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題名 模糊數在一般條件下之對稱梯形逼近
Computational analysis of symmetric trapezoidal approximations for fuzzy numbers under general conditions作者 林欣亭
Lin, Hsin-Ting貢獻者 陳隆奇<br>葉啟村
Chen, Lung-Chi<br>Yeh, Chi-Tsun
林欣亭
Lin, Hsin-Ting關鍵詞 模糊數
對稱三角形逼近
對稱梯形逼近
Fuzzy numbers
Symmetric triangular approximation
Symmetric trapezoidal approximation日期 2023 上傳時間 1-Feb-2024 11:25:15 (UTC+8) 摘要 本篇博士論文主要探討模糊數在一般條件下之對稱梯形逼近。Ban和Coroianu在2016年的《Soft Computing》期刊中提出模糊數在一般條件下之對稱三角形逼近的概念，本研究深入研究模糊數的對稱梯形逼近，特別在一般的條件下，這是前者未曾涵蓋的範疇。我們完整計算出對稱梯形逼近的解析解，並深入探討這種逼近方法的各項性質。同時，我們也研究了對稱梯形逼近退化為對稱三角逼近的條件。最後，論文提供了有關期望值和模糊性等關鍵參數的實例，以探討逼近過程中數值誤差的優勢。這份研究的貢獻在於針對模糊數的逼近提供更為實用、有效的逼近方式。
In their publication in Soft Computing [Soft Comput 20:1249–1261, 2016], Ban and Coroianu introduced the concept of symmetric triangular approximation under a general condition, along with extensive calculations and a computational formula. However, their conclusions did not support the derivation of the symmetric trapezoidal approximation. In this study, calculations for the symmetric trapezoidal approximations of fuzzy numbers are conducted under general conditions. Additionally, the properties of identity, translation, and scale invariance, as well as additivity of the derived approximation operators, are explored. The conditions that lead to the degeneration from the nearest symmetric trapezoidal approximation to the symmetric triangular approximation are also investigated. Furthermore, applications and numerical examples related to significant parameters such as value, expected value, and ambiguity are provided. Finally, quantitative improvements in the approximation process are examined using several illustrative examples.參考文獻 1. Abbasbandy, S., Ahmady, E., & Ahmady, N. (2010). Triangular approximations of fuzzy numbers using α-weighted valuations. Soft Computing, 14(1), 71–79. 2. Abbasbandy, S., & Amirfakhrian, M. (2006). The nearest approximation of a fuzzy quantity in parametric form. Applied Mathematics and Computation, 172, 624–632. 3. Abbasbandy, S., & Amirfakhrian, M. (2006). The nearest trapezoidal form of a generalized left-right fuzzy number. International Journal of Approximate Reasoning, 43, 166–178. 4. Abbasbandy, S., & Asady, B. (2004). The nearest trapezoidal fuzzy number to a fuzzy quantity. Applied Mathematics and Computation, 156, 381–386. 5. Allahviranloo, T., & Firozja, M. A. (2007). Note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets and Systems, 158, 755–756. 6. Ban, A. I. (2008). Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval. Fuzzy Sets and Systems, 159, 1327–1344. 7. Ban, A. I. (2011). Remarks and corrections to the triangular approximations of fuzzy numbers using α-weighted valuations. Soft Computing, 15, 351–361. 8. Ban, A. I., Brândaş, A., Coroianu, L., Negruţiu, C., & Nica, O. (2011). Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value. Computers & Mathematics with Applications, 61, 1379–1401. 9. Ban, A. I., & Coroianu, L. (2012). Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity. International Journal of Approximate Reasoning, 53, 805–836. 10. Ban, A. I., & Coroianu, L. (2014). Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets and Systems, 257, 3–22. 11. Ban, A. I., & Coroianu, L. (2015). Existence, uniqueness, calculus and properties of triangular approximations of fuzzy numbers under a general condition. International Journal of Approximate Reasoning, 62, 1–26. 12. Ban, A. I., & Coroianu, L. (2016). Symmetric triangular approximations of fuzzy numbers under a general condition and properties. Soft Computing, 20, 1249–1261. 13. Chanas, S. (2001). On the interval approximation of a fuzzy number. Fuzzy Sets and Systems, 122, 353–356. 14. Coroianu, L. (2011). Best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval. Fuzzy Sets and Systems, 165, 81–97. 15. Coroianu, L. (2012). Lipschitz functions and fuzzy number approximations. Fuzzy Sets and Systems, 200, 116–135. 