Abstract
In this study, we work on the traveling salesperson problems and bottleneck traveling salesperson problems that have special matrix structures and lead to polynomially solvable cases. We extend the problems to multiple objectives and investigate the properties of the nondominated points. We develop a pseudo-polynomial time algorithm to find a nondominated point for any number of objectives. Finally, we propose an approach to generate all nondominated points for the biobjective case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aneja Y.P., Nair K.P.: Bicriteria transportation problem. Manage. Sci. 25(1), 73–78 (1979)
Baki M.F.: A new asymmetric pyramidally solvable class of the traveling salesman problem. Oper. Res. Lett. 34(6), 613–620 (2006)
Burkard R.E., Deineko V.G.: On the Euclidean TSP with permuted Van Der Veen Matrix. Inf. Process. Lett. 91, 259–262 (2004)
Burkard R.E., Sandholzer W.: Efficiently solvable special cases of bottleneck traveling salesman problems. Dis. Appl. Math. 32, 61–76 (1991)
Burkard R.E., Klinz B., Rudolf R.: Perspectives of Monge properties in optimization. Dis. Appl. Math. 70, 95–161 (1996)
Burkard R.E., Deineko V.G., Van Dal R., Van der Veen J.A.A., Woeginger G.J.: Well-solvable special cases of the traveling salesman problem: a survey. SIAM Rev. 40(3), 496–546 (1998)
Ehrgott M., Gandibleux X.: Multiobjective combinatorial optimization—theory, methodology, and applications. In: Ehrgott, M. (eds) Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys, pp. 369–444. Kluwer, Secaucus (2002)
Gilmore R.C., Lawler E.L., Shmoys D.B.: Well-solved special cases. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds) The Traveling Salesman Problem, pp. 87–143. Wiley, Chichester (1985)
Gutin G., Yeo A., Zverovitch A.: Exponential neighborhoods and domination analysis for the TSP. In: Gutin, G., Punnen, A.P. (eds) The Traveling Salesman Problem and Its Variations, pp. 223–256. Kluwer, Dordrecht (2002)
Kabadi S.N.: Polynomially solvable cases of the TSP. In: Gutin, G., Punnen, A.P. (eds) The Traveling Sales- man Problem and Its Variations, pp. 489–583. Kluwer, Dordrecht (2002)
Oda Y.: An asymmetric analog of van der Veen conditions and the traveling salesman problem (II). Eur. J. Oper. Res. 138(1), 43–62 (2002)
Özpeynirci Ö., Köksalan M.: Multiobjective traveling salesperson problem on Halin graphs. Eur. J. Oper. Res. 196(1), 155–161 (2009)
Steuer R.E.: Multiple Criteria Optimization: Theory, Computation and Application. Wiley, New York (1986)
Van Dal R., Van der Veen J.A.A., Sierksma G.: Small and large TSP—2 polynomially solvable cases of the traveling salesman problem. Eur. J. Oper. Res. 69(1), 107–120 (1993)
Van der Veen J.A.A.: A new class of pyramidally solvable symmetric traveling salesman problems. SIAM J. Dis. Math. 7, 585–592 (1994)
Van der Veen J.A.A., Sierksma G., Van Dal R.: Pyramidal tours and the traveling salesman problem. Eur. J. Oper. Res. 51(1), 90–102 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Özpeynirci, Ö., Köksalan, M. Pyramidal tours and multiple objectives. J Glob Optim 48, 569–582 (2010). https://doi.org/10.1007/s10898-009-9505-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9505-0