Classical TSP Research Articles

Coordinated last-mile deliveries with trucks and drones: A comparative study of operational modes

Throughout the recent decade, drone delivery technology has been developed to address problems such as last-mile delivery and recurrent traffic jams in metropolitan regions. To resolve the relatively short travel range and limited capacity issues of drones, researchers have proposed to further enhance the utility of drones by deploying them with ground vehicles (e.g., trucks) in tandem. This new concept leads to a new transportation planning problem, i.e., coordinated delivery of trucks and drones (CDTD). In this paper, we conduct a comparative study of three representative operational models: the flying sidekick traveling salesman problem (FSTSP), the traveling salesman problem with drone (TSP-D), and the parallel drone scheduling traveling salesman problem (PDSTSP). Metrics including delivery efficiency and vehicle utilization rate are evaluated for the three models under various scenarios, to extract insights on which operational model is advantageous for which scenario. We also benchmark the algorithms for CDTD by various metrics including result quality, runtime and scalability, and conduct sensitivity analysis to assess the performance of the system under various system parameters, including drone speed, drone range and customer geographic distribution. Finally, a case study is conducted to demonstrate the superior performance of CDTD model as compared to classical TSP model based on real-world data. Experimental results demonstrate that the FSTSP and TSP-D highlight superior performance over PDSTSP and truck-only TSP when the majority of customers are out of the flight range of drone and the customer demand is clustered, as a result of the synchronization and compensation for limitations between the two types of vehicles.

Traveling Salesman Problem Solving Based on Hybrid Particle Swarm Optimization Algorithm

Genetic algorithms all the methods used to solve the Traveling Agent (TSP) problem, the use of genetic algorithms on TSP problems is more common than other methods, and it is also possible to achieve global search and obtain the best values. Compared with the traditional genetic algorithm, the particle swarm algorithm is not only easy to be applied, but also can be widely used because of the function of rapid search, but at the same time, the shortcomings of the traditional standard particle swarm algorithm are more prominent, and it is easy to fall into the situation of optimal solution of local problems. So first of all, we use the classical TSP algorithm to propose a mixed particle swarm algorithm, which solves the TSP problem by using the selection, crossover, variation and other operators of the traditional genetic algorithm, and through the conclusion of the solution, it can be shown that the method not only greatly improves the search function of the traditional standard particle swarm, but also greatly improves the convergence speed while obtaining the approximate optimal solution.

Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension

In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [ 14 ], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V , and a vehicle of bounded capacity Q . The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d v and the total demand of each tour must be no more than Q . Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed \(ε \gt \frac{1}{3000}\) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [ 5 ] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [ 15 ] presented an approximation scheme with time n log O(1/ε) n for Euclidean plane ℝ 2 . No other approximation scheme is known for any other class of metrics (without further restrictions on Q ). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n O(log 6 n/ε 5 ) .

Hybrid search with neighborhood reduction for the multiple traveling salesman problem

We present an effective hybrid algorithm with neighborhood reduction for solving the multiple traveling salesman problem (mTSP). This problem aims to optimize one of the two objectives: to minimize the total traveling distance (the minsum mTSP) or to minimize the longest tour (the minmax mTSP). The proposed algorithm hybridizes inter-tour optimization with an efficient neighborhood search based on tabu search and intra-tour optimization using the traveling salesman heuristic EAX. A dedicated neighborhood reduction strategy is introduced to avoid the examination of non-promising candidate solutions and thus speed up the neighborhood search. Results of extensive computational experiments are shown on 41 popular instances from several sources and 36 new large instances. Comparisons with five state-of-the-art methods in the literature demonstrate a high competitiveness of the proposed algorithm. Additional experiments on applying a classical TSP heuristic to the minsum mTSP instances show excellent results.

Sinüs Kosinüs Algoritmasının Gezgin Satıcı Problemi Üzerinde Parametre Analizi

Sine Cosine Algorithm (SCA) is a fairly new algorithm developed in 2016 by Mirjalili, likewise Black Hole Algorithm (BHA), Whale Optimization Algorithm (WOA), Artificial Atom Algorithm (A3) and Physarum-Energy Optimization Algorithm (PEO) proposed in 2013, 2016, 2018 and 2019, respectively. Due to new ideas in SCA, a few of publications have been published on SCA. SCA was applied on continuous and discrete optimization problems. In addition, there exist remarkable implementations of SCA in the field of engineering, science and technology. In this work, a parameters analysis of SCA has been done on a classical TSP (Berlin52-CTSP) and randomly generated TSP (RTSP). In order to do parameters analysis, major parameters have been changed gradually. For classical TSP, symmetric data has been taken from TSPLIB (TSP Library in net). The results are given as best, mean, worst solutions, std. deviation and CPU time for CTSP and RTSP. Besides, figures and tables demonstrate the effect of parameters for solving TSP. After adequate experimentation, based on trial-and-error methodology, optimal parameters and best ever solutions have been found. As a result, the findings indicate that major parameters of SCA influence the performance of that algorithm significantly.

