Algorithm For Traveling Salesman Problem Research Articles

Improvement of the Nearest Neighbor Heuristic Search Algorithm for Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is classified as a non-deterministic polynomial (NP) hard problem, which has found widespread application in several scientific and technological domains. Due to its NP-hard nature, it is very hard to solve effectively and efficiently. Despite this rationale, a multitude of optimization approaches have been proposed and developed by scientists and researchers during the last several decades. Among these several algorithms, heuristic approaches are deemed appropriate for addressing this intricate issue. One of the simplest and most easily implementable heuristic algorithms for TSP is the nearest neighbor algorithm (NNA). However, its solution quality suffers owing to randomness in the optimization process. To address this issue, this study proposes a deterministic NNA for solving symmetric TSP. It is an improved version of NNA, which starts with the shortest edge consisting of two cities and then repeatedly includes the closest city on the route until an effective route is established. The simulation is conducted on 20 benchmark symmetric TSP datasets obtained from TSPLIB. The simulation results provide evidence that the improved NNA outperforms the basic NNA throughout most of the datasets in terms of solution quality as well as computational time.

LEADERS AND FOLLOWERS ALGORITHM FOR TRAVELING SALESMAN PROBLEM

Leaders and Followers algorithm is a metaheuristics algorithm. In solving continuous optimization, this algorithm is proved to be better than other well-known algorithms, such as Genetic Algorithm and Particle Swarm Optimization. This paper aims to apply the Leaders and Followers algorithm for the Traveling Salesman Problem (TSP), a well-known combinatorial optimization problem to minimize distance. There are some modifications in order to fit the algorithm in TSP problems. Some most-used-problems in TSP are used to test this algorithm. The result is that the Leaders and Followers algorithm performs well, stable, and guarantees the optimality of the obtained solution in TSP with fewer than 20 cities. In TSP with a bigger number of cities, the proposed algorithm is not stable and might has difficulties in finding the optimal solutions.

A Novel Crossover based Discrete Artificial Algae Algorithm for Solving Traveling Salesman Problem

The Artificial Algae Algorithm (AAA) is a newly proposed metaheuristic algorithm that is inspired by microalgae behaviors. This algorithm has been proposed for solving continuous optimization problems and achieved good results for the continuous problems. In addition, binary versions of AAA are proposed in the literature. This paper presents a discrete version of AAA, which is named Discrete Artificial Algae Algorithm (DAAA). For discretization of AAA, Crossover operators (one-point and uniform) are used in the processes (helical movement, evolutionary process, and adaptation). In this study, in addition to crossover operators, transformation operators such as swapping, insertion, symmetry, and reversion are also used. DAAA’s performance was analyzed on a well-known discrete optimization problem called the Traveling Salesman Problem (TSP). DAAA was tested on thirty-two Benchmark instances of the TSP. These instances were small-sized, medium-sized, and large-sized. Firstly, the AAA processes (evolutionary process, adaptation, and helical movement) with the combination of nearest neighbor and transformation operators were tested for selected benchmark instances and this testing was called Process Analysis. After this process Analysis the best processes with which to continue were selected, and after this decision comparisons with other algorithms were started. The main comparison is between discrete Social Spider Algorithm (DSSA) and DAAA, and DAAA outperformed DSSA on most of the problems. Further, DAAA’s performance on some of the benchmark instances was compared with some of the well-known algorithms for TSP. In this comparison, DAAA has achieved better results than many other algorithms. Experimental results show that DAAA has the capability of solving discrete optimization problems and outperforming other algorithms.

Extended TANYAKUMU Labelling Method to Compute Shortest Paths in Directed Networks

Shortest path problem (SPP) has various applications in areas such as telecommunications, transportation and emergency services, and postal services among others. As a result, several algorithms have been developed to solve the SPP and related problems. The current paper extends the TANYAKUMU labelling method for solving the Travelling salesman problem (TSP) to solve SPP in directed transportation networks. Numerical illustrations are used to prove the validity of the proposed method. The main contributions of this paper are as follows: (i) modification of TSP algorithm to solve single source SPP, (ii) the developed method numerically evaluated on four increasingly complex problems of sizes 11×11, 21×21, 23×23 and 26×26 and lastly (iii) the solutions obtained from solving these four problems are compared with those obtained by Minimum incoming weight label (MIWL) algorithm. The proposed algorithm computed the same shortest paths as the MIWL algorithm on all four problems.

Real-time Fast Selection System with Object Recognition and TSP algorithms

The stage before the conversion of agricultural products into post-harvest consumer products is the process of separating the raw products into appropriate classes. Today, this difficult manual separating process is a process in which a large number of workers work at an intense pace on the product line and the workforce is intensively spent. Disruptions in separating as a result of carelessness cause product loss, loss of time and cost increases. In this study, as an alternative to manual separating processes, a real-time separating system, which detects the products in the factory band with object recognition methods and enables fast positioning of the separating tool on the products, works simultaneously with object recognition and traveling salesman problem algorithms has been created. In this way, a low-budget separating system is recommended for large selecting processes with a time- and cost-effective selecting model. In the study, the creation of a real-time fast separating system with the support of the traveling salesman algorithm, performance evaluation and research and findings on the fast separating model are presented.

Learning-Augmented Algorithms for Online TSP on the Line

We study the online Traveling Salesman Problem (TSP) on the line augmented with machine-learned predictions. In the classical problem, there is a stream of requests released over time along the real line. The goal is to minimize the makespan of the algorithm. We distinguish between the open variant and the closed one, in which we additionally require the algorithm to return to the origin after serving all requests. The state of the art is a 1.64-competitive algorithm and a 2.04-competitive algorithm for the closed and open variants, respectively. In both cases, a tight lower bound is known. In both variants, our primary prediction model involves predicted positions of the requests. We introduce algorithms that (i) obtain a tight 1.5 competitive ratio for the closed variant and a 1.66 competitive ratio for the open variant in the case of perfect predictions, (ii) are robust against unbounded prediction error, and (iii) are smooth, i.e., their performance degrades gracefully as the prediction error increases. Moreover, we further investigate the learning-augmented setting in the open variant by additionally considering a prediction for the last request served by the optimal offline algorithm. Our algorithm for this enhanced setting obtains a 1.33 competitive ratio with perfect predictions while also being smooth and robust, beating the lower bound of 1.44 we show for our original prediction setting for the open variant. Also, we provide a lower bound of 1.25 for this enhanced setting.

An Optimization Method for CNC Laser Combination Cutting of Irregular Plate Remainders

The key research question in this study is how to cut pieces in irregular plate remainders, because there are many irregular plate leftovers created during the CNC (Computer Numerical Control) process. This will increase material utilization and allow plate leftovers to be reused. One of the issues being researched is how to arrange plate remainders on the surface of the CNC machine; this issue is known as combination layout optimization. The other issue being researched is combination cutting-path optimization of plate remainders, which aims to determine the cutting path of parts of plate remainders. A genetic algorithm based on the gravity-center NFP (No-Fit Polygon) method was applied to optimize the layout pattern, and then the geometric coordinates of a part included in one plate remainder after packing were obtained by geometric transformation with the help of a three-layer graphic data correlation model, which quickly identified the inside and outside contours of parts. A colony algorithm based on the mathematical model of cutting-path optimization was used to optimize the cutting path of the parts in the plate remainders. Finally, some simulation tests were performed to illustrate the feasibility and effectiveness of the proposed method. The results of the algorithm for packing irregular shapes for some instances show that our algorithm outperforms the other algorithms. On most instances, the average plate utilization ratio using our algorithm, after running 20 times, is improved by 1% to 9% in comparison to the best plate utilization ratio using the tree search algorithm. The best idle travel of an example achieved by the algorithm in this paper is 7632 mm after running the cutting-path optimization algorithm 20 times, while that of the traditional equivalent TSP (Traveling Salesman Problem) algorithm is 11,625 mm, which significantly demonstrates the efficiency of the algorithm.

A Graph Pointer Network-Based Multi-Objective Deep Reinforcement Learning Algorithm for Solving the Traveling Salesman Problem

Traveling Salesman Problems (TSPs) have been a long-lasting interesting challenge to researchers in different areas. The difficulty of such problems scales up further when multiple objectives are considered concurrently. Plenty of work in evolutionary algorithms has been introduced to solve multi-objective TSPs with promising results, and the work in deep learning and reinforcement learning has been surging. This paper introduces a multi-objective deep graph pointer network-based reinforcement learning (MODGRL) algorithm for multi-objective TSPs. The MODGRL improves an earlier multi-objective deep reinforcement learning algorithm, called DRL-MOA, by utilizing a graph pointer network to learn the graphical structures of TSPs. Such improvements allow MODGRL to be trained on a small-scale TSP, but can find optimal solutions for large scale TSPs. NSGA-II, MOEA/D and SPEA2 are selected to compare with MODGRL and DRL-MOA. Hypervolume, spread and coverage over Pareto front (CPF) quality indicators were selected to assess the algorithms’ performance. In terms of the hypervolume indicator that represents the convergence and diversity of Pareto-frontiers, MODGRL outperformed all the competitors on the three well-known benchmark problems. Such findings proved that MODGRL, with the improved graph pointer network, indeed performed better, measured by the hypervolume indicator, than DRL-MOA and the three other evolutionary algorithms. MODGRL and DRL-MOA were comparable in the leading group, measured by the spread indicator. Although MODGRL performed better than DRL-MOA, both of them were just average regarding the evenness and diversity measured by the CPF indicator. Such findings remind that different performance indicators measure Pareto-frontiers from different perspectives. Choosing a well-accepted and suitable performance indicator to one’s experimental design is very critical, and may affect the conclusions. Three evolutionary algorithms were also experimented on with extra iterations, to validate whether extra iterations affected the performance. The results show that NSGA-II and SPEA2 were greatly improved measured by the Spread and CPF indicators. Such findings raise fairness concerns on algorithm comparisons using different fixed stopping criteria for different algorithms, which appeared in the DRL-MOA work and many others. Through these lessons, we concluded that MODGRL indeed performed better than DRL-MOA in terms of hypervolumne, and we also urge researchers on fair experimental designs and comparisons, in order to derive scientifically sound conclusions.

Toolpath optimisation for drilling process using shortest distance permutation algorithms

ABSTRACT The toolpath optimisation has become the need of the hour due to fourth industrial revolution. The article presents an investigation on the toolpath optimisation for drilling process. Two novel algorithms have been introduced and implemented, namely Random Start of Shortest Distance Permutation (RSSDP) and First Point Start Shortest Distance Permutation (FPSSDP). The outcome has been compared with Shortest Distance Location (SDL) algorithm and Travelling Salesman Problem algorithm (TSP). MATLAB is used to implement the algorithms. Moreover, virtual simulation is carried out in CATIA V5 software. The results have shown significant reduction in the non-productive machining time.

