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**Modeling uncertain variables of the weighted average operation by fuzzy vectors.**
*(English)*
Zbl 1241.68117

Summary: The paper deals with the fuzzy extension of the weighted average operation. First, we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors. We show that, in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now, fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values. Next, we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights. We derive a general formula and we study its special forms. The advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation. This fact is illustrated by several examples. Finally, we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average.

### MSC:

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

### Keywords:

fuzzy weighted average; fuzzy vector; separability of fuzzy vectors; normalized fuzzy weights; fuzzy probabilities; multiple criteria decision making
Full Text:
DOI

### References:

[1] | Baas, S.; Kwakernaak, H., Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica, 13, 47-58, (1977) · Zbl 0363.90010 |

[2] | De Campos, L.M.; Heute, J.F.; Moral, S., Probability intervals: a tool for uncertain reasoning, International journal of uncertainty, fuzziness and knowledge-based systems, 2, 167-196, (1994) · Zbl 1232.68153 |

[3] | Dong, W.M.; Wong, F.S., Fuzzy weighted averages and implementation of the extension principle, Fuzzy sets and systems, 21, 183-199, (1987) · Zbl 0611.65100 |

[4] | Dubois, D.; Prade, H., The use of fuzzy numbers in decision analysis, (), 309-321 |

[5] | Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE transactions on automatic control, 26, 4, 926-936, (1981) |

[6] | Dubois, D.; Kerre, E.; Mesiar, R.; Prade, H., Fuzzy interval analysis, (), 483-582 |

[7] | Fullér, R.; Majlender, P., On weighted possibilistic Mean and variance of fuzzy numbers, Fuzzy sets and systems, 136, 363-374, (2003) · Zbl 1022.94032 |

[8] | Fullér, R.; Majlender, P., On interactive fuzzy numbers, Fuzzy sets and systems, 143, 355-369, (2004) · Zbl 1056.94023 |

[9] | Fullér, R.; Majlender, P., Correction to: on interactive fuzzy numbers [fuzzy sets and systems 143 (2004) 355-369], Fuzzy sets and systems, 152, 159, (2005) |

[10] | Guh, Y.-Y.; Hong, C.C.; Wang, K.M.; Lee, E.S., Fuzzy weighted average: a max – min paired elimination method, Computers & mathematics with applications, 32, 115-123, (1996) · Zbl 0873.90111 |

[11] | Guh, Y.-Y.; Hon, Ch.-Ch.; Lee, E.S., Fuzzy weighted average: the linear programming approach via charness and cooper’s rule, Fuzzy sets and systems, 117, 157-160, (2001) · Zbl 1032.91041 |

[12] | Hisdal, E., Conditional possibilities independence and noninteraction, Fuzzy sets and systems, 1, 283-297, (1978) · Zbl 0393.94050 |

[13] | Juang, C.-H.; Huang, X.H.; Elton, D.J., Fuzzy information processing by Monte Carlo simulation technique, Civil engineering and environmental systems, 8, 19-25, (1991) |

[14] | Klir, G.J.; Pan, Y., Constrained fuzzy arithmetic: basic questions and some answers, Soft computing, 2, 100-108, (1998) |

[15] | Kwakernaak, H., Fuzzy random variables – vol. I: definitions and theorems, Information sciences, 15, 1-29, (1978) · Zbl 0438.60004 |

[16] | Kwakernaak, H., Fuzzy random variables – part II: algorithms and examples in discrete case, Information sciences, 17, 253-278, (1979) · Zbl 0438.60005 |

[17] | Liou, t.S.; Wang, M.J.J.; Elton, D.J., Fuzzy weighted average: an improved algorithm, Fuzzy sets and systems, 49, 307-315, (1992) · Zbl 0796.90069 |

[18] | Mashinchi, M., An application of fuzzy set representation, Scientia iranica, 2, 4, 341-346, (1996) · Zbl 0958.03035 |

[19] | Nahmias, S., Fuzzy variables in the random environment, (), 165-180 |

[20] | Negoita, C.V.; Ralescu, D.A., Representation theorems for fuzzy concepts, Kybernetes, 4, 169-174, (1975) · Zbl 0352.02044 |

[21] | Pan, Y.; Klir, G.J., Bayesian inference based on interval probabilities, Journal of intelligence and fuzzy systems, 5, 193-203, (1997) |

[22] | Pan, Y.; Yuan, B., Bayesian inference of fuzzy probabilities, International journal of general systems, 26, 1-2, 73-90, (1997) · Zbl 0974.28501 |

[23] | O. Pavlačka, Fuzzy methods of decision making, Dissertation thesis, Palacký University Olomouc, 2007 (in Czech). |

[24] | O. Pavlačka, J. Talašová, The fuzzy weighted average operation in decision making models, in: Proceedings of the 24th International Conference Mathematical Methods in Economics, Plzeň, 13th-15th September 2006, Plzeň 2006, pp. 419-426. |

[25] | O. Pavlačka, J. Talašová, Application of the Fuzzy Weighted Average of Fuzzy Numbers in Decision Making Models, in: U. Bodenhofer, V. Novák, M. Štěpnička (Eds.), New Dimensions in Fuzzy Logic and Related technologies, Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, Ostravská univerzita, Ostrava, vol. II, September 11-14, 2007, pp. 455-462. |

[26] | O. Pavlačka, J. Talašová, Application of Fuzzy Vectors of Normalized Weights in Decision Making Models, in: J.P. Carvalho, D. Dubois, U. Kaymak and J.M.C. Sousa (Eds.), Proceedings of the Joint 2009 IFSA World Congress and 2009 EUSFLAt Conference, Lisbon, Portugal, July 20-24, 2009, pp. 495-500. |

[27] | Pavlačka, O.; Talašová, J., Fuzzy vectors as a tool for modeling uncertain multidimensional quantities, Fuzzy sets and systems, 161, 1585-1603, (2010) · Zbl 1186.90144 |

[28] | Talašová, J., Nefrit – multicriteria decision making based on fuzzy approach, Central European journal of operations research, 8, 4, 297-319, (2000) · Zbl 0981.90035 |

[29] | Talašová, J.; Bebčáková, I., Fuzzification of aggregation operators based on Choquet integral, Aplimat – journal of applied mathematics, 1, 1, 463-474, (2008) |

[30] | Talašová, J.; Pavlačka, O., Fuzzy probability spaces and their applications in decision making, Austrian journal of statistics, 35, 2-3, 347-356, (2006) |

[31] | Viertl, R., Statistical methods for non-precise data, (1996), CRC Press Boca Raton, Florida |

[32] | Wang, Y.-M.; Elhag, t. M.S., On the normalization of interval and fuzzy weights, Fuzzy sets and systems, 157, 2456-2471, (2006) · Zbl 1171.68764 |

[33] | Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004 |

[34] | Weichselberger, K.; Pőhlmann, S., A methodology for uncertainty in knowledge-based systems, (1990), Springer-Verlag New York · Zbl 0705.68097 |

[35] | Yager, R.R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE transactions on systems, man, and cybernetics, 3, 1, 183-190, (1988) · Zbl 0637.90057 |

[36] | L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning I, II, Information Sciences 8 (1975) 199-249, 301-357; III, Information Sciences 9 (1975) 43-80. · Zbl 0397.68071 |

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