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Fuzzy Optimization and Decision Making

, Volume 12, Issue 4, pp 357–372 | Cite as

Fuzzy optimality based decision making under imperfect information without utility

  • Rafik A. Aliev
  • Witold Pedrycz
  • Akif V. Alizadeh
  • Oleg H. Huseynov
Article

Abstract

In the realm of decision making under uncertainty, the general approach is the use of the utility theories. The main disadvantage of this approach is that it is based on an evaluation of a vector-valued alternative by means of a scalar-valued quantity. This transformation is counterintuitive and leads to loss of information. The latter is related to restrictive assumptions on preferences underlying utility models like independence, completeness, transitivity etc. Relaxation of these assumptions results into more adequate but less tractable models. In contrast, humans conduct direct comparison of alternatives as vectors of attributes’ values and don’t use artificial scalar values. Although vector-valued utility function-based methods exist, a fundamental axiomatic theory is absent and the problem of a direct comparison of vectors remains a challenge with a wide scope of research and applications. In the realm of multicriteria decision making there exist approaches like TOPSIS and AHP to various extent utilizing components-wise comparison of vectors. Basic principle of such comparison is the Pareto optimality which is based on a counterintuitive assumption that all alternatives within a Pareto optimal set are considered equally optimal. The above mentioned mandates necessity to develop new decision approaches based on direct comparison of vector-valued alternatives. In this paper we suggest a fuzzy Pareto optimality (FPO) based approach to decision making with fuzzy probabilities representing linguistic decision-relevant information. We use FPO concept to differentiate “more optimal” solutions from “less optimal” solutions. This is intuitive, especially when dealing with imperfect information. An example is solved to show the validity of the suggested ideas.

Keywords

Decision making Imprecise probabilities Fuzzy \(k\)-optimality  Degree of optimality 

