Previous |  Up |  Next


OWA operator; ordered weighted averaging operator; gradual number; gradual interval; fuzzy interval; linear order; total order; multi-expert decision making; type-2 fuzzy set
A new concept in fuzzy sets theory, namely that of gradual element, was introduced recently. It is known that the set of gradual real numbers is not ordered linearly. We restrict our attention to a discrete case and propose a class of linear orders for discrete gradual real numbers. Then, using idea of the so-called admissible order of intervals, we present a class of linear orders for discrete gradual intervals. Once we have the linear orders it is possible to define OWA operator for discrete gradual real numbers and OWA operator for discrete gradual intervals. Recall that gradual intervals also encompass fuzzy intervals, hence our results are applicable to the setting of fuzzy intervals. Our approach is illustrated on a multi-expert decision making problem.
[1] Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A guide for practitioners. Studies in Fuzziness and Soft Computing 221 (2007), 261-269. DOI 10.1007/978-3-540-73721-6_5
[2] Bustince, H., Barrenechea, E., Calvo, T., James, S., Beliakov, G.: Consensus in multi-expert decision making problems using penalty functions defined over a Cartesian product of lattices. Inform. Fusion 17 (2014), 56-64. DOI 10.1016/j.inffus.2011.10.002
[3] Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets and Systems 220 (2013), 69-77. DOI 10.1016/j.fss.2012.07.015 | MR 3042258 | Zbl 1284.03242
[4] Bustince, H., Galar, M., Bedregal, B., Kolesárová, A., Mesiar, R.: A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Trans. Fuzzy Systems 21 (2013), 1150-1162. DOI 10.1109/tfuzz.2013.2265090
[5] Castillo, O., Melin, P.: A review on interval type-2 fuzzy logic applications in intelligent control. Inform. Sciences 279 (2014), 615-631. DOI 10.1016/j.ins.2014.04.015 | MR 3212110
[6] Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets and Systems 97 (1998), 33-48. DOI 10.1016/s0165-0114(96)00339-9 | MR 1618276 | Zbl 0932.91012
[7] Dubois, D., Kerre, E., Mesiar, R., Prade, H.: Fuzzy interval analysis. In: Fundamentals of Fuzzy Sets (D. Dubois and H. Prade, eds.), The Handbooks of Fuzzy Sets Series, Vol. 7, Springer US 2000, pp. 483-581. DOI 10.1007/978-1-4615-4429-6_11 | MR 1890240 | Zbl 0988.26020
[8] Dubois, D., Prade, H.: A review of fuzzy set aggregation connectives. Inform. Sciences 36 (1985), 85-121. DOI 10.1016/0020-0255(85)90027-1 | MR 0813766 | Zbl 0582.03040
[9] Dubois, D., Prade, H.: Gradual elements in a fuzzy set. Soft Computing 12 (2008), 165-175. DOI 10.1007/s00500-007-0187-6 | Zbl 1133.03026
[10] Fortin, J., Dubois, D., Fargier, H.: Gradual numbers and their application to fuzzy interval analysis. IEEE Trans. Fuzzy Systems 16 (2008), 388-402. DOI 10.1109/tfuzz.2006.890680
[11] Grabisch, M., Marichal, J. L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge 2009. DOI 10.1017/cbo9781139644150 | MR 2538324 | Zbl 1206.68299
[12] Herrera, F., Martínez, L.: A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making. IEEE Trans. Systems Man and Cybernetics, Part B: Cybernetics 31 (2001), 227-234. DOI 10.1109/3477.915345
[13] Karnik, N. N., Mendel, J. M.: Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122 (2001), 327-348. DOI 10.1016/s0165-0114(00)00079-8 | MR 1854822 | Zbl 1010.03047
[14] Kosiński, W., Prokopowicz, P., Rosa, A.: Defuzzification functionals of ordered fuzzy numbers. IEEE Trans. Fuzzy Systems 21 (2013), 1163-1169. DOI 10.1109/tfuzz.2013.2243456
