Traveling Salesman Problem Instances Research Articles

Discrete Salp Swarm Algorithm for symmetric traveling salesman problem.

In the Salp Swarm Algorithm (SSA), the update mechanism is inspired by the unique chain movement of the salp swarm. Numerous versions of SSA were already put forward to deal with various optimization problems, but there are very few discrete versions among them. d-opt is improved based on the 2-opt algorithm: a decreasing factor d is introduced to control the range of neighborhood search; TPALS are modified by Problem Aware Local Search (PALS) based on the characteristics of Travelling Salesman Problem (TSP); The second leader mechanism increases the randomness of the algorithm and avoids falling into the local optimal solution to a certain extent. We also select six classical crossover operators to experiment and select Subtour Exchange Crossover (SEC) and the above three mechanisms to integrate them into the SSA algorithm framework to form Discrete Salp Swarm Algorithm (DSSA). In addition, DSSA was tested on 23 known TSP instances to verify its performance. Comparative simulation studies with other advanced algorithms are conducted and from the results, it is observed that DSSA satisfactorily solves TSP.

Heuristic smoothing ant colony optimization with differential information for the traveling salesman problem

The traveling salesman problem (TSP) is an NP complete problem with potential applications. Thus far, numerous approaches have been proposed to solve the TSP including exact, heuristic and metaheuristic methods. Among them, the ant colony optimization (ACO) algorithm, belonging to metaheuristic methods, has proven to be an efficient method for solving the TSP. However, the phenomena of rapid convergence to local optima and unsatisfactory computational accuracy distinctly limit the performance of ACO. This study therefore proposes a novel ACO algorithm (HSDACO) to compensate for these shortcomings. In the HSDACO algorithm, the involvement of heterogeneous population automation at each iteration generates better candidate solutions with maintaining parameter diversity. Then, the implementation of three smoothing techniques with the 2-Opt method guarantees the effective assessment of candidate solutions in favor of higher quality. Next, the introduction of a differential information updating mechanism, using differential edge information obtained from the candidate solutions, enables the evaporation operation of the pheromone trail to obtain more reasonable guidance. Subsequently, the design of evolutionary state estimation and adjustment is used to monitor the population under correct guidance and regulate the evolutionary state effectively and efficiently. Finally, the HSDACO algorithm is evaluated on public TSP benchmark instances and compared with the other state-of-the-art algorithms. Experimental results show that the proposed HSDACO achieves substantial improvement for mid-scale TSP instances and overwhelming advantages for small-scale TSP instances, in terms of solution accuracy and convergence speed.

A hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system for the travelling salesman problem

AbstractThe ant colony optimization (ACO) is one efficient approach for solving the travelling salesman problem (TSP). Here, we propose a hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system (SSMFAS) to address the TSP. The state-adaptive slime mold (SM) model with two targeted auxiliary strategies emphasizes some critical connections and balances the exploration and exploitation ability of SSMFAS. The consideration of fractional-order calculus in the ant system (AS) takes full advantage of the neighboring information. The pheromone update rule of AS is modified to dynamically integrate the flux information of SM. To understand the search behavior of the proposed algorithm, some mathematical proofs of convergence analysis are given. The experimental results validate the efficiency of the hybridization and demonstrate that the proposed algorithm has the competitive ability of finding the better solutions on TSP instances compared with some state-of-the-art algorithms.

Solving a Multi-Conveyance Travelling Salesman Problem using an Ant Colony Optimization Method

Objectives: A well-known NP-complete problem is the travelling salesman problem (TSP). It has numerous engineering and scientific applications. In this article, we have proposed a multi-conveyance TSP where different conveyances are present to travel from one city to another city. This is an extension to classical TSP. In this TSP, the salesman visits all the cities only once during his/her tour, using different conveyances to travel from one city to another. The cost of travelling between cities using various modes of conveyance varies. The objective of this research is to find the minimum cost tour using an ant colony optimization (ACO) based approach by satisfying the constraints of the proposed multi-conveyance TSP. Method: The considered TSP has been solved using a novel ACO technique. The proposed ACO is adopted with the Roulette-wheel selection and “tuning solution” techniques. We have used a few benchmark datasets from TSPLIB to check the effectiveness of the proposed algorithm. The experimental findings for a few benchmarks TSP instances show that almost always, the proposed ACO is able to find a better result. Then, we used some redefined and randomly generated datasets for experiments. The experimental outcomes for various input datasets are also very encouraging. Findings: The goal of the proposed TSP is to find a complete tour with a minimum cost without exceeding the total travel cost and total travel time. Thus, there are two novel constraints in the classical TSP. Novelty: A unique aspect of the proposed research is the use of several conveyance facilities and a fixed total travel time and cost. This is new, as these three factors are integrated into a single TSP model. Keywords: Travelling Salesman Problem; Travel Cost; Travel Time; MultiConveyance TSP; Ant Colony Optimization

