Clustered Traveling Salesman Problem Research Articles

Formulations for the clustered traveling salesman problem with [formula omitted]-relaxed priority rule

In the classical Traveling Salesman Problem, the order in which the vertices are visited has no restrictions. The only condition imposed is that each vertex is visited only once. In some real situations this condition may not be sufficient to represent the problem, as there are cases in which the vertex visit order becomes extremely important. In other words, it is also necessary to consider a priority between the vertices. To deal with these situations, some formulations are proposed in the literature and are based on a rule called d-relaxed priority rule that captures the trade-off between total distance and vertex priorities. In this paper, the formulations from the literature are improved using valid inequalities and different formulations based on precedence variables are proposed. Computational results, based on data from the literature, are presented to demonstrate the competitiveness of the proposed approaches.

Two heuristic approaches for clustered traveling salesman problem with d-relaxed priority rule

Two heuristic approaches for clustered traveling salesman problem with d-relaxed priority rule

Heuristics using a variable neighborhood approach for the clustered traveling salesman problem

In this work are proposed new heuristic algorithms to solve the Clustered Traveling Salesman Problem (CTSP). The CTSP is an extension of the Traveling Salesman Problem (TSP), where the vertices are partitioned into disjoint clusters and the goal is to find a minimum cost Hamiltonian cycle, such that all vertices of each cluster must be visited consecutively. Two algorithms using Iterated Local Search with the Variable Neighborhood Random Descent were implemented to solve the CTSP. The proposed heuristic algorithms were tested in the types of different granularity instances with the diversified number of vertices and clusters. Computational results show that the heuristic algorithms reached the best performance in comparison to the literature algorithms and they are competitive to an exact method using the Parallel CPLEX software.

New mixed integer linear programming models and an iterated local search for the clustered traveling salesman problem with relaxed priority rule

New mixed integer linear programming models and an iterated local search for the clustered traveling salesman problem with relaxed priority rule

A multi-parent genetic algorithm for solving longitude–latitude-based 4D traveling salesman problems under uncertainty

In this study, we propose a mathematical model of a 4D clustered traveling salesman problem (CTSP) to address the cost-effective security and risk-related difficulties associated with the TSP. We used a multiparent-based memetic genetic algorithm to optimize paths between all clusters and proposed unique heuristic approaches to create clusters and reconnect them. We constructed a 4D CTSP considering multiple routes between two locations and multiple available vehicles on each route. Travel expenses and risks impact every itinerary; however, the behaviors of these costs and risks are always uncertain. We inspected various standard benchmark problems from (TSPLIB) using the proposed calculations. Real-life problems in the tourism industry motivate a longitude–latitude-based CTSP with risk constraints. Thus, we determined the risk of each path based on longitude and latitude. The contributions of this study are twofold: developing a genetic algorithm and heuristics based on mathematical modeling of a real problem.

Selective clustered traveling salesman problem

In this study, we introduce the Selective Clustered Travelling Salesman Problem, an extension of the well-known Travelling Salesman Problem where customers are grouped in clusters, and a profit is associated with each customer. The purpose of this problem is to find the most beneficial tour within a certain time budget, which consists of a subset of clusters and all nodes in each cluster visited on the tour. We formulate the problem as a mixed integer linear programming odel and develop a metaheuristic, using three constructive algorithms, by proposing a problem-specific neighbourhood structures to solve the problem effectively. The proposed algorithm has a sort of large variable neighbourhood search structure. Computational tests are made on benchmark instances with up to 400 vertices. Results show that the mathematical formulation is able to find the optimal solutions of all instances up to 358 vertices. Also, it is found that the proposed algorithm in spite of one parameter unlike the metaheuristics have several, significantly reduces the solution time, and besides, it gives high quality solutions especially for large-size problems.

Achieving feasibility for clustered traveling salesman problems using PQ‐trees

AbstractLet be a hypergraph, where is a set of vertices and is a set of clusters , , such that the clusters in are not necessarily disjoint. This article considers the feasibility clustered traveling salesman problem, denoted by . In the we aim to decide whether a simple path exists that visits each vertex exactly once, such that the vertices of each cluster are visited consecutively. We focus on hypergraphs with no feasible solution path and consider removing vertices from clusters, such that the hypergraph with the new clusters has a feasible solution path for . The algorithm uses a PQ‐tree data structure and runs in linear time.

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.

Solving the clustered traveling salesman problem with ‐relaxed priority rule

AbstractThe clustered traveling salesman problem with a prespecified order on the clusters, a variant of the well‐known traveling salesman problem, is studied in the literature. In this problem, delivery locations are divided into clusters with different urgency levels and more urgent locations must be visited before less urgent ones. However, this could lead to an inefficient route in terms of traveling cost. This priority‐oriented constraint can be relaxed by a rule called ‐relaxed priority that provides a trade‐off between transportation cost and emergency level. Given a positive integer , at any point along the route, the ‐relaxed rule allows the vehicle to visit locations with priority , before visiting all locations in class , where is the highest priority class among all unvisited locations. Our research proposes two approaches to solve the problem with ‐relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A metaheuristic method based on the framework of iterated local search with problem‐tailored operators is also introduced to find approximate solutions. Experimental results show the effectiveness of our methods.

