The Precedence Constrained Traveling Salesman Problem

Operations Research Letters, 2005

We propose a new formulation for the asymmetric traveling salesman problem, with and without precedence relationships, which employs a polynomial number of subtour elimination constraints that imply an exponential subset of certain relaxed Dantzig-Fulkerson-Johnson subtour constraints. Promising computational results are presented, particularly in the presence of precedence constraints.

Computational Optimization and Applications, 2000

In this article we consider a variant of the classical asymmetric traveling salesman problem (ATSP), namely the ATSP in which precedence constraints require that certain nodes must precede certain other nodes in any feasible directed tour. This problem occurs as a basic model in scheduling and routing and has a wide range of applications varying from helicopter routing (Timlin, Master's

2002

Operations Research Letters, 1999

We consider the Ordered Cluster Traveling Salesman Problem OCTSP. In this problem, a vehicle starting and ending at a given depot must visit a set of n points. The points are partitioned into K , K n, prespeci ed clusters. The vehicle must rst visit the points in cluster 1, then the points in cluster 2, : : : , and nally the points in cluster K so that the distance traveled is minimized. We present a 5 3-approximation algorithm for this problem which runs in On 3 time. We show that our algorithm can also be applied to the path version of the OCTSP: the Ordered Cluster Traveling Salesman Path Problem OCTSPP. Here the di erent starting and ending points of the vehicle may o r m a y not be prespeci ed. For this problem, our algorithm is also a 5 3-approximation algorithm.

The minimum-travel-cost algorithm is a dynamic programming algorithm to compute an exact and deterministic lower bound for the general case of the traveling salesman problem (TSP). The algorithm is presented with its mathematical proof and asymptotic analysis. It has a (n 2) complexity. A program is developed for the implementation of the algorithm and successfully tested among well known TSP problems.

Traveling Salesman Problem, Theory and Applications, 2010

International Transactions in Operational Research, 2020

The Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters, a variant of the well-known traveling salesman problem is studied in literature. In this problem, delivery locations are divided into clusters with different urgency levels and more urgent locations must be visited before less urgent ones. However, this could lead to an inefficient route in terms of traveling cost. This priority-oriented constraint can be relaxed by a rule called d-relaxed priority that provides a trade-off between transportation cost and emergency level. Our research proposes two approaches to solve the problem with d-relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A meta-heuristic method based on the framework of Iterated Local Search with problem-tailored operators is also introduced to find approximate solutions. Experimental results show the effectiveness of our methods. Keywords. Clustered traveling salesman problem, d-relaxed priority rule, mixed integer programming, iterated local search. the locations are supposed to have the same degrees of urgencies, i.e., they can be visited in any order. However, in a number of real-world routing applications, different levels of priorities at the delivery locations need to be taken into account in routing plans. For example, as a result of a natural disaster such as a storm, earthquake, tsunami, or hurricane, there are demands at many locations for relief supplies such as food, bottled water, blankets, or medical packs. Some locations are in more urgent need of supplies than other locations due to the relative position of the source of disasters, the damage status, or its importance (schools, hospitals, and government institutions should be considered as more important). Locations requiring the same level of urgency can be clusterized into groups. And the priority of a group during the relief process has to be considered, e.g., higher priority groups should be visited before others. In the example above, the priorities indicate the importance (or urgency) of the demand at each location. Typically, priority 1 nodes must be served before priority 2 nodes, priority 2 nodes must be served before priority 3 nodes, and so on. Such a problem is called the Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters (CTSP-PO) and has been studied in [22, 17]. However, this rule is strict with respect to the priority and can lead to an inefficient route in terms of traveling cost. It may be relevant to visit some lower priority nodes while serving higher priority nodes. In [4, 5], the authors proposed a simple, but elegant rule called d-relaxed priority that provides flexibility to the decision maker in terms of capturing trade-offs between total distance and node priorities. In [5] and Chapter 14 of [6], the d-relaxed priority rule is defined as follows. Given a positive number d, at any point of the route, if p is the highest priority class among all unvisited locations, the relaxed rule allows the vehicle to visit locations with priority p, p + 1, ..., p + d before visiting all locations in class p. By changing the value of d, we can flexibly control to focus more on economic aspect or urgency level. Indeed, if we consider the 0-relaxed priority rule (i.e., d = 0), all the higher priority nodes must be visited before lower priority nodes. The problem is a CTSP-PO, the strictest version w.r.t priority. On the other hand, if d is set to g − 1, where g is the number of priorities, the problem becomes a typical TSP, all the node priorities being ignored.