Approximately optimal construction of parallel algorithm portfolios by evolutionary intelligence

<p indent="0mm">As a high-performance general-purpose form of parallel solvers, parallel algorithm portfolios (PAPs) have recently shown great performance for solving decision, counting, continuous, and discrete optimization problems. However, the manual construction of PAPs is laborious work, which typically requires substantial domain knowledge and human effort. To address this issue, this paper proposes AutoPAP, an evolutionary intelligence-based method for PAP construction. Overall, AutoPAP follows the (<italic>n</italic>+1) evolutionary framework, i.e., in each generation, <italic>n</italic> candidate algorithms are generated, and the best one among them is retained in the PAP. Considering that the algorithm configuration space is typically very large, this study designs a highly effective mutation operator to improve the practical performance of AutoPAP and further theoretically proves that AutoPAP can achieve (1−1/<italic>e</italic>) approximation. Finally, to validate the effectiveness of AutoPAP, this study uses it to build a PAP, namely, TSP_PAP, for traveling salesman problems (TSPs). The test results show that TSP_PAP significantly outperforms state-of-the-art TSP solvers EAX and LKH in terms of efficiency (runtime) and effectiveness (solution quality). On 128 TSP instances of sizes ranging from 1000 to 30000, compared with LKH and EAX, TSP_PAP can reduce the average runtime by at least 45.71% and lower the average deviation ratio by at least 87.50%, indicating the huge potential of AutoPAP in automatic algorithm design and evolution.

Visibility Adaptation in Ant Colony Optimization for Solving Traveling Salesman Problem

Ant Colony Optimization (ACO) is a practical and well-studied bio-inspired algorithm to generate feasible solutions for combinatorial optimization problems such as the Traveling Salesman Problem (TSP). ACO is inspired by the foraging behavior of ants, where an ant selects the next city to visit according to the pheromone on the trail and the visibility heuristic (inverse of distance). ACO assigns higher heuristic desirability to the nearest city without considering the issue of returning to the initial city or starting point once all the cities are visited. This study proposes an improved ACO-based method, called ACO with Adaptive Visibility (ACOAV), which intelligently adopts a generalized formula of the visibility heuristic associated with the final destination city. ACOAV uses a new distance metric that includes proximity and eventual destination to select the next city. Including the destination in the metric reduces the tour cost because such adaptation helps to avoid using longer links while returning to the starting city. In addition, partial updates of individual solutions and 3-Opt local search operations are incorporated in the proposed ACOAV. ACOAV is evaluated on a suite of 35 benchmark TSP instances and rigorously compared with ACO. ACOAV generates better solutions for TSPs than ACO, while taking less computational time; such twofold achievements indicate the proficiency of the individual adoption techniques in ACOAV, especially in AV and partial solution update. The performance of ACOAV is also compared with the other ten state-of-the-art bio-inspired methods, including several ACO-based methods. From these evaluations, ACOAV is found as the best one for 29 TSP instances out of 35 instances; among those, optimal solutions have been achieved in 22 instances. Moreover, statistical tests comparing the performance revealed the significance of the proposed ACOAV over the considered bio-inspired methods.

Learning to Solve Travelling Salesman Problem with Hardness-Adaptive Curriculum

Various neural network models have been proposed to tackle combinatorial optimization problems such as the travelling salesman problem (TSP). Existing learning-based TSP methods adopt a simple setting that the training and testing data are independent and identically distributed. However, the existing literature fails to solve TSP instances when training and testing data have different distributions. Concretely, we find that different training and testing distribution will result in more difficult TSP instances, i.e., the solution obtained by the model has a large gap from the optimal solution. To tackle this problem, in this work, we study learning-based TSP methods when training and testing data have different distributions using adaptive-hardness, i.e., how difficult a TSP instance can be for a solver. This problem is challenging because it is non-trivial to (1) define hardness measurement quantitatively; (2) efficiently and continuously generate sufficiently hard TSP instances upon model training; (3) fully utilize instances with different levels of hardness to learn a more powerful TSP solver. To solve these challenges, we first propose a principled hardness measurement to quantify the hardness of TSP instances. Then, we propose a hardness-adaptive generator to generate instances with different hardness. We further propose a curriculum learner fully utilizing these instances to train the TSP solver. Experiments show that our hardness-adaptive generator can generate instances ten times harder than the existing methods, and our proposed method achieves significant improvement over state-of-the-art models in terms of the optimality gap. The codes are publicly available.

