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Interactive Multiobjective Mixed-Integer Optimization Using Dominance-Based Rough Set Approach

  • Salvatore Greco
  • Benedetto Matarazzo
  • Roman Słowiński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)

Abstract

We present a new methodology for dealing with interactive multiobjective optimization in case of mixed-integer variables. The preference information elicited by the Decision Maker (DM) in course of the interaction is processed using the Dominance-based Rough Set Approach (DRSA). This permits to ask the DM simple questions and obtain in return a decision model expressed in terms of easily understandable “if..., then...” decision rules. In each iteration, the current set of decision rules is presented to the DM with the proposal of selecting one of them considered the most representative. The selected decision rule specifies some minimal requirements that the DM desires to be achieved by the objective functions. This information is translated into a set of constraints which are added to the original problem restricting the space of feasible solutions. Moreover, we introduce one simple but effective algorithm, called bound-and-cut, that efficiently reduces the set of feasible values of the integer variables. This process continues iteratively until the part of the Pareto front that is interesting for the DM can be exhaustively explored with respect to the integer variables. The bound-and-cut algorithm can be embedded in an Evolutionary Multiobjective Optimization (EMO) method, which permits to compute a reasonable approximation of the considered part of the Pareto front. A subset of representative solutions can be selected from this approximation and presented to the DM in the dialogue phase of each iteration.

Keywords

Decision Maker Feasible Solution Pareto Front Multiobjective Optimization Integer Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alves, M.J., Climaco, J.: A review of interactive methods for multiobjective integer and mixed-integer programming. European Journal of Operatinal Research 180, 99–115 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-Price: Column Generation for Huge Integer Programs. Operations Research 46(3), 316–329 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.): Multiobjective Optimization. LNCS, vol. 5252. Springer, Heidelberg (2008)Google Scholar
  4. 4.
    Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester (2001)zbMATHGoogle Scholar
  5. 5.
    Evans, G.W.: An overview of techniques for solving multi-objective mathematical programs. Management Science 30, 1263–1282 (1984)Google Scholar
  6. 6.
    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64, 275–278 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European Journal of Operational Research 129, 1–47 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Greco, S., Matarazzo, B., Słowiński, R.: Dominance-Based Rough Set Approach to Interactive Multiobjective Optimization. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization: Interactive and Evolutionary Approaches. LNCS, vol. 5252, pp. 121–155. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Land, A.H., Doig, A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991)zbMATHGoogle Scholar
  11. 11.
    Słowiński, R., Greco, S., Matarazzo, B.: Rough set based decision support. In: Burke, E.K., Kendall, G. (eds.) Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, ch. 16, pp. 475–527. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Słowiński, R., Greco, S., Matarazzo, B.: Rough Sets in Decision Making. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 7753–7786. Springer, Heidelberg (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Salvatore Greco
    • 1
  • Benedetto Matarazzo
    • 1
  • Roman Słowiński
    • 2
    • 3
  1. 1.Faculty of EconomicsUniversity of CataniaCataniaItaly
  2. 2.Institute of Computing SciencePoznań University of TechnologyPoznańPoland
  3. 3.Systems Research InstitutePolish Academy of SciencesWarsawPoland

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