Approximation Algorithms for Not Necessarily Disjoint Clustered TSP

Let $G=(V,E)$ be a complete undirected graph with vertex set $V$, edge set $E$ and let $H = \langle G,\mathcal{S} \rangle $ be a hypergraph, where $\mathcal{S}$ is a set of not necessarily disjoint clusters $S_1,\ldots,S_m$, $S_i \subseteq V$ $\forall i \in \{ 1,\ldots,m \}$. The clustered traveling salesman problem CTSP is to compute a shortest Hamiltonian path that visits each one of the vertices once, such that the vertices of each cluster are visited consecutively. In this paper, we present a 4-approximation algorithm for the general case. When the intersection graph is a path, we present a 5/3-approximation algorithm. When the clusters' sizes are all bounded by a constant and the intersection graph is connected, we present an optimal polynomial time algorithm.

A two-phase local search algorithm for the ordered clustered travelling salesman problem

The ordered clustered travelling salesman problem (OCTSP) is an extension of the classical travelling salesman problem where the set of vertices is partitioned into clusters with a prespecified order. The objective is to find a minimum cost Hamiltonian tour such that the vertices in any cluster are visited contiguously and the clusters are visited in the given order. This paper proposes a two-phase approach using the 2-Opt heuristic and the guided local search (GLS) metaheuristic for obtaining approximate solutions of high-quality to the OCTSP. The first phase attempts to find an optimal Hamiltonian cycle for each cluster. The resulted cycles will be concatenated in the second phase, which then focuses on finding an optimal Hamiltonian cycle for the OCTSP. Computational results confirm the effectiveness of the proposed approach in terms of solution quality and computational time, in comparison with a hybrid genetic algorithm (HGA) on symmetric instances obtained from the travelling salesman problem instance library (TSPLIB).

A two-phase local search algorithm for the ordered clustered travelling salesman problem

The ordered clustered travelling salesman problem (OCTSP) is an extension of the classical travelling salesman problem where the set of vertices is partitioned into clusters with a prespecified order. The objective is to find a minimum cost Hamiltonian tour such that the vertices in any cluster are visited contiguously and the clusters are visited in the given order. This paper proposes a two-phase approach using the 2-Opt heuristic and the guided local search (GLS) metaheuristic for obtaining approximate solutions of high-quality to the OCTSP. The first phase attempts to find an optimal Hamiltonian cycle for each cluster. The resulted cycles will be concatenated in the second phase, which then focuses on finding an optimal Hamiltonian cycle for the OCTSP. Computational results confirm the effectiveness of the proposed approach in terms of solution quality and computational time, in comparison with a hybrid genetic algorithm (HGA) on symmetric instances obtained from the travelling salesman problem instance library (TSPLIB).

New hybrid heuristic algorithm for the clustered traveling salesman problem

Abstract In this study, we present a new hybrid heuristic algorithm for the Clustered Traveling Salesman Problem (CTSP). The CTSP is NP -hard because it includes the well-known Traveling Salesman Problem as a special case. In the CTSP, the vertices are partitioned into clusters and all the vertices of each cluster have to be visited contiguously. The algorithm is based on iterated metaheuristics, including local search, greedy randomized adaptive search procedure, and variable neighborhood random descent. Our new hybrid heuristic algorithm uses several variable neighborhood structures, which are selected in a random order. The new algorithm helps to intensify the search by using local search operators and diversification with a constructive heuristic and a perturbation method. The results of computational tests are reported for different classes of instances based on data with different levels of granularity, i.e., with different numbers of vertices and clusters. The results obtained by our new hybrid heuristic were compared with those produced using four previously reported heuristics and an exact method. The computational results demonstrate that the new hybrid heuristic obtained better results when it was compared with a hybrid heuristic that uses variable neighborhood descent. The new hybrid heuristic also outperformed previously described algorithms and it was competitive within a reasonable amount of computational time.

A note on approximation algorithms of the clustered traveling salesman problem

In an earlier paper (Bao and Liu [1]), we considered a version of the clustered traveling salesman problem (CTSP), in which both the starting and ending vertex of each cluster are free to be selected, and proposed a 2.167-approximation algorithm. In this note, we first improve this approximation ratio to 1.9 by introducing a new method to define the inter-node lengths for all the nodes in Step 2 of Algorithm A of Bao and Liu [1]. Based on the above method, we then provide a 2.5-approximation algorithm for another version of CTSP where the starting vertex of each cluster is given while the ending vertex is free to be selected, which improves the previous approximation ratio of 2.643 of Guttmann-Beck et al. [5].