[15] Lizasoain, I., Moreno, C.: OWA operators defined on complete lattices. Fuzzy Sets and Systems 224 (2013), 36-52. DOI 10.1016/j.fss.2012.10.012 | MR 3068107 | Zbl 1284.03246
[16] Lodwick, W. A., Untiedt, E. A.: A comparison of interval analysis using constraint interval arithmetic and fuzzy interval analysis using gradual numbers. In: Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS 2008, pp. 1-6. DOI 10.1109/nafips.2008.4531302
[17] Martin, T. P., Azvine, B.: The X-mu approach: Fuzzy quantities, fuzzy arithmetic and fuzzy association rules. In: IEEE Symposium on Foundations of Computational Intelligence (FOCI), 2013, pp. 24-29. DOI 10.1109/foci.2013.6602451
[18] Melin, P., Castillo, O.: A review on type-2 fuzzy logic applications in clustering, classification and pattern recognition. Applied Soft Computing J. 21 (2014), 568-577. DOI 10.1016/j.asoc.2014.04.017
[19] Mesiar, R., Kolesárová, A., Calvo, T., Komorníková, M.: A review of aggregation functions. In: Fuzzy Sets and Their Extensions: Representation, Aggregation and Models (H. Bustince et al., eds.), Springer, Berlin 2008, pp. 121-144. DOI 10.1007/978-3-540-73723-0_7 | Zbl 1147.68081
[20] Mizumoto, M., Tanaka, K.: Some properties of fuzzy sets of type 2. Information and Control 31 (1976), 312-340. DOI 10.1016/s0019-9958(76)80011-3 | MR 0449947 | Zbl 0331.02042
[21] Moore, E. R.: Methods and Applications of Interval Analysis. SIAM 1979. DOI 10.1137/1.9781611970906 | MR 0551212 | Zbl 0417.65022
[22] Moore, E. R., Lodwick, W. A.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135 (2003), 5-9. DOI 10.1016/s0165-0114(02)00246-4 | MR 1977533 | Zbl 1015.03513
[23] Ochoa, G., Lizasoain, I., Paternain, D., Bustince, H., Pal, N. R.: Some properties of lattice OWA operators and their importance in image processing. In: Proc. IFSA-EUSFLAT 2015, pp. 1261-1265. DOI 10.2991/ifsa-eusflat-15.2015.178
[24] Roubens, M.: Fuzzy sets and decision analysis. Fuzzy Sets and Systems 90 (1997), 199-206. DOI 10.1016/s0165-0114(97)00087-0 | MR 1486262 | Zbl 0921.90007
[25] Sánchez, D., Delgado, M., Vila, M. A., Chamorro-Martínez, J.: On a non-nested level-based representation of fuzziness. Fuzzy Sets and Systems 192 (2012), 159-175. DOI 10.1016/j.fss.2011.07.002 | MR 2878560 | Zbl 1238.68164
[26] Takáč, Z.: Aggregation of fuzzy truth values. Inform. Sciences 271 (2014), 1-13. DOI 10.1016/j.ins.2014.02.116 | MR 3191831
[27] Takáč, Z.: A linear order and OWA operator for discrete gradual real numbers. In: Proc. IFSA-EUSFLAT 2015, pp. 260-266. DOI 10.2991/ifsa-eusflat-15.2015.39
[28] Walker, C. L., Walker, E. A.: The algebra of fuzzy truth values. Fuzzy Sets and Systems 149 (2005), 309-347. DOI 10.1016/j.fss.2003.12.003 | MR 2116888 | Zbl 1152.03331
[29] Xu, Z. S., Da, Q. L.: The uncertain OWA operator. Int. J. Intelligent Systems 17 (2002), 569-575. DOI 10.1002/int.10038 | Zbl 1016.68025
[30] Yager, R. R.: On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Transactions on Systems Man and Cybernetics 18 (1988), 183-190. DOI 10.1109/21.87068 | MR 0931863 | Zbl 0637.90057
[31] Zhou, S. M., Chiclana, F., John, R. I., Garibaldi, J. M.: Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers. Fuzzy Sets and Systems 159 (2008), 3281-3296. DOI 10.1016/j.fss.2008.06.018 | MR 2467606 | Zbl 1187.68619
[32] Zhou, S. M., Chiclana, F., John, R. I., Garibaldi, J. M.: Alpha-level aggregation: A practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments. IEEE Tran. Knowledge Data Engrg. 23 (2011), 1455-1468. DOI 10.1109/tkde.2010.191
Partner of
EuDML logo