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Keywords:
relation; poset; order reversing involutions; weakly orthogonal poset; transitivity
relation; poset; order reversing involutions; weakly orthogonal poset; transitivity
Summary:
In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of particular order reversing involution on the co-domain poset. We prove that for every set of alternatives there is a poset valued preference whose cut relations are all relations on this domain. We also deal with particular transitivity of such preferences.
References:
[1] Bezdek, J., Spillman, B., Spillman, R.: A fuzzy relation space for group decision theory. Fuzzy Sets Systems 1 (1978), 255-268. DOI 10.1016/0165-0114(78)90017-9 | MR 0508976 | Zbl 0398.90009
[2] Birkhoff, G.: Lattice Theory. (Third edition. AMS Colloquium Publications, Vol. XXV, 1967. MR 0227053
[3] Chajda, I.: An algebraic axiomatization of orthogonal posets. Soft Computing 18 (2013), 1, 1-4). DOI 10.1007/s00500-013-1047-1
[4] Cignoli, R., Esteva, F.: Commutative, integral bounded residuated lattices with an added involution. Annals of Pure and Applied Logic 161 (2009), 2, 150-160. DOI 10.1016/j.apal.2009.05.008 | MR 2552735 | Zbl 1181.03061
[5] Dasgupta, M., Deb, R.: Fuzzy choice functions. Soc. Choice Welfare 8 (1991), 2, 171-182. DOI 10.1007/bf00187373 | MR 1115895 | Zbl 0717.90004
[6] David, H.: The Method of Paired Comparisons. Griffin's Statistical Monographs and Courses, Vol. 12, Charles Griffin and D. Ltd., 1963. MR 0174105 | Zbl 0665.62075
[7] Baets, B. De, Meyer, H. De: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152 (2005), 2, 249-270. MR 2138509 | Zbl 1114.91031
[8] Baets, B. De, Meyer, H. De, Schuymer, B. De: Cyclic Evaluation of Transitivity of Reciprocal Relations. Social Choice and Welfare 26 (2006), 217-238. DOI 10.1016/j.fss.2004.11.002 | MR 2226508 | Zbl 1158.91338
[9] Doignon, J.-P., Monjardet, B., Roubens, M., Vincke, Ph.: Biorder families, valued relations, and preference modelling. J. Math. Psych. 30 (1986), 435-480. DOI 10.1016/0022-2496(86)90020-9 | MR 0868774 | Zbl 0612.92020
[10] Dutta, B., Laslier, J.F.: Comparison functions and choice correspondences. Soc. Choice Welfare 16 (1999), 513-532. DOI 10.1007/s003550050158 | MR 1713186 | Zbl 1066.91535
[11] Fan, Z.-P., X, X. Chen: Consensus measures and adjusting inconsistency of linguistic preference relations in group decision making. Lecture Notes in Artificial Intelligence, Springer-Verlag 3613 (2005), 130-139. DOI 10.1007/11539506_16
[12] Fan, Z.-P., Jiang, Y. P.: A judgment method for the satisfying consistency of linguistic judgment matrix. Control and Decision 19 (2004), 903-906.
[13] Fishburn, P. C.: Binary choice probabilities: on the varieties of stochastic transitivity. J. Math. Psychology 10 (1973), 327-352. DOI 10.1016/0022-2496(73)90021-7 | MR 0327330 | Zbl 0277.92008
[14] Flachsmeyer, J.: Note on orthocomplemented posets II. In: Proc. 10th Winter School on Abstract Analysis (Z. Frolík, ed.), Circolo Matematico di Palermo, Palermo 1982. pp. 67-74. MR 0683769 | Zbl 0535.06003
[15] Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers 1994. DOI 10.1007/978-94-017-1648-2 | Zbl 0827.90002
[16] García-Lapresta, J., Llamazares, B.: Aggregation of fuzzy preferences: some rules of the mean. Soc. Choice Welfare 17 (2000), 673-690. DOI 10.1007/s003550000048 | MR 1778698 | Zbl 1069.91518
[17] García-Lapresta, J., Llamazares, B.: Majority decisions based on difference of votes. J. Math. Economics 35 (2001), 463-481. DOI 10.1016/s0304-4068(01)00055-6 | MR 1838607 | Zbl 0987.91022
[18] Goguen, J. A.: $L$-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. DOI 10.1016/0022-247x(67)90189-8 | MR 0224391 | Zbl 0145.24404
[19] Herrera, F.: A sequential selection process in group decision making with linguistic assessment. Inform. Sci. 85 (1995), 223-239. DOI 10.1016/0020-0255(95)00025-k
[20] Herrera, F., Herrera-Viedma, E.: Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets and Systems 115 (2000), 67-82. DOI 10.1016/s0165-0114(99)00024-x | MR 1776304 | Zbl 1073.91528
[21] Herrera, F., Martínez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems 8 (2000), 746-752. DOI 10.1109/91.890332 | MR 1784646
