Multiple Traveling Salesman Problem (mTSP)
The Multiple Traveling Salesman Problem (\(m\)TSP) is a generalization of the Traveling Salesman Problem (TSP) in which more than one salesman is allowed. Given a set of cities, one depot where \(m\) salesmen are located, and a cost metric, the objective of the \(m\)TSP is to determine a tour for each salesman such that the total tour cost is minimized and that each city is visited exactly once by only one salesman.
Problem Statement
The Multiple Traveling Salesman Problem (\(m\)TSP) is a generalization of the Traveling Salesman Problem (TSP) in which more than one salesman is allowed. Given a set of cities, one depot (where \(m\) salesmen are located), and a cost metric, the objective of the \(m\)TSP is to determine a set of routes for \(m\) salesmen so as to minimize the total cost of the \(m\) routes. The cost metric can represent cost, distance, or time. The requirements on the set of routes are:
- All of the routes must start and end at the (same) depot.
- Each city must be visited exactly once by only one salesman.
The \(m\)TSP is a relaxation of the vehicle routing problem (VRP); if the vehicle capacity in the VRP is a sufficiently large value so as not to restrict the vehicle capacity, then the problem is the same as the \(m\)TSP. Therefore, all of the formulations and solution approaches for the VRP are valid for the \(m\)TSP. The \(m\)TSP is a generalization of the TSP; if the value of \(m\) is 1, then the \(m\)TSP problem is the same as the TSP. Therefore, all of the formulations and solution approaches for the \(m\)TSP are valid for the TSP.
Bektas (2006) lists a number of variations on the \(m\)TSP.
- Multiple depots : Instead of one depot, the multi-depot \(m\)TSP has a set of depots, with \(m_j\) salesmen at each depot \(j\). In the fixed destination version, a salesman returns to the same depot from which he started. In the non-fixed destination version, a salesman does not need to return to the same depot from which he started but the same number of salesmen must return as started from a particular depot. The multi-depot \(m\)TSP is important in robotic applications involving ground and aerial vehicles. For example, see Oberlin et al. (2009).
- Specifications on the number of salesmen : The number of salesmen may be a fixed number \(m\), or the number of salesmen may be determined by the solution but bounded by an upper bound \(m\).
- Fixed charges : When the number of salesmen is not fixed, there may be a fixed cost associated with activating a salesman. In the fixed charge version of the \(m\)TSP, the overall cost to minimize includes the fixed charges for the salesmen plus the costs for the tours.
- Time windows : As with the TSP and the VRP, there is a variation of the \(m\)TSP with time windows. Associated with each node is a time window during which the node must be visited by a tour. The \(m\)TSPTW has many applications, such as school bus routing and airline scheduling.
Mathematical Formulation
Here we present an assignment-based integer programming formulation for the \(m\)TSP.
Consider a graph \(G=(V,A)\), where \(V\) is the set of \(n\) nodes, and \(A\) is the set of edges. Associated with each edge \((i,j) \in A\) is a cost (or distance) \(c_{ij}\). We assume that the depot is node 1 and there are \(m\) salesmen at the depot. We define a binary variable \(x_{ij}\) for each edge \((i,j) \in A\); \(x_{ij}\) takes the value 1 if edge \((i,j)\) is included in a tour and \(x_{ij}\) takes the value 0 otherwise. For the subtour elimination constraints, we define an integer variable \(u_i\) to denote the position of node \(i\) in a tour, and we define a value \(p\) to be the maximum number of nodes that can be visited by any salesman.
Objective Minimize \( \sum_{(i,j) \in A} c_{ij} x_{ij}\)
Constraints Ensure that exactly \(m\) salesmen depart from node 1 \(\sum_{j \in V: (1,j) \in A} x_{1j} = m\)
Ensure that exactly \(m\) salesmen return to node 1 \(\sum_{j \in V: (j,1) \in A} x_{j1} = m\)
Ensure that exactly one tour enters each node \(\sum_{i \in V: (i,j) \in A} x_{ij} = 1, \forall j \in V\)
Ensure that exactly one tour exits each node \(\sum_{j \in V: (i,j) \in A} x_{ij} = 1, \forall i \in V\)
Include subtour elimination constraints (Miller-Tucker-Zemlin) \(u_i – u_j + p \cdot x_{ij} \leq p-1, \forall 2 \leq i \neq j \leq n\)
The literature includes a number of alternative formulations. Some of the alternatives to the two-index variable, assignment-based formulation are a two-index variable formulation with the original subtour elimination constraints (see Laporte and Nobert, 1980), a three-index variable formulation (see Bektas, 2006), and a \(k\)-degree center tree-based formulation (see Christofides et al., 1981 and Laporte, 1992).
