scholarly journals On Tightly Bounding the Dubins Traveling Salesman's Optimum

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Satyanarayana G. Manyam ◽  
Sivakumar Rathinam
Keyword(s):  
Lower Bound ◽  
Traveling Salesman Problem ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Motion Constraints ◽  
Euclidean Traveling Salesman Problem ◽  
Problem Instances ◽  
Feasible Solutions ◽  
The Cost ◽  
Dubins Traveling Salesman Problem

The Dubins traveling salesman problem (DTSP) has generated significant interest over the last decade due to its occurrence in several civil and military surveillance applications. This problem requires finding a curvature constrained shortest path for a vehicle visiting a set of target locations. Currently, there is no algorithm that can find an optimal solution to the DTSP. In addition, relaxing the motion constraints and solving the resulting Euclidean traveling salesman problem (ETSP) provide the only lower bound available for the DTSP. However, in many problem instances, the lower bound computed by solving the ETSP is far below the cost of the feasible solutions obtained by some well-known algorithms for the DTSP. This paper addresses this fundamental issue and presents the first systematic procedure for developing tight lower bounds for the DTSP.

scholarly journals Simple Constructive, Insertion, and Improvement Heuristics Based on the Girding Polygon for the Euclidean Traveling Salesman Problem

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 5 ◽  
Author(s):  
Víctor Pacheco-Valencia ◽  
José Alberto Hernández ◽  
José María Sigarreta ◽  
Nodari Vakhania
Keyword(s):  
Traveling Salesman Problem ◽  
Real Life ◽  
Traveling Salesman ◽  
Computational Time ◽  
Problem Instance ◽  
Dimensional Euclidean Space ◽  
Benchmark Problem ◽  
Small Constant ◽  
Euclidean Traveling Salesman Problem ◽  
Problem Instances

The Traveling Salesman Problem (TSP) aims at finding the shortest trip for a salesman, who has to visit each of the locations from a given set exactly once, starting and ending at the same location. Here, we consider the Euclidean version of the problem, in which the locations are points in the two-dimensional Euclidean space and the distances are correspondingly Euclidean distances. We propose simple, fast, and easily implementable heuristics that work well, in practice, for large real-life problem instances. The algorithm works on three phases, the constructive, the insertion, and the improvement phases. The first two phases run in time O ( n 2 ) and the number of repetitions in the improvement phase, in practice, is bounded by a small constant. We have tested the practical behavior of our heuristics on the available benchmark problem instances. The approximation provided by our algorithm for the tested benchmark problem instances did not beat best known results. At the same time, comparing the CPU time used by our algorithm with that of the earlier known ones, in about 92% of the cases our algorithm has required less computational time. Our algorithm is also memory efficient: for the largest tested problem instance with 744,710 cities, it has used about 50 MiB, whereas the average memory usage for the remained 217 instances was 1.6 MiB.

scholarly journals An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 143 ◽  
Author(s):  
Furqan Essani ◽  
Sajjad Haider
Keyword(s):  
Petri Nets ◽  
Traveling Salesman Problem ◽  
Feasible Solution ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Colored Petri Nets ◽  
Cost Matrix ◽  
Multiple Traveling Salesman Problem ◽  
Hard Problems ◽  
The Cost

The Multiple Traveling Salesman Problem is an extension of the famous Traveling Salesman Problem. Finding an optimal solution to the Multiple Traveling Salesman Problem (mTSP) is a difficult task as it belongs to the class of NP-hard problems. The problem becomes more complicated when the cost matrix is not symmetric. In such cases, finding even a feasible solution to the problem becomes a challenging task. In this paper, an algorithm is presented that uses Colored Petri Nets (CPN)—a mathematical modeling language—to represent the Multiple Traveling Salesman Problem. The proposed algorithm maps any given mTSP onto a CPN. The transformed model in CPN guarantees a feasible solution to the mTSP with asymmetric cost matrix. The model is simulated in CPNTools to measure two optimization objectives: the maximum time a salesman takes in a feasible solution and the collective time taken by all salesmen. The transformed model is also formally verified through reachability analysis to ensure that it is correct and is terminating.

scholarly journals Comparing greedy constructive heuristic subtour elimination methods for the traveling salesman problem

2020 ◽  
Vol 4 (2) ◽  
pp. 167-182
Author(s):  
Petar Jackovich ◽  
Bruce Cox ◽  
Raymond R. Hill
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Computational Results ◽  
Content Type ◽  
Constructive Heuristic ◽  
Subtour Elimination ◽  
Constructive Heuristics ◽  
Problem Instances ◽  
Feasible Solutions ◽  
The Traveling Salesman Problem