Solving the clustered traveling salesman problem via traveling salesman problem methods.

The Clustered Traveling Salesman Problem (CTSP) is a variant of the popular Traveling Salesman Problem (TSP) arising from a number of real-life applications. In this work, we explore a transformation approach that solves the CTSP by converting it to the well-studied TSP. For this purpose, we first investigate a technique to convert a CTSP instance to a TSP and then apply powerful TSP solvers (including exact and heuristic solvers) to solve the resulting TSP instance. We want to answer the following questions: How do state-of-the-art TSP solvers perform on clustered instances converted from the CTSP? Do state-of-the-art TSP solvers compete well with the best performing methods specifically designed for the CTSP? For this purpose, we present intensive computational experiments on various benchmark instances to draw conclusions.

A computational optimization research on ant colony optimization for the traveling salesman problem

Abstract The traveling salesman problem (TSP) is one of typical combinatorial optimization problems. Ant colony optimization (ACO) is an effective method to solve the traveling salesman problem, but there are some non-negligible shortcomings hidden in the original algorithm. The primary objective of this research is to optimize the ACO to produce quality work throughout solving TSP. To this end, the hybrid SOS-MMAS algorithm is proposed. Concretely, apply the advanced Max-Min Ant System (MMAS) as the basic algorithm to raise task scheduling efficiency, meanwhile introduce symbiotic organisms search (SOS) into the MMAS to optimize the key parameters. Experiments were carried out on typical TSP instances of different scales, and the SOS-ACO and ACO algorithms were compared with SOS-MMAS, which proved the excellent performance of SOS-MMAS in solving TSP. Rationality of the algorithm design and high performance has been illuminated by experimentation. In addition, the model also could serve to suggest further research of TSP or other related areas.

Adaptive gradient descent enabled ant colony optimization for routing problems

The design of Ant Colony Optimization (ACO) has been inspired by the foraging behavior of ant colonies. ACO is one of the most widely used metaheuristic algorithms applicable to various optimization problems. Nevertheless, the deficiency of ACO is early maturity and slow convergence, usually leading to dissatisfactory performance. In this research, a new type of ACO is proposed with innovative adaptive learning mechanism is devised (the algorithm is called ADACO). In the paper, first, the ACO algorithm for Traveling Salesman Problem (TSP) is modeled in the framework of reinforcement learning (RL) and learns a best policy by stochastic gradient descent (SGD). Then, an adaptive gradient descent strategy, which can exploit the update history of per-dimensional pheromones to achieve intelligent convergence, is integrated into the ACO algorithm as ADACO. A parallel computation process is implemented in the process to expedite computational efficiency. Finally, through case studies in TSP and Capacitated Vehicle Routing Problem (CVRP), ADACO is validated. ADACO is trialed on various sizes of TSP and CVRP instances and compared with other state-of-the-art algorithms. Results show that ADACO is competitive in terms of accuracy (better in most cases), stability (statistically significant) and adaptability (support various types of problems). The results also elucidate that the algorithm maintains good computational efficiency owing to its parallel implementation. It can be concluded that ADACO is effectively applicable to routing problems with good performance.

Improving Ant Colony Optimization efficiency for solving large TSP instances

Ant Colony Optimization (ACO) is a family of nature-inspired metaheuristics often applied to finding approximate solutions to difficult optimization problems. Despite being significantly faster than exact methods, the ACOs can still be prohibitively slow, especially if compared to basic problem-specific heuristics. As recent research has shown, it is possible to significantly improve the performance through algorithm refinements and careful parallel implementation benefiting from multi-core CPUs and dedicated accelerators. In this paper, we present a novel ACO variant, namely the Focused ACO (FACO). One of the core elements of the FACO is a mechanism for controlling the number of differences between a newly constructed and a selected previous solution. The mechanism results in a more focused search process, allowing to find improvements while preserving the quality of the existing solution. An additional benefit is a more efficient integration with a problem-specific local search. Computational study based on a range of the Traveling Salesman Problem instances shows that the FACO outperforms the state-of-the-art ACOs when solving large TSP instances. Specifically, the FACO required less than an hour of an 8-core commodity CPU time to find high-quality solutions (within 1% from the best-known results) for TSP Art Instances ranging from 100000 to 200000 nodes.

