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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Appadoo, S., Bhatt, S., & Bector, C. (2008). Application of possibility theory to investment decisions. Fuzzy Optimization and Decision Making, 7, 35–57.
Bector, C. R., & Chandra, S. (2005). Fuzzy mathematical programming and fuzzy matrix games, volume 169 of studies in fuzziness and soft computing. Berlin: Springer.
Bellman, R. E., & Zadeh, L. A. (1970). Decision-marking in a fuzzy environment. Management Science, 17(4), B141–B164.
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.
Delgado, M., Verdegay, J. L., & Vila, M. (1987). Imprecise costs in mathematical programming problems. Control and Cybernetics, 16(2), 113–121.
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.
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.
Hock, W., & Schittkowski, K. (1981). Test examples for nonlinear programming codes, volume 187 of lecture notes in economics and mathematical systems. Berlin: Spring.
Ida, M. (2003). Portfolio selection problem with interval coefficients. Applied Mathematics Letters, 16, 709–713.
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.
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.
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.
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.
Negoita, C. V., & Ralescu, D. A. (1975). Applications of fuzzy sets to systems analysis. Stuttgard: Birkhauser Verlag.
Rommelfanger, H., Hanuscheck, R., & Wolf, J. (1989). Linear programming with fuzzy objectives. Fuzzy Sets and Systems, 29, 31–48.
Schittkowski, K. (1987). More test examples for nonlinear programming codes, volume 282 of lecture notes in economics and mathematical systems. Berlin: Springer.
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.
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.
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.
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.
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.
Wang, S., & Zhu, S. (2002). On fuzzy portfolio selection problems. Fuzzy Optimization and Decision Making, 1, 361–377.
Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Silva, R.C., Cruz, C. & Verdegay, J.L. Fuzzy costs in quadratic programming problems. Fuzzy Optim Decis Making 12, 231–248 (2013). https://doi.org/10.1007/s10700-013-9153-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-013-9153-1