Optimization of Warehouse Picking Based on TSP

Modern warehouse management has entered the era of information intelligence, which requires the establishment of a set of accurate, reasonable and efficient warehouse management methods, to improve the efficiency of workers. In this paper, a series of TSP (Traveling salesman problem) algorithms are used to further optimize the process efficiency of goods removal from the warehouse. Based on Freudian algorithm, the distance problem between cargo grid and recheck station was solved in the first problem. Founded on the genetic algorithm in TSP, specific application problems were solved in two, three and four questions. For problem 1, by observing the relation between the coordinates given and the actual route, all the coordinates of cargo grid were divided into two kinds: odd and even columns, and the two kinds of coordinates were processed, respectively. After coordinate processing, the distance matrix between each lattice can be obtained by using Freudian algorithm. At the same time, due to the small number of recheck stations and their regular distribution, the distance matrix between each recheck station and each cargo grid can be obtained by manually classifying the coordinates of the recheck stations and applying Freud algorithm again. Output all the matrices in the same EXCEL table, the distance matrix between the 3013 elements can be obtained. (See Appendix 1 for the distance matrix and Appendix 1 for the algorithm). For problem 2, this problem is a unidirectional TSP problem by macro analysis. Firstly, the distance matrix of the required point was called from the solution of problem 1, which was imported into LINGO program to establish 0-1 decision variables, and the objective function is established to solve according to the principles of single-direction connection and loop-breaking, to obtain the connection sequence and loop distance. Select the cargo lattice back to the initial recheck station and connect it with the nearest recheck station, to complete the task of breaking the ring, solving the connection sequence and the total distance length. The outbound time can be divided into three parts: (1) journey time; (2) Pick up time; (3) Packing time, group calculation, sum can get the result. The total distance is 382.5m (about 1254.92 ft) and the total minimum time is 462 seconds (about 7 and a half minutes). For problem 3, distance matrix of the required point was called from the solution of problem 1, and imported into LINGO program, afterwards, 0-1 decision variable and objective function were established according to the principles of single-direction connection and loop-breaking to solve, and then the connection sequence and loop distance was obtained. Select the cargo lattice back to the initial recheck station and connect it with the nearest recheck station, to complete the task of breaking the ring, solving the connection sequence and the total distance length. Compared with Question 2, Question 3 specifies the available recheck station and the total time needed to complete the shipment. During the calculation of each task order, it was classified and discussed (double-starting point operation), and the overall optimization was carried out according to the starting point and end point of each task order, and the total shortest time was 2288.6 seconds (about 38 minutes) (See the attachment for the results of this question). For question 4, queuing theory is needed. To reduce the total outbound time, it is necessary to give priority to orders with abbreviated time. Firstly, the shortest outbound time of 49 task orders should be calculated, and the time should be arranged in order from short to long. After that, according to the order of the task list, nine pickers pick goods in a certain order. Hence, a variable should be introduced to preserve the status of the recheck table and the remaining time currently occupied.

Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a \(\theta \)-cyclic point: \(x_e \in \{0,\theta , 1-\theta , 1\}\), where the support graph is subcubic and each vertex is incident to at least one edge with x-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with x-value 1 from the \(\frac{3}{2}\) of Christofides algorithm to \(\frac{3}{2}-\frac{\theta }{10}\) while keeping the usage of edges with fractional x-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a \(\frac{2}{3}\)-uniform point: \(x_e \in \{0, \frac{2}{3}\}\), we give a \(\frac{17}{12}\)-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios \(\frac{3}{2}\) of Christofides algorithm and \(\frac{4}{3}\) implied by the famous “four-thirds conjecture”.

Accelerate Incremental TSP Algorithms on Time Evolving Graphs with Partitioning Methods

In time-evolving graphs, the graph changes at each time interval, and the previously computed results become invalid. We addressed this issue for the traveling salesman problem (TSP) in our previous work and proposed an incremental algorithm where the TSP tour is computed from the previous result instead of the whole graph. In our current work, we have mapped the TSP problem to three partitioning methods named vertex size attribute, edge attribute, and k-means; then, we compared the TSP tour results. We have also examined the effect of increasing the number of partitions on the total computation time. Through our experiments, we have observed that the vertex size attribute performs the best because of a balanced number of vertices in each partition.

