臺灣博碩士論文加值系統

模糊理論被廣泛應用在管理與工程科學問題上，也因此突顯了模糊數學運算操作的重要性。本論文主要目的是在於探討模糊運算的程序操作，進而提出運算程序的簡化操作規則，有效縮短運算時間。本文主要分成三個部份進行探討，分別為簡化規則、模糊加權平均與模糊函數最佳化。
首先，Alpha截集模糊運算的問題就類似數學運算中的區間計算一樣，模糊參數(模糊變數)在運算時都可能發生結果值的高估計算問題，同時，區域解也有可能存在模糊參數的支集裡，進而引起結果值低估計算的不合理現象，其引發原因分別為模糊函數中運算元(模糊參數)的重複出現，或是運算子出現負號，或是有些模糊參數所屬區間值橫跨由負到正所引起。而為了克服這些問題，已經有許多研究者提出解決方法。然而，在這些解決方法的運算操作程序中，所需計算組合次數會隨著參數的遞增而變得非常龐大且運算費時。因此，本研究將以頂點方法(vertex method)為基礎，藉由各種觀察綜整歸納，進而提出運算簡化規則，縮減計算所需的計算組合；而整個簡化規則運作機制，則是透過模糊參數(模糊變數)型態的區分，將函數切割成許多子函數的概念而達成，並於各子函數(或是函數)中使用簡化規則。而經由例子說明，執行模糊運算操作時，計算組合已非先前的所有可能性組合，而是僅需少數計算組合，證明簡化規則的運用對於降低計算組合次數有顯著的改善。
再者，模糊加權平均演算法被當成模糊數的函數運算，函數公式中的加權參數即是重複出現的模糊數，也會引起計算結果值的不合理問題。本文提出一個替代式演算法，除對現有的離散型模糊加權平均演算法比較外，也與線性規劃方法進行效率比較。而本研究所提的演算法與現存的二元搜尋法有所區別，整個演算法運作機制是藉由引用全部準則參數當成候選人的劃分方法，並利用加權作不同基準(benchmark)的更換調整作業程序，達到計算出函數最後的區間值域。從理論的最差情況(worst-case)，我們所提的演算法在比較複雜度稍劣於Guu方法;更進一步，我們執行各演算方法的數例實驗比較，從電腦執行的CPU時間與函數評估次數相比，本文所提的方法均勝過現存的方法，證明本文所提的方法是一個有效率的演算法。
最後，在Alpha截集法模糊運算與頂點方法為基礎下，本文也探討模糊函數公式中使用一個模糊決策變數，當決策變數出現區域解問題時，函數與決策變數之間應如何作最佳化調整變的很重要，本文提出一個能處理此問題的程序操作，建議使用於模糊函數最佳化問題，內文並舉例說明其操作程序。

The fuzzy theory has been used to solve various problems in management science and engineering. Hence, it becomes important to use fuzzy arithmetic operations. The main purpose of this dissertation is to investigate the procedures of fuzzy arithmetic operation. Furthermore, it provides simplifying rules of fuzzy arithmetic to save required time of arithmetic operation efficiently. We divided this article into three parts to investigate: simplifying rules, fuzzy weighted average (FWA) and fuzzy function optimization.
First, the problems of Alpha-cut fuzzy arithmetic have been shown, like in interval arithmetic, that distinct states of fuzzy parameters (or fuzzy variable values) may be chosen and produce an overestimated fuzziness. Meanwhile, local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number’s result. Previously approaches to overcoming these problems have appeared in the literature. Yet, the computational burden of these approaches got even heavier. Thus, this article is based on the vertex method in the literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method. These rules are devised through a function partitioned into sub-functions, distinguishing the types of fuzzy parameter/variable occurrences, and types of sub-functions or functions with the various observations. The improved efficiency has been found able to significantly reduce the combination (vertex) test of the vertex method for the fuzzy parameters’ Alpha cut endpoints possibly to only a few fuzzy parameters’ endpoint combinations.
Moreover, fuzzy weighted average as function of fuzzy numbers, is suitable for the problem of multiple occurrences of fuzzy parameters. We have reviewed and compared discrete algorithms for the FWAs in both theoretical comparison and numerical comparisons. An alternative efficient algorithm is also proposed. The algorithm introduces an all-candidate (criteria ratings) weights-replaced benchmark adjusting procedure other than a binary (dichotomy) search in the existing methods. In the number of element comparisons, Lee and Park’s algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used. However, from criterion of average CPU time and average number of evaluations, the alternative algorithm is efficiently, the results outperform than other FWA methods. It has been demonstrated efficient by the proposed alterative algorithm.
Finally, a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument (a single decision variable) is examined base on the -cut arithmetic and the vertex method. When a variable appeared a local solution problem, it becomes important to adjust between function and variable. In this article, a proper and useful preliminary algorithm is proposed. Numerical examples with results are also provided.

CONTENTS
摘 要 I
ABSTRACT II
誌 謝 III
ACKNOWLEDGEMENTS IV
CONTENTS V
LIST OF TABLES VII
LIST OF FIGURES VIII
Chapter 1. INTRODUCTION 1
1.1 Research Background and Motivation 1
1.2 Research Purposes 3
1.3 Organization of the Dissertation 4
Chapter 2. LITERATURES REVIEW 5
2.1 Fuzzy Sets and Fuzzy Numbers 5
2.2 Fuzzy Arithmetic Operations 7
2.3 Fuzzy Weighted Average 10
2.4 Fuzzy Function 13
Chapter 3. SIMPLIFYING RULES OF FUZZY ARITHMETIC 15
3.1 The Problems of -Cut Arithmetic 15
3.1.1 The Problem of Overestimation and Local Extrema 16
3.1.2 Observation of Function Partitioning and the Framework 18
3.2 The Sets of Simplifying Rules 20
3.2.1 Pre-Determining the Endpoint Assignments of Fuzzy Parameters / Variable Values: Extending Condition 2.1 20
3.2.2 Pre-Determining the Endpoint Assignments of Unrepeated Fuzzy Parameters/Variables in the Mixed Cases 25
3.2.3 All Fuzzy Parameters/Variables Unrepeated in a Sub-function /Function 32
3.3 Numerical Example 34
3.4 Summary 37
Chapter 4. SIMPLIFYING ALGORITHM FOR FUZZY WEIGHTED AVERAGE 38
4.1 Theoretical Background 38
4.2 Existing Algorithms for the Fuzzy Weighted Averages 42
4.3 The Algorithm Concept of Convergent Benchmark Adjustment 44
4.4 The Proposed Alternative Algorithm 50
4.5 Theoretical (Worst Case) Comparison 53
4.6 Numerical Experiments and Discussions 54
4.7 Summary 60
Chapter 5. FUZZY FUNCTION OPTIMIZATION WITH A FUZZY VARIABLE 68
5.1 The Concept of Fuzzy Function Optimization 68
5.2 The Proposed Algorithm for FFO 70
5.3 Illustrative Example 73
5.4 Summary 74
Chapter 6. CONCLUSIONS AND FUTURE WORKS 76
6.1 Research Findings and Concluding Remarks 76
6.2 Future Works 77
APPENDIX A (The Vertex Method) 79
APPENDIX B (Proof of the computational complexity of the alternative algorithm (the worst case)) 79
REFERENCES 81

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