References

  1. Aliev, R. A., Alizadeh, A. V., Guirimov, B. G., & Huseynov, O. H. (2010). Precisiated information-based approach to decision making with imperfect information. In Proceedings of the ninth international conference on application of fuzzy systems and soft computing, 2010, ICAFS-2010 (pp. 91–103). Prague, Czech Republic.Google Scholar
  2. Billot, A. (1995). An existence theorem for fuzzy utility functions: A new elementary proof. Fuzzy Sets and Systems, 74, 271–276.MathSciNetCrossRefMATHGoogle Scholar
  3. Borisov, A. N., Alekseyev, A. V., Merkuryeva, G. V., Slyadz, N. N., & Gluschkov, V. I. (1989). Fuzzy information processing in decision making systems. Moscow: Radio i Svyaz (in Russian).Google Scholar
  4. Enea, M., & Piazza, T. (2004). Project selection by constrained fuzzy AHP. Fuzzy Optimization and Decision Making, 3(1), 39–62.CrossRefMATHGoogle Scholar
  5. Farina, M., & Amato, P. (2004). A fuzzy definition of ”optimality” for many-criteria optimization problems. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 34(3), 315–326.CrossRefGoogle Scholar
  6. Gil, M. A., & Jain, P. (1992). Comparison of experiments in statistical decision problems with fuzzy utilities. IEEE Transactions on Systems, Man, and Cybernetics, 22(4), 662–670.MathSciNetCrossRefMATHGoogle Scholar
  7. Gilboa, I., & Schmeidler, D. (1989). Maximin expected utility with a non-unique prior. Journal of Mathematical Economics, 18, 141–153.MathSciNetCrossRefMATHGoogle Scholar
  8. Herrera, F., Alonso, S., Chiclana, F., & Herrera-Viedma, E. (2009). Computing with words in decision making: Foundations, trends and prospects. Fuzzy Optimization and Decision Making, 8(4), 337–364.MathSciNetCrossRefGoogle Scholar
  9. Herrera, F., & Herrera-Viedma, E. (2000). Linguistic decision analysis: Steps for solving decision problems under linguistic information. Fuzzy Sets and Systems, 115, 67–82.MathSciNetCrossRefMATHGoogle Scholar
  10. Huang, W.-C., & Chen, C.-H. (2005). Using the ELECTRE II method to apply and analyze the differentiation theory. In Proceedings of the Eastern Asia Society for Transportation Studies, (Vol. 5, pp. 2237–2249).Google Scholar
  11. Hwang, C. L. (1981). Multiple attribute decision making. Lecture Notes in Economics and Mathematical Systems. Berlin: Springer-Verlag.CrossRefGoogle Scholar
  12. Jahanshahloo, G. R., Lotfi, F., & Hosseinzadeh, I. M. (2006). An algorithmic method to extend TOPSIS for decision-making problems with interval data. Appl. Math. Comput., 175(2), 1375–1384.CrossRefMATHGoogle Scholar
  13. Kapoor, V., & Tak, S. S. (2005). Fuzzy application to the analytic hierarchy process for robot selection. Fuzzy Optimization and Decision Making, 4(3), 209–234.MathSciNetCrossRefMATHGoogle Scholar
  14. Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.MathSciNetCrossRefMATHGoogle Scholar
  15. Lawry, J. (2001). An alternative approach to computing with words. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9(Suppl), 3–16.MathSciNetCrossRefMATHGoogle Scholar
  16. Lee, E. S., & Li, R.-J. (1998). Comparison of fuzzy numbers based on the probability measure of fuzzy events. Computers and Mathematics with Applications, 15(10), 887–896.MathSciNetCrossRefGoogle Scholar
  17. Li, D.-F. (2007). A fuzzy closeness approach to fuzzy multi-attribute decision making. Fuzzy Optimization and Decision Making, 6(3), 237–254.MathSciNetCrossRefGoogle Scholar
  18. Liu, W. J., & Zeng, L. (2008). A new TOPSIS method for fuzzy multiple attribute group decision making problem. Journal of Guilin University of Electronic Technology, 28(1), 59–62.Google Scholar
  19. Liu, P., & Wang, M. (2011). An extended VIKOR method for multiple attribute group decision making based on generalized interval-valued trapezoidal fuzzy numbers. Scientific Research and Essays, 6(4), 766–776.Google Scholar
  20. Lu, J., Zhang, G., Ruan, Da, & Wu, F. (2007). Multi-objective group decision making. In: Methods, Software and Applications with Fuzzy Set Techniques. Series in Electrical and Computer Engineering (vol. 6). London: Imperial College Press.Google Scholar
  21. Martin, O., & Klir, G. J. (2006). On the problem of retranslation in computing with perceptions. International Journal of General Systems, 35(6), 655–674.MathSciNetCrossRefMATHGoogle Scholar
  22. Martinez, L., Ruan, D., & Herrera, F. (2010). Computing with words in decision support systems: An overview on models and applications. International Journal of Computational Intelligence Systems, 3(4), 382–395.Google Scholar
  23. Mathieu-Nicot, B. (1986). Fuzzy expected utility. Fuzzy Sets and Systems, 20(2), 163–173.MathSciNetCrossRefMATHGoogle Scholar
  24. Mendel, J. M., Zadeh, L. A., Yager, R. R., Lawry, J., Hagras, H., & Guadarrama, S. (2010). What computing with words means to me. IEEE Computational Intelligence Magazine, 5(1), 20–26.CrossRefGoogle Scholar
  25. Ohnishi, S., Dubois, D., Prade, H., Yamanoi, T., et al. (2008). A fuzzy constraint-based approach to the analytic hierarchy process. In B. Bouchon-Meunier (Ed.), Uncertainty and intelligent information systems (pp. 217–228). Singapore: World Scientific.CrossRefGoogle Scholar
  26. Opricovic, S., & Tzeng, G. H. (2007). Extended VIKOR method in comparison with outranking methods. European Journal of Operational Research, 178(2), 514–529.CrossRefMATHGoogle Scholar
  27. Opricovic, S., & Tzeng, G. H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2), 445–455.CrossRefMATHGoogle Scholar
  28. Park, J. H., Cho, H. J., & Kwun, Y. C. (2011). Extension of the VIKOR method for group decision making with interval-valued intuitionistic fuzzy information. Fuzzy Optimization and Decision Making, 10(3), 233–253.MathSciNetCrossRefMATHGoogle Scholar
  29. Ramík, J., & Korviny, P. (2010). Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161, 1604–1613.MathSciNetCrossRefMATHGoogle Scholar
  30. Roy, B. (1996). Multicriteria methodology for decision aiding. Dordrecht: Kluwer Academic Publishers.CrossRefMATHGoogle Scholar
  31. Roy, B., & Berlier, B. (1972). La Metode ELECTRE II. Dublin: Sixieme Conference internationale de rechearche operationelle.Google Scholar
  32. Savage, L. J. (1954). The foundations of statistics. New York: Wiley.MATHGoogle Scholar
  33. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.MathSciNetCrossRefMATHGoogle Scholar
  34. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323.CrossRefMATHGoogle Scholar
  35. Vahdani, B., & Zandieh, M. (2010). Selecting suppliers using a new fuzzy multiple criteria decision model: The fuzzy balancing and ranking method. International Journal of Production Research, 48(18), 5307–5326.CrossRefMATHGoogle Scholar
  36. Wang, Y. M., & Elhag, T. M. S. (2006). Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment. Expert Systems with Applications, 31(2), 309–319.CrossRefGoogle Scholar
  37. Yoon, K. (1987). A reconciliation among discrete compromise solutions. Journal of the Operational Research Society, 38(3), 272–286.Google Scholar
  38. Zadeh, L. A. (2006). Generalized theory of uncertainty (GTU)—principal concepts and ideas. Computational Statistics and Data Analysis, 51, 15–46.MathSciNetCrossRefMATHGoogle Scholar
  39. Zadeh, L.A., Aliev, R.A., Fazlollahi, B., Alizadeh, A.V., Guirimov, B.G., & Huseynov, O.H. (2009). Decision Theory with Imprecise Probabilities. Contract on “Application of Fuzzy Logic and Soft Computing to communications, planning and management of uncertainty”. Technical report, Berkeley, Baku, 95 p. http://www.raliev.com/report.pdf
  40. Zeleny, M. (1982). Multiple criteria decision making. New York: McGraw-Hill.MATHGoogle Scholar
  41. Zhu, H. P., Zhang, G. J., & Shao, X. Y. (2007). Study on the application of fuzzy TOPSIS to multiple criteria group decision making problem. Industrial Engineering and Management, 1, 99–102.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Rafik A. Aliev
    • 1
  • Witold Pedrycz
    • 2
    • 3
  • Akif V. Alizadeh
    • 4
  • Oleg H. Huseynov
    • 4
  1. 1.Azerbaijan State Oil AcademyBakuAzerbaijan
  2. 2.Department of Electrical and Computer EngineeringUniversity of AlbertaEdmontonCanada
  3. 3.System Research InstitutePolish Academy of SciencesWarsawPoland
  4. 4.Azerbaijan Association of “Zadeh’s Legacy and Artificial Intelligence”BakuAzerbaijan

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