To solve this integer linear programming problem, we can use one of the NEOS Server solvers in the Mixed Integer Linear Programming (MILP) category. Each MILP solver has one or more input formats that it accepts.
Click here for a GAMS model for the bayg29 instance from the TSPLIB. The formulation was provided by Erwin Kalvelagen in a blog post here .
If we submit this model to FICO-Xpress with a CPU time limit of 1 hour, we obtain a solution with a total cost of 2176 (gap of 4.3%) and the following tours:
- Tour 1: i13 – i4 – i10 – i20 – i2 – i13 cost = 60 + 39 + 25 + 49 + 87 = 260
- Tour 2: i13 – i18 – i14 – i17 – i22 – i11 – i15 – i13 cost = 128 + 32 + 51 + 47 + 63 + 68 + 86 = 475
- Tour 3: i13 – i24 – i8 – i28 – i12 – i6 – i1 – i13 cost = 56 + 48 + 64 + 71 + 46 + 60 + 82 = 427
- Tour 4: i13 – i29 – i3 – i26 – i9 – i5 – i21 – i13 cost =160 + 60 + 78 + 57 + 42 + 50 + 82 = 529
- Tour 5: i13 – i19 – i25 – i7 – i23 – i27 – i16 – i13 cost = 71 + 52 + 72 + 111 + 74 + 48 + 57 = 485
- Bektas, T. 2006. The multiple traveling salesman problem: an overview of formulations and solution procedures. OMEGA: The International Journal of Management Science 34 (3), 209-219.
- Christofides, N., A. Mingozzi, and P. Toth. 1981. Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations. Mathematical Programming 20 , 255-282.
- Laporte, G. 1992. The vehicle routing problem: an overview of exact and approximate algorithms. European Journal of Operational Research 59 , 345-358.
- Laporte, G. and Y. Nobert. 1980. A cutting planes algorithm for the \(m\)-salesmen problem. Journal of the Operational Research Society 31 , 1017-1023.
- Oberlin, P., S. Rathinam, and S. Darbha. 2009. A transformation for a heterogeneous, multi-depot, multiple traveling salesman problem. In Proceedings of the American Control Conference , 1292-1297, St. Louis, June 10 – 12, 2009.
Advertisement
Multi-depot Multiple TSP: a polyhedral study and computational results
- Published: 17 November 2011
- Volume 207 , pages 7–25, ( 2013 )
Cite this article
- Enrique Benavent 1 &
- Antonio Martínez 1
1544 Accesses
37 Citations
Explore all metrics
We study the Multi-Depot Multiple Traveling Salesman Problem (MDMTSP), which is a variant of the very well-known Traveling Salesman Problem (TSP). In the MDMTSP an unlimited number of salesmen have to visit a set of customers using routes that can be based on a subset of available depots. The MDMTSP is an NP-hard problem because it includes the TSP as a particular case when the distances satisfy the triangular inequality. The problem has some real applications and is closely related to other important multi-depot routing problems, like the Multi-Depot Vehicle Routing Problem and the Location Routing Problem. We present an integer linear formulation for the MDMTSP and strengthen it with the introduction of several families of valid inequalities. Certain facet-inducing inequalities for the TSP polyhedron can be used to derive facet-inducing inequalities for the MDMTSP. Furthermore, several inequalities that are specific to the MDMTSP are also studied and proved to be facet-inducing. The partial knowledge of the polyhedron has been used to implement a Branch-and-Cut algorithm in which the new inequalities have been shown to be very effective. Computational results show that instances involving up to 255 customers and 25 possible depots can be solved optimally using the proposed methodology.
This is a preview of subscription content, log in via an institution to check access.
Access this article
Price includes VAT (Russian Federation)
Instant access to the full article PDF.
Rent this article via DeepDyve
Institutional subscriptions
Similar content being viewed by others
Alternative formulations and improved bounds for the multi-depot fleet size and mix vehicle routing problem.
A new matheuristic approach for the multi-depot vehicle routing problem with inter-depot routes
Exact and Compact Formulation of the Fixed-Destination Travelling Salesman Problem by Cycle Imposement Through Node Currents
Applegate, D., Bixby, R. E., Chvátal, V., & Cook, W. (2006). The traveling salesman problem: a computational study . New Jersey: Princeton University Press.
