Methods for Lower Approximation Reduction in Inconsistent Decision Table Based on Tolerance Relation

Affiliation(s)

School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China.

School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China.

ABSTRACT

It is well known that most of information systems are based on tolerance relation instead of the classical equivalence relation because of various factors in real-world. To acquire brief decision rules from the information systems, lower approximation reduction is needed. In this paper, the lower approximation reduction is proposed in inconsistent information systems based on tolerance relation. Moreover, the properties are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to lower reductions can be provided in the complicated information systems.

It is well known that most of information systems are based on tolerance relation instead of the classical equivalence relation because of various factors in real-world. To acquire brief decision rules from the information systems, lower approximation reduction is needed. In this paper, the lower approximation reduction is proposed in inconsistent information systems based on tolerance relation. Moreover, the properties are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to lower reductions can be provided in the complicated information systems.

Cite this paper

X. Zhang and W. Xu, "Methods for Lower Approximation Reduction in Inconsistent Decision Table Based on Tolerance Relation,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 144-148. doi: 10.4236/am.2013.41024.

X. Zhang and W. Xu, "Methods for Lower Approximation Reduction in Inconsistent Decision Table Based on Tolerance Relation,"

References

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[2] M. Kryszkiewicz, “Comparative Studies of Alternative Type of Knowledge Reduction in Inconsistent Systems,” International Journal of Intelligent Systems, Vol. 16, No. 1, 2001, pp. 105-120. doi:10.1002/1098-111X(200101)16:1<105::AID-INT8>3.0.CO;2-S

[3] Y. Leuang, W. Z. Wu and W. X. Zhang, “Knowledge Acquisition in Incomplete Information Systems: A Rough Set Approach,” European Journal of Operational Research, Vol. 168, No. 1, 2006, pp. 164-180. doi:10.1016/j.ejor.2004.03.032

[4] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publishers, Boston, 1991.

[5] W. X. Zhang, W. Z. Wu, J. Y. Liang and D. Y. Li, “Theory and Method of Rough Sets,” Science Press, Beijing, 2001.

[6] W. Z. Wu, Y. Leuang and J. S. Mi, “On Characterizations of (I,T)-Fuzzy Rough Approximation Operators,” Fuzzy Sets and Systems, Vol. 154, No. 1, 2005, pp. 76-102. doi:10.1016/j.fss.2005.02.011

[7] W. Z. Wu, M. Zhang, H. Z. Li and J. S. Mi, “Knowledge Reduction in Random Information Systems via Dempster-Shafer Theory of Evidence,” Information Sciences, Vol. 174, No. 3-4, 2005, pp. 143-164. doi:10.1016/j.ins.2004.09.002

[8] M. Zhang, L. D. Xu, W. X. Zhang and H. Z. Li, “A Rough Set Approach to Knowledge Reduction Based on Inclusion Degree and Evidence Reasoning Theory,” Expert Systems, Vol. 20, No. 5, 2003, pp. 298-304.

[9] S. Greco, B. Matarazzo and R. Slowingski, “Rough Approximation of a Preference Relation by Dominance Relatioin ICS Research Report 16/96, Warsaw University of Technology, 1996,” European Journal of Operational Research, Vol. 117, 1999, pp. 63-83.

[10] S. Greco, B. Matarazzo and R. Slowingski, “A New Rough Set Approach to Multicriteria and Multiattribute Classification,” In: L. Polkowsik and A. Skowron, Eds., Rough Sets and Current Trends in Computing (RSCTC’98), Springer-Verlag, Berlin, 1998, pp. 60-67.

[11] S. Greco, B. Matarazzo and R. Slowingski, “A New Rough Sets Approach to Evaluation of Bankruptcy Risk,” In: X. Zopounidis, Ed., Operational Tools in the Management of Financial Risks, Kluwer, Dordrecht, 1999, pp. 121-136.

[12] S. Greco, B. Matarazzo and R. Slowingski, “Rough Sets Theory for Multicriteria Decision Analysis,” European Journal of Operational Research, Vol. 129, No. 1, 2001, pp. 11-47. doi:10.1016/S0377-2217(00)00167-3

[13] S. Greco, B. Matarazzo and R. Slowingski, “Rough Sets Methodology for Sorting Problems in Presence of Multiple Attributes and Criteria,” European Journal of Operational Research, Vol. 138, No. 2, 2002, pp. 247-259. doi:10.1016/S0377-2217(01)00244-2

[14] K. Dembczynski, R. Pindur and R. Susmaga, “Generation of Exhaustive Set of Rules within Dominance-Based Rough Set Approach,” Electronic Notes Theory Computer Science, Vol. 82, No. 4, 2003, p. 1.

[15] K. Dembczynski, R. Pindur and R. Susmaga, “Dominance-Based Rough Set Classifier without Induction of Decision Rules,” Electronic Notes Theory Computer Science, Vol. 82, No. 4, 2003, pp. 84-95

[16] Y. Sai, Y. Y. Yao and N. Zhong, “Data Analysis and Mining in Ordered Information Tables,” Proceedings of the 2001 IEEE International Conference on Data Mining, New York, 2001, pp. 497-504.