Lower bounds on the size of general branch-and-bound trees

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form \(\pi ^{\top } x \le \pi _0 \,\vee \, \pi ^{\top }x \ge \pi _0 + 1\), where \(\pi \) is an integer vector and \(\pi _0\) is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists even when we add Gaussian noise to the coefficients of the cross-polytope, thus showing that a polynomial-size “smoothed analysis” upper bound is not possible. The results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. (Linear Algebra Appl 114:455–499, 1989), who proved lower bounds for the Chvátal–Gomory rank.

A Carnivorous Plant Algorithm With Heuristic Decoding Method for Traveling Salesman Problem

The traveling salesman problem (TSP) is one of the most extensively studied problems in the combinatorial optimization area and still presents unsolved challenges due to its NP-hard attribute. Although many real-coded algorithms are available for TSP, they still have some performance challenges in the switch from continuous space to discrete space and perform at low convergence speed. This paper proposes a real-coded carnivorous plant algorithm with a heuristic decoding method (CPA-HDM) to solve the traveling salesman problem (TSP), which exhibits good convergence speed and solution accuracy. In this improved method, a new heuristic decoding method (HDM) is designed, which helps to map continuous variables to discrete ones without losing information, maintain population diversity, and enhance the solution quality after decoding. To balance the algorithm’s search capability and enhance the probability of preferable individuals generated, an adaptive attraction probability (AAP), an improved growth model of carnivorous plants (IGMOCP), and a position update method of prey (IPUMOP) are developed. Aiming to reduce the probability of premature and prevent search stagnation, an improved reproduction strategy (IRS) and an adaptive combination perturbation are reconstructed. Finally, a local search algorithm is employed to improve the exploitation capability. To verify its validation, CPA-HDM is compared with six algorithms, for solving 28 TSP instances. The simulation results and statistical analyses demonstrate the superior performance of the proposed algorithm.

Finding the best tour for travelling salesman problem using artificial ecosystem optimization

This paper presents a new method based on the artificial ecosystem optimization (AEO) algorithm for finding the shortest tour of the travelling salesman problem (TSP). Wherein, AEO is a newly developed algorithm based on the idea of the energy flow of living organisms in the ecosystem consisting of production, consumption and decomposition mechanisms. In order to improve the efficiency of the AEO for the TSP problem, the 2-opt movement technique is equipped to enhance the quality of the solutions created by the AEO. The effectiveness of AEO for the TSP problem has been verified on four TSP instances consisting of the 14, 30, 48 and 52 cities. Based on the calculated results and the compared results with the previous methods, the proposed AEO method is one of the effective approaches for solving the TSP problem.

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.

Ant Colony Optimization with Warm-Up

The Ant Colony Optimization (ACO) is a probabilistic technique inspired by the behavior of ants for solving computational problems that may be reduced to finding the best path through a graph. Some species of ants deposit pheromone on the ground to mark some favorable paths that should be used by other members of the colony. Ant colony optimization implements a similar mechanism for solving optimization problems. In this paper a warm-up procedure for the ACO is proposed. During the warm-up, the pheromone matrix is initialized to provide an efficient new starting point for the algorithm, so that it can obtain the same (or better) results with fewer iterations. The warm-up is based exclusively on the graph, which, in most applications, is given and does not need to be recalculated every time before executing the algorithm. In this way, it can be made only once, and it speeds up the algorithm every time it is used from then on. The proposed solution is validated on a set of traveling salesman problem instances, and in the simulation of a real industrial application for the routing of pickers in a manual warehouse. During the validation, it is compared with other ACO adopting a pheromone initialization technique, and the results show that, in most cases, the adoption of the proposed warm-up allows the ACO to obtain the same or better results with fewer iterations.