Multiregion Inspection by Combining Clustered Traveling Salesman Tours With Sampling-Based Motion Planning

This paper develops an efficient approach to generate a collision-free and dynamically feasible trajectory that enables a robotic vehicle to inspect the entire workspace or a subset consisting of one or several regions. The approach makes it possible to specify constraints on the order in which the regions should be inspected by using colors to ensure that regions with the same color are inspected as a group. A key aspect is the transformation of the multiregion inspection into a clustered traveling salesman problem (CTSP). This is achieved by generating several inspection points on the medial axis of each region to increase the visibility and by grouping the inspection points into clusters. We also develop a fast branch-and-bound CTSP solver to find low-cost clustered tours. These tours guide sampling-based motion planning, as it expands a motion tree in search for a collision-free and dynamically feasible trajectory to carry out the multiregion inspection. The approach is evaluated in simulation using a snake-like robot model to carry out inspection tasks in complex environments. Comparisons to related work show significant speedups.

Solving the Free Clustered TSP Using a Memetic Algorithm

The free clustered travelling salesman problem (FCTSP) is an extension of the classical travelling salesman problem where the set of vertices is partitioned into clusters, and the task is to find a minimum cost Hamiltonian tour such that the vertices in any cluster are visited contiguously. This paper proposes the use of a memetic algorithm (MA) that combines the global search ability of Genetic Algorithm with local search to refine solutions to the FCTSP. The effectiveness of the proposed algorithm is examined on a set of TSPLIB instances with up to 318 vertices and clusters varying between 2 and 50 clusters. Moreover, the performance of the MA is compared with a Genetic Algorithm and a GRASP with path relinking. The computational results confirm the effectiveness of the MA in terms of both solution quality and computational time.

A HYBRID HEURISTIC ALGORITHM FOR THE CLUSTERED TRAVELING SALESMAN PROBLEM

This paper proposes a hybrid heuristic algorithm, based on the metaheuristics Greedy Randomized Adaptive Search Procedure, Iterated Local Search and Variable Neighborhood Descent, to solve the Clustered Traveling Salesman Problem (CTSP). Hybrid Heuristic algorithm uses several variable neighborhood structures combining the intensification (using local search operators) and diversification (constructive heuristic and perturbation routine). In the CTSP, the vertices are partitioned into clusters and all vertices of each cluster have to be visited contiguously. The CTSP is 𝒩𝒫-hard since it includes the well-known Traveling Salesman Problem (TSP) as a special case. Our hybrid heuristic is compared with three heuristics from the literature and an exact method. Computational experiments are reported for different classes of instances. Experimental results show that the proposed hybrid heuristic obtains competitive results within reasonable computational time.

Métodos heurísticos usando busca local aleatória em vizinhança variável para o problema do caixeiro viajante com grupamentos

Nesse artigo, são propostos novos métodos heurísticos para resolver o Problema do Caixeiro Viajante com Grupamentos (PCVG). O PCVG é uma generalização do Problema do Caixeiro Viajante (PCV), onde os vértices são particionados em grupos disjuntos e o objetivo é encontrar um ciclo hamiltoniano de custo mínimo tal que os vértices de cada grupo são visitados de forma contígua. Nós desenvolvemos dois Métodos de Descida Aleatória em Vizinhança Variável com Iterated Local Search para resolver o PCVG. Os métodos heurísticos propostos foram testados em tipos de instâncias com dados em níveis diferentes de granularidade para o número de vértices e grupos. Resultados computacionais mostraram que os métodos heurísticos se sobrepõem aos métodos recentes existentes na literatura e são competitivos com um método exato usando o software CPLEX Paralelo.

The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic Algorithm

The ordered clustered travelling salesman problem is a variation of the usual travelling salesman problem in which a set of vertices (except the starting vertex) of the network is divided into some prespecified clusters. The objective is to find the least cost Hamiltonian tour in which vertices of any cluster are visited contiguously and the clusters are visited in the prespecified order. The problem is NP-hard, and it arises in practical transportation and sequencing problems. This paper develops a hybrid genetic algorithm using sequential constructive crossover, 2-opt search, and a local search for obtaining heuristic solution to the problem. The efficiency of the algorithm has been examined against two existing algorithms for some asymmetric and symmetric TSPLIB instances of various sizes. The computational results show that the proposed algorithm is very effective in terms of solution quality and computational time. Finally, we present solution to some more symmetric TSPLIB instances.

Métodos heurísticos usando grasp, reconexão de caminhos e busca em vizinhança variável para o problema do caixeiro viajante com grupamentos

O Problema do Caixeiro Viajante com Grupamentos (PCVG) é uma generalização do Problema do Caixeiro Viajante (PCV) em que o conjunto de vértices é particionado em grupos disjuntos e o objetivo é encontrar um ciclo Hamiltoniano de custo mínimo tal que os vértices em cada grupo são visitados na forma contígua. O PCVG é NP-difícil e nesse contexto são propostos métodos heurísticos para o PCVG usando GRASP, Reconexão de Caminhos e Método de Descida em Vizinhança Variável (MDVV). Os métodos heurísticos foram testados usando instancias Euclidianas com até 2000 vértices e grupos variando entre 4 a 15 vértices. Os testes computacionais foram executados para comparar o desempenho dos métodos heurísticos com um algoritmo exato usando o software CPLEX Paralelo. Os resultados computacionais mostraram que o método heurístico híbrido usando MDVV sobressaiu em relação aos outros métodos.