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Bilevel Biobjective Pseudo Boolean Programming Problems

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Journal of Mathematical Modelling and Algorithms

Abstract

In the present paper a kind of bilevel programming problem in 0–1 variables, based on the mathematical model attached by us to a concrete portfolio optimization problem, is analyzed. The upper level function is to be maximized, while the lower level function (which is a biobjective function) is to be maximized-minimized in the lexicographic sense. The core idea of this paper is to present a way for solving the proposed bilevel problem by reducing it to a finite number of couples of linear pseudo boolean programming problems. One of these last types of problems is an assignment problem. Also, in this paper the notion of a max  − p min  −  max point is introduced and several necessary and sufficient optimal conditions (useful for solving the problems gained after applying the splitting technique) are proved. Furthermore, an algorithm for solving the initial problem is proposed and an example is given.

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References

  1. Bandopadhyaya, L.: Cost-time trade-off in three-axial sums’ transportation problem. J. Aust. Math. Soc. 35, 498–505 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berthold, T., Heinz, S., Pfetsch, M.E.: Solving pseudo-boolean problems with SCIP. http://opus4.kobv.de/opus4-zib/files/1067/ZR-08-12.pdf. Accessed 26 September 2010

  3. Bertsekas, D.P.: A new algorithm for the assignment problem. Math. Program. 21, 152–171 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bracken, J., McGill, J.: Mathematical programs with optimization problems in the constraints. Oper. Res. 21, 37–44 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brotcorne, L., Labbé, M., Marcotte, P., Savard, G.: Joint design and pricing on a network. Oper. Res. 56, 1104–1115 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Candler, W., Norton, R.: Multilevel Programming. Technical Report 20, World Bank Development Research Center, Washington D.C. (1977)

  7. Duca, D., Lupşa, L.: Bilevel transportation problems. Rev. Anal. Numer. Teor. Aprox. 30(1), 25–34 (2001)

    MATH  Google Scholar 

  8. Faisca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, E.N.: A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. Comput. Mater. Sci. 6(4), 377–397 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Gao, Y., Lu, J., Zhang, G., Gao, S.: A bilevel model for railway train set organizing optimization. htpp://Lu_Jie.pdf; ISKE2007-Lu_Jie.pdf; www.atlantis-press.com. Accessed 15 October 2010

  10. Gaur, A., Arora, S.R.: Multi-level multi-objective integer linear programming problem. Advanced Modelling and Optimization 2, 297–322 (2008)

    MathSciNet  Google Scholar 

  11. Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer-Verlag, New York (1968)

    Book  MATH  Google Scholar 

  12. Kuhn, H.W.: Variants of the Hungaria method for assignement problems. Nav. Res. Logist. Q. 3, 253–258 (1956)

    Article  Google Scholar 

  13. Li, Y., Chen, S.: Optimal traffic signal control for an M × N traffic network. J. Ind. Manag. Optim. 4, 661–672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, Y., Mei, J.: Optimality conditions in bilevel multiobjective programming problems. Southeast Asian Bull. Math. 33, 79–87 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Lotito, P.A., Mancinelli, E.M., Quadrat, J.P., Wynter, L.: A global descent heuristic for bilevel network pricing. http://www-rocq.inria.fr/metalau/quadrat/bpalgocomplet.pdf (2004). Accessed 15 October 2010

  16. Manquinho, V., Marques-Silva, J.: On applying cutting planes in DLL-based algorithms for pseudo-boolean optimization. Technical Report RT /003/05-CDIL, INESC-ID (2005)

  17. Neamtiu, L.: A medical resources allocation problem. Results Math. 53(3–4), 341–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sahu, A., Tapadar, R.: Solving the assignment problem using genetic algorithm and simulated annealing. IAENG Int. J. Appl. Math. 36(1), 37–40 (2007)

    MathSciNet  Google Scholar 

  19. Stackelberg, H.F.: Marktform und Gleichgewicht. Springer-Verlag, Berlin (1934)

    Google Scholar 

  20. Tuns (Bode), O.R.: Bilevel E Cost-Time-P programming problems. Proceedings of the 2011 International Conference on Computational and Mathematical Methods in Science and Engineering 3, 1168–1179 (2011)

    Google Scholar 

  21. Wang, Z.W., Nagasawa, H., Nishiyama, N.: An algorithm for a multiobjective, multilevel linear programming. J. Oper. Res. Soc. Jpn. 39, 176–187 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Mihail Kogălniceanu Str., no. 1, Cluj-Napoca, 400084, Romania

    Oana Ruxandra Tuns (Bode)

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  1. Oana Ruxandra Tuns (Bode)
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Correspondence to Oana Ruxandra Tuns (Bode).

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Tuns (Bode), O.R. Bilevel Biobjective Pseudo Boolean Programming Problems. J Math Model Algor 11, 325–344 (2012). https://doi.org/10.1007/s10852-012-9188-2

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  • DOI: https://doi.org/10.1007/s10852-012-9188-2

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