模糊集合之凸性與廣義凸性及其應用

CONVEXITY AND GENERALIZED CONVEXITY OF FUZZY SETS WITH APPLICATIONS

陳傑(Jye Chen)

指導教授 ： 丁慶榮

元智大學/工程學院/工業工程與管理學系/博士(2003年)

摘要

凸性的觀念及其延伸之各式廣義凸性的觀念，對作業研究及應用數學領域中數量及質量方面的研究均極為重要。而在模糊集合理論也有許多學者探討這些凸性及廣義凸性的觀念。在先前其他學者的研究成果激勵下，並有鑑於凸性觀念的重要，本論文的主要目的在探討模糊集合的凸性及擬凸性的觀念。論文中提出了模糊集合半嚴格擬凸性的一個全新概念。本研究並提供了模糊集合在下半連續性的情形下，其凸性判定的準則。另外，本論文證明了在上半連續性的情形下，半嚴格擬凸性的模糊集合，係介於擬凸性模糊集合及嚴格擬凸性模糊集合之間。本論文亦證明了對半嚴格擬凸性模糊集合及嚴格擬凸性模糊集合兩個族群而言，每一個區域性的最大值均是全域的最大值。此外，本研究亦提出了利用等高集來描述擬單調模糊集合之特徵的方式。
本論文的第二個目的在研究模糊集合的Φ-凸性及Φ-擬凸性的觀念。論文中提出了一些模糊集合Φ-凸性及Φ-擬凸性的特徵化定理。在這些模糊集合Φ-凸性及Φ-擬凸性觀念的基礎上，引伸出一些模糊過程，例如：Φ-凸性模糊過程及Φ-擬凸性模糊過程等，同時，亦探討了這些模糊過程的一些基本特性。再者，這些模糊過程與圖之間的相關性亦在本文探討之列。

關鍵字

Φ-凸性模糊集合 ； Φ-凸性模糊化過程 ； 凸性 ； 廣義凸性 ； 模糊集合 ； 凸性模糊集合

並列摘要

The concept of convexity and its various generalization is important for quantitative and qualitative studies in operations research and applied mathematics. This was considered by many authors also in fuzzy set theory. Motivated both by earlier research works and by the importance of the concept of convexity, the primary objective of this research is to study the concept of convexity and quasiconvexity of fuzzy sets. A new concept of semistrictly quasiconvex fuzzy sets is proposed. A criterion for convex fuzzy sets under lower semicontinuity is given. It is proved in the upper semicontinuous case, that the class of semistrictly quasiconvex fuzzy sets lies between the quasiconvex and strictly quasiconvex classes. It is also proved for both families of semistrictly quasiconvex and strictly quasiconvex fuzzy sets,
that every local maximizer is also a global one. In addition, a characterization of quasimonotonic fuzzy sets in terms of level sets is given.
A secondary objective is to study the concept of Φ-convexity and Φ-quasiconvexity of fuzzy sets. Characterizations for Φ-convex and Φ-quasiconvex fuzzy sets are given. Based on the concept of Φ-convex and Φ-quasiconvex fuzzy sets, fuzzy processes such as Φ-convex fuzzy processes and Φ-quasiconvex fuzzy processes are proposed and their basic properties are studied. Furthermore, the relationship between these fuzzy processes and their graphs is investigated.

並列關鍵字

Φ-convex fuzzy sets ； Φ-convex fuzzy processes ； convexity ； generalized convexity ； fuzzy sets ； convex fuzzy sets

參考文獻

[1] E. Ammar, "Some properties of convex fuzzy sets and convex fuzzy cones," Fuzzy Sets and Systems 106, pp.381-386 (1999).

[2] E. Ammar, J. Metz, "On fuzzy convexity and parametric fuzzy optimization," Fuzzy Sets and Systems 49, pp.135-141 (1992) .

[3]J. M. Apostol, Mathematical Analysis, 2nd ed. Addison-Wesley Publishing Company, (1975).

[6] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed. Wiley, (1993).

[7] R. E. Bellman, L. A. Zadeh, "Decision making in a fuzzy environment," Management Sciences 17, pp.141-164 (1970).