[17] M. W. Shao and W. X. Zhang, “Dominance Relation and Rules in an Incomplete Ordered Information System,” International Journal of Intelligent Systems, Vol. 20, No. 1, 2005, pp. 13-27.

[18] W. Z. Wu, Y. Leung and W. X. Zhang, “Connections between Rough Set Theory and Dempster-Shafer Theory of Evidence,” International Journal of General Systems, Vol. 31, No. 4, 2002, pp. 405-430. doi:10.1080/0308107021000013626

[1] Z. Pawlak, “Rough Sets,” International Journal of Computer and Information Science, Vol. 11, No. 5, 1982, pp. 341-356. doi:10.1007/BF01001956

[2] M. Kryszkiewicz, “Comparative Studies of Alternative Type of Knowledge Reduction in Inconsistent Systems,” International Journal of Intelligent Systems, Vol. 16, No. 1, 2001, pp. 105-120. doi:10.1002/1098-111X(200101)16:1<105::AID-INT8>3.0.CO;2-S

[3] Y. Leuang, W. Z. Wu and W. X. Zhang, “Knowledge Acquisition in Incomplete Information Systems: A Rough Set Approach,” European Journal of Operational Research, Vol. 168, No. 1, 2006, pp. 164-180. doi:10.1016/j.ejor.2004.03.032

[4] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publishers, Boston, 1991.

[5] W. X. Zhang, W. Z. Wu, J. Y. Liang and D. Y. Li, “Theory and Method of Rough Sets,” Science Press, Beijing, 2001.

[6] W. Z. Wu, Y. Leuang and J. S. Mi, “On Characterizations of (I,T)-Fuzzy Rough Approximation Operators,” Fuzzy Sets and Systems, Vol. 154, No. 1, 2005, pp. 76-102. doi:10.1016/j.fss.2005.02.011

[7] W. Z. Wu, M. Zhang, H. Z. Li and J. S. Mi, “Knowledge Reduction in Random Information Systems via Dempster-Shafer Theory of Evidence,” Information Sciences, Vol. 174, No. 3-4, 2005, pp. 143-164. doi:10.1016/j.ins.2004.09.002

[8] M. Zhang, L. D. Xu, W. X. Zhang and H. Z. Li, “A Rough Set Approach to Knowledge Reduction Based on Inclusion Degree and Evidence Reasoning Theory,” Expert Systems, Vol. 20, No. 5, 2003, pp. 298-304.

[9] S. Greco, B. Matarazzo and R. Slowingski, “Rough Approximation of a Preference Relation by Dominance Relatioin ICS Research Report 16/96, Warsaw University of Technology, 1996,” European Journal of Operational Research, Vol. 117, 1999, pp. 63-83.

[10] S. Greco, B. Matarazzo and R. Slowingski, “A New Rough Set Approach to Multicriteria and Multiattribute Classification,” In: L. Polkowsik and A. Skowron, Eds., Rough Sets and Current Trends in Computing (RSCTC’98), Springer-Verlag, Berlin, 1998, pp. 60-67.

[11] S. Greco, B. Matarazzo and R. Slowingski, “A New Rough Sets Approach to Evaluation of Bankruptcy Risk,” In: X. Zopounidis, Ed., Operational Tools in the Management of Financial Risks, Kluwer, Dordrecht, 1999, pp. 121-136.

[12] S. Greco, B. Matarazzo and R. Slowingski, “Rough Sets Theory for Multicriteria Decision Analysis,” European Journal of Operational Research, Vol. 129, No. 1, 2001, pp. 11-47. doi:10.1016/S0377-2217(00)00167-3

[13] S. Greco, B. Matarazzo and R. Slowingski, “Rough Sets Methodology for Sorting Problems in Presence of Multiple Attributes and Criteria,” European Journal of Operational Research, Vol. 138, No. 2, 2002, pp. 247-259. doi:10.1016/S0377-2217(01)00244-2

[14] K. Dembczynski, R. Pindur and R. Susmaga, “Generation of Exhaustive Set of Rules within Dominance-Based Rough Set Approach,” Electronic Notes Theory Computer Science, Vol. 82, No. 4, 2003, p. 1.

[15] K. Dembczynski, R. Pindur and R. Susmaga, “Dominance-Based Rough Set Classifier without Induction of Decision Rules,” Electronic Notes Theory Computer Science, Vol. 82, No. 4, 2003, pp. 84-95

[16] Y. Sai, Y. Y. Yao and N. Zhong, “Data Analysis and Mining in Ordered Information Tables,” Proceedings of the 2001 IEEE International Conference on Data Mining, New York, 2001, pp. 497-504.

[17] M. W. Shao and W. X. Zhang, “Dominance Relation and Rules in an Incomplete Ordered Information System,” International Journal of Intelligent Systems, Vol. 20, No. 1, 2005, pp. 13-27.

[18] W. Z. Wu, Y. Leung and W. X. Zhang, “Connections between Rough Set Theory and Dempster-Shafer Theory of Evidence,” International Journal of General Systems, Vol. 31, No. 4, 2002, pp. 405-430. doi:10.1080/0308107021000013626