Advertisement

Possibilistic Framework for Multi-Objective Optimization Under Uncertainty

  • Oumayma Bahri
  • Nahla Ben Amor
  • El-Ghazali TalbiEmail author
Chapter
  • 648 Downloads
Part of the Operations Research/Computer Science Interfaces Series book series (ORCS, volume 62)

Abstract

Optimization under uncertainty is an important line of research having today many successful real applications in different areas. Despite its importance, few works on multi-objective optimization under uncertainty exist today. In our study, we address combinatorial multi-objective problem under uncertainty using the possibilistic framework. To this end, we firstly propose new Pareto relations for ranking the generated uncertain solutions in both mono-objective and multi-objective cases. Secondly, we suggest an extension of two well-known Pareto-base evolutionary algorithms namely, SPEA2 and NSGAII. Finally, the extended algorithms are applied to solve a multi-objective Vehicle Routing Problem (VRP) with uncertain demands.

Keywords

Multi-objective optimization Uncertainty Possibilty theory Evolutionary algorithms Vehicle routing problem 

References

  1. 1.
    S. Asma, E.-G. Talbi, M. Aider, A. Liefooghe, Multi-objective local search with epistemic uncertainty: application to multi-objective vehicle routing problem with uncertain demands, in ISOR’11 (2011), pp. 1–22Google Scholar
  2. 2.
    M. Babbar, A. Lakshmikantha, D.E. Goldberg, A modified NSGA-II to solve noisy multiobjective problems, in Genetic and Evolutionary Computation Conference (GECCO’03). Lecture Notes in Computer Science, Chicago, IL (Springer, Berlin, 2003), pp. 2723–2727Google Scholar
  3. 3.
    M. Basseur, A. Liefooghe, L. Jourdan, E.-G. Talbi, ParadisEO-MOEO: a framework for evolutionary multi-objective optimization, in Evolutionary Multi-Criterion Optimization (Springer, Berlin, 2007), pp. 386–400Google Scholar
  4. 4.
    J. Brito, J.A. Morino, J.L. Verdegay, Fuzzy optimization in vehicle routing problems, in IFSA-EUSFLAT (2009)Google Scholar
  5. 5.
    C.A.C. Coello, G.B. Lamont, Applications of Multi-objective Evolutionary Algorithms (World Scientific, Singapore, 2004)CrossRefGoogle Scholar
  6. 6.
    C.A.C. Coello, G.B. Lamont, D.A. Van Veldhuisen, Evolutionary Algorithms for Solving Multi-objective Problems (Springer, New York, 2007)Google Scholar
  7. 7.
    K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms (Wiley, New York, 2001)Google Scholar
  8. 8.
    K. Deb, N. Srinivas, Multiobjective optimization using nondominated sorting in genetic algorithms. IEEE Trans. Evol. Comput. 2(3), 221–248 (1994)Google Scholar
  9. 9.
    K. Deb, S. Agrawal, A. Pratap, T. Meyarivan, A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2000)CrossRefGoogle Scholar
  10. 10.
    D. Dubois, H. Prade, An introductory survey of possibility theory and its recent developments. J. Jpn. Soc. Fuzzy Theory Syst. 10, 21–42 (1998)Google Scholar
  11. 11.
    C.M. Fonseca, P.J. Fleming, Genetic algorithms for multiobjective optimization: formulation, discussion and generalization, in Proceedings of the Fifth International Conference on Genetic Algorithms (1993), pp. 416–423Google Scholar
  12. 12.
    C.K. Goh, K.C. Tan, Evolutionary Multi-objective Optimization in Uncertain Environments: Issues and Algorithms (Springer, Heidelberg, 2009)Google Scholar
  13. 13.
    G. Goncalves, T. Hsu, J. Xu, Vehicle routing problem with time windows and fuzzy demands: an approach based on the possibility theory. Int. J. Adv. Oper. Manage. Inderscience 4, 312–330 (2009)Google Scholar
  14. 14.
    J. Horn, N. Nafpliotis, D.E. Goldberg, A niched Pareto genetic algorithm for multiobjective optimization, in Proceedings of the First IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Intelligence, vol. 1 (1994), pp. 82–87Google Scholar
  15. 15.
    E. Hughes, Evolutionary multi-objective ranking with uncertainty and noise, in Proceedings of the First International Conference on Evolutionary Multi-Criterion Optimization, London (Springer, Berlin, 2001), pp. 329–343Google Scholar
  16. 16.
