臺灣博碩士論文加值系統

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

Adamo, J. M. (1980), "Fuzzy Decision Trees", Fuzzy Sets and Systems, Vol. 4, pp.207-219.
Amaya Cruz, G. P., and Beliakov, G. (1996), "Approximate Reasoning and Interpretation of Laboratory Tests in Medical Diagnostics", Cybernetics and Systems, Vol. 26, pp.713-729.
Baas, S. M., and Kwakernaak, H. (1977), "Rating and Ranking of Multiple-Aspect Alternatives Using Fuzzy Sets", Automatica, Vol. 13, pp.47-58.
Baldwin, J. F., and Guild, N. C. F. (1979), "Comparison of Fuzzy Sets on the Same Decision Space", Fuzzy Sets and Systems, Vol. 2, pp.213-231.
Bellman, R. E., and Zadeh, L. A. (1970), "Decision-Making in a Fuzzy Environment", Management Science, Vol. 17, No. 4, pp.B-141-B-164.
Boender, C. G. E., de Graan, J. G., and Lootsma, F. A. (1989), "Multi-Criteria Decision Analysis with Fuzzy Pairwise Comparisons", Fuzzy Sets and Systems, Vol. 29, pp.133-143.
Bortolan, G., and Degani, R. (1985), "A Review of Some Methods for Ranking Fuzzy Subsets", Fuzzy Sets and Systems, Vol. 15, pp.1-19.
Buckley, J. J. (1984), "The Multiple Judge，Multiple Criteria Ranking Problem：A Fuzzy Set Approach", Fuzzy Sets and Systems, Vol. 13, pp.25-37.
Buckley,J.J. (1985), "Ranking Alternatives Using Fuzzy Numbers", Fuzzy Sets and Systems, Vol. 15, pp.21-31.
Chang, P. L., and Chen, Y. C. (1994), "An Application of Fuzzy Set Theory to Postal Establishment Investment Strategy Selection under Fuzzy Environment", Management and Systems, Vol. 1, No. 2, pp.125-144.
Chang, T. C., Hasegawa, K., and Ibbs, C. W. (1991), "The Effect of Membership Function on Fuzzy Reasoning", Fuzzy Sets and Systems, Vol. 44, pp.169-186.
Chang Y. H., Lin S. L., and Shen C. C. (1997), "Applications of Fuzzy Multiobjective Programming on Bus Operations Plan", Journal of the Chinese Fuzzy Systems Association, Vol. 3, No. 1, pp.65-80.
Chen, S. H. (1985), "Ranking Fuzzy Numbers with Maximizing Set and Minimizing Set", Fuzzy Sets and Systems, Vol. 17, pp.113-129.
Chen, S. H. (1998), "Operations of Fuzzy Numbers with Step Form Membership Function Using Function Principle", Journal of Information Sciences, Vol. 108, pp.149-155.
Cheng, C. H. (1998), "A New Approach for Ranking Fuzzy Numbers by Distance Method", Fuzzy Sets and Systems, Vol. 95, pp.307-317.
Delgado, M., Verdegay, J. L., and Vila, M. A. (1988), "A Procedure for Ranking Fuzzy Numbers Using Fuzzy Relations", Fuzzy Sets and Systems, Vol. 26, pp.49-62.
de Campos Ibanez, L. M., and Gonzalez Munoz, A. (1989), "A Subjective Approach for Ranking Fuzzy Numbers", Fuzzy Sets and Systems, Vol. 29, pp.145-153.
Dubois, D. and Prade, H. (1983), "Ranking Fuzzy Numbers in the Setting of Possibility Theory", Information Science, Vol. 30, pp.183-224.
George, J. K., and Yuan, B. (1995), Fuzzy Sets and Fuzzy Logic: Theory and Application, N. J.: Prentice-Hall.
George, J. K., Ute, S. C., and Yuan, B. (1997), Fuzzy Sets Theory : Foundations and Application, N. J.: Prentice-Hall.
Heilpern. S. (1997), "Representation and Application of Fuzzy Numbers", Fuzzy Sets and Systems, Vol. 91, pp.259-268,
Herrera, F., and Verdegay, J. L. (1997), "Fuzzy Sets and Operations Research", Fuzzy Sets and Systems, Vol. 90, pp.207-218.
Hsu, H. M., and Chen, C. T. (1994), "Fuzzy Hierarchical Weight Analysis Model for Multicriteria Decision Problem", Journal of the Chinese Institute of Industrial Engineers, Vol. 11, No. 3, pp.129-136.
