Large-scale Traveling Salesman Problem Research Articles

An Improved Ant Colony Optimization Based on an Adaptive Heuristic Factor for the Traveling Salesman Problem

The traveling salesman problem (TSP) is a typical combinatorial optimization problem, which is often applied to sensor placement, path planning, etc. In this paper, an improved ACO algorithm based on an adaptive heuristic factor (AHACO) is proposed to deal with the TSP. In the AHACO, three main improvements are proposed to improve the performance of the algorithm. First, the k-means algorithm is introduced to classify cities. The AHACO provides different movement strategies for different city classes, which improves the diversity of the population and improves the search ability of the algorithm. A modified 2-opt local optimizer is proposed to further tune the solution. Finally, a mechanism to jump out of the local optimum is introduced to avoid the stagnation of the algorithm. The proposed algorithm is tested in numerical experiments using 39 TSP instances, and results shows that the solution quality of the AHACO is 83.33% higher than that of the comparison algorithms on average. For large-scale TSP instances, the algorithm is also far better than the comparison algorithms.

Development of Deer Hunting linked Earthworm Optimization Algorithm for solving large scale Traveling Salesman Problem

Traveling Salesman Problem (TSP) has been seen in diverse applications, which is proven to be NP-complete in most cases. Even though there are multiple heuristic techniques, the problem is still a complex combinatorial optimization problem. The candidate solutions are chosen by considering only a set of high values of the objective function which may not lead to the best solutions. Hence, this paper develops a hybrid optimization algorithm, named Earthworm-based DHOA (EW-DHOA) to solve the TSP problem by finding an optimal solution. The proposed EW-DHOA is developed by integrating the two well-performing meta-heuristic algorithms, such as Deer Hunting Optimization Algorithm (DHOA) and Earthworm Optimization Algorithm (EWA). The EW-DHOA intends to optimize the constraint as the number of cities traveled by the salesman in terms of an optimal path. The main process for attaining this objective is to minimize the distance traveled by the salesman concerning the entire cities. The effectiveness of the proposed hybrid meta-heuristic algorithm is validated over the benchmark dataset. Finally, the experimental results show that the convergence of the proposed hybrid optimization will be better while solving TSP with less computational complexity, and improved significantly in attaining optimal results.

Two-layered ant colony system to improve engraving robot’s efficiency based on a large-scale TSP model

Laser engraving is an essential tool of automatic drawings and 3D printers. When the laser engraving tasks become large and complicated, engraving process will be time-consuming. To improve the time and energy efficiency, the trajectory optimization of laser engraving is studied. The trajectory of laser engraving robot is modelled as a large-scale traveling salesman problem (TSP), by converting grayscale images into halftone images. To solve the large-scale TSP, two-layered ant colony system (ACS) is newly proposed to combine k-means, top-layer ACS, and bottom-layer ACS. Finally, we use the presented algorithm to optimize the path of four engraving instances which include tens of thousands of discrete points. Experimental results show that this method can reduce laser engraving time by about 50% compared with traditional engraving methods.

Improving Initial Population for Genetic Algorithm using the Multi Linear Regression Based Technique (MLRBT)

Genetic algorithms (GAs) are powerful heuristic search techniques that are used successfully to solve problems for many different applications. Seeding the initial population is considered as the first step of the GAs.
 In this work, a new method is proposed, for the initial population seeding called the Multi Linear Regression Based Technique (MLRBT). That method divides a given large scale TSP problem into smaller sub-problems and the technique works frequently until the sub-problem size is very small, four cities or less. Experiments were carried out using the well-known Travelling Salesman Problem (TSP) instances and they showed promising results in improving the GAs' performance to solve the TSP.

Pearson correlation coefficient-based pheromone refactoring mechanism for multi-colony ant colony optimization

To solve the problem of falling into local optimum and poor convergence speed in large Traveling Salesman Problem (TSP), this paper proposes a Pearson correlation coefficient-based Pheromone refactoring mechanism for multi-colony Ant Colony Optimization (PPACO). First, the dynamic guidance mechanism is introduced to dynamically adjust the pheromone concentration on the path of the maximum and minimum spanning tree, which can effectively balance the diversity and convergence of the algorithm. Secondly, the frequency of communication between colonies is adjusted adaptively according to a criterion based on the similarity between the minimum spanning tree and the optimal solution. Besides, the pheromone matrix of the colony is reconstructed according to the Pearson correlation coefficient or information entropy to help the algorithm jump out of the local optimum, thus improving the accuracy of the solution. These strategies greatly improve the adaptability of the algorithm and ensure the effectiveness of the interaction. Finally, the experimental results indicate that the proposed algorithm could improve the solution accuracy and accelerate the convergence speed, especially for large-scale TSP instances.

