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研究生:吳光耀
研究生(外文):Kuang-Yao Wu
論文名稱:最佳排列問題與模糊線性規劃之研究及其在主生產排程案例之應用
論文名稱(外文):Optimization in Permutation Problems and Fuzzy Linear Programs with Applications to a Case of Master Production Scheduling
指導教授:王小璠王小璠引用關係
指導教授(外文):Hsiao-Fan Wang
學位類別:博士
校院名稱:國立清華大學
系所名稱:工業工程與工程管理學系
學門:工程學門
學類:工業工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:145
中文關鍵詞:最佳化方法排列問題模糊線性規劃基因演算法偏好式模式化一般化線性分式規劃主生產排程
外文關鍵詞:optimization approachpermutation problemfuzzy linear programminggenetic algorithmpreference modelinggeneralized linear fractional programmingmaster production scheduling
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  • 被引用被引用:2
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最佳化是一個決策過程,在其過程中要找出最好的方案以利達成我們所欲的目標。依據變數、關係式與評估準則的類型,最佳化問題可謂多樣化。在本論文中,我們考慮兩類源自於一個主生產排程案例的最佳化問題,它們分別稱之為排列問題最佳化(Optimization in permutation problems,簡稱PO)與模糊線性規劃最佳化(Optimization in fuzzy linear programs,簡稱FLO)。兩者皆在應用面與理論面有顯著的重要性。既是最佳化方法,本研究亦即要探討PO與FLO的模式結構與演算法。兩者的發展要點如下所述:
(1) 排列性質已被認知是一個在組合問題中常見且具挑戰的要項。本文列出一個PO的一般式以便能夠表達不同排列問題的結構性與複雜度。因為其計算之複雜,最近的研究乃轉向以基因演算法(Genetic algorithms)來解決此一問題。雖然基因演算法已有文獻證實能對整體求解空間的搜尋有助益,但此演算法卻缺乏微調的能力來獲得整體最佳解。因此,本研究整合基因演算法與鄰近搜尋法,進而發展一個混合式基因演算法以對PO問題求解。在我們分析此整合方式中,特別正視介於基因演算法與鄰近搜尋法的正反互補性。
(2) 在實際應用領域中,某些不確定性是非隨機的。針對這些固有的不確定性,模糊集合的概念於是被提出。當以主觀性隸屬函數來表達模糊的資料,模糊線性規劃即是融入個人偏好看法來擴增線性規劃的既有能力。雖然一些研究已經發展模糊線性規劃的最佳化方法,但尚未有研究處理有關如何明確地表達個人的偏好性或整體係數的寬放性並含求解程序。因此,本研究中我們發展一個以偏好方式(Preference approach)為基礎的FLO模式,此模式能融入個人的樂觀或悲觀態度,以及容許所有係數給予寬放。因為其非線性模式的複雜因素,耗時的求解亦是發展FLO的一項核心議題。無論如何,藉由研究其內隱的線性特質,我們發展一個以基底轉換的演算法來求解我們所提的FLO問題。
本研究結果已經顯示能有效率地應用在此主生產排程案例中。就此案例的實驗結果顯示我們所提的混合式基因演算法優於其他被比較的方法,尤其是處理較大維度的問題上。同時,本結果也證實了應用偏好方式到模糊最佳化的必要,以及揭示了所提基底式演算法相較於Dinkelbach式演算法與二分逼近法的優越性。
Optimization is a process of decision making which aims to finding the best alternative in order to achieve the goals as concerned. Regarding the kinds of variables, relations and the performance criterion, optimization problems are manifold. In this dissertation, we consider two kinds of optimization problems motivