Determination of Adequate Fuzzy Inequalities for their Usage in Fuzzy Query Languages

Authors: Marcel Shirvanian, Wolfram Lippe

Abstract:

Although the usefulness of fuzzy databases has been pointed out in several works, they are not fully developed in numerous domains. A task that is mostly disregarded and which is the topic of this paper is the determination of suitable inequalities for fuzzy sets in fuzzy query languages. This paper examines which kinds of fuzzy inequalities exist at all. Afterwards, different procedures are presented that appear theoretically appropriate. By being applied to various examples, their strengths and weaknesses are revealed. Furthermore, an algorithm for an efficient computation of the selected fuzzy inequality is shown.

Keywords: Fuzzy Databases, Fuzzy Inequalities, Fuzzy QueryLanguages, Fuzzy Ranking.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070091

References:

[1] S. Abbasbandy and B. Asasy, Ranking of Fuzzy Numbers by Sign Distance. Information Sciences, vol. 176(16), 2006, pp. 2405-2416.
[2] J. M. Adamo, Fuzzy Decision Trees. Fuzzy Sets and Systems, vol. 4(3), 1980, pp. 207-219.
[3] S. M. Baas and H. Kwakernaak, Rating and Ranking of Multiple-Aspect Alternatives Using Fuzzy Sets. Automatica, vol. 13(1), 1977, pp. 47-58.
[4] J. F. Baldwin and N. C. F. Guild, Comparison of Fuzzy Sets on the Same Decision Space. Fuzzy Sets and Systems, vol. 2(3), 1979, pp. 213-231.
[5] G. Bortolan and R. Degani, A Review of Some Methods for Ranking Fuzzy Subsets. Fuzzy Sets and Systems, vol. 15(1), 1985, pp. 1-19.
[6] P. Bosc and O. Pivert, About Projection-Selection-Join-Queries Addressed to Possibilistic Relational Databases. IEEE Transactions on Fuzzy Systems, vol. 13(1), 2005, pp. 124-139.
[7] B. P. Buckles and F. E. Petry, Query Languages for Fuzzy Databases. Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, Edited by J. Kacprzyk and R. R. Yager, Cologne, 1985, pp. 241- 252.
[8] J. J. Buckley, A Fuzzy Ranking of Fuzzy Numbers. Fuzzy Sets and Systems, vol. 33(1), 1989, pp. 119-121.
[9] J. J. Buckley and S. Chanas, A Fast Method of Ranking Alternatives Using Fuzzy Numbers. Fuzzy Sets and Systems, vol. 30(3), 1989, pp. 337-338.
[10] R. De Caluwe, R. Vandenberghe, N. Van Gyseghem and A. Van Schooten, Integrating Fuzziness in Database Models. Fuzziness in Database Management Systems, Edited by P. Bosc and J. Kacprzyk, Heidelberg, 1995, pp. 71-113.
[11] L. M. De Campos-Ibanez and A. Gonzales-Munoz, A Subjective Approach for Ranking Fuzzy Numbers. Fuzzy Sets and Systems, vol. 29(2), 1989, pp. 145-153.
[12] G. Canfora and L. Troiano, Fuzzy Ordering of Fuzzy Numbers. Proceedings of the IEEE International Conference on Fuzzy Systems, vol. 2, Budapest, 2004, pp. 669-674.
[13] J. R. Chang, C. H. Cheng and C. Y. Kuo, Conceptual Procedure of Ranking Fuzzy Numbers Based on Adaptive Two-Dimensions Dominance. Soft Computing - A Fusion of Foundations, Methodologies and Applications, vol. 10(2), 2006, pp. 94-103.
[14] G. Chen, Fuzzy Data Modeling: Perspectives, Problems and Solutions. Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications, Edited by E. E. Kerre, Gent, 1991, pp. 294-343.
[15] L. H. Chen and H. W. Lu, An Approximate Approach for Ranking Fuzzy Numbers Based on Left and Right Dominance. Computers and Mathematics with Applications, vol. 41(12), 2001, pp. 1589-1602.
[16] S. H. Chen, Ranking Fuzzy Numbers with Maximizing Set and Minimizing Set. Fuzzy Sets and Systems, vol. 17(2), 1985, pp. 113-129.
[17] S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision Making - Methods and Applications, Berlin, 1992.
[18] C. H. Cheng, A New Approach for Ranking Fuzzy Numbers by Distance Method. Fuzzy Sets and Systems, vol. 95(3), 1998, pp. 307-317.
[19] F. Choobineh and H. Li, An Index for Ordering Fuzzy Numbers. Fuzzy Sets and Systems, vol. 54(3), 1993, pp. 287-294.
[20] T. C. Chu and C. T. Tsao, Ranking Fuzzy Numbers with an Area between the Centroid Point and Original Point. Computers and Mathematics with Applications, vol. 43(1-2), 2002, pp. 111-117.
[21] M. Delgado, J. L. Verdegay and M. A. Vila, A Procedure for Ranking Fuzzy Numbers Using Fuzzy Relations. Fuzzy Sets and Systems, vol. 26(1), 1988, pp. 49-62.
[22] D. Dubois and H. Prade, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences, vol. 30(3), 1983, pp. 183-224.
[23] D. Dubois and H. Prade, A Unified View of Ranking Techniques for Fuzzy Numbers. Proceedings of the IEEE International Conference on Fuzzy Systems, vol. 3, Seoul, 1999, pp. 1328-1333.
[24] J. Efstathiou and R. M. Tong, Ranking Fuzzy Sets Using Linguistic Preference Relations. Proceedings of the International Symposium on Multiple-Valued Logic, Evanston, 1980, pp. 137-142.
[25] P. Fortemps and M. Roubens, Ranking and Defuzzification Methods Based on Area Compensation. Fuzzy Sets and Systems, vol. 82(3), 1996, pp. 319-330.
[26] J. Galindo, A. Urrutia and M. Piattini, Fuzzy Databases: Modeling, Design and Implementation, Hershey, 2006.
[27] R. Jain, A Procedure for Multiple-Aspect Decision-Making Using Fuzzy Sets. International Journal of Systems Science, vol. 8(1), 1977, pp. 1-7.
[28] E. E. Kerre, The Use of Fuzzy Set Theory in Electrocardiological Diagnostics. Approximate Reasoning in Decision Analysis, Edited by M. M. Gupta and E. Sanchez, Amsterdam, 1982, pp. 277-282.
[29] K. Kim and K. S. Park, Ranking Fuzzy Numbers with Index of Optimism. Fuzzy Sets and Systems, vol. 35(2), 1990, pp. 143-150.
[30] W. Kolodziejczyk, Orlovsky-s Concept of Decision-Making with Fuzzy Preference Relation-Further Results. Fuzzy Sets and Systems, vol. 19(1), 1986, pp. 11-20.
[31] E. S. Lee and R. J. Li, Comparison of Fuzzy Numbers Based on the Propability Measure of Fuzzy Events. Computers and Mathematics with Applications, vol. 15(10), 1988, pp. 887-896.
[32] Z. Lian, B. Jiao and X. Gu, A Novel Method of Ranking Fuzzy Numbers for Decision-Making Problems. International Conference on Intelligent Systems Design and Applications, vol. 1, Jinan, 2006, pp. 354-360.
[33] T. S. Liou and M. J. J. Wang, Ranking Fuzzy Numbers with Integral Value. Fuzzy Sets and Systems, vol. 50(3), 1992, pp. 247-255.
[34] Z. M. Ma and L. Yan, Updating Extended Possibility-Based Fuzzy Relational Databases. International Journal of Inteligent Systems, vol. 22(3), 2007, pp. 237-258.
[35] S. Mabuchi, An Approach to the Comparison of Fuzzy Subsets with an a-Cut Dependent Index. IEEE Transactions on Systems, Man and Cybernetics, vol. 18(2), 1988, pp. 264-272.
[36] C. S. McCahon, Fuzzy Set Theory Applied to Production and Inventory Control, Ph.D. Thesis, University of Kansas State, 1987.
[37] J. M. Medina, O. Pons and M. A. Vila, GEFRED. A Generalized Model of Fuzzy Relational Databases. Ver. 1.1. Information Sciences, vol. 76(1- 2), 1994, pp. 87-109.
[38] S. Murakami, S. Maeda and S. Imamura, Fuzzy Decision Analysis on the Development of Centralized Regional Energy Control System. Proceedings of the IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, 1983, pp. 363-368.
[39] K. Nakamura, Preference Relations on a Set of Fuzzy Utilities as a Basis for Decision Making. Fuzzy Sets and Systems, vol. 20(2), 1986, pp. 147-162.
[40] H. Prade and C. Testemale, Generalizing Database Relational Algebra for the Treatment of Incomplete or Uncertain Information and Vague Queries. Information Sciences, vol. 34(4), 1984, pp. 115-143.