Multi-core-, multi-thread-based optimization algorithm for large-scale traveling salesman problem

With the rapid development of general hardware technology, microcomputers with multi-core CPUs have been widely applied in commercial services and household usage in the last ten years. Multi-core chips could, theoretically, lead to much better performance and computational efficiency than single-core chips. But so far, they have not shown general advantages for users, other than for operating systems and some specialized software. It is not easy to transform traditional single-core-based algorithms into multi-core-, multi-thread-based algorithms that can greatly improve efficiency, because of difficulties in computation and scheduling of hardware kernels, and because some programming languages cannot support multi-core, multi-thread programming. Therefore, a kind of multi-core-, multi-thread-based fast algorithm was designed and coded with Delphi language for the medium- and large-scale traveling salesman problem instances from TSPLIB, which can fully speed up the searching process without loss of quality. Experimental results show that the algorithm proposed can, under the given hardware limitations, take full advantage of multi-core chips and effectively balance the conflict between increasing problem size and computational efficiency and thus acquire satisfactory solutions.

Dıscrete socıal spıder algorıthm for the travelıng salesman problem

Heuristic algorithms are often used to find solutions to real complex world problems. These algorithms can provide solutions close to the global optimum at an acceptable time for optimization problems. Social Spider Algorithm (SSA) is one of the newly proposed heuristic algorithms and based on the behavior of the spider. Firstly it has been proposed to solve the continuous optimization problems. In this paper, SSA is rearranged to solve discrete optimization problems. Discrete Social Spider Algorithm (DSSA) is developed by adding explorer spiders and novice spiders in discrete search space. Thus, DSSA's exploration and exploitation capabilities are increased. The performance of the proposed DSSA is investigated on traveling salesman benchmark problems. The Traveling Salesman Problem (TSP) is one of the standard test problems used in the performance analysis of discrete optimization algorithms. DSSA has been tested on a low, middle, and large-scale thirty-eight TSP benchmark datasets. Also, DSSA is compared to eighteen well-known algorithms in the literature. Experimental results show that the performance of proposed DSSA is especially good for low and middle-scale TSP datasets. DSSA can be used as an alternative discrete algorithm for discrete optimization tasks.

Dynamic Adaptation of Genetic Algorithm for Solving Routing Problems on Large Scale Systems

This paper is devoted to the implementation of the dynamic adaptation procedure for the genetic algorithm used for solving large-scale travelling salesman problem. This procedure serves to obtain more profitable solutions by a fixed operating time. In order to evaluate effectiveness of new approach computational experiments were performed on well-known problem instances from TSPLib library. As a result, generated solutions reduce the length of the routing plans in considered problem instances compare to classical genetic heuristics. By that, we show how to use the property of time inconsistency of heuristics to get better solutions. New criteria for estimating the efficiency of heuristics algorithms called experimental level of time consistency is introduced.

Multiple k −opt evaluation multiple k −opt moves with GPU high performance local search to large-scale traveling salesman problems

The 2-opt, 3-opt or k–opt heuristics are classical local search algorithms for traveling salesman problems (TSP) in combinatorial optimization area, while sequential k–opt complete neighborhood examination takes polynomial time complexity which is timeconsuming to approach large scale TSP instances. This paper introduces a reasonable methodology called “multiple k–opt evaluation, multiple k–opt moves” that allows to simultaneously execute, without interference, massive k −opt moves that are globally found on the same TSP tour, as well as keep high performance GPU (Graphics Processing Unit) parallel 2-/3-opt evaluation with characteristic of “data parallelism, decentralized control and O(1) local memory for each GPU thread”. The methodology is reasonable since intervention of a sequential O(N) time complexity tour reversal operation is unavoidable for each k −opt move when using array of ordered coordinates as TSP tour data structure for high performance GPU k −opt local search that considers coalesced memory access and usage of limited on-chip shared memory. Innovation work includes two parts, a sequential non-interacted k-opt moves’ set partition algorithm that takes linear time complexity; a new TSP tour representation, array of ordered coordinates-index, that unveils how to combine the advantages of using doubly linked list and array of ordered coordinates data structures for iterative parallel k −opt local search based on GPU CUDA. We test this methodology on 22 national TSP instances with up to 71009 cities and with brute initial tour solution. Average maximum 997 non-interacted 2-opt moves are found and executed on the same tour of ch71009.tsp instance after one iteration of complete $\frac {N*(N-1)}{2}$ 2-opt checks working in parallel on GPU. And the proposed iterative GPU parallel 2-opt methodology executes average 306631 2-opt moves while only iterates 786 tour reversal operations, in comparison with methods that have to execute tour reversal operation after each 2-opt move. Experimental comparisons show that our proposed methodology gets huge acceleration over both classical sequential and a current state-of-the-art GPU parallel 2-opt implementation.

