subsets within the set of feasible group assignments, enabling efficient search of the solution space. The online implementation of the algorithm is discussed and simulation results confirm the efficiency of the proposed algorithm. Results from recent flight tests are also provided.