Classical Traveling Salesman Problem Research Articles

A Graph Neural Network Assisted Monte Carlo Tree Search Approach to Traveling Salesman Problem

We tackle the classical traveling salesman problem (TSP) by combining a graph neural network and Monte Carlo Tree Search. We adopt a greedy algorithm framework to derive a promising tour by adding the vertices successively. A graph neural network is trained to capture graph motifs and interactions between vertices, and then to give the prior probability of selecting a vertex at every step. Instead of making decisions directly based on the output of graph neural networks, we combine the graph neural network with Monte Carlo Tree Search to provide a more reliable policy as the output of the latter is the feedback information by fusing the prior probability with the scouting exploration. Without much heuristic designing, our approach outperforms recent state-of-the-art learning-based methods on the TSP. Experimental results demonstrate that the proposed method can be generalized to instances with more vertices than those used during the training.

Approximate and exact algorithms for an energy minimization traveling salesman problem

Energy saving is a great challenge for clean transportation. In this paper, we study the Energy Minimization Traveling Salesman Problem (EMTSP), which is a generation of the classical Traveling Salesman Problem (TSP), and an important theoretical basis and a special case of the Energy Minimization Vehicle Routing Problem (EMVRP). The objective of the studied problem is to minimize the sum of the product of load (including curb weight of the vehicle) and traveled distances. An approximation algorithm based on the Christofides’s Heuristic is proposed and its worst-case ratio bound is proven. A branch and bound (B&B) algorithm integrated with a fast 1-tree based lower bound is developed to obtain optimal solutions. The results of computational experiments show the efficiency and the effectiveness of the B&B algorithm, as well as the heuristic methods.

A Simple Approach for the Solution of Fuzzy Multi Objective Travelling Salesman Problem

We propose a simple method for solving multi objective travelling salesman problem whose decision parameters are expressed as triangular fuzzy number. We apply arithmetic operation and ranking for the parametric form of triangular fuzzy number. The proposed gives preference solution to the decision maker without change to classical travelling salesman problem. A numerical example is provided to show the efficiency of the propose method.

The multicommodity traveling salesman problem with priority prizes: a mathematical model and metaheuristics

The classic Traveling Salesman Problem (TSP) only considers the costs involved in the routes and does not differentiate products or customers. Logistic companies face conflict between operational costs, customers with different categories of products, and customer satisfaction, which is directly related to delivery time. This paper presents a new mathematical model for a TSP with variable costs and priority prizes, taking into account the customer’s product and preference values. This problem is denoted as the Multicommodity Traveling Salesman Problem with Priority Prizes (MTSPPP). Two versions of the Biased Random-Key Genetic Algorithm (BRKGA) are proposed to solve medium and large instances of the MTSPPP. Computational tests were performed, using modified instances based on classical TSP instances. The proposed methods have proved to be efficient in solving the MTSPPP.

Maximizing the profit per unit time for the travelling salesman problem

This paper deals with a new extension of the Travelling Salesman Problem (TSP) in which the objective is to maximize the profit per unit time. A travelling salesman has to complete a tour comprised of n cities. The tour starts and ends at one of the cities and must pass through all n cities. There are several travelling options between every pair of cities which differ in their costs and durations. When completing the tour, the travelling salesman will receive a predefined known income. The salesman's profit per unit time is defined as the profit gained after completing the tour (calculated by subtracting the route's costs from the profit) divided by the total time required to complete the tour.This problem is a new practical and more complex extension of the well-studied TSP due to the multiple travelling options. The paper defines the problem and presents an iterative solution procedure, which converges after a limited number of iterations, while each iteration having the same time complexity of solving the classical TSP optimally or heuristically.

The Steiner Traveling Salesman Problem and its extensions

This paper considers the Steiner Traveling Salesman Problem, an extension of the classical Traveling Salesman Problem on an incomplete graph where not all vertices have demand. Some extensions including several depots or location decisions are introduced, modeled and solved. A compact integer linear programming formulation is proposed for each problem, where the routes are represented with two-index decision variables, and parity conditions are modeled using cocircuit inequalities. Exact branch-and-cut algorithms are developed for all formulations. Computational results obtained confirm the good performance of the algorithms. Instances with up to 500 vertices are solved optimally.

Variants of the Traveling Salesman Problem

Abstract This paper includes an introduction to the concept of spreadsheet optimization and modeling as it specifically applies to combinatorial problems. One of the best known of the classic combinatorial problems is the “Traveling Salesman Problem” (TSP). The classic Traveling Salesman Problem has the objective of minimizing some value, usually distance, while defining a sequence of locations where each is visited once. An additional requirement is that the tour ends in the same location where the tour started. Variants of the classic Traveling Salesman Problem are developed including the Bottleneck TSP and the Variation Bottleneck TSP.