Dynamic Density Clustering Ant Colony Algorithm With Filtering Recommendation Backtracking Mechanism

Ant colony system has a good performance in solving Traveling Salesman Problem (TSP), but it tends to fall into local optimum and is deficient in convergence speed. To address this problem, a dynamic density clustering ant colony algorithm with a filtering recommendation backtracking mechanism is proposed (DBACS). Firstly, a dynamic density clustering strategy is proposed to accelerate the convergence speed of the algorithm and improve the quality of the solution. Under this strategy, the search radius is expanded dynamically to merge adjacent classes, so as to form the differential pheromone distribution. The splicing paths between each class are adjusted through the ant colony algorithm to achieve better performance. Secondly, a recommendation backtracking mechanism based on collaborative filtering is proposed to increase the diversity of the population, thus helping the algorithm jump out of the local optimum. With the help of the collaborative filtering algorithm, some dense data points are recommended for pheromone dynamic backtracking, which can not only help algorithm jump out of the local optimum, but also help the algorithm accelerate convergence. Simulation results show that the improved algorithm can obtain a better solution and higher stability. Especially in solving large-scale TSP, the accuracy of the solution is significantly improved.

Discrete pigeon-inspired optimization algorithm with Metropolis acceptance criterion for large-scale traveling salesman problem

Abstract Pigeon-inspired optimization (PIO) algorithm, which is a newly proposed swarm intelligence algorithm, has been mainly applied to continuous optimization problems. In this paper, a discrete PIO (DPIO) algorithm, which uses the Metropolis acceptance criterion of simulated annealing algorithm, is proposed for Traveling Salesman Problems (TSPs). A new map and compass operator with comprehensive learning ability is designed to enhance DPIO's exploration ability. A new landmark operator, which has cooperative learning ability and can learn from the heuristic information of TSP instance, is designed to improve DPIO's exploitation ability. Aim to enhance its ability to escape from premature convergence, the Metropolis acceptance criterion is used to decide whether to accept newly produced solutions. Systematic experiments were performed to analyze the behaviours of the map and compass operator and the landmark operator. The performance of DPIO algorithm was tested on 33 large-scale TSP instances from TSPLIB with city number from 1000 to 85900. Simulation results show that the proposed algorithm is effective and is competitive with most other state-of-the-art meta-heuristic algorithms.

Discrete symbiotic organism search with excellence coefficients and self-escape for traveling salesman problem

Traveling salesman problem (TSP) is one of the well-known NP-hard problems in combinatorial optimization. The optimal solution of large-scale TSP is difficult to find with exact algorithms in acceptable computation time. Thus, heuristic algorithms are widely studied to find the near-optimal or satisfactory solutions. The discrete symbiotic organism search (DSOS) is enhanced with excellence coefficients and self-escape strategy (ECSDSOS) for TSP. The excellence coefficients let ECSDSOS choose shorter edges (routes) for generating better local paths. The local optimization capability is improved so that ECSDSOS is accelerated for finding satisfactory solutions. The self-escape strategy improves the organism versatility in order to prevent ECSDSOS from being trapped into local minima. The premature convergence is constrained and the global optimization capability is improved. ECSDSOS is tested with certain real-world TSP instances in TSPLIB. The results are compared with those computed with basic DSOS and several other heuristic algorithms. Under the same preconditions, ECSDSOS shows competitive optimization capacities for computing better solutions for TSP than basic DSOS and some other heuristic algorithms.

