Elsevier

Omega

Volume 34, Issue 3, June 2006, Pages 209-219
Omega

The multiple traveling salesman problem: an overview of formulations and solution procedures

https://doi.org/10.1016/j.omega.2004.10.004Get rights and content

Abstract

The multiple traveling salesman problem (mTSP) is a generalization of the well-known traveling salesman problem (TSP), where more than one salesman is allowed to be used in the solution. Moreover, the characteristics of the mTSP seem more appropriate for real-life applications, and it is also possible to extend the problem to a wide variety of vehicle routing problems (VRPs) by incorporating some additional side constraints. Although there exists a wide body of the literature for the TSP and the VRP, the mTSP has not received the same amount of attention. The purpose of this survey is to review the problem and its practical applications, to highlight some formulations and to describe exact and heuristic solution procedures proposed for this problem.

Introduction

A generalization of the well-known traveling salesman problem (TSP) is the multiple traveling salesman problem (mTSP), which consists of determining a set of routes for m salesmen who all start from and turn back to a home city (depot). Although the TSP has received a great deal of attention, the research on the mTSP is limited. The purpose of this paper is to review the existing literature on the mTSP, with an emphasis on practical applications, integer programming formulations (ILPFs) and solution procedures devised specifically for this problem.

The rest of the paper will proceed as follows: The following section formally defines the problem and presents some important variations. Section 3 describes practical applications of the mTSP reported in the literature and explores its connections with other type of problems. Integer programming formulations, exact and heuristic solution procedures are presented in Sections 4–6, respectively. Section 7 describes the transformation-based approaches to solve the problem. Finally, Section 8 presents some concluding results and further remarks.

Section snippets

Problem definition and variations

The mTSP can in general be defined as follows: Given a set of nodes, let there be m salesmen located at a single depot node. The remaining nodes (cities) that are to be visited are called intermediate nodes. Then, the mTSP consists of finding tours for all m salesmen, who all start and end at the depot, such that each intermediate node is visited exactly once and the total cost of visiting all nodes is minimized. The cost metric can be defined in terms of distance, time, etc. Possible

Applications and connections with other problems

This section is further subdivided into three parts. The first part describes the main practical applications of the mTSP. The second part points out the relationships of the mTSP with other problems. The third part specifically deals with the connections between the mTSP and the well-known VRP.

Integer programming formulations

Different types of integer programming formulations are proposed for the mTSP. Before presenting them, some technical definitions are as follows. The mTSP is defined on a graph G=(V,A), where V is the set of n nodes (vertices) and A is the set of arcs (edges). Let C=(cij) be a cost (distance) matrix associated with A. The matrix C is said to be symmetric when cij=cji,(i,j)A and asymmetric otherwise. If cij+cjkcik,i,j,kV, C is said to satisfy the triangle inequality.

Various integer

Exact algorithms

The first approach to solve the mTSP directly, without any transformation to the TSP (see Section 7) is due to Laporte and Nobert [37], who propose an algorithm based on the relaxation of some constraints of the problem. The motivation for such a direct approach is that a transformed problem would include many equivalent suboptimal solutions and thus would be harder to solve. The problem they consider is an mTSP with a fixed cost f associated with each salesman, activated whenever a salesman is

Heuristic solution procedures

One of the first heuristics addressing the problem of m-tours in TSP with some side conditions is due to Russell [44], although the solution procedure is based on transforming the problem to a single TSP on an expanded graph. The algorithm is an extended version of the Lin and Kernighan [45] heuristic originally developed for the TSP. Another heuristic based on an exchange procedure for the mTSP is given by Potvin et al. [46].

A parallel processing approach to solve the mTSP using evolutionary

Transformations to the TSP

One of the approaches used for solving the mTSP is to transform the problem to a standard TSP, thus being able to use any algorithm developed for the latter to be used to obtain a solution to the former. One of the first transformations for the single depot mTSP is due to Gorenstein [6]. He proposes that a TSP with m salesmen can be solved using an augmented TSP with (m-1) additional home cities, where infinite costs are assigned to home-to-home distances to prohibit such travels and zero costs

Conclusion

The multiple traveling salesman problem is an important problem in terms of both theoretical and practical reasons. First of all, it generalizes the traveling salesman problem (TSP) and can be studied to achieve a better understanding of the TSP from a theoretical point of view. On the other hand, by incorporating additional side constraints such as capacity, distance and time windows restrictions, it could easily be extended to a variety of vehicle routing problems (VRPs). A natural

Acknowledgements

The author would like to thank Imdat Kara for suggesting the topic. Thanks are also due to Ben Lev and the two anonymous referees for their valuable comments, which helped to improve the presentation of the subject.

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