Geometric and LP-based heuristics for angular travelling salesman problems in the plane

The angular-metric travelling salesman problem (AngleTSP) asks for a Hamiltonian cycle for given points in the Euclidean plane mimimizing the sum of all turning angles. Setting as objective the linear combination of these angles with the distances of the classical TSP, gives rise to the angular-distance-metric travelling salesman problem (AngleDistanceTSP).Since both problems are NP-hard, we first introduce a wide range of heuristic approaches, motivated by the typical geometric structure of optimal solutions. In particular, we exploit lens-shaped neighbourhoods of edges and a decomposition of the graph into layers of convex hulls, which are then merged into a tour. Secondly, we consider an ILP model based on the more general quadratic travelling salesman problem (QTSP). By rounding its fractional solutions we obtain a collection of subtours, paths and isolated points, which are then combined into a tour by various strategies, all of them involving auxiliary ILP models. Finally, different improvement heuristics are proposed, most notably a matheuristic which locally reoptimizes the solution for rectangular sectors of the given point set by an ILP approach.Results of extensive computational experiments using benchmark instances from the literature illustrate the Pareto-efficient frontier of algorithms in a (running time, objective value)-space. It turns out that our new methods clearly dominate previously published heuristics.

MODELING AND SOLVING THE TRAVELING SALESMAN PROBLEM WITH PRIORITY PRIZES

This paper addresses the Traveling Salesman Problem with Priority Prizes (TSPPP), an extension of the classical TSP in which the order of the node visits is taken into account in the objective function. A prize p ki is received by the traveling salesman when node i is visited in the k-th order of the route, while a travel cost c ij is incurred when the salesman travels from node i to node j . The aim of the TSPPP is to find the maximum profit n-node tour. The problem can be seen as a TSP variant with a more general objective function, aiming at solutions that in some way consider the quality of customer service and the delivery priorities and costs. A natural representation for the TSPPP is here grounded in the point of view of Koopmans and Beckmann approach, according to which the problem is seem as a special case of the quadratic assignment problem (QAP). Given the novelty of this TSP variant, we propose different mixed integer programming models to appropriately represent the TSPPP, some of them based on the QAP. Computational experiments are also presented when solving the MIP models with a well-known optimization software, as well as with a tabu search algorithm.

Approximation Schemes for the Generalized Traveling Salesman Problem

The Generalized Traveling Salesman Problem (GTSP) is defined by a weighted graph G = (V,E,w) and a partition of its vertex set into k disjoint clusters V = V1 ∪... ∪ Vk. It is required to find a minimum-weight cycle that contains exactly one vertex of each cluster. We consider a geometric setting of the problem (we call it the EGTSP-k-GC), in which the vertices of the graph are points in the plane, the weight function corresponds to the Euclidean distances between the points, and the partition into clusters is specified implicitly by means of a regular integer grid with step 1. In this setting, a cluster is a subset of vertices lying in the same cell of the grid; the arising ambiguity is resolved arbitrarily. Even in this special setting, the GTSP remains intractable, generalizing in a natural way the classical planar Euclidean TSP. Recently, a $$(1.5 + 8\sqrt 2 + \varepsilon )$$ -approximation algorithm with complexity depending polynomially both on the number of vertices n and on the number of clusters k has been constructed for this problem. We propose three approximation schemes for this problem. For each fixed k, all the schemes are polynomial and the complexity of the first two is linear in the number of nodes. Furthermore, the first two schemes remain polynomial for k = O(log n), whereas the third scheme is polynomial for k = n − O(log n).

Optimization for Vehicle Routing Based on Taxi Pick-up Models

With the increase of traffic congestion, public transport is gradually rising and energy saving and environmental protection is more and more advocated. According to the taxi pick-up model, the paper will optimize the path of a taxi running and analysis to figure out that the cost of the total useless cost depends on the empty running cost of the taxi for a certain period of time. On this basis, the original taxi pick-up problem is transformed into a classic TSP problem. The corresponding mathematical model is established and an improved adaptive genetic algorithm is designed.

Linear Models and Computational Experiments for the Quadratic TSP

We consider the Symmetric Quadratic Traveling Salesman Problem (SQTSP), which is a generalization of the classical TSP where each sequence of two consecutive edges in the tour gives rise to a certain cost value. For the standard linearization we apply a purely integral subtour elimination strategy which outperforms the usual fractional separation routine in computational experiments, even if strengthened inequalities are added. The maximization version of the problem is introduced and turns out to benefit from this strengthening. Finally, a new geometry-based linearization with only a linear number of additional variables is presented for the Angular Metric TSP and variants thereof. It is faster than the other approaches for medium-sized instances of one of the variants.

