# 臺灣博碩士論文加值系統

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 在模糊理論的理論發展與應用領域，模糊數排序是很重要的研究議題。儘管許多學者陸續提出模糊數排序方法，然而在應用上均有所限制或不足。本論文的研究目的是針對模糊數的排序方法，發展一模糊數排序方法架構，使既有的排序方法可以根據此一架構來解構、分析與檢驗，而排序方法的使用者也可以根據其不同的要求，挑擇不同的模組組合成一排序方法。本論文所發展的架構可用來解釋一些既有的方法，也驗證了此架構的可行性與可靠性。本論文並提出發展模糊數排序方法所應具備的特性，以及改善既有方法或發展新方法的策略。
 The comparison or ranking of fuzzy numbers has been a very important topic. Many researchers have proposed different methods for ranking fuzzy numbers. However, there are some limitations or insufficiency on the application of these ranking methods. The objective of this study is to develop a generic framework of fuzzy ranking methods so that the ranking methods can be decomposed into modules and be examined systematically. Thus, the users can choose different modules to generate specific fuzzy ranking method depending on different demands. To examine the feasibility and reliability, we used different ranking methods to illustrate the proposed framework. We also discussed the objectives for developing fuzzy ranking methods and proposed the strategies for improving existing methods and developing new methods.
 Content i List of Figures iii List of Tables v Chapter 1: Introduction 1 1.1 Background and significance 1 1.2 Research objectives of this thesis 1 1.3 Overview of this thesis 2 Chapter 2: Literature Review 3 2.1 The concept of fuzzy sets 3 2.2 The fundamentals of fuzzy sets 7 2.2.1 Definition and notation of fuzzy sets 7 2.2.2 Basic concepts of fuzzy sets theory 12 2.2.3 Operations on fuzzy sets 14 2.2.4 The representations of fuzzy sets with 16 2.2.5 The types of membership function 18 2.2.6 Fuzzy numbers 25 2.2.7 Arithmetic operations on intervals 32 2.2.8 Arithmetic operations on fuzzy numbers 33 2.2.9 The methods to construct membership functions of fuzzy sets 37 2.3 Ranking methods of fuzzy numbers 39 Chapter 3: A Framework of Modularized Fuzzy Ranking Methods and Illustrations 42 3.1 A framework of modularized fuzzy ranking methods 42 3.2 Illustrations of the proposed framework 49 3.2.1 Ranking method of Baas and Kwakernaak (1977) 49 3.2.2 Ranking method of Jain (1977) 55 3.2.3 Ranking method of Adamo (1980) 58 3.2.4 Ranking method of Yager (1981) 61 3.2.5 Ranking method of Chen (1985) 73 3.2.6 Ranking method of Lee and Li (1988) 78 3.2.7 Ranking method of de Campos Ibanez and Gonzalez Munoz (1989) 82 3.2.8 Ranking method of Kim and Park (1990) 90 3.2.9 Ranking method of Liou and Wang (1992) 95 3.2.10 Ranking method of Cheng (1998) 99 Chapter 4: Strategy for the framework of modularized fuzzy ranking methods 110 Chapter 5: Conclusion 117 References 118 List of Figures Figure 2.1The relation of classical mathematics, statistics, and fuzzy sets theory 5 Figure 2.2Fuzzification of the threshold increase the sensitivity and specificity (Amaya Cruz and Beliakov, 1996) 6 Figure 2.3The of a fuzzy set 11 Figure 2.4The standard fuzzy complement 15 Figure 2.5The standard fuzzy intersection 15 Figure 2.6The standard fuzzy union 15 Figure 2.7Illustration of decomposition theorem of fuzzy sets 18 Figure 2.8Four examples of fuzzy numbers (George et al., 1997) 28 Figure 2.9A convex and normal fuzzy set (Kaufmann and Gupta, 1991) 29 Figure 2.10A comparison of a real number and a crisp interval with a fuzzy number and a fuzzy interval, respectively (George and Yuan, 1995). 30 Figure 2.11Basic types of fuzzy numbers (George and Yuan, 1995) 31 Figure 2.12Limitation of fuzzy ranking methods in types of fuzzy numbers 41 Figure 3.1A Framework of Modularized Ranking Methods for Fuzzy Numbers 43 Figure 3.2The continuously weighting function with respect to the grade of membership 47 Figure 3.3An example of membership function of the conditional fuzzy set proposed by Baas and Kwakernaak (1977) 53 Figure 3.4An example of the membership function of ranking fuzzy set proposed by Baas and Kwakernaak (1977) 54 Figure 3.5The ratings of two compared fuzzy numbers from index proposed by Baas and Kwakernaak (1977) 54 Figure 3.6The ratings of fuzzy numbers based on the concept of maximizing set of Jain (1977) 58 Figure 3.7The ratings of two fuzzy numbers from ranking function of Adamo (1980) 61 Figure 3.8A discrete fuzzy subset for illustration (Yager, 1981) 65 Figure 3.9Continuous fuzzy subsets for illustration (Yager, 1981) 70 Figure 3.10Different fuzzy numbers evaluated by Chen (1985) 73 Figure 3.11Definition of right utility value of each fuzzy number of ranking method of Chen (1985) 76 Figure 3.12Definition of left utility value of each fuzzy number of ranking method of Chen (1985) 77 Figure 3.13Three input fuzzy numbers for illustration 86 Figure 3.14The probability density distribution of P on 89 Figure 3.15The ratings of two fuzzy numbers evaluated by Kim and Park (1990); a = , b = , c = , d = 95 Figure 4.1An example of two fuzzy numbers can not be discriminated 112
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