Trajectory Design for Distributed Estimation in UAV-Enabled Wireless Sensor Network

In this paper, we study an unmanned aerial vehicle (UAV)-enabled wireless sensor network, where a UAV is dispatched to collect the sensed data from distributed sensor nodes (SNs) for estimating an unknown parameter. It is revealed that in order to minimize the mean square error for the estimation, the UAV should collect the data from as many SNs as possible, based on which an optimization problem is formulated to design the UAV's trajectory subject to its practical mobility constraints. Although the problem is nonconvex and NP-hard, we show that the optimal UAV trajectory consists of connected line segments only. With this simplification, an efficient suboptimal solution is proposed by leveraging the classic traveling salesman problem and applying convex optimization techniques. Simulation results show that the proposed trajectory design achieves a significant performance gain in terms of the number of SNs whose data are successfully collected, as compared with other benchmark schemes.

A note on computational aspects of the Steiner traveling salesman problem

AbstractThe Steiner traveling salesman problem (StTSP) is a variant of the classical traveling salesman problem (TSP). In the StTSP, we are given a graph with edge distances, and a set of terminal nodes, which are a subset of all nodes. The goal is to find a minimum distance closed walk to visit each terminal node at least once. Two recent articles proposed solution approaches to the problem, namely, on the exact side, Letchford et al. proposed compact integer linear programming models, and on the heuristic side, Interian and Ribeiro proposed greedy randomized adaptive search procedure, enhanced with a local search. Using the exact approach, instances with up to 250 nodes could be solved to optimality, with runtimes up to 1400 seconds, and the heuristic approach was used to tackle instances with up to 3353 nodes, with runtimes up to 8500 seconds. In this note, we show that by transforming the problem to the classical TSP, and using a state‐of‐the‐art TSP solver, all instances from the literature can be solved to optimality within 20 seconds, most of them within a second. We provide optimal solution values for 14 instances, where the optimal solution was not known.

A population algorithm based on randomized tabu thresholding for the multi-commodity pickup-and-delivery traveling salesman problem

The multi-commodity pickup-and-delivery traveling salesman problem (m-PDTSP) is a variant of the classic traveling salesman problem with pickup and delivery. It arises from a number of real-life applications where a capacitated vehicle services a set of customers that provide or require certain amounts of different commodities. In this paper, we present a simple but highly effective population based algorithm for m-PDTSP (denoted as PRTTA) that uses a randomized tabu thresholding algorithm for local improvement. Extensive experiments on a large set of benchmark instances show that the proposed approach competes very favorably with the existing methods from the literature.

A Steiner Zone Variable Neighborhood Search Heuristic for the Close-Enough Traveling Salesman Problem

This paper addresses a variant of the Traveling Salesman Problem (TSP) known as the Close-Enough Traveling Salesman Problem (CETSP). A salesman does not have to visit the exact location of each customer. Instead, the salesman needs only to get close enough to each customer. This feature allows savings from the classic TSP solution, but also adds complexity to the problem. We develop a Steiner Zone Variable Neighborhood Search heuristic (SZVNS) to solve the CETSP. We conduct computational experiments on 842 instances. SZVNS often produces optimal solutions to the small instances where optimal solutions are known. On the larger instances, SZVNS produces shorter or similar solutions in less time when compared to the solutions produced by state-of-the-art heuristics.

UAV-Enabled Radio Access Network: Multi-Mode Communication and Trajectory Design

In this paper, we consider an unmanned aerial vehicle (UAV)-enabled radio access network (RAN) with the UAV acting as an aerial platform to communicate with a set of ground users (GUs) in a variety of modes of practical interest, including data collection in the uplink, data transmission in the downlink, and data relaying between GUs involving both the uplink and downlink. Under this general framework, two UAV operation scenarios are considered: periodic operation, where the UAV serves the GUs in a periodic manner by following a certain trajectory repeatedly, and one-time operation where the UAV serves the GUs with one single fly and then leaves for another mission. In each scenario, we aim to minimize the UAV periodic flight duration or mission completion time, while satisfying the target rate requirement of each GU via a joint UAV trajectory and communication resource allocation design approach. Iterative algorithms are proposed to find efficient locally optimal solutions by utilizing successive convex optimization and block coordinate descent techniques. Moreover, as the quality of the solutions obtained by the proposed algorithms critically depends on the initial UAV trajectory adopted, we propose new methods to design the initial trajectories for both operation scenarios by leveraging the existing results for solving the classic Traveling Salesman Problem (TSP) and Pickup-and-Delivery Problem (PDP). Numerical results show that the proposed trajectory initialization designs lead to significant performance gains compared to the benchmark initialization based on circular trajectory.

An iterated local search for the Traveling Salesman Problem with release dates and completion time minimization

In the Traveling Salesman Problem (TSP) with release dates and completion time minimization an uncapacitated vehicle delivers to customers goods which arrive at the depot over time. A customer cannot be served before the demanded goods arrive at the depot. A release date is associated with each customer which represents the time at which the goods requested by the customer arrive at the depot. The vehicle may perform multiple routes, all starting and ending at the depot. The release dates of the customers served in each route must be not larger than the time at which the route starts. The objective of the problem is to minimize the total time needed to serve all customers, given by the sum of the traveling time and the waiting time at the depot. The waiting time is due to the fact that the vehicle has to wait at the depot until the latest release date of the customers it is going to serve in the next route. We introduce some properties, propose a mathematical programming formulation and present a heuristic approach based on an iterated local search where the perturbation is performed by means of a destroy-and-repair method. Two alternative repair operators, one simple and fast and the other based on a mathematical programming model, are proposed, which give rise to two variants of the heuristic. The mathematical formulation is used to find the optimal solution on instances with up to 20 customers, built from benchmark instances for the classical TSP. Comparison with optimal solutions shows that both algorithms provide high-quality solutions. Tests are also made on larger instances to compare the performance of the two variants of the heuristic.

