臺灣博碩士論文加值系統

距離、相似度及熵在我們日常生活的多項領域中具有其重要角色，其中距離及相似度已廣泛地用於區分不同集合或事物上，而熵也被用於模糊集的模糊度的量測上。在文獻中，針對猶豫模糊集已有多種不同的距離及相似度的公式被提出了，然而，其中多數要不是不夠充分，不然就是會產生不好的結果。在本篇論文中，我們提出了以Hausdorff矩量在猶豫模糊集上架構新式的距離及相似度測量，我們首先利用Hausdorff矩量以直覺的方式提出計算距離的簡易方法，其中具有兩個主要特性：(1)對較短的猶豫模糊元件不需要再加最小、最大或任何值使其與較長的猶豫模糊元件同長；(2)對猶豫模糊元件不需要按照大小次序排列。此兩特性使我們避免掉產生不好的結果。接著，我們再將所提出的距離擴展為相似度，最後，我們利用Hausdorff來量測一個猶豫模糊集以及它的 補集間的距離，作為兩者間區隔的大小，並利用此量測來架構猶豫模糊集的熵，我們也建立了新的熵原理，並證明所提出的熵滿足所有原理，進一步並將此熵做了更多的擴展，反應出其各自的優勢；更多特性的建立，還有更多的例子被用來將所提出的距離、相似度及熵與目前已有的方法做比較。在應用方面，我們將距離應用在多準則決策，將相似度應用在聚類上，我們也將所提出的熵應用在TOPSIS上，並擴展其為猶豫模糊TOPSIS，作為處理多準則決策的另一種重要工具。經由比較以及應用的例子，我們發現關於猶豫模糊集上所提出新的距離、相似度以及熵確實比已存在的方法具有其簡單性、直覺性、準確性以及應用性等優勢。

Distance, similarity and entropy play an indispensable role in almost every field of our daily life settings. Distance and similarity measures are widely used to differentiate between two sets or objects. While entropy measures the fuzziness in a fuzzy set. Different distance and similarity
measures have been proposed for hesitant fuzzy sets (HFSs) in the literature, but either they are in sufficient or not reflect desirable results. In this manuscript, the construction of new distance and similarity measures between HFSs based on Hausdorff metric is proposed. We first present a novel and simple method for calculating a distance between HFSs based on Hasudorff metric in a suitable and intuitive way. Two main features of the proposed approach are: (1) not necessary to add a minimum value, a maximum value or any value to the shorter one of hesitant fuzzy elements (HFEs) for extending it to the larger one of HFEs; and (2) no need to arrange HFEs either in ascending or descending order. This is because adding such values and arrangements of elements will not put any impact on final results. We then extend distance to similarity measure between HFSs. Next, measuring uncertainty for an HFS is computed by an amount of distinction between an HFS and its complement. Hausdorff metric is used to calculate a distance between an HFS and its complement which assists us to construct novel entropy of HFSs. An axiomatic definition of entropy measure for HFSs is also given in this dissertation. The proposed entropy is proved to satisfy all axioms. Furthermore, more generalizations of the proposed entropy allow us to onstruct
different entropy measures of HFSs which reflect that the closer of an HFS to its complement shows less distinction between them and produces the larger entropy measure of the HFS, and also the more distinction between them gives smaller amount of uncertainty. Furthermore, we claim some
properties and also several examples are presented to compare our proposed distance, similarity and entropy measures with existing methods. We apply the proposed distance of HFSs to multi-criteria decision making and the similarity measure of HFSs to clustering. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method is used to construct hesitant fuzzy (TOPSIS) based on the proposed entropy measure to solve multicriteria decision making problems.
Finally, expository examples are utilized to manifest simplicity, practicability and effectiveness of our proposed distance, similarity and entropies as compared to existing methods. The comparison results demonstrate that the proposed distance, similarity and entropy measures are much simpler, intuitive and better than most existing methods.

Table of Contents

摘要
I
Abstract II
Dedication IV
Acknowledgement V
Table of Contents VII
List of Figures X
List of Tables XI
Chapter 1 Introduction 1
1.1 Back ground and motivation 1
1.2 Intention of research 4
1.3 Dissertation organization 5
Chapter 2 Introduction to Hesitant Fuzzy Sets 6
2.1 Preliminaries of Hesitant Fuzzy sets 6
2.1.1 Hesitant fuzzy sets 6
2.2 Review of some existing distance and similarity measures for HFSs 8

8

2.3 Review of some existing entropies for hesitant fuzzy sets 10
2.4 Hasudorff metric 13
2.5 Hesitancy with the amount of distinction between an HFS and its complement 15
Chapter 3 New Distance and Similarity Measures of Hesitant Fuzzy Sets
Based on Hausdorff Metric with Applications to Multicriteria Decision Making
and Clustering 20
3.1 Introduction 20
3.2 Hausdorff metric for hesitant fuzzy sets 22
3.3 New distance and similarity measures between HFSs based on Hausdorff metric 23
3.4 Numerical examples and comparisons 29
3.5 Applications to multicriteria decision making and clustering 34
3.5.1 Application to multicriteria decision making 35
3.5.2 Application to clustering 41
3.6. Conclusions 43
Chapter 4 Entropy for Hesitant Fuzzy Sets Based on Hausdorff Metric with
Construction of Hesitant Fuzzy TOPSIS 44
4.1 Introduction 44
4.2. New entropy of hesitant fuzzy sets based on Hausdorff metric 47
4.3 Numerical examples and comparison 52
4. 4 Construction of a new hesitant fuzzy TOPSIS 55
4.5. Conclusions 66

9

Chapter 5 Summary 68
5.1 Conclusions 68
5.2 Future prospectives and challenges 70
References 71

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