Advertisement

Fuzzy Optimization and Decision Making

, Volume 12, Issue 3, pp 231–248 | Cite as

Fuzzy costs in quadratic programming problems

  • Ricardo C. Silva
  • Carlos Cruz
  • José L. Verdegay
Article

Abstract

Although quadratic programming problems are a special class of nonlinear programming, they can also be seen as general linear programming problems. These quadratic problems are of the utmost importance in an increasing variety of practical fields. As, in addition, ambiguity and vagueness are natural and ever-present in real-life situations requiring operative solutions, it makes perfect sense to address them using fuzzy concepts formulated as quadratic programming problems with uncertainty, i.e., as Fuzzy Quadratic Programming problems. This work proposes two novel fuzzy-sets-based methods to solve a particular class of Fuzzy Quadratic Programming problems which have vagueness coefficients in the objective function. Moreover, two other linear approaches are extended to solve the quadratic case. Finally, it is shown that the solutions reached from the extended approaches may be obtained from two proposed parametric multiobjective approaches.

Keywords

Fuzzy set Decision making Fuzzy mathematical optimization Quadratic programming Efficient solutions 

Notes

Acknowledgments

The authors want to thank the financial support from the agency FAPESP (project number 2010/51069-2) and the Spanish projects CEI BioTic GENIL from the MICINN, as well as TIN2011-27696-C02-01, P11-TIC-8001, TIN2008-06872-C04-04, and TIN2008-01948.

References

  1. Appadoo, S., Bhatt, S., & Bector, C. (2008). Application of possibility theory to investment decisions. Fuzzy Optimization and Decision Making, 7, 35–57.MathSciNetMATHCrossRefGoogle Scholar
  2. Bector, C. R., & Chandra, S. (2005). Fuzzy mathematical programming and fuzzy matrix games, volume 169 of studies in fuzziness and soft computing. Berlin: Springer.Google Scholar
  3. Bellman, R. E., & Zadeh, L. A. (1970). Decision-marking in a fuzzy environment. Management Science, 17(4), B141–B164.MathSciNetMATHCrossRefGoogle Scholar
  4. Cruz, C., Silva, R. C., & Verdegay, J. L. (2011). Extending and relating different approaches for solving fuzzy quadratic problems. Fuzzy Optimization and Decision Making, 10, 193–210.MathSciNetMATHCrossRefGoogle Scholar
  5. Delgado, M., Verdegay, J. L., & Vila, M. (1987). Imprecise costs in mathematical programming problems. Control and Cybernetics, 16(2), 113–121.MathSciNetMATHGoogle Scholar
  6. Delgado, M., Verdegay, J. L., & Vila, M. (1990). Relating different approaches to solve linear programming problems with imprecise costs. Fuzzy Sets and Systems, 37, 33–42.MathSciNetMATHCrossRefGoogle Scholar
  7. Floudas, C. A., Pardalos, P. M., Adjiman, C., Esposito, W. R., Gümüs, Z. H., Harding, S. T., et al. (1999). Handbook of test problems in local and global optimization, volume 33 of nonconvex optimization and its applications. Dordrecht: Kluwer.CrossRefGoogle Scholar
  8. Hock, W., & Schittkowski, K. (1981). Test examples for nonlinear programming codes, volume 187 of lecture notes in economics and mathematical systems. Berlin: Spring.Google Scholar
  9. Ida, M. (2003). Portfolio selection problem with interval coefficients. Applied Mathematics Letters, 16, 709–713.Google Scholar
  10. Jiménez, F., Cadenas, J., Sánchez, G., Gómez-skarmeta, A., & Verdegay, J. L. (2006). Multi-objective evolutionary computation and fuzzy optimization. International Journal of Approximate Reasoning, 43(1), 59–75.MathSciNetMATHCrossRefGoogle Scholar
  11. Lai, Y. J., & Hwang, C. L. (1992). Fuzzy mathematical programming: Methods and applications, volume 394 of lecture notes in economics and mathematical systems. Berlin: Springer.Google Scholar
  12. Liang, R.-H., & Liao, J.-H. (2007). A fuzzy-optimization approach for generation scheduling with wind and solar energy systems. IEEE Transactions on Power Systems, 22(4), 1665–1674.CrossRefGoogle Scholar
  13. Mollov, S., Babuska, R., Abonyi, J., & Verbruggen, H. B. (2004). Effective optimization for fuzzy model predictive control. IEEE Transactions on Fuzzy Systems, 12(5), 661–675.CrossRefGoogle Scholar
  14. Negoita, C. V., & Ralescu, D. A. (1975). Applications of fuzzy sets to systems analysis. Stuttgard: Birkhauser Verlag.MATHGoogle Scholar
  15. Rommelfanger, H., Hanuscheck, R., & Wolf, J. (1989). Linear programming with fuzzy objectives. Fuzzy Sets and Systems, 29, 31–48.MathSciNetMATHCrossRefGoogle Scholar
  16. Schittkowski, K. (1987). More test examples for nonlinear programming codes, volume 282 of lecture notes in economics and mathematical systems. Berlin: Springer.Google Scholar
  17. Silva, R. C., Cruz, C., Verdegay, J. L., & Yamakami, A. (2010a). A survey of fuzzy convex programming models. In L. Weldon & J. Kacprzyk (Eds.), Fuzzy optimization: Recent advances and applications, volume 254 of studies in fuzziness and soft computing. Berlin: Springer.Google Scholar
  18. Silva, R. C., Verdegay, J. L., & Yamakami, A. (2010b). A parametric convex programming approach applied in portfolio selection problem with fuzzy costs. In 2010 IEEE international fuzzy systems conference, Barcelona, FUZZ-IEEE 2010.Google Scholar
  19. Tanaka, H., Hichihashi, H., & Asai, K. (1984). A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers. Control and Cybernetics, 13(1), 1–10.MATHGoogle Scholar
  20. Tang, J., & Wang, D. (1997). An interactive approach based on a genetic algorithm for a type of quadratic programming problems with fuzzy objective and resources. Computers & Operations Research, 24(5), 413–422.MathSciNetMATHCrossRefGoogle Scholar
  21. Wang, S., & Singh, C. (2008). Balancing risk ad cost in fuzzy economic dispatch including wind power penetration based on particle swarm optimization. Electric Power Systems Research, 78, 1361–1368.CrossRefGoogle Scholar
  22. Wang, S., & Zhu, S. (2002). On fuzzy portfolio selection problems. Fuzzy Optimization and Decision Making, 1, 361–377.MathSciNetMATHCrossRefGoogle Scholar
  23. Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ricardo C. Silva
    • 1
  • Carlos Cruz
    • 2
  • José L. Verdegay
    • 2
  1. 1.Institute of Science and TechnologyFederal University of São PauloSão José dos CamposBrazil
  2. 2.Department of Computer Science and Artificial IntelligenceCITIC–University of GranadaGranadaSpain

Personalised recommendations