16. Delgado, M., Vila, M. A., & Voxman, W. (1998). On a canonical representation of a fuzzy number. Fuzzy Sets and Systems, 93, 125–135. 17. Diamond, P., & Kloeden, P. (1994). Metric spaces of fuzzy sets. Theory and applications. World Scientific, Singapore. 18. Dubois, D., & Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9, 613–626. 19. Dubois, D., & Prade, H. (1987). The mean value of a fuzzy number. Fuzzy Sets and Systems, 24, 279–300. 20. Grzegorzewski, P. (1998). Metrics and orders in the space of fuzzy numbers. Fuzzy Sets and Systems, 97, 83–94. 21. Grzegorzewski, P. (2002). Nearest interval approximation of a fuzzy number. Fuzzy Sets and Systems, 130, 321–330. 22. Grzegorzewski, P. (2008). Trapezoidal approximations of fuzzy numbers preserving the expected interval – algorithms and properties. Fuzzy Sets and Systems, 159, 1354–1364. 23. Grzegorzewski, P. (2008). New algorithms for trapezoidal approximation of fuzzy numbers preserving the expected interval. In: Magdalena L, Ojeda M, Verdegay JL (eds) Proceedings on information processing and management of uncertainty in knowledge-based system conference, Malaga, pp 117–123. 24. Grzegorzewski, P., & Mrówka, E. (2005). Trapezoidal approximations of fuzzy numbers. Fuzzy Sets and Systems, 153, 115–135. 25. Grzegorzewski, P., & Mrówka, E. (2007). Trapezoidal approximations of fuzzy numbers-revisited. Fuzzy Sets and Systems, 158, 757–768. 26. Heilpern, S. (1992). The expected value of a fuzzy number. Fuzzy Sets and Systems, 47, 81–86. 27. Li, J., Wang, Z.-X., & Yue, Q. (2012). Triangular approximation preserving the centroid of fuzzy numbers. International Journal of Computer Mathematics, 89, 810–821. 28. Ma, M., Kandel, A., & Friedman, M. (2000). A new approach for defuzzication. Fuzzy Sets and Systems, 111, 351–356. 29. Nasibov, E. N., & Peker, S. (2008). On the nearest parametric approximation of a fuzzy number. Fuzzy Sets and Systems, 159, 1365–1375. 30. Rockafeller, R. T. (1970). Convex analysis. Princeton University Press, Princeton, NJ. 31. Yeh, C.-T. (2007). A note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets and Systems, 158, 747–754. 32. Yeh, C.-T. (2008). On improving trapezoidal and triangular approximations of fuzzy numbers. International Journal of Approximate Reasoning, 48, 297–313. 33. Yeh, C.-T. (2008). Trapezoidal and triangular approximations preserving the expected interval. Fuzzy Sets and Systems, 159, 1345–1353. 34. Yeh, C.-T. (2009). Weighted trapezoidal and triangular approximations of fuzzy numbers. Fuzzy Sets and Systems, 160, 3059–3079. 35. Yeh, C.-T. (2011). Weighted semi-trapezoidal approximations of fuzzy numbers. Fuzzy Sets and Systems, 165, 61–80. 36. Yeh, C.-T. (2017). Existence of interval, triangular, and trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets and Systems, 310, 1–13. 37. Zeng, W., & Li, H. (2007). Weighted triangular approximation of fuzzy numbers. International Journal of Approximate Reasoning, 46, 137–150. 38. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. 39. Lin, H.-T. (2023). Symmetric trapezoidal approximations of fuzzy numbers under a general condition. Soft Computing, 1–15. 描述 博士
國立政治大學
應用數學系
101751501資料來源 http://thesis.lib.nccu.edu.tw/record/#G0101751501 資料類型 thesis dc.contributor.advisor 陳隆奇<br>葉啟村 zh_TW dc.contributor.advisor Chen, Lung-Chi<br>Yeh, Chi-Tsun en_US dc.contributor.author (Authors) 林欣亭 zh_TW dc.contributor.author (Authors) Lin, Hsin-Ting en_US dc.creator (作者) 林欣亭 zh_TW dc.creator (作者) Lin, Hsin-Ting en_US dc.date (日期) 2023 en_US dc.date.accessioned 1-Feb-2024 11:25:15 (UTC+8) - dc.date.available 1-Feb-2024 11:25:15 (UTC+8) - dc.date.issued (上傳時間) 1-Feb-2024 11:25:15 (UTC+8) - dc.identifier (Other Identifiers) G0101751501 en_US dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/149593 - dc.description (描述) 博士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description (描述) 101751501 zh_TW dc.description.abstract (摘要) 本篇博士論文主要探討模糊數在一般條件下之對稱梯形逼近。