[22] Herrera, F., Herrera-Viedma, E., Mart{í}nez, L.: A fusion approach for managing multi-granularity linguistic terms sets in decision making. Fuzzy Sets and Systems 114 (2000), 43-58. DOI 10.1016/s0165-0114(98)00093-1 | MR 1776304
[23] Herrera, F., Herrera-Viedma, E., Verdegay, J. L.: A Model of Consensus in group decision making under linguistic assessments. Fuzzy Sets and Systems 78 (1996), 73-87. DOI 10.1016/0165-0114(95)00107-7 | MR 1376216
[24] Herrera, F., Herrera-Viedma, E., Verdegay, J. L.: Direct approach processes in group decision making using linguistic OWA operators. Fuzzy Sets and Systems 79 (1996), 175-190. DOI 10.1016/0165-0114(95)00162-x | MR 1388390 | Zbl 0870.90007
[25] Herrera-Viedma, E., Herrera, F., Chiclana, F.: A consensus model for multiperson decision making with different preference structures. IEEE Trans. Systems, Man and Cybernetics 32 (2002), 394-402. DOI 10.1109/tsmca.2002.802821
[26] Kacprzyk, J., Fedrizzi, M., Nurmi, H.: Group decision making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems 49 (1992), 21-31. DOI 10.1016/0165-0114(92)90107-f | MR 1177944 | Zbl 0768.90003
[27] Kacprzyk, J., Nurmi, H., (eds.), M. Fedrizzi: Consensus under Fuzziness. Kluwer Academic Publishers, Boston 1996. Zbl 0882.00024
[28] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Boston - London - Dordrecht 2000. DOI 10.1007/978-94-015-9540-7 | MR 1790096 | Zbl 1087.20041
[29] Lahiri, S.: Axiomatic characterizations of threshold choice functions for comparison functions. Fuzzy Sets and Systems 132 (2002), 77-82. DOI 10.1016/s0165-0114(01)00240-8 | MR 1936216 | Zbl 1042.91016
[30] Menger, K.: Probabilistic theories of relations. Proc. Nat. Acad. Sci. (Math.) 37 (1951), 178-180. DOI 10.1073/pnas.37.3.178 | MR 0042080 | Zbl 0042.37103
[31] Monjardet, B.: A generalisation of probabilistic consistency: linearity conditions for valued preference relations. In: Non-conventional Preference Relations in Decision Making (J. Kacprzyk and M. Roubens, eds.), Lecture Notes in Economics and Mathematical Systems, Vol. 301, Springer-Verlag, 1988. DOI 10.1007/978-3-642-51711-2_3 | MR 1133647
[32] Nurmi, H.: Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets Systems 6 (1981), 249-259. DOI 10.1016/0165-0114(81)90003-8 | MR 0635346 | Zbl 0465.90006
[33] Nurmi, H.: Comparing Voting Systems. Reidel, Dordrecht 1987. DOI 10.1007/978-94-009-3985-1
[34] Ovchinnikov, S.: Similarity relations, fuzzy partitions, and fuzzy orderings. Fuzzy Sets and Systems 40 (1991), 1, 107-126. DOI 10.1016/0165-0114(91)90048-u | MR 1103658 | Zbl 0725.04003
[35] Pták, P., Pulmannová, P.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht 1991. MR 1176314 | Zbl 0743.03039
[36] Roberts, F.: Homogeneous families of semiorders and the theory of probabilistic consistency. J. Math. Psych. 8 (1971), 248-263. DOI 10.1016/0022-2496(71)90016-2 | MR 0292534 | Zbl 0223.92017
[37] Roubens, M., Vincke, Ph.: Preference Modelling. Springer-Verlag, Berlin 1985. DOI 10.1007/978-3-642-46550-5 | MR 0809182 | Zbl 0612.92020
[38] Šešelja, B., Tepavčević, A.: On a construction of codes by $P$-fuzzy sets. Review of Research, Fac. of Sci., Univ. of Novi Sad, Math. Ser. 20 2 (1990), 71-80. MR 1158427 | Zbl 0749.94027
[39] Šešelja, B., Tepavčević, A.: Partially ordered and relational valued fuzzy relations I. Fuzzy Sets and Systems 72 (1995), 2, 205-213. DOI 10.1016/0165-0114(94)00352-8 | MR 1335535 | Zbl 0844.04008
[40] Šešelja, B., Tepavčević, A.: Completion of ordered structures by cuts of fuzzy sets. An overview. Fuzzy Sets and Systems 136 (2003), 1-19. DOI 10.1016/s0165-0114(02)00365-2 | MR 1978466 | Zbl 1020.06005
[41] Šeselja, B., Tepavčević, A.: Representing ordered structures by fuzzy sets. An overview. Fuzzy Sets and Systems 136 (2003), 21-39. DOI 10.1016/s0165-0114(02)00366-4 | MR 1978467 | Zbl 1026.03039
[42] Switalski, Z.: Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems 107 (1999), 187-190. DOI 10.1016/s0165-0114(97)00313-8 | MR 1702851 | Zbl 0938.91019
[43] Yager, R. R.: An approach to ordinal decision making. Int. J. Approx. Reasoning 12 (1995), 237-261. DOI 10.1016/0888-613x(94)00035-2 | MR 1327857 | Zbl 0870.68137
[44] Zadeh, L. A.: Fuzzy sets. Information and Control 8 (1965), 338-353. DOI 10.1016/s0019-9958(65)90241-x | MR 0219427 | Zbl 0942.00007
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