[41] J. Ramik and J. Rimanek, Inequality Relation between Fuzzy Numbers and its Use in Fuzzy Optimization. Fuzzy Sets and Systems, vol. 16(2), 1985, pp. 123-138.
[42] H. Rommelfanger, Rangordnungsverfahren fuer unscharfe Mengen - Ein kritischer Vergleich mit empirisch ermittelten Praeferenzaussagen. ORSpectrum, vol. 8(4), 1986, pp. 219-228.
[43] J. J. Saade and H. Schwarzlander, Ordering Fuzzy Sets over the Real Line: An Approach Based on Decision Making under Uncertainty. Fuzzy Sets and Systems, vol. 50(3), 1992, pp. 237-246.
[44] M. Setnes and V. Cross, Compatibility-Based Ranking of Fuzzy- Numbers. Proceedings of the International Conference of the North American Fuzzy Information Processing Society, Syracuse, 1997, pp. 305-310.
[45] S. Shenoi and A. Melton, An Extended Version of the Fuzzy Relational Database Model. Information Sciences, vol. 52(1), 1990, pp. 35-52.
[46] X. Tang and G. Chen, A Complete Set of Fuzzy Relational Algebraic Operators in Fuzzy Relational Databases. Proceedings of the IEEE International Conference on Fuzzy Systems, vol. 1, Budapest, 2004, pp. 565-569.
[47] R. M. Tong and P. P. Bonissone, Linguistic Solutions to Fuzzy Decision Problems. Studies in the Management Science, Edited by H. J. Zimmermann, L. A. Zadeh and B. R. Gaines, vol. 20, North-Holland, 1984, pp. 323-334.
[48] L. Tran and L. Duckstein, Comparison of Fuzzy Numbers Using a Fuzzy Distance Measure. Fuzzy Sets and Systems, vol. 130(3), 2002, pp. 331- 341.
[49] T. Y. Tseng and C. M. Klein, New Algorithm for the Ranking Procedure in Fuzzy Decision-Making. IEEE Transactions on Systems, Man and Cybernetics, vol. 19(5), 1989, pp. 1289-1296.
[50] Y. Tsukamoto, P. N. Nikiforuk and M. M. Gupta, On the Comparison of Fuzzy Sets Using Fuzzy Chopping. Proceedings of the Triennial World Congress of the International Federation of Automatic Control (Control Science and Technology for the Progress of Society), vol. 5 (Process Control), Kyoto, 1981, pp. 46-51.
[51] M. Umano, Freedom-0: A Fuzzy Database System. Fuzzy Information and Decision Processes, Edited by M. M. Gupta and E. Sanchez, Amsterdam, 1982, pp. 339-347.
[52] M. Umano and S. Fukami, Fuzzy Relational Algebra for Possibility- Distribution-Fuzzy-Relational Model of Fuzzy Data. Journal of Intelligent Information Systems, vol. 3(1), 1994, pp. 7-27.
[53] X. Wang and E. E. Kerre, On the Classification and the Dependencies of the Ordering Methods. Fuzzy Logic Foundations and Industrial Applications, Edited by D. Ruan, Dordrecht, 1996, pp. 73-90.
[54] X. Wang and E. E. Kerre, Reasonable Properties for the Ordering of Fuzzy Quantities (I). Fuzzy Sets and Systems, vol. 118(3), 2001, pp. 375- 385.
[55] X. Wang and E. E. Kerre, Reasonable Properties for the Ordering of Fuzzy Quantities (II). Fuzzy Sets and Systems, vol. 118(3), 2001, pp. 387-405.
[56] S. R. Watson, J. J. Weiss and M. L. Donnell, Fuzzy Decision Analysis. IEEE Transactions on Systems, Man and Cybernetics, vol. 9(1), 1979, pp. 1-9.
[57] R. R. Yager, Ranking Fuzzy Subsets over the Unit Interval. Proceedings of the International Conference on Decision and Control, San Diego, 1978, pp. 1435-1437.
[58] R. R. Yager, On Choosing between Fuzzy Subsets. Kybernetes, vol. 9(2), 1980, pp. 151-154.
[59] R. R. Yager, A Procedure for Ordering Fuzzy Subsets of the Unit Interval. Information Sciences, vol. 24(2), 1981, pp. 143-161.
[60] Y. Yuan, Criteria for Evaluating Fuzzy Ranking Methods. Fuzzy Sets and Systems, vol. 43(2), 1991, pp. 139-157.
[61] M. Zemankova-Leech and A. Kandel, Fuzzy Relational Data Bases - A Key to Expert Systems, Cologne, 1984.
[62] Q. Zhu and E. S. Lee, Comparison and Ranking of Fuzzy Numbers. Fuzzy Regression Analysis, Edited by J. Kacprzyk and M. Fedrizzi, Warsaw, 1992, pp. 21-44.