Abstract
In this paper, we consider the Prize Collecting Travelling Salesman Problem (PCTSP),which is a variant of the Travelling Salesman Problem (TSP), where a tour visiting eachnode at most once in a given graph has to be computed such that a prize is associated witheach node and a penalty has to be paid for every unvisited node: moreover, a knapsackconstraint guarantees that a sufficiently large prize is collected. We develop a Lagrangianheuristic and obtain an upper bound in the form of a feasible solution starting from a lowerbound to the problem recently proposed in the literature. We evaluate these bounds utilizingboth randomly generated instances and real ones with very satisfactory results.
Similar content being viewed by others
References
F. Baccus, P. Cowling, N. Vaessen and L. Van Neron, Optimal rolling mill planning at Usines Gustane Boel, Steel Times /Steel Times International, 1995, pp. S2–S4.
E. Balas, The prize collecting travelling salesman problem, Networks 19(1989)621 –636.
E. Balas, The prize collecting travelling salesman problem: II. Polyhedral results, Networks 25 (1995)199–216.
E. Balas and C.H. Martin, ROLL-A-ROUND: Software package for scheduling the rounds of a rolling mill, Copyright Balas and Martin Associates, 104 Maple Heights Road, Pittsburgh, USA, 1985.
E. Balas and C.H. Martin, Combinatorial optimization in steel rolling, DIMACS /RUTCOR Workshop on Combinatorial Optimization in Science and Technology, Rutgers University, USA, April 1991.
G. Carpaneto, M. Dell'Amico and P. Toth, Exact solution of large-scale asymmetric travelling salesman problem, ACM Transaction on Mathematical Software 2(1995)394–409.
P. Cowling, Optimization in steel hot rolling, in: Optimization in Industry 3, ed. A. Sciomachen, Wiley, 1995, pp. 55–66.
M. Dell' Amico, F. Maffioli and P. Värbrand, On prize-collecting tours and the asymmetric travelling salesman problem, International Transactions in Operational Research 2(1995)297–308.
M. Fischetti and P. Toth, An additive approach for the optimal solution of the prize collecting travelling salesman problem, in: Vehicle Routing: Methods and Studies, eds. B.L. Golden and A Assad, Elsevier Science, 1988, pp. 319–343.
M. Fischetti, J. Salazar, J. González and P. Toth, A branch-and-cut algorithm for the symmetric generalized travelling salesman problem, DEIS Research Report, Bologna, Nov. 1993.
B.L. Golden, L. Levy and R. Vohra, The orienteering problem, Naval Research Logistics Quarterly 34(1987)307–318.
B.L. Golden, Q. Wang and L. Liu, A multifaceted heuristic for the orienteering problem, Naval Research Logistics Quarterly 35(1988)359 –366.
G. Laporte and S. Martello, The selective travelling salesman problem, Discrete Applied Mathematics 26(1990)797–809.
T. Volgenant and R. Jonker, On some generalizations of the travelling salesman problem, Journal of the Operational Research Society 38(1987)1073–1079.
Rights and permissions
About this article
Cite this article
Dell'Amico, M., Maffioli, F. & Sciomachen, A. A Lagrangian heuristic for the Prize CollectingTravelling Salesman Problem. Annals of Operations Research 81, 289–306 (1998). https://doi.org/10.1023/A:1018961208614
Issue Date:
DOI: https://doi.org/10.1023/A:1018961208614