Purpose This paper aims to define the class of fragment constructive heuristics used to compute feasible solutions for the traveling salesman problem (TSP) into edge-greedy and vertex-greedy subclasses. As these subclasses of heuristics can create subtours, two known methodologies for subtour elimination on symmetric instances are reviewed and are expanded to cover asymmetric problem instances. This paper introduces a third novel subtour elimination methodology, the greedy tracker (GT), and compares it to both known methodologies. Design/methodology/approach Computational results for all three subtour elimination methodologies are generated across 17 symmetric instances ranging in size from 29 vertices to 5,934 vertices, as well as 9 asymmetric instances ranging in size from 17 to 443 vertices. Findings The results demonstrate the GT is the fastest method for preventing subtours for instances below 400 vertices. Additionally, a distinction between fragment constructive heuristics and the subtour elimination methodology used to ensure the feasibility of resulting solutions enables the introduction of a new vertex-greedy fragment heuristic called ordered greedy. Originality/value This research has two main contributions: first, it introduces a novel subtour elimination methodology. Second, the research introduces the concept of ordered lists which remaps the TSP into a new space with promising initial computational results.

scholarly journals A Data-Guided Lexisearch Algorithm for the Asymmetric Traveling Salesman Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Zakir Hussain Ahmed
Keyword(s):  
Traveling Salesman Problem ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Computational Time ◽  
Cost Matrix ◽  
Asymmetric Traveling Salesman Problem ◽  
Path Representation ◽  
Random Instances ◽  
Representation Method ◽  
The Cost

A simple lexisearch algorithm that uses path representation method for the asymmetric traveling salesman problem (ATSP) is proposed, along with an illustrative example, to obtain exact optimal solution to the problem. Then a data-guided lexisearch algorithm is presented. First, the cost matrix of the problem is transposed depending on the variance of rows and columns, and then the simple lexisearch algorithm is applied. It is shown that this minor preprocessing of the data before the simple lexisearch algorithm is applied improves the computational time substantially. The efficiency of our algorithms to the problem against two existing algorithms has been examined for some TSPLIB and random instances of various sizes. The results show remarkably better performance of our algorithms, especially our data-guided algorithm.

OPTIMAL FEASIBLE SOLUTIONS TO A ROAD FREIGHT TRANSPORTATION PROBLEM

2020 ◽  
Vol 5 (1) ◽  
pp. 456
Author(s):  
Tolulope Latunde ◽  
Joseph Oluwaseun Richard ◽  
Opeyemi Odunayo Esan ◽  
Damilola Deborah Dare
Keyword(s):  
Transportation Problem ◽  
Optimal Solution ◽  
Transportation Cost ◽  
North West ◽  
Life Problems ◽  
Basic Feasible Solutions ◽  
Feasible Solutions ◽  
Road Freight Transportation ◽  
The Cost ◽  
Cost Of Transportation

For twenty decades, there is a visible ever forward advancement in the technology of mobility, vehicles and transportation system in general. However, there is no "cure-all" remedy ideal enough to solve all life problems but mathematics has proven that if the problem can be determined, it is most likely solvable. New methods and applications will keep coming to making sure that life problems will be solved faster and easier. This study is to adopt a mathematical transportation problem in the Coca-Cola company aiming to help the logistics department manager of the Asejire and Ikeja plant to decide on how to distribute demand by the customers and at the same time, minimize the cost of transportation. Here, different algorithms are used and compared to generate an optimal solution, namely; North West Corner Method (NWC), Least Cost Method (LCM) and Vogel’s Approximation Method (VAM). The transportation model type in this work is the Linear Programming as the problems are represented in tables and results are compared with the result obtained on Maple 18 software. The study shows various ways in which the initial basic feasible solutions to the problem can be obtained where the best method that saves the highest percentage of transportation cost with for this problem is the NWC. The NWC produces the optimal transportation cost which is 517,040 units.

scholarly journals ON SAMPLING BASED METHODS FOR THE DUBINS TRAVELING SALESMAN PROBLEM WITH NEIGHBORHOODS

2015 ◽  
Vol 2 (2) ◽  
pp. 57-61
Author(s):  
Petr Váňa ◽  
Jan Faigl
Keyword(s):  
Path Planning ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Computational Time ◽  
Solution Quality ◽  
Goal Region ◽  
Dubins Vehicle ◽  
Proposed Modification ◽  
Dubins Traveling Salesman Problem

In this paper, we address the problem of path planning to visit a set of regions by Dubins vehicle, which is also known as the Dubins Traveling Salesman Problem Neighborhoods (DTSPN). We propose a modification of the existing sampling-based approach to determine increasing number of samples per goal region and thus improve the solution quality if a more computational time is available. The proposed modification of the sampling-based algorithm has been compared with performance of existing approaches for the DTSPN and results of the quality of the found solutions and the required computational time are presented in the paper.

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