Ban和Coroianu在2016年的《Soft Computing》期刊中提出模糊數在一般條件下之對稱三角形逼近的概念，本研究深入研究模糊數的對稱梯形逼近，特別在一般的條件下，這是前者未曾涵蓋的範疇。我們完整計算出對稱梯形逼近的解析解，並深入探討這種逼近方法的各項性質。同時，我們也研究了對稱梯形逼近退化為對稱三角逼近的條件。最後，論文提供了有關期望值和模糊性等關鍵參數的實例，以探討逼近過程中數值誤差的優勢。這份研究的貢獻在於針對模糊數的逼近提供更為實用、有效的逼近方式。 zh_TW dc.description.abstract (摘要) In their publication in Soft Computing [Soft Comput 20:1249–1261, 2016], Ban and Coroianu introduced the concept of symmetric triangular approximation under a general condition, along with extensive calculations and a computational formula. However, their conclusions did not support the derivation of the symmetric trapezoidal approximation. In this study, calculations for the symmetric trapezoidal approximations of fuzzy numbers are conducted under general conditions. Additionally, the properties of identity, translation, and scale invariance, as well as additivity of the derived approximation operators, are explored. The conditions that lead to the degeneration from the nearest symmetric trapezoidal approximation to the symmetric triangular approximation are also investigated. Furthermore, applications and numerical examples related to significant parameters such as value, expected value, and ambiguity are provided. Finally, quantitative improvements in the approximation process are examined using several illustrative examples. en_US dc.description.tableofcontents 1 Introductions (p1-2) 2 Preliminaries (p3-9) 3 Symmetric triangular approximation under general conditions (p10-12) 4 Symmetric trapezoidal approximation under general conditions (p13-34) 5 Existence and uniqueness of symmetric trapezoidal approximation for fuzzy numbers under general conditions (p35-39) 6 Properties (p40-51) 7 Exploring the degeneration from symmetric trapezoidal to triangular approximations (p52-55) 8 Applications (p56-62) 9 Conclusion (p63-64) Acknowledgement (p65) Bibliography (p66-69) zh_TW dc.format.extent 1187666 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0101751501 en_US dc.subject (關鍵詞) 模糊數 zh_TW dc.subject (關鍵詞) 對稱三角形逼近 zh_TW dc.subject (關鍵詞) 對稱梯形逼近 zh_TW dc.subject (關鍵詞) Fuzzy numbers en_US dc.subject (關鍵詞) Symmetric triangular approximation en_US dc.subject (關鍵詞) Symmetric trapezoidal approximation en_US dc.title (題名) 模糊數在一般條件下之對稱梯形逼近 zh_TW dc.title (題名) Computational analysis of symmetric trapezoidal approximations for fuzzy numbers under general conditions en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) 1. Abbasbandy, S., Ahmady, E., & Ahmady, N. (2010). Triangular approximations of fuzzy numbers using α-weighted valuations. Soft Computing, 14(1), 71–79. 2. Abbasbandy, S., & Amirfakhrian, M. (2006). The nearest approximation of a fuzzy quantity in parametric form. Applied Mathematics and Computation, 172, 624–632. 3. Abbasbandy, S., & Amirfakhrian, M. (2006). The nearest trapezoidal form of a generalized left-right fuzzy number. International Journal of Approximate Reasoning, 43, 166–178. 4. Abbasbandy, S., & Asady, B. (2004). The nearest trapezoidal fuzzy number to a fuzzy quantity. Applied Mathematics and Computation, 156, 381–386. 5. Allahviranloo, T., & Firozja, M. A. (2007). Note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets and Systems, 158, 755–756. 6. Ban, A. I. (2008). Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval. Fuzzy Sets and Systems, 159, 1327–1344. 7. Ban, A. I. (2011). Remarks and corrections to the triangular approximations of fuzzy numbers using α-weighted valuations. Soft Computing, 15, 351–361. 8. Ban, A. I., Brândaş, A., Coroianu, L., Negruţiu, C., & Nica, O. (2011). Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value. Computers & Mathematics with Applications, 61, 1379–1401. 9. Ban, A. I., & Coroianu, L. (2012). Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity. International Journal of Approximate Reasoning, 53, 805–836. 10. Ban, A. I., & Coroianu, L. (2014). Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets and Systems, 257, 3–22. 11. Ban, A. I., & Coroianu, L. (2015). Existence, uniqueness, calculus and properties of triangular approximations of fuzzy numbers under a general condition. International Journal of Approximate Reasoning, 62, 1–26. 12. Ban, A. I., & Coroianu, L. (2016). Symmetric triangular approximations of fuzzy numbers under a general condition and properties. Soft Computing, 20, 1249–1261. 13. Chanas, S. (2001). On the interval approximation of a fuzzy number. Fuzzy Sets and Systems, 122, 353–356. 14. Coroianu, L. (2011). Best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval. Fuzzy Sets and Systems, 165, 81–97. 15. Coroianu, L. (2012). Lipschitz functions and fuzzy number approximations. Fuzzy Sets and Systems, 200, 116–135. 16. Delgado, M., Vila, M. A., & Voxman, W. (1998). On a canonical representation of a fuzzy number. Fuzzy Sets and Systems, 93, 125–135. 17. Diamond, P., & Kloeden, P. (1994). Metric spaces of fuzzy sets. Theory and applications. World Scientific, Singapore. 18. Dubois, D., & Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9, 613–626. 19. Dubois, D., & Prade, H. (1987). The mean value of a fuzzy number. Fuzzy Sets and Systems, 24, 279–300. 20. Grzegorzewski, P. (1998). Metrics and orders in the space of fuzzy numbers. Fuzzy Sets and Systems, 97, 83–94. 21. Grzegorzewski, P. (2002). Nearest interval approximation of a fuzzy number. Fuzzy Sets and Systems, 130, 321–330. 22. Grzegorzewski, P. (2008). Trapezoidal approximations of fuzzy numbers preserving the expected interval – algorithms and properties. Fuzzy Sets and Systems, 159, 1354–1364. 23. Grzegorzewski, P. (2008). New algorithms for trapezoidal approximation of fuzzy numbers preserving the expected interval. In: Magdalena L, Ojeda M, Verdegay JL (eds) Proceedings on information processing and management of uncertainty in knowledge-based system conference, Malaga, pp 117–123. 24. Grzegorzewski, P., & Mrówka, E. (2005). Trapezoidal approximations of fuzzy numbers. Fuzzy Sets and Systems, 153, 115–135. 25. Grzegorzewski, P., & Mrówka, E. (2007). Trapezoidal approximations of fuzzy numbers-revisited. Fuzzy Sets and Systems, 158, 757–768. 26. Heilpern, S. (1992). The expected value of a fuzzy number. Fuzzy Sets and Systems, 47, 81–86. 27. Li, J., Wang, Z.-X., & Yue, Q. (2012). Triangular approximation preserving the centroid of fuzzy numbers. International Journal of Computer Mathematics, 89, 810–821. 28. Ma, M., Kandel, A., & Friedman, M. (2000). A new approach for defuzzication. Fuzzy Sets and Systems, 111, 351–356. 29. Nasibov, E. N., & Peker, S. (2008). On the nearest parametric approximation of a fuzzy number. Fuzzy Sets and Systems, 159, 1365–1375. 30. Rockafeller, R. T. (1970). Convex analysis. Princeton University Press, Princeton, NJ. 31. Yeh, C.-T. (2007). A note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets and Systems, 158, 747–754. 32. Yeh, C.-T. (2008). On improving trapezoidal and triangular approximations of fuzzy numbers. International Journal of Approximate Reasoning, 48, 297–313. 33. Yeh, C.-T. (2008). Trapezoidal and triangular approximations preserving the expected interval. Fuzzy Sets and Systems, 159, 1345–1353. 34. Yeh, C.-T. (2009). Weighted trapezoidal and triangular approximations of fuzzy numbers. Fuzzy Sets and Systems, 160, 3059–3079. 35. Yeh, C.-T. (2011). Weighted semi-trapezoidal approximations of fuzzy numbers. Fuzzy Sets and Systems, 165, 61–80. 36. Yeh, C.-T. (2017). Existence of interval, triangular, and trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets and Systems, 310, 1–13. 37. Zeng, W., & Li, H. (2007). Weighted triangular approximation of fuzzy numbers. International Journal of Approximate Reasoning, 46, 137–150. 38. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. 39. Lin, H.-T. (2023). Symmetric trapezoidal approximations of fuzzy numbers under a general condition. Soft Computing, 1–15. zh_TW