Hsu, H. M. (1997), "Fuzzy Credibility Relation Method for Multiple Criteria Decision-Making Problems", Information Sciences, Vol. 96, pp.79-91.
Hsu, T. H., and Lee, T. C. (1997), "The Fuzzy Product Portfolio Model", Journal of Management, Vol. 14, No. 2, pp.263-286.
Jain, R. (1977), "A Procedure for Multiple-Aspect Decision Making Using Fuzzy Sets", International Journal of Systems Science, Vol. 8, pp.1-7.
Kaufmann, A., and Gupta, M. M. (1991), Introduction to Fuzzy Arithmetic: Theory and Applications, N. Y.: Van Nostrand Reinhold.
Kim, K., and Park, K. S. (1990), "Ranking Fuzzy Numbers with Index of Optimism", Fuzzy Sets and Systems, Vol. 35, pp.143-150.
Lee, E. S., and Li, R. J. (1988), "Comparison of Fuzzy Numbers Based on the Probability Measure of Fuzzy Events", Computers and Mathematics with Applications, Vol. 15, No. 10, pp.887-896.
Liou, T. S., and Wang, M. J. J. (1992), "Ranking Fuzzy Numbers with Integral Value", Fuzzy sets and Systems, Vol. 50, pp.247-255.
Roubens, M. (1997), "Fuzzy Sets and Decision Analysis", Fuzzy Sets and Systems, Vol. 90, pp.199-206.
Sugimoto, M., and Yasunobu, S. (1996), "Supporting System to Evaluate Physical Performance Based on Fuzzy Reasoning", Japanese Journal of Fuzzy Theory and Systems, Vol. 8, No. 1, pp.61-69.
Teng, J. Y., and Tzeng, G. H. (1998), "Transportation Investment Project Selection Using Fuzzy Multiobjective Programming", Fuzzy Sets and Systems, Vol. 96, pp.259-280.
Thole, U., Zimmermann, H. J., and Zysno, P. (1979), "On The Suitability of Minimum and Product Operators for the Intersection of Fuzzy Sets", Fuzzy Sets and Systems, Vol. 2, pp.167-180.
Wang, H. F., and Hsu, H. M. (1991), "Sensitivity Analysis of Fuzzy Relation Equations", International Journal of General Systems, Vol. 19, pp.155-169.
Wang, H. F., and Hsu, H. M. (1992), "An Alternative Approach to the Resolution of Fuzzy Relation Equations", Fuzzy Sets and Systems, Vol. 45, pp.203-213.
Wang, M. J., and Liang, G. S. (1995), "Benefit/Cost Analysis Using Fuzzy Concept", The Engineering Economist, Vol. 40, No. 4, pp.359-376.
Watson, S. R., Weiss, J. J., and Donnell, M. L (1979), "Fuzzy Decision Analysis", IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-9, No. 1, pp.1-9.
Yager, R. R. (1981), "A Procedure for Ordering Fuzzy Subsets of the Unit Interval", Information Sciences, Vol. 24, pp.143-161.
Yager, R. R. (1984), "Approximate Reasoning as a Basis for Rule-Based Expert Systems", IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-14, No. 4, pp.636-643.
Yager, R. R. (1997), "Fuzzy Logics and Artificial Intelligence", Fuzzy Sets and Systems, Vol. 90, pp.193-198.
Yeh, H. P. (1998), "Decision Making with Fuzzy Alternatives: A New Method Based on Decision Maker''s Fuzziness Attitude", a dissertation submitted in partial fulfillment of the requirements for the degree of doctor of philosophy, department of industrial engineering, University of Wisconsin-Madison.
Zadeh, L. A. (1965), "Fuzzy Sets", Information and Control, Vol. 8, pp.338-353.
Zadeh, L. A. (1975), "The Concept of Linguistic Variable and its Application to Approximate Reasoning-Ⅰ", Information Sciences, Vol. 8, pp.199-249.
Zadeh, L. A. (1975), "The Concept of Linguistic Variable and its Application to Approximate Reasoning-Ⅱ", Information Sciences, Vol. 8, pp.301-357.
Zadeh, L. A. (1975), "The Concept of Linguistic Variable and its Application to Approximate Reasoning-Ⅲ", Information Sciences, Vol. 8, pp.43-80.
Zadeh, L. A. (1997), "Toward a Theory of Fuzzy Information Granulation and its Centrality in Human Reasoning and Fuzzy Logic", Fuzzy Sets and Systems, Vol. 90, pp.111-127.