Enhanced List-Based Simulated Annealing Algorithm for Large-Scale Traveling Salesman Problem

List-based simulated annealing (LBSA) algorithm is a novel simulated annealing algorithm where list-based cooling scheme is used to control the change of parameter temperature. Aiming to improve the efficiency of the LBSA algorithm for large-scale optimization problems, this paper proposes an enhanced LBSA (ELBSA) algorithm for solving large-scale traveling salesman problem (TSP). The ELBSA algorithm can drive more sampling at more suitable temperatures and from more promising neighborhoods. Specifically, heuristic augmented sampling strategy is used to ensure that more neighbors are from promising neighborhoods, systematic selection strategy is proposed to guarantee that each component of the current solution has a chance to be improved, and variable Markov chain length (VMCL), based on arithmetic sequence, is used to sample more neighbors at more suitable temperatures. Extensive experiments were performed to show the contribution of the heuristic augmented sampling strategy, and to verify the advantage of using systematic selection and VMCL. Comparative experiments, which were conducted on a wide range of large-scale TSP instances, show that the ELBSA algorithm is better than or competitive with most other state-of-the-art metaheuristics.

Entropy-Based Dynamic Heterogeneous Ant Colony Optimization

To solve large-scale traveling salesman problem (TSP) with better performance, this paper proposes an entropy-based dynamic heterogeneous ant colony optimization (EDHACO). The allotropic mechanism and the heterogeneous colonies model are proposed to balance the convergence and the solution diversity. First, entropy is used to measure the diversity, and the entropy-based allotropic mechanism with three communication strategies can improve the adaptability of EDHACO. Then, the heterogeneous colonies with complementary advantages are proposed to balance the convergence speed and the diversity of the algorithm. Besides, two operators are proposed to improve the performance of the algorithm. The adaptive 3-opt operator can be used to accelerate the convergence, and the dynamic-pheromone-reset operator can be introduced to avoid trapping in a local optimum. Finally, EDHACO is applied to solve TSPs, and the experimental results suggest that it has better performance with higher stability and precision in TSP instances, especially in the large-scale TSP instances.

Multi-Colony Ant Colony Optimization Based on Generalized Jaccard Similarity Recommendation Strategy

Ant Colony Optimization has achieved good results in solving Traveling Salesman Problem (TSP), it has a tendency to fall into local optima and the convergence speed is limited. To address this problem, multi-colony ant colony optimization based on the generalized Jaccard similarity recommendation strategy (JCACO) is proposed. Firstly, two classical ant populations, Ant Colony System and Max-Min Ant System are selected to form heterogeneous multi-colony. Secondly, attribute-based collaborative filtering recommendation mechanism is proposed to balance the diversity and convergence of the algorithm, three strategies have been implemented under this recommendation mechanism: The attribute cross-learning strategy is used to highlight the effect of excellent attributes and improve the attribute comprehensive performance; According to the diversity results of the population measured by information entropy, the attribute recommendation learning strategy is used to enrich the diversity of the population adaptively; The pheromone reward strategy is implemented on the public path to accelerate the convergence speed; Among which, according to the generalized Jaccard similarity coefficient, the most suitable communication object is recommended in order to achieve the best learning efficiency. Finally, when the algorithm stagnates, the elite reverse learning mechanism is used to jump out of the local optimum. Experimental results show that JCACO has good performance and high stability in TSP instances, especially in large-scale TSP instances.

On computational complexity of the constructive-optimizer neural network for the traveling salesman problem

Abstract The authors formerly proposed the constructive-optimizer neural network (CONN) for the traveling salesman problem (TSP) to provide the best compromise between the solution quality and convergence speed. However, the computational complexity of CONN were cautiously reported as o(n3). In this paper, by using a probabilistic analysis approach, we prove that the real computational complexity of CONN is of O(n2logn). Three sets of benchmark TSPs from TSPLIB were used to evaluate the performance of CONN. We demonstrated that a polynomial of order n2logn provided the best fit to the CPU time of CONN versus the number of TSP cities. Also, CONN was further compared with a large number of state-of-the-art neural networks in terms of both solution quality and CPU time. We demonstrated that for ordinary TSPs, CONN may provide the best tradeoff between the CPU time and solution quality while for very large-scale TSPs, the memetic self-organizing map may be preferred.

An Improved Genetic Algorithm with a New Initialization Mechanism Based on Regression Techniques

