Some Simple Applications of the Travelling Salesman Problem

The travelling salesman problem arises in many different contexts. In this paper we report on typical applications in computer wiring, vehicle routing, clustering and job-shop scheduling. The formulation as a travelling salesman problem is essentially the simplest way to solve these problems. Most applications originated from real world problems and thus seem to be of particular interest. Illustrated examples are provided with each application.     

Heuristic for the Asymmetric Travelling Salesman Problem

The paper describes a heuristic algorithm for the asymmetric travelling salesman problem. The procedure is a generalization of Webb's Simple Loss 1 Method and is of the sequential type, i.e. in each step one link of the tour is fixed. The method proved to produce "high quality" near optimal solutions in a computing time, which only grows proportional to n^1.82.     

旅行推销员问题的算法综述

本文综述了旅行推销员问题 (TSP)近几十年来的算法研究进展 ,给出了一些主要算法的求解思想及其时间复杂度     

An approximation algorithm for a symmetric Generalized Multiple Depot, Multiple Travelling Salesman Problem

In this paper, we present an algorithm with an approximation factor of 2 for a Generalized, Multiple Depot, Multiple Travelling Salesman Problem (GMTSP) when the costs are symmetric and satisfy the triangle inequality. The algorithm requires finding a degree constrained minimum spanning tree which we compute using a Lagrangian relaxation.     

New lower bounds for the Symmetric Travelling Salesman Problem

In this paper new lower bounds for the Symmetric Travelling Salesman Problem are proposed and combined in additive bounding procedures. Efficient implementations of the algorithms are given; in particular, fast procedures for computing the linear programming reduced costs of the Shortest Spanning Tree (SST) Problem and for finding all ther-SST of a given graph, are presented. Computational results on randomly generated test problems are reported.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

Solving the Asymmetric Travelling Salesman Problem with time windows by branch-and-cut

Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real–world application we aim at is the control of a stacker crane in a warehouse.?We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered – at least for our special application – and that the heuristics provide acceptable solutions. Received: August 1999 / Accepted: September 2000?Published online April 12, 2001     

Visual Interactive (Computer) Solutions for the Travelling Salesman Problem

A visual interactive method of improving solutions for the travelling salesman problem is described. The travelling or multiple travelling salesman problem, when constraints are included, forms the core of the local delivery routing problem. The approach described in this note may be modified to give a visual interactive method of investigating practical physical distribution problem situations.     

Branch-and-bound for the Precedence Constrained Generalized Traveling Salesman Problem

The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) combines the Generalized Traveling Salesman Problem (GTSP) and the Sequential Ordering Problem (SOP). We present a novel branching technique for the GTSP which enables the extension of a powerful pruning technique. This is combined with some modifications of known bounding methods for related problems. The algorithm manages to solve problem instances with 12–26 groups within a minute, and instances with around 50 groups which are denser with precedence constraints within 24 h.     

A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem

This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.?First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP. Received: September 5, 1997 / Accepted: November 15, 1999?Published online February 23, 2000     

The Two-Period Travelling Salesman Problem Applied to Milk Collection in Ireland

We describe a new extension to the Symmetric Travelling Salesman Problem (STSP) in which some nodes are visited inboth of 2 periods and the remaining nodes are visited in either 1 ofthe periods. A number of possible Integer Programming Formulationsare given. Valid cutting plane inequalities are defined for thisproblem which result in an, otherwise prohibitively difficult, modelof 42 nodes becoming easily solvable by a combination of cuts andBranch-and-Bound. Some of the cuts are entered in a pool andonly used when it is automatically verified that they are violated.Other constraints which are generalisations of the subtour and combinequalities for the single period STSP, are identified manuallywhen needed. Full computational details of solution process aregiven.     

Heuristic algorithms for the 2-period balanced Travelling Salesman Problem in Euclidean graphs

In the 2-period Travelling Salesman Problem some nodes, called double nodes, are visited in both of two periods while the remaining ones, called single nodes, are visited in either one of the periods. In this paper we study the case in which a balance constraint is also introduced. We require that the difference between the number of visited nodes in the two periods must be below a fixed threshold. Moreover, we suppose that distances between nodes are Euclidean. The problem is NP-hard, and exact methods, now available, appear inadequate. Here, we propose three heuristics. Computational experiences and a comparison between the algorithms are also given.     

An iterated local search algorithm for the Travelling Salesman Problem with Pickups and Deliveries

The Travelling Salesman Problem with Pickups and Deliveries (TSPPD) consists in designing a minimum cost tour that starts at the depot, provides either a pickup or delivery service to each of the customers and returns to the depot, in such a way that the vehicle capacity is not exceeded in any part of the tour. In this paper, the TSPPD is solved by considering a metaheuris-tic algorithm based on Iterated Local Search with Variable Neighbourhood Descent and Random neighbourhood ordering. Our aim is to propose a fast, flexible and easy to code algorithm, also capable of producing high quality solutions. The results of our computational experience show that the algorithm finds or improves the best known results reported in the literature within reasonable computational time.     

Lineare Charakterisierungen von Travelling Salesman Problemen

Zusammenfassung Wir betrachten die Formulierung des asymmetrischen Travelling Salesman Problems als lineares Programm und leiten mehrere Klassen neuer Ungleichungen ab, die eineschärfere Charakterisierung des Travelling Salesman Polytopen (konvexe Hülle der Touren) in Form von Ungleichungen ergeben.Es zeigt sich, daß einige der neuen Ungleichungen und auch einige der bekannten Kurzzyklus-Bedingungen tatsächlich Facetten des Travelling Salesman Polytopen sind, d.h. daß sie zu der Klasse von Ungleichungen gehören, die die konvexe Hülle aller Touren einesn-Städte Problems in eindeutiger Weise charakterisiert.

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Summary We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield asharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e. the convex hull of tours.In fact some of the new inequalities as well as some of the well-known subtour elimination constraints are indeedfacets of the travelling salesman polytope, i.e. belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.

     

Algorithms for Large-scale Travelling Salesman Problems

The paper describes a new algorithm to produce r-optimal tours for the travelling salesman problem. This algorithm is faster than the original r-optimal method, and computation times increase much less rapidly with problem size. The new algorithm makes it possible to solve large-scale travelling salesman problems and examples are given for problems varying in size from 100 to 500 cities. The paper also discusses r-optimality in terms of multistage 2-optimality.     

An Insert/Delete Heuristic for the Travelling Salesman Subset-Tour Problem with One Additional Constraint

The Travelling Salesman Subset-tour Problem (TSSP) differs from the well-known Travelling Salesman Problem (TSP) in that the salesman is not required to visit every city. Many problems, such as the prize collecting TSP, the orienteering problem, and the time constrained TSP, can be viewed as TSSPs with one additional constraint (TSSP + 1). In this paper, we present a heuristic approach for the TSSP + I class of problems. The general philosophy of our approach is to maintain tour feasibility with respect to the TSSP subproblem. This allows us to begin our approach with any TSSP tour. In each step, the insertion or deletion of a city is performed either to reduce infeasibility in the additional constraint or to improve upon the minimization objective. We present computational results that show our approach is superior to approaches using just insertion, and thus demonstrate the importance of considering the possible deletion of cities.     

Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.     

A Boundary Method for Planar Travelling Salesman Problems

An efficient method is described for obtaining near-optimal solutions to planar travelling salesman problems. Firstly, the convex boundary of points is determined and this is subsequently deformed by an iterative process until all the points have been included in the tour. Results are good on some trial problems.