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A BranchandCut Algorithm for the Symmetric Generalized Travelling Salesman Problem
, 1995
"... We consider a variant of the classical symmetric Travelling Salesman Problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GT ..."
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Cited by 75 (4 self)
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We consider a variant of the classical symmetric Travelling Salesman Problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GTSP), and finds practical applications in routing, scheduling and locationrouting. In a companion paper [5] we modeled GTSP as an integer linear program, and studied the facial structure of two polytopes associated with the problem. Here we propose exact and heuristic separation procedures for some classes of facetdefining inequalities, which are used within a branchandcut algorithm for the exact solution of GTSP. Heuristic procedures are also described. Extensive computational results for instances taken from the literature and involving up to 442 nodes are reported.
A RandomKey Genetic Algorithm for the Generalized Traveling Salesman Problem
 European Journal of Operational research
, 2004
"... The Generalized Traveling Salesman Problem is a variation of the well known Traveling Salesman Problem in which the set of nodes is divided into clusters; the objective is to find a minimumcost tour passing through one node from each cluster. We present an effective heuristic for this problem. The ..."
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Cited by 47 (0 self)
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The Generalized Traveling Salesman Problem is a variation of the well known Traveling Salesman Problem in which the set of nodes is divided into clusters; the objective is to find a minimumcost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.
Generalized network design problems
 European Journal of Operational Research
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A discrete particle swarm optimization algorithm for the generalized traveling salesman problem
 GECCO ’07: Proceedings of the 9th annual conference on Genetic and evolutionary computation, 2007, 158–167. Conclusion 15
"... Dividing the set of nodes into clusters in the wellknown traveling salesman problem results in the generalized traveling salesman problem which seeking a tour with minimum cost passing through only a single node from each cluster. In this paper, a discrete particle swarm optimization is presented t ..."
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Cited by 14 (0 self)
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Dividing the set of nodes into clusters in the wellknown traveling salesman problem results in the generalized traveling salesman problem which seeking a tour with minimum cost passing through only a single node from each cluster. In this paper, a discrete particle swarm optimization is presented to solve the problem on a set of benchmark instances. The discrete particle swarm optimization algorithm exploits the basic features of its continuous counterpart. It is also hybridized with a local search, variable neighborhood descend algorithm, to further improve the solution quality. In addition, some speedup methods for greedy node insertions are presented. The discrete particle swarm optimization algorithm is tested on a set of benchmark instances with symmetric distances up to 442 nodes from the literature. Computational results show that the discrete particle optimization algorithm is very promising to solve the generalized traveling salesman problem.
Effective neighborhood structures for the generalized traveling salesman problem
 Evolutionary Computation in Combinatorial Optimisation – EvoCOP 2008, volume 4972 of LNCS
, 2008
"... Abstract. We consider the generalized traveling salesman problem in which a graph with nodes partitioned into clusters is given. The goal is to identify a minimum cost round trip visiting exactly one node from each cluster. For solving difficult instances of this problem heuristically, we present a ..."
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Cited by 11 (2 self)
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Abstract. We consider the generalized traveling salesman problem in which a graph with nodes partitioned into clusters is given. The goal is to identify a minimum cost round trip visiting exactly one node from each cluster. For solving difficult instances of this problem heuristically, we present a new Variable Neighborhood Search (VNS) approach that utilizes two complementary, large neighborhood structures. One of them is the already known generalized 2opt neighborhood for which we propose a new incremental evaluation technique to speed up the search significantly. The second structure is based on node exchanges and the application of the chained LinKernighan heuristic. A comparison with other recently published metaheuristics on TSPlib instances with geographical clustering indicates that our VNS, though requiring more time than two genetic algorithms, is able to find substantially better solutions.
A BranchandCut Algorithm for the Undirected Traveling Purchaser Problem
 Networks
, 1998
"... The purpose of this article is to present a branchandcut algorithm for the undirected Traveling Purchaser Problem which consists of determining a minimumcost route through a subset of markets, where the cost is the sum of travel and purchase costs. The problem is formulated as an integer linear p ..."
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Cited by 10 (2 self)
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The purpose of this article is to present a branchandcut algorithm for the undirected Traveling Purchaser Problem which consists of determining a minimumcost route through a subset of markets, where the cost is the sum of travel and purchase costs. The problem is formulated as an integer linear program and several families of valid inequalities are derived to strengthen the linear relaxation. The polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facetdefining. A branchandcut procedure is developed and tested over several classes of randomly generated instances. Results show that the proposed algorithm outperform all previous approaches and can solve optimally instances containing up to 200 markets.
Transformations of generalized ATSP into ATSP
 Operations Research Letters
"... The Generalized Traveling Salesman Problem (GTSP) is stated as follows. Given a weighted complete digraph K ∗ n and a partition V1,..., Vk of its vertices, find a minimum weight cycle containing exactly one vertex from each set Vi, i = 1,..., k. We study transformations from GTSP to TSP. The ’exact ..."
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Cited by 10 (3 self)
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The Generalized Traveling Salesman Problem (GTSP) is stated as follows. Given a weighted complete digraph K ∗ n and a partition V1,..., Vk of its vertices, find a minimum weight cycle containing exactly one vertex from each set Vi, i = 1,..., k. We study transformations from GTSP to TSP. The ’exact ’ NoonBean transformation is investigated in computational experiments. We study the ’nonexact ’ FischettiSalazarToth (FST) transformation and its two modifications in computational experiments and theoretically using domination analysis. One of our conclusions is that one of the modifications of the FST transformation is better than the original FST transformation in the worst case in terms of domination analysis.
The Generalized Minimum Spanning Tree Problem: Polyhedral Analysis and BranchandCut Algorithm
, 2002
"... This article presents a branchandcut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum cost tree including exactly one vertex per cluster. Applications of the GMS ..."
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Cited by 9 (4 self)
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This article presents a branchandcut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum cost tree including exactly one vertex per cluster. Applications of the GMSTP are encountered in telecommunications. An integer linear programming formulation is presented and new classes of valid inequalities are developed, several of which are proved to be facet defining. A branchandcut algorithm and a tabu search heuristic are developed. Extensive computational experiments show that instances involving up to 160 or 200 vertices can be solved to optimality, depending on whether edge costs are Euclidean or random.
Integer linear programming formulation of the generalized vehicle routing problem
 In Proceedings of the 5th EURO/INFORMS Joint International Meeting
, 2003
"... The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formu ..."
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Cited by 5 (0 self)
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The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formulation of the GVRP with O(n 2) binary variables and O(n 2) constraints is presented. It is shown that, under specific circumstances, the proposed model reduces to the wellknown routing problems. The computational performance of the models solved using a commercial code on test problems are also presented.