Transformation operators based grey wolf optimizer for travelling salesman problem

In the field of swarm intelligence, the Grey Wolf Optimizer (GWO) is a popular algorithm based on leadership hierarchy. Primarily, GWO was proposed to solve continuous optimization problem. However, in recent years, GWO has been extensively explored to deal with a wide variety of real world problem regardless the nature of problem. GWO has received a lot of attention from researchers because of its advantages over other swarm intelligence approaches and its simplicity. The classical GWO is redesigned in this paper by incorporating the swap, shift and symmetry transformation operators to solve permutation-coded travelling salesman problem (TSP), and it is named as transformation operator based grey wolf optimizer (TO-GWO). In TO-GWO, each wolf represents a possible solution of TSP and using swap, shift and symmetry operators wolves interact with leader wolves in order to obtain optimal solution for TSP. In order to improve the proposed algorithm’s local search capability when solving discrete problems, 2-opt algorithm have also been adapted. The TO-GWO is implemented in MATLAB environment. In this study, the TO-GWO is tested over 50 TSP instances. Also, the results of proposed algorithm are compared with 12 state-of-the-art algorithms for TSP instances with various numbers of cities in order to evaluate its performance. For the majority of the TSP instances used in the experiment, the TO-GWO significantly outperforms other algorithms in terms of quality of solutions and efficiency.

A New Constructive Heuristic Driven by Machine Learning for the Traveling Salesman Problem

Recent systems applying Machine Learning (ML) to solve the Traveling Salesman Problem (TSP) exhibit issues when they try to scale up to real case scenarios with several hundred vertices. The use of Candidate Lists (CLs) has been brought up to cope with the issues. A CL is defined as a subset of all the edges linked to a given vertex such that it contains mainly edges that are believed to be found in the optimal tour. The initialization procedure that identifies a CL for each vertex in the TSP aids the solver by restricting the search space during solution creation. It results in a reduction of the computational burden as well, which is highly recommended when solving large TSPs. So far, ML was engaged to create CLs and values on the elements of these CLs by expressing ML preferences at solution insertion. Although promising, these systems do not restrict what the ML learns and does to create solutions, bringing with them some generalization issues. Therefore, motivated by exploratory and statistical studies of the CL behavior in multiple TSP solutions, in this work, we rethink the usage of ML by purposely employing this system just on a task that avoids well-known ML weaknesses, such as training in presence of frequent outliers and the detection of under-represented events. The task is to confirm inclusion in a solution just for edges that are most likely optimal. The CLs of the edge considered for inclusion are employed as an input of the neural network, and the ML is in charge of distinguishing when such edge is in the optimal solution from when it is not. The proposed approach enables a reasonable generalization and unveils an efficient balance between ML and optimization techniques. Our ML-Constructive heuristic is trained on small instances. Then, it is able to produce solutions—without losing quality—for large problems as well. We compare our method against classic constructive heuristics, showing that the new approach performs well for TSPLIB instances up to 1748 cities. Although ML-Constructive exhibits an expensive constant computation time due to training, we proved that the computational complexity in the worst-case scenario—for the solution construction after training—is O(n2logn2), n being the number of vertices in the TSP instance.

A Multi Ant Colony Optimization Approach For The Traveling Salesman Problem

In this paper, we propose a new approach to solving the Traveling Salesman Problem (TSP), for which no exact algorithm is known that allows to find a solution in polynomial time. The proposed approach is based on optimization by ants. It puts several colonies in competition for improved solutions (in execution time and solution quality) to large TSP instances, and allows to efficiently explore the range of possible solutions. The results of our experiments show that the approach leads to better results compared to other heuristics from the literature, especially in terms of the quality of solutions obtained and execution time.

Solving traveling salesman problem using hybridization of rider optimization and spotted hyena optimization algorithm

Traveling Salesman Problem (TSP) is the combinatorial optimization problem, where a salesman starting from a home city travels all the other cities and returns to the home city in the shortest possible path. TSP is a popular problem because the instances of TSP can be applied to solve real-world problems, the implication of which turns TSP into a typical test bench for performance evaluation of novel algorithms. In the current years, different optimization algorithms inspired by biological groups have become very familiar. A combined intelligence of diverse social insects like bees, ants, birds, termites, fish, etc. has been analyzed to introduce multiple meta-heuristic algorithms in the field of swarm intelligence. The main intent of this paper is to develop a hybrid algorithm for solving the TSP robustly and effectively. In order to attain this challenging point, the objective model considered in this research work is the minimization of the distance of the salesman traveling through entire cities. Here, the optimal solution pertains to solve the TSP is to minimize the distance travelled by the salesman, which is determined based on the new hybrid optimization algorithm. This proposal plans to integrate the two well-performing optimization algorithms like Rider Optimization Algorithm (ROA) and Spotted Hyena Optimizer algorithm (SHO) to frame the new algorithm, Spotted Hyena-based Rider Optimization (S-ROA). Finally, the experimental results obtained by the hybrid algorithm to solve these TSP cases are benchmarked against the results obtained by using state-of-the-art algorithms and prove the competitive performance of the proposed model.