The multiple traveling salesman problem: an overview of formulations and solution procedures
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 , where V is the set of n nodes (vertices) and A is the set of arcs (edges). Let be a cost (distance) matrix associated with A. The matrix C is said to be symmetric when and asymmetric otherwise. If , 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 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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