Enhancing QoS and Residual Energy by Using of Grid-Size Clustering, K-Means, and TSP Algorithms With MDC in LEACH Protocol

Some recent researches have shown that the energy consumption problem caused by data collection in a wireless sensor network (WSN) based on a static data collector is a main threat to the network lifetime. However, with the progress of the mobile terminal technology, the implementation of mobile data collectors (MDCs) has become more popular in large-scale WSNs, but it remains a big problem to improve the Quality of Service (QoS) criteria and minimize the energy consumption at the same time. However, most existing systems based on MDCs do not successfully strike a balance between routing energy consumption and QoS. In addition, most WSN protocols fail to maintain their impact when the network topology changes. Thus, for a dynamic WSN, it is important to support an intelligent MDC to continue data propagation despite the inevitable changes in the WSN topology. Considering all the above challenges, we propose a new intelligent MDC based on the traveling salesman problem (TSP) to determine the optimal path traveled by the MDC for energy efficiency and latency. Specifically, our proposed Mobile Data Collectors-Traveling Salesman Problem-Low Energy Adaptive Clustering Hierarchy-K-Means (MDC-TSP-LEACH-K) protocol uses K-Means and Grid clustering algorithm to decrease energy consumption in the cluster head (CH) election phase. Additionally, MDC is utilized as an intermediate between CH and the sink to further enhance the QoS of WSNs, to reduce delays while collecting data, and improve the transmission phase of the LEACH protocol.

ITÖ algorithm with local search for large scale multiple balanced traveling salesmen problem

In the fields such as computer science, intelligent transport systems and logistics scheduling, many problems can be reflected as the variants of traveling salesman problem (TSP). In recent years, although there are many achievements and developments on the study of intelligent optimization algorithms for TSP and its variants, the challenge and problem are still existed: while the scale of problem is large, there is the dimension disaster problem; the optimization algorithms for TSP variants are easy to fall into the local optimum. Multiple balanced traveling salesmen problem (MBTSP) is a variant of TSP, it can be used in the fields such as optimizing multiple gas turbine engines. Since MBTSP is NP-hard problem, many intelligent optimization algorithms, such as genetic algorithm and ITÖ algorithm, have been used to solve it. However, while the scale of MBTSP is large, the traditional algorithms are easy to fall into local optimum. Aiming at the problem, this paper extends the scale of MBTSP to large scale scenes, and proposes a novel ITÖ algorithm (NITÖ) based on crossover operator and local search for large scale MBTSP. For NITÖ algorithm, the drift operator and volatility operator of ITÖ algorithm are redesigned, which are carried out by improved crossover operator and local search. The extensive experiments show that NITÖ can demonstrate better solution quality than the compared state-of-the-art algorithms.

Efficient Solid Waste Management in Prai Industrial Area through GIS using Dijkstra and Travelling Salesman Problem Algorithms

The fourth industrial revolution (IR 4.0) supports new solid waste management and effective routing system for collection and transport of solid wastes, especially in achieving Penang 2030 vision to become a pollution free smart city. This study will enhance Seberang Perai Municipal Council (MBSP) solid waste routing system in Prai industrial area by implementing Dijkstra and Travelling Salesman Problem (TSP) algorithms using Geographic Information System version 10.1. The route optimization study involved 24 companies in Phase I, Phase II, and Phase IV of Prai industrial area. The authority is currently using only one route to transfer the waste-to-waste transfer station. The Dijkstra algorithm can optimize alternative route 1 distance by 19.74% whereby alternative route 2 ended up with extra distance by 3.73% compared to existing single route used by MBSP. The forward Dijkstra algorithm involves single direction route with cleaning depot (source) as starting point and waste transfer station (destination) as ending point. TSP algorithm is having advantage with return direction route. The alternative route 1 evaluated through TSP algorithm gave shorter distance by 6.61% compared to existing route. Alternative route 1 evaluated through Dijkstra algorithm is potential to save fuel cost by 19.75%. Existing route carries 9.2% per year of transportation carbon emission level. The alternative route 1 assessed through Dijkstra and TSP algorithms reported lower carbon emission level at 7.4% per year and 8.6% per year, respectively. Findings of this study can help in improving MBSP’s routing system and realize Penang 2030 vision.

Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances

For the traveling salesman problem (TSP), the existing supervised learning based algorithms suffer seriously from the lack of generalization ability. To overcome this drawback, this paper tries to train (in supervised manner) a small-scale model, which could be repetitively used to build heat maps for TSP instances of arbitrarily large size, based on a series of techniques such as graph sampling, graph converting and heat maps merging. Furthermore, the heat maps are fed into a reinforcement learning approach (Monte Carlo tree search), to guide the search of high-quality solutions. Experimental results based on a large number of instances (with up to 10,000 vertices) show that, this new approach clearly outperforms the existing machine learning based TSP algorithms, and significantly improves the generalization ability of the trained model.

Combining Reinforcement Learning with Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problem

We address the Traveling Salesman Problem (TSP), a famous NP-hard combinatorial optimization problem. And we propose a variable strategy reinforced approach, denoted as VSR-LKH, which combines three reinforcement learning methods (Q-learning, Sarsa and Monte Carlo) with the well-known TSP algorithm, called Lin-Kernighan-Helsgaun (LKH). VSR-LKH replaces the inflexible traversal operation in LKH, and lets the program learn to make choice at each search step by reinforcement learning. Experimental results on 111 TSP benchmarks from the TSPLIB with up to 85,900 cities demonstrate the excellent performance of the proposed method.

Distributed and incremental travelling salesman algorithm on time-evolving graphs

Travelling salesman problem (TSP) is a graph problem that has been widely used in many applications, especially for transportation and logistics. Because TSP is a NP hard problem, minimizing the complexity of TSP algorithms is an important problem. Many heuristic algorithms have been proposed to compute the TSP tours of a given static graph. But limited studies have been done on time-evolving graph (TEG) where the graph can change over time due to update events, such as weight changes on edges or vertices. It is a more challenging problem because the speed of TSP computations must be high enough to catch up the graph update frequency. In this paper, we make the very first attempt to minimize the computation time of solving TSP on time-evolving graphs. By exploring parallel computing power and reusing previous computing results, we proposed a distributed and incremental TSP algorithm which can be implemented on vertex-centric parallel graph computing frameworks to efficiently find TSP tours on large changing graphs. Our incremental algorithm can maintain shortest TSP tour with minimum amount of recomputation. Through our experimental evaluation, we have shown incremental TSP algorithm is 98% faster than distributed algorithm.

Implementation of Traveling salesman problem Algorithm for Scheduling and Shortest Distance Optimization

PD Mitra Makmur Perkasa is a company that distributes bottled drinking water. The decline in service levels in the timeliness of delivery is one of the problems experienced by the company. Based on the analysis conducted, the cause is due to irregular and unsystematic scheduling. To overcome this problem, we need a system that can manage systematic scheduling and provide optimal mileage recommendations to streamline delivery times and maximize the number of deliveries. These papers aim to build a web-based information system for scheduling the delivery of bottled drinking water using the waterfall system development method and the heuristic TSP (travelling salesman problem) method. TSP used to determine travel routes and produce recommendations for the shortest distance to be traversed and to maximize the amount of drinking water that can be delivered optimally. The results TSP method can be applied to a web-based scheduling information system for the delivery of bottled drinking water. It will have an impact on increasing service and maintaining, as well as increasing the number of customers. The resulting testing with the structural and functional BlackBox test results declared appropriate and functional.

Complexity indices for the traveling salesman problem continued

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs G with distances between cities as edge weights. A complexity index is an invariant of an instance I by which we predict the execution time of an exact TSP algorithm for I. In the paper [5] we have considered some short edge subgraphs of G and defined several new invariants related to their connected components. Extensive computational experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1,100] show that there exists correlation between the sequences of selected invariants and the sequence of execution times of the well-known TSP Solver Concorde. In this paper we extend these considerations for instances up to 100 vertices.