    E.J. Hughes, Constraint handling with uncertain and noisy multi-objective evolution, in Proceedings of the 2001 Congress on Evolutionary Computation, Seoul (2001)Google Scholar
  17. 17.
    Y. Jin, J. Branke, Evolutionary optimization in uncertain environments: a survey. IEEE Trans. Evol. Comput. 9(3), 303–317 (2005)CrossRefGoogle Scholar
  18. 18.
    J.D. Knowles, D.W. Corne, Approximating the nondominated front using the Pareto archived evolution strategy. IEEE Trans. Evol. Comput. 8(2), 149–172 (2000)Google Scholar
  19. 19.
    A.N. Kolmogorov, Foundations of the Theory of Probability, 2nd edn. (Chelsea Pub Co., New York, 1960)Google Scholar
  20. 20.
    M.H. Laarabi, R. Sacile, A. Boulmakoul, E. Garbolino, Ranking triangular fuzzy numbers using fuzzy set inclusion index, in Fuzzy Logic and Applications (Springer, Cham, 2013), pp. 100–108Google Scholar
  21. 21.
    P. Limbourg, Multi-objective optimization of problems with epistemic uncertainty, in Evolutionary Multi-Criterion Optimization (Springer, Berlin, 2005), pp. 413–427Google Scholar
  22. 22.
    P. Limbourg, E.S. Daniel, An optimization algorithm for imprecise multi-objective problem functions. Evol. Comput. 1, 459–466 (2005)Google Scholar
  23. 23.
    B. Oumayma, B.A. Nahla, E.-G. Talbi, A possibilistic framework for solving multi-objective problems under uncertainty: definition of new Pareto optimality, in IPDPSW 2013 (2013), pp. 405–414Google Scholar
  24. 24.
    L.F. Paquete, T. Stutzle, Stochastic local search algorithms for multiobjective combinatorial optimization: methods and analysis, in Handbook of Approximation Algorithms and Metaheuristics, vol. 13 (Chapman & Hall/CRC Boca Raton, 2007)Google Scholar
  25. 25.
    G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, 1976)Google Scholar
  26. 26.
    P. Smets, Constructing the pignistic probability function in a context of uncertainty, in Proceeding 5th Conference on Uncertainty in Artificial intelligence, Windsor (1989), pp. 29–40Google Scholar
  27. 27.
    M.M. Solomon, Algorithms for the vehicle routing and scheduling problem with time window constraints. Oper. Res. 35(2), 254–265 (1987)CrossRefGoogle Scholar
  28. 28.
    D. Sulieman, L. Jourdan, E.-G. Talbi, Using multiobjective metaheuristics to solve VRP with uncertain demands, in IEEE Congress on Evolutionary Computation (2010), pp. 1–8Google Scholar
  29. 29.
    E.-G. Talbi, Metaheuristics: From Design to Implementation (Wiley, New York, 2009)CrossRefGoogle Scholar
  30. 30.
    J. Teich, Pareto-front exploration with uncertain objectives, in Evolutionary Multi-Criterion Optimization (EMO2001). Lecture Notes in Computer Science, vol. 1993 (2001), pp. 314–328Google Scholar
  31. 31.
    P. Toth, D. Vigo, The Vehicle Routing Problem (SIAM, Philadelphia, 2002)CrossRefGoogle Scholar
  32. 32.
    Z. Wang, F. Tian, A note of the expected value and variance of fuzzy variables. Int. J. Nonlinear Sci. 9(4), 486–492 (2010)Google Scholar
  33. 33.
    L.A. Zadeh, Fuzzy sets. Inf. Control 16, 338–353 (1965)CrossRefGoogle Scholar
  34. 34.
    L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 100, 9–34 (1999)CrossRefGoogle Scholar
  35. 35.
    E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)CrossRefGoogle Scholar
  36. 36.
    E. Zitzler, M. Laumans, L. Thiele, SPEA2: improving the strength Pareto evolutionary algorithm. Technical Report 103, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich, Zurich, May 2001Google Scholar
  37. 37.
    E. Zitzler, L. Thiele, M. Laumanns, C.M. Foneseca, D. Grunert, Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7, 117–132 (2003)CrossRefGoogle Scholar
  38. 38.
    E. Zitzler, L. Thiele, J.D Knowles, A tutorial on the performance assessment of stochastic multiobjective optimizers, in Proceeding of the Third International Conference on Evolutionary Multi-Criterion Optimization (2005)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Oumayma Bahri
    • 1
  • Nahla Ben Amor
    • 2
  • El-Ghazali Talbi
    • 3
    Email author
  1. 1.LARODEC and INRIA Lab.University Lille 1LilleFrance
  2. 2.LARODEC LaboratoryISG TunisLe BardoTunisia
  3. 3.INRIA Laboratory, CRISTAL/CNRSUniversity Lille 1LilleFrance

Personalised recommendations