Solving the TSP using Traditional Computing Approach

From the last decade, even though there have been sudden advances in present technology in all areas, there exist some real-world NP composite problems that still escape scientists. The Travel salesman Problem is no exception. As it is an NPHard problem, lots of divergent solutions have been created to determine in shortest possible time, the optimal solution. Traditional algorithms are one of the oldest suggested solutions which present successful solutions that are to a larger extent optimal except in few occasions which may be close to the optimal. In this paper, a variant of the classical TSP, Random TSP (RTSP) is computed using various traditional algorithms. Their performances are evaluated with emphasis on length of tour and the algorithm effectiveness. Also, this paper presents the comparison among the algorithms based on a variety of parameters that facilitated to decide the superior algorithm with regards to their needs. General Terms Travelling Salesman Problem, Dynamic Programming, Branch and Bound and Nearest Neighbor algorithm

FPGA on FPGA: Implementation of Fine-grained Parallel Genetic Algorithm on Field Programmable Gate Array

optimization problems have complex search space, which either increase the solving problem time or finish searching without obtaining the best solution. Genetic Algorithm (GA) is an optimization technique used in solving many practical problems in science, engineering, and business domains. Parallel Genetic Algorithm (PGA) has been widely used to increase speed of GA, especially after the spread of parallel platforms such as GPUs, FPGA, and Multi-Core Processors. In this paper, we introduce a type of PGA called Fine-grained Parallel Genetic Algorithm, which has the advantages of maintaining better population diversity, and inhibiting premature. Fine-grained PGA is implemented on Field Programmable Gate Array, and the system is used to solve the classical TSP problem. The results show the advantages of the Fine-grained PGA over sequential GA, and the advantages of Field Programmable Gate Array as a parallel platform. KeywordsGenetic algorithm, FPGA, TSP, Parallel Processing.

Intelligent Optimization Study of Traveling Salesman Problem

Ant colony optimization algorithm is a new simulated evolutionary algorithm by simulating the ant routing behavior of the natural world. In this paper, the basic principle of ant colony algorithm is applied to the classic TSP problem. Simulation examples show that the ant colony algorithm has better robustness and can quickly find the optimal solution.

The hierarchical traveling salesman problem

The distribution of relief aid is a complex problem where the operations have to be managed efficiently due to limited resources. We present a routing problem for relief operations whose primary goal is to satisfy demand for relief supplies at many locations taking into account the urgency of each demand. We have a single vehicle of unlimited capacity. Each node (location) has a demand and a priority. The priority indicates the urgency of the demand. Typically, nodes with the highest priorities need to be visited before lower priority nodes. We describe a new and interesting model for humanitarian relief routing that we call the hierarchical traveling salesman problem (HTSP). We compare the HTSP and the classical TSP in terms of worst-case behavior. We obtain a simple, but elegant result that exhibits the fundamental tradeoff between efficiency (distance) and priority and we provide several related observations and theorems.

Biogeography migration algorithm for traveling salesman problem

PurposeBiogeography‐based optimization algorithm is a new kind of optimization algorithm based on biogeography. It is designed based on the migration strategy of animals to solve the problem of optimization. The purpose of this paper is to present a new algorithm – biogeography migration algorithm for traveling salesman problem (TSPBMA). A new special migration operator is designed for producing new solutions.Design/methodology/approachThe paper gives the definition of TSP and models of TSPBMA; introduces the algorithm of TSPBMA in detail and gives the proof of convergence in theory; provides simulation results of TSPBMA compared with other optimization algorithms for TSP and presents some concluding remarks and suggestions for further work.FindingsThe TSPBMA is tested on some classical TSP problems. The comparison results with the other nature‐inspired optimization algorithms show that TSPBMA is useful for TSP combination optimization. Especially, the designed migration operator is very effective for TSP solving. Although the proposed TSPBMA is not better than ant colony algorithm in the respect of convergence speed and accuracy, it provides a new way for this kind of problem.Originality/valueThe migration operator is a new strategy for solving TSPs. It has never been used by any other evolutionary algorithm or swarm intelligence before TSPBMA.