Google Scholar
Baldacci, R., & Mingozzi, A. (2009). A unified exact method for solving different classes of vehicle routing problems. Mathematical Programming Series A , 120 (2), 347–380.
Article Google Scholar
Bektas, T. (2006). The multiple traveling salesman problem: an overview of formulations and solution procedures. The International Journal of Management Science , 34 , 1449–1458.
Belenguer, J. M., Benavent, E., Prins, C., Prodhon, C., & Wolfler-Calvo, R. (2011). A branch and cut method for the capacitated location-routing problem. Computers & Operations Research , 8 (6), 931–941.
Benavent, E., & Martínez, A. (2011). A polyhedral study of the multi-depot multiple TSP. Internal report, Universitát de Valencia. http://www.uv.es/~benavent/MDMTSP/report_MDMTSP.pdf .
Chvátal, V. (1973). Edmonds polytopes and weakly hamiltonian graphs. Mathematical Programming , 5 , 29–40.
Cordeau, J. F., Gendreau, M., & Laporte, G. (1997). A tabu search heuristic for periodic and multi-depot vehicle routing problems. Networks , 30 , 105–119.
Grötschel, M., & Holland, O. (1987). A cutting-plane algorithm for minimum perfect 2-matchings. Computing , 39 , 327–344.
Grötschel, M., & Padberg, M. W. (1979). On the symmetric traveling salesman problem I: inequalities. Mathematical Programming Studies , 16 , 265–280.
GuoXing, Y. (1995). Transformation of multi-depot multi-salesmen problem to the travelling salesman problem. European Journal of Operational Research , 81 , 557–560.
Gutin, G., & Punnen, A. P. (2002). The traveling salesman problem and its variations . Dordrecht: Kluwer Academic.
Kara, I., & Bektas, T. (2006). Integer linear programming formulations of multiple salesman problems and its variations. European Journal of Operational Research , 174 , 1449–1458.
Labbé, M., Martín, I. R., & Salazar, J. J. (2004). A branch-and-cut algorithm for the plant-cycle location problem. The Journal of the Operational Research Society , 55 , 513–520.
Laporte, G., Nobert, Y., & Taillefer, S. (1988). Solving a family of multi-depot vehicle routing and location-routing problems. Transportation Science , 22 (3), 161–172.
Laporte, G., Nobert, Y., & Arpin, D. (1986). An exact algorithm for solving a capacitated location-routing problem. Annals of Operations Research , 6 , 283–310.
Laporte, G., Nobert, Y., & Arpin, D. (1984). Optimal solutions to capacitated multi-depot vehicle routing problems. Congressus Numerantium , 44 , 283–292.
Malik, W., Rathinam, S., & Darbha, S. (2007). An approximation algorithms for a symmetric generalized multiple depot, multiple travelling salesman problem. Operations Research Letters , 35 , 747–753.
Naddef, D., & Rinaldi, G. (1993). The graphical relaxation: a new framework for the symmetric traveling salesman polytope. Mathematical Programming Series A , 58 , 53–88.
Padberg, M. W., & Rinaldi, G. (1990). Facet identification for the symmetric traveling salesman polytope. Mathematical Programming Series A , 47 , 219–257.
Parragh, S. N. (2010). Solving a real-world service technician routing and scheduling problem. Presented at seventh Triennial symposium on transportation analysis . TRISTAN VII, Tromso, Norway, June 20–25.
Queyranne, M., & Wang, Y. (1993). Hamiltonian path and symmetric travelling salesman polytopes. Mathematical Programming Series A , 58 , 89–110.
Prins, C., Prodhon, C., Ruiz, A., Soriano, P., & Calvo, R. W. (2007). Solving the Capacitated LRP by a cooperative Lagrangean relaxation-granular tabu search heuristic. Transportation Science , 41 (4), 470–483.
Rathinam, S., Sengupta, R., & Darbha, S. (2007). A resource allocation algorithm for multiple vehicle systems with non-holonomic constraints. IEEE Transactions on Automation Science and Engineering , 4 (1), 98–104.
Reinelt, G. (1991). TSPLIB: a travelling salesman problem library. ORSA Journal on Computing , 3 , 376–384.
Yadlapalli, S., Malik, W. A., Darbha, S., & Pachter, M. (2009). A Lagrangian-based algorithm for a multiple depot, multiple traveling salesmen problem. Nonlinear Analysis: Real World Applications , 10 , 1990–1999.