Genetic algorithm (GA) is one of the well-known techniques from the area of evolutionary computation that plays a significant role in obtaining meaningful solutions to complex problems with large search space. GAs involve three fundamental operations after creating an initial population, namely selection, crossover, and mutation. The first task in GAs is to create an appropriate initial population. Traditionally GAs with randomly selected population is widely used as it is simple and efficient; however, the generated population may contain poor fitness. Low quality or poor fitness of individuals may lead to take long time to converge to an optimal (or near-optimal) solution. Therefore, the fitness or quality of initial population of individuals plays a significant role in determining an optimal or near-optimal solution. In this work, we propose a new method for the initial population seeding based on linear regression analysis of the problem tackled by the GA; in this paper, the traveling salesman problem (TSP). The proposed Regression-based technique divides a given large scale TSP problem into smaller sub-problems. This is done using the regression line and its perpendicular line, which allow for clustering the cities into four sub-problems repeatedly, the location of each city determines which category/cluster the city belongs to, the algorithm works repeatedly until the size of the subproblem becomes very small, four cities or less for instance, these cities are more likely neighboring each other, so connecting them to each other creates a somehow good solution to start with, this solution is mutated several times to form the initial population. We analyze the performance of the GA when using traditional population seeding techniques, such as the random and nearest neighbors, along with the proposed regression-based technique. The experiments are carried out using some of the well-known TSP instances obtained from the TSPLIB, which is the standard library for TSP problems. Quantitative analysis is carried out using the statistical test tools: analysis of variance (ANOVA), Duncan multiple range test (DMRT), and least significant difference (LSD). The experimental results show that the performance of the GA that uses the proposed regression-based technique for population seeding outperforms other GAs that uses traditional population seeding techniques such as the random and the nearest neighbor based techniques in terms of error rate, and average convergence.

A Hierarchical Algorithm Based on Density Peaks Clustering and Ant Colony Optimization for Traveling Salesman Problem

This paper proposed a hierarchical hybrid algorithm for traveling salesman problem (TSP) according to the idea of divide-and-conquer. The TSP problem is decomposed into a few subproblems with small-scale nodes by density peaks clustering algorithm. Every subproblem is resolved by ant colony optimization algorithm, this is the lower layer. The center nodes of all subproblems constitute a new TSP problem, which forms the upper layer. All local tours of these subproblems are joined to generate the initial global tour in the same order that the center nodes are traversed in the upper layer TSP problem. Finally, the global tour is optimized by k-Opt algorithms. Thirty benchmark instances taken from TSPLIB are divided into three groups on the basis of problem size: small-scale, large-scale, and very large-scale. The experimental result shows that the proposed algorithm can obtain the solutions with higher accuracy and stronger robustness, and significantly reduce runtime, especially for the very large-scale TSP problem.

Parameter Optimization in GA for Large-scale Traveling Salesman Problem

Traveling Salesman Problems (TSP) is a typical problem in computer science and it is also the most concerned problem for researchers in combinatorial optimization. Therefore, how to solve the optimal loop of TSP problem is a hot topic in researchers. At present, the genetic algorithm has been mature in solving the TSP problem, but the research on the optimization of genetic algorithm parameters is still relatively less. Therefore, in this paper, we analyzed the relevant parameters of the genetic algorithm, optimized the parameters in the application of TSP problem, strengthened its optimization ability, and gave a calculation framework: Firstly, we sampled and analyzed the parameters by using the experimental design method and found the most influential parameters. Secondly, we established and optimized the proxy model and obtained the optimal parameters. Finally, we verified the results with examples. The experiment proved that this method improved the efficiency of solving TSP problem greatly.

Simulated annealing based symbiotic organisms search optimization algorithm for traveling salesman problem

Symbiotic Organisms Search (SOS) algorithm is an effective new metaheuristic search algorithm, which has recently recorded wider application in solving complex optimization problems. SOS mimics the symbiotic relationship strategies adopted by organisms in the ecosystem for survival. This paper, presents a study on the application of SOS with Simulated Annealing (SA) to solve the well-known traveling salesman problems (TSPs). The TSP is known to be NP-hard, which consist of a set of (n−1)!/2 feasible solutions. The intent of the proposed hybrid method is to evaluate the convergence behaviour and scalability of the symbiotic organism's search with simulated annealing to solve both small and large-scale travelling salesman problems. The implementation of the SA based SOS (SOS-SA) algorithm was done in the MATLAB environment. To inspect the performance of the proposed hybrid optimization method, experiments on the solution convergence, average execution time, and percentage deviations of both the best and average solutions to the best known solution were conducted. Similarly, in order to obtain unbiased and comprehensive comparisons, descriptive statistics such as mean, standard deviation, minimum, maximum and range were used to describe each of the algorithms, in the analysis section. The Friedman's Test (with post hoc tests) was further used to compare the significant difference in performance between SOS-SA and the other selected state-of-the-art algorithms. The performances of SOS-SA and SOS are evaluated on different sets of TSP benchmarks obtained from TSPLIB (a library containing samples of TSP instances). The empirical analysis’ results show that the quality of the final results as well as the convergence rate of the new algorithm in some cases produced even more superior solutions than the best known TSP benchmarked results.