Joint order batching and picker routing in single and multiple-cross-aisle warehouses using cluster-based tabu search algorithms

The organization of order picking operations is one of the most critical issues in warehouse management. In this paper, novel tabu search (TS) algorithms integrated with a novel clustering algorithm are proposed to solve the order batching and picker routing problems jointly for multiple-cross-aisle warehouse systems. A clustering algorithm that generates an initial solution for the TS algorithms is developed to provide fast and effective solutions to the order-batching problem. Unlike most common picker routing heuristics, we model the routing problem of pickers as a classical TSP and propose efficient Nearest Neighbor+Or-opt and Savings+2-Opt heuristics to meet the specific features for the problem. Various problem instances including the number of orders, weight of items, and picking coordinates are generated randomly, and detailed numerical experiments are carried out to evaluate the performances of the proposed methods. In conclusion, the TS algorithms come out to be the most efficient methods in terms of solution quality and computational efficiency.

The Parameterized Approximability of TSP with Deadlines

Modern algorithm theory includes numerous techniques to attack hard problems, such as approximation algorithms on the one hand and parameterized complexity on the other hand. However, it is still uncommon to use these two techniques simultaneously, which is unfortunate, as there are natural problems that cannot be solved using either technique alone, but rather well if we combine them. The problem to be studied here is not only natural, but also practical: Consider TSP, generalized as follows. We impose deadlines on some of the vertices, effectively constraining them to be visited prior to a given point of time. The resulting problem DlTSP (a special case of the well-known TSP with time windows) inherits its hardness from classical TSP, which is both well known from practice and renowned to be one of the hardest problems with respect to approximability: Within polynomial time, not even a polynomial approximation ratio (let alone a constant one) can be achieved (unless P = NP). We will show that DlTSP is even harder than classical TSP in the following sense. Classical TSP, despite its hardness, admits good approximation algorithms if restricted to metric (or near-metric) inputs. Not so DlTSP (and hence, neither TSP with time windows): We will prove that even for metric inputs, no constant approximation ratio can ever be achieved (unless P = NP). This is where parameterization becomes crucial: By combining methods from the field of approximation algorithms with ideas from the theory of parameterized complexity, we apply the concept of parameterized approximation. Thereby, we obtain a 2.5-approximation algorithm for DlTSP with a running time of k! · poly(|G|), where k denotes the number of deadlines. Furthermore, we prove that there is no fpt-algorithm with an approximation guarantee of 2-ε for any ε > 0, unless P = NP. Finally, we show that, unlike TSP, DlTSP becomes much harder when relaxing the triangle inequality. More precisely, for an arbitrary small violation of the triangle inequality, DlTSP does not admit an fpt-algorithm with approximation guarantee ((1-ε)/2)|V| for any ε > 0, unless P = NP.

Locating median cycles in networks

In the median cycle problem the aim is to determine a simple cycle through a subset of vertices of a graph involving two types of costs: a routing cost associated with the cycle itself, and the cost of assigning vertices not on the cycle to visited vertices. The objective is to minimize the routing cost, subject to an upper bound on the total assignment cost. This problem arises in the location of a circular-shaped transportation and telecommunication infrastructure. We present a mixed integer linear model, and strengthen it with the introduction of additional classes of non-trivial valid inequalities. Separation procedures are developed and an exact branch-and-cut algorithm is described. Computational results on instances from the classical TSP library and randomly generated ones confirm the efficiency of the proposed algorithm. An application related to the city of Milan (Italy) is also solved within reasonable computation time.

Exact And Heuristic Procedures For The Traveling Salesman Problem With Precedence Constraints, Based On Dynamic Programming

The Traveling Salesman Problem with Precedence Constraints is to find an hamiltonian tour of minimum cost in a graph G = (X,A) of n vertices, starting from vertex 1, visiting every vertex that must precede i before i (i = 2,3,..., n) and returning to vertex 1. This problem is NP-hard and arises in practical transportation and sequencing problems. In this paper we describe a new bounding procedure and a new optimal algorithm, based on dynamic programming. Computational results are given for randomly generated test problems, including the dial-a-ride problem with the classical TSP objective function.

Further results on the probabilistic traveling salesman problem

In 1985, Jaillet introduced the probabilistic traveling salesman problem (PTSP), a variant of the classical TSP in which only a subset of the nodes may be present in any given instance of the problem. The goal is to find an a priori tour of minimal expected length, with the strategy of visiting the present nodes in a particular instance in the same order as they appear in the a priori tour. In this paper we reexamine the PTSP using a variety of theoretical and computational approaches. We sharpen the best known bounds for the PTSP, derive several asymptotic relations, and compare from various veiwpoints the PTSP with the re-optimization strategy, i.e., finding an optimal tour in every problem instance. When a Euclidean metric is used and the nodes are uniformly distributed in the unit square, a heuristic for the PTSP is shown to be very close to the re-optimization strategy. We examine some PTSP heuristics with provable worst-case performance, and address the question of finding constant-guarantee heuristics. Implementations of various heuristics, some based on sorting and some on local optimality, permit us to discuss the qualitative and quantitative properties of computational problems with up to 5000 nodes.