Yadlapalli, S., Malik, W. A., Rathinam, S., & Darbha, S. (2007). A Lagrangian-based algorithm for a combinatorial motion planning problem. In Proceedings of the 46th conference on decision and control , New Orleans, Dec. 12–14.
Download references
Acknowledgements
The contributions by Enrique Benavent and Antonio Martínez have been partially supported by the Ministerio de Ciencia e Innovación of Spain through projects MTM2009-14039-C06-02 and DPI2008-02700, respectively and co financed by FEDER funds.
Author information
Authors and affiliations.
Departamento de Estadística e Investigación Operativa, Universitat de València, C/Dr. Moliner, 50, 46100, Burjassot, Valencia, Spain
Enrique Benavent & Antonio Martínez
You can also search for this author in PubMed Google Scholar
Corresponding author
Correspondence to Enrique Benavent .
Rights and permissions
Reprints and permissions
About this article
Benavent, E., Martínez, A. Multi-depot Multiple TSP: a polyhedral study and computational results. Ann Oper Res 207 , 7–25 (2013). https://doi.org/10.1007/s10479-011-1024-y
Download citation
Published : 17 November 2011
Issue Date : August 2013
DOI : https://doi.org/10.1007/s10479-011-1024-y
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative
- Multiple depot traveling salesman problem
- Polyhedral study
- Branch-and-cut
- Find a journal
- Publish with us
- Track your research
A transformation for a Heterogeneous, Multiple Depot, Multiple Traveling Salesman Problem
Ieee account.
- Change Username/Password
- Update Address
Purchase Details
- Payment Options
- Order History
- View Purchased Documents
Profile Information
- Communications Preferences
- Profession and Education
- Technical Interests
- US & Canada: +1 800 678 4333
- Worldwide: +1 732 981 0060
- Contact & Support
- About IEEE Xplore
- Accessibility
- Terms of Use
- Nondiscrimination Policy
- Privacy & Opting Out of Cookies
A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. © Copyright 2024 IEEE - All rights reserved. Use of this web site signifies your agreement to the terms and conditions.
IMAGES
VIDEO
COMMENTS
The Multiple Traveling Salesman Problem (MTSP) is among the most interesting combinatorial optimization problems because it is widely adopted in real-life applications, including robotics, transportation, networking, etc. Although the importance of this optimization problem, there is no survey dedicated to reviewing recent MTSP contributions. ...
Multiple Travelling Salesman Problem (MTSP) is an extension of the famous Travelling Salesman Problem (TSP) that visiting each city exactly once with no sub-tours (Gerhard, Citation 1994). MTSP involves assigning m salesmen to n cities, and each city must be visited by a salesman while requiring a minimum total cost.
The Multiple Traveling Salesman Problem (mTSP) is a generalization of the Traveling Salesman Problem (TSP) in which more than one salesman is allowed. Given a set of cities, one depot where (m) salesmen are located, and a cost metric, the objective of the (m)TSP is to determine a tour for each salesman such that the total tour cost is minimized and that each city is visited exactly once by ...
1. Mathematical Problem. In this notebook, we show how to solve the Multiple Traveling Salesman Problem (mTSP) using CVXPY. The problem considers m traveling salesmen. To solve it, I'm going to use the Miller-Tucker-Zemlin formulation, which follows: The cities are identified with the numbers 1, …, n, with which we define:
The Multiple Traveling Salesman Problem (MTSP) is a generalization of the well-known Traveling Salesman Problem (TSP), where multiple salesmen are involved to visit a given number of cities exactly once and return to the initial position with the minimum traveling cost. MTSP is highly related to other optimization problems such as Vehicle ...
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 ...
The Multiple Travelling Salesman Problem (MTSP) is among the most interesting combinatorial optimization problems because it is widely adopted in real-life applications, including robotics, transportation, networking, etc. Although the importance of this optimization problem, there is no survey dedicated to reviewing recent MTSP contributions. In this paper, we aim to fill this gap by ...
The multiple traveling salesman problem (mTSP) is a NP-hard combinatorial optimi-zation problem. It has many real-world applications, for example, the School Bus Routing Prob-lem, and the Pickup and Delivery Problem. In the mTSP, a set of routes is assigned to m salesmen
The single depot Multiple Traveling Salesman Problem (MTSP) is a generalization of the well-known Traveling Salesman Problem (TSP) where given a depot, a set of targets, and multiple salesman stationed at the depot, the objective is to find a tour for each salesman through a subset of targets that starts and ends at the depot, such that each target is a part of exactly one tour while ...