2017

The multiple traveling salesman problem (mTSP) is a NP-hard combinatorial optimization problem. It has many real-world applications, for example, the School Bus Routing Problem, and the Pickup and Delivery Problem. In the mTSP, a set of routes is assigned to m salesmen who all start from and return to a home city(depot). In this problem, each other node is located in exactly one tour, the number of nodes visited by a salesman lies within a predetermined interval, and the overall cost of visiting all nodes is minimized. In this study, we discuss how to use constraint programming (CP) to formulate and solve mTSP by applying interval variables, global constraints and domain filtering algorithms. We propose a CP model for the mTSP. The CPmTSP was tested on a set of benchmark instances from the TSPLIB. Solutions of the CPmTSP are compared to the ILP-CPLEX of mTSP model and other algorithms (ACO, SW+ASelite, GELS-GA and Enhanced GA) in the literature. The computational results indicate th...

In this paper, some of the main known algorithms for the traveling salesman problem are surveyed. The paper is organized as follows: 1) definition; 2) applications; 3) complexity analysis; 4) exact algorithms; 5) heuristic algorithms; 6) conclusion.

Information Sciences, 1997

The Generalized Traveling Salesman Problem (GTSP) is a useful model for problems involving decisions of selection and sequence. The problem is defined on a directed graph in which the nodes have been pregrouped into m mutually exclusive and ejdiaustive nodesets. Arcs are defined only between nodes belonging to different nodesets with each arc having an associated cost. The GTSP is the problem of finding a minimum cost m-aic directed cycle which includes exactly one node from each nodeset. In ttiis paper, we show how to efficiently transform a GTSF into a standard asymmetric Traveling Salesman Problem (TSP) over the same number of nodes. The transformation allows certain routing problems which involve discrete alternatives to be modeled using the TSF framework. One such problem is the heterogenous Multiple Traveling Salesman Problem (MTSP) for which we provide a GTSP formulation.

New formulations are presented for the Travelling Salesman problem, and their relationship to previous formulations is investigated. The new formulations are extended to include a variety of transportation scheduling problems, such as the Multi-Travelling Salesman problem, the Delivery problem, the School Bus problem and the Dial-a-Bus problem. A Benders decomposition procedure is applied on the new formulations and the resulting computational rocedure is seen to be identical to previous methods for solving the Travelling Salesman problem. Based on the Lagrangean Relaxation method, a new procedure is suggested for generating lagrange multipliers for a subgradient optimization procedure.

ACM Transactions on Mathematical Software, 1995

The Fortran code CDT, implementing an algorithm for the asymmetrw traveling salesman problem, m presented. The method is based on the Assignment Problem relaxation and on a sub tour ellml nation branching scheme. The effectiveness of the implementation derives from reduction procedures and parametric solutlon of the relaxed problems associated with the nodes of the branch-decision tree,

SIAM Journal on Computing, 1977

Several polynomial time algorithms finding "good," but not necessarily optimal, tours for the traveling salesman problem are considered. We measure the closeness of a tour by the ratio of the obtained tour length to the minimal tour length. For the nearest neighbor method, we show the ratio is bounded above by a logarithmic function of the number of nodes. We also provide a logarithmic lower bound on the worst case. A class of approximation methods we call insertion methods are studied, and these are also shown to have a logarithmic upper bound. For two specific insertion methods, which we call nearest insertion and cheapest insertion, the ratio is shown to have a constant upper bound of 2, and examples are provided that come arbitrarily close to this upper bound. It is also shown that for any n => 8, there are traveling salesman problems with n nodes having tours which cannot be improved by making n/4 edge changes, but for which the ratio is 2(1-l/n).

2007

In this paper, we present a network flow-based, polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). Computational results are discussed.

2007

In this paper we introduce three greedy algorithms for the traveling salesman problem. These algorithms are unique in that they use arc tolerances, rather than arc weights, to decide whether or not to include an arc in a solution. We report extensive computational experiments on benchmark instances that clearly demonstrate that our tolerance-based algorithms outperform their weight-based counterpart. Along with other papers dealing with the Assignment Problem, this paper indicates that the potential for using tolerance-based algorithms for various optimization problems is high and motivates further investigation of the approach. We recommend one of our algorithms as a significantly better alternative to the weight-based greedy, which is often used to produce initial TSP tours.