Routing and scheduling of vehicles and crews: The state of the art

http://web.ist.utl.pt/~ist11038/CD_Casquilho/TSP1992EJOR_Laporte.pdf

The traveling salesman problem: An overview of exact and approximate algorithms

We develop local search algorithms for routing problems with time windows. The presented algorithms are based on thek-interchange concept. The presence of time windows introduces feasibility constraints, the checking of which normally requires O(N) time. Our method reduces this checking effort to O(1) time. We also consider the problem of finding initial solutions. A complexity result is given and an insertion heuristic is described.

/pdf/local-search-in-routing-problems-with-time-windows-3f12unrhh2.pdf

Local search in routing problems with time windows

In this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality. Our implementation is based on a fast LP-solver (IBM's MPSX) and makes effective use of polyhedral results on the symmetric travelling salesman polytope. We describe the important ingredients of our code and give an extensive documentation of its computational performance.

/pdf/solution-of-large-scale-symmetric-travelling-salesman-fxmr3zn1mt.pdf

Solution of large-scale symmetric travelling salesman problems

Many algorithms have been developed for the optimal solution of the asymmetric travelling salesman problem: the most efficient ones are based on the subtour elimination approach. This paper presents a breadth-first branch and bound algorithm which differs from the method of Smith, Srinivasan and Thompson in the selection of the subtour to be split, in the ordering of the arcs in the selected subtour, in the computation of different partial lower bounds and in different data structures to facilitate the updating of the cost matrix. Extensive computational results considering random problems with up to 240 vertices are presented for various ranges of the coefficients of the cost matrix.

Some New Branching and Bounding Criteria for the Asymmetric Travelling Salesman Problem

In this paper we develop and computationally test three implicit enumeration algorithms for solving the asymmetric traveling salesman problem. All three algorithms use the assignment problem relaxation of the traveling salesman problem with subtour elimination similar to the previous approaches by Eastman, Shapiro and Bellmore and Malone. The present algorithms, however, differ from the previous approaches in two important respects: (i) lower bounds on the objective function for the descendants of a node in the implicit enumeration tree are computed without altering the assignment solution corresponding to the parent node – this is accomplished using a result based on “cost operators”, (ii) a LIFO ( Last In , First Out) depth first branching strategy is used which considerably reduces the storage requirements for the implicit enumeration approach. The three algorithms differ from each other in the details of implementing the implicit enumeration approach and in terms of the type of constraint used for eliminating subtours. Computational experience with randomly generated test problems indicates that the present algorithms are more efficient and can solve larger problems compared to (i) previous subtour elimination algorithms and (ii) the 1-arborescence approach of Held and Karp (as implemented by T. H. C. Smith) for the asymmetric traveling salesman problem. Computational experience is reported for up to 180 node problems with costs (distances) in the interval (1,1000) and up to 200 node problems with bivalent costs.

Computational Performance of Three Subtour Elimination Algorithms for Solving Asymmetric Traveling Salesman Problems.

We propose here a LIFO implicit enumeration search algorithm for the symmetric traveling salesman problem which uses the 1-tree relaxation of Held and Karp. The proposed algorithm has significantly smaller memory requirements than Held and Karp's branch-and-bound algorithm. Computational experience with this algorithm and an improved version of Held and Karp's algorithm is reported and on the basis of the sample it can be stated that the proposed algorithm is faster and generates many fewer subproblems than Held and Karp's algorithm.

G. L. Thompson

Papers

Computational Performance of Three Subtour Elimination Algorithms for Solving Asymmetric Traveling Salesman Problems.

A Lifo Implicit Enumeration Search Algorithm for the Symmetric Traveling Salesman Problem Using Held and Karp's 1-Tree Relaxation