Real-time Collision Avoidance and Path Optimizer for Semi-autonomous UAVs.

Whilst UAV offers a potentially cheaper and more localized observation platform than current satellite or land-based approaches, it requires an advance path planner to reveal its true potential, particularly in real-time missions. Manual control by human will have limited line-of-sights and prone to errors due to careless and fatigue. A good alternative solution is to equip the UAV with semi-autonomous capabilities that able to navigate via a pre-planned route in real-time fashion. In this paper, we propose an easy-and-practical path optimizer based on the classical Travelling Salesman Problem and adopts a brute force search method to re-optimize the route in the event of collisions using range finder sensor. The former utilizes a Simple Genetic Algorithm and the latter uses Nearest Neighbour algorithm. Both algorithms are combined to optimize the route and avoid collision at once. Although many researchers proposed various path planning algorithms, we find that it is difficult to integrate on a basic UAV model and often lacks of real-time collision detection optimizer. Therefore, we explore a practical benefit from this approach using on-board Arduino and Ardupilot controllers by manually emulating the motion of an actual UAV model prior to test on the flying site. The result showed that the range finder sensor provides a real-time data to the algorithm to find a collision-free path and eventually optimized the route successfully.

Multi-restart iterative search for the pickup and delivery traveling salesman problem with FIFO loading

The pickup and delivery traveling salesman problem with FIFO loading (TSPPDF) is a variant of the classic traveling salesman problem with pickup and delivery arising from several practical applications where services have to be carried out in the first-in-first-out fashion. In this paper, we present a multi-restart iterative search approach (MIS) for TSPPDF based on a combined use of 6 move operators. The basic idea behind MIS is to alternate between a threshold-based exploration phase, during which degrading solutions are considered as long as they satisfy a quality threshold, and a descent-based improvement phase that only accepts solutions that improve on the quality of the current solution. A dedicated restart strategy is applied to generate a new starting point for the next round of the iterative search as soon as the search is deemed trapped into a deep local optimum. Extensive experiments on a set of 42 benchmark instances from the literature show that the proposed approach competes very favorably with the existing methods from the literature. In particular, it reports new upper bounds (improved best-known solutions) for 20 instances, while matching the best-known result for the remaining instances. The key components of MIS are analyzed to shed light on their impact to the overall algorithmic performance.

Fuzzy Hungarian Method for Solving Intuitionistic Fuzzy Travelling Salesman Problem

The travelling salesman problem is to identify the shortest route that the salesman journey all the places and return the starting place with minimum cost. We develop a fuzzy version of Hungarian algorithm for the solution of intuitionistic fuzzy travelling salesman problem using triangular intuitionistic fuzzy numbers without changing them to classical travelling salesman problem. The purposed method is easy to empathize and to implement for finding solution of intuitionistic travelling salesman problem happening in real life situations. To illustrate the proposed method numerical example are provided.

A hybrid approach for biobjective optimization

A large number of the real world planning problems which are today solved using Operations Research methods are actually multiobjective planning problems, but most of them are solved using singleobjective methods. The reason for converting, i.e. simplifying, multiobjective problems to singleobjective problems is that no standard multiobjective solvers exist and specialized algorithms need to be programmed from scratch.In this article we will present a hybrid approach, which operates both in decision space and in objective space. The approach enables massive efficient parallelization and can be used to a wide variety of biobjective Mixed Integer Programming models. We test the approach on the biobjective extension of the classic traveling salesman problem, on the standard datasets, and determine the full set of nondominated points. This has only been done once before (Florios and Mavrotas, 2014), and in our approach we do it in a fraction of the time.

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).

A comparative study of improved GA and PSO in solving multiple traveling salesmen problem

Multiple traveling salesman problem (MTSP) is a generalization of the classic traveling salesman problem (TSP). Compared to TSP, MTSP is more common in real-life applications. In this paper, in order to solve the minsum MTSP with multiple depots, closed path, and the requirement of minimum number of cities each salesman should visit, we propose two partheno genetic algorithms (PGA). One is a PGA with roulette selection and elitist selection in which four new kinds of mutation operation are proposed. The other one, named IPGA, binds the selection and mutation together. A new selection operator and a more comprehensive mutation operator are used. The new mutation operator is based on the four kinds of mutation operation in PGA, which eliminates the mutation probability. For comparative analysis, we also adopt particle swarm optimization algorithm (PSO) and one state of the art method (invasive weed optimization algorithm from literature) to solve MTSP. The algorithms are validated with publicly available TSPLIB benchmarks. The performance is discussed and evaluated through a series of comparative experiments. IPGA is demonstrated to be superior in solving MTSP.