1 INTRODUCTION. The multiple travelling salesman problem (mTSP) is an optimisation problem that is widely applied in various fields, such as hot rolling scheduling [], multi-robot task allocation and unmanned aerial vehicle trajectory [2, 3].Given n cities, one depot (where m salesmen are located) and a cost metric, the objective of the mTSP is to determine a set of routes for m salesmen and ...
the MTSP (Multiple Travelling Salesmen Problem), i.e., the Vehicle Routing Problem without capacity constraints [13, 1]. This problem consists of two connected subproblems: alloca-tion and routing. This article studies the (de-)centralisation of the former. In this context, our research question may be stated
The multiple-travelling salesman problem (MTSP) is a computationally complex combinatorial optimisation problem, with several theoretical and real-world applications. However, many state-of-the-art heuristic approaches intended to specifically solve MTSP, do not obtain satisfactory solutions when considering an optimised workload balance. In this article, we propose a method specifically ...
as multiple traveling salesman problem with specified timeframe (mTSPTW). The application of mTSPTW can be very well seen in the aircraft scheduling problems. Other constraints : Constraints can be on the number of nodes each salesman can visits, maximum or minimum distance a salesman travels or any other constraints. The mTSP is generally
1. Introduction. The multiple traveling salesmen problem (mTSP) is an extended variant of the well-known traveling salesmen problem (TSP).In the mTSP, n nodes are assigned to m salespeople to generate m tours, where each tour starts and ends in a predefined depot. Each node has to be visited only once by exactly one salesperson and each salesperson should be assigned at least one node to visit ...
The multiple Traveling Salesman Problem (mTSP) is a complex combinatorial optimization problem, which is a generalization of the well-known Traveling Salesman Problem (TSP), where one or more salesmen can be used in the solution. The optimization task can be described as follows: given a fleet of vehicles, a common depot and several requests by ...
The travelling salesman problem, also known as the travelling salesperson problem (TSP), asks the following question: ... would produce a solution whose length is provably bounded by a multiple of the optimal length, and in doing so would create lower bounds for the problem; these lower bounds would then be used with branch-and-bound approaches
Multi-Depot Multiple Traveling Salesman Problem (MDMTSP) is a generalization of the well-known Traveling Salesman Problem (TSP), which consists of determining a set of routes for the salesmen that jointly visit a set of given clients, such that each salesman starts from and returns to one depot among a set of available depots and the total cost of the routes is minimized.
The multiple travelling salesman problem (MTSP), is the generalization of TSP, which aims to determine m − routes for ' m ' salesmen to cover a set of n − cities exactly once where each route starts and ends at a depot such that the total distance is least. In this, the number of cities in each route of the optimal solution may be ...
The multiple travelling salesman problem (mTSP) is an opti-misation problem that is widely applied in various fields, such as hot rolling scheduling [1], multi‐robot task allocation and unmanned aerial vehicle trajectory [2, 3]. Given n cities, one depot (where m salesmen are located) and a cost metric, the objective of the mTSP is to ...
Abstract. The fixed charge symmetric multiple traveling salesmen problem with n cities and m salesmen is transformed to a standard symmetric traveling salesman problem with n + m − 1 cities. The asymmetric problem with m = 2 and a different base city for each salesman is also transformed to a standard asymmetric traveling salesman problem ...
A fundamental variant of the classical traveling salesman problem (TSP) is the so-called multiple TSP (mTSP), where a set of m salesmen jointly visit all cities from a set of n cities. The mTSP models many important real-life applications, in particular for vehicle routing problems. An extensive survey by Bektas ( Omega34 (3), 2006) lists a ...
This paper presents a transformation of a Heterogeneous, Multiple Depot, Multiple Traveling Salesman Problem (HMDMTSP) into a single, Asymmetric, Traveling Salesman Problem (ATSP). As a result, algorithms available for the single salesman problem can be used to solve the HMDMTSP. To show the effectiveness of the transformation, the well known ...
The Multiple Travelling Salesman Problem (MTSP) is among the most in-teresting combinatorial optimization problems because it is widely adopted in real-life applications, including robotics, transportation, networking, etc. Although the importance of this optimization problem, there is no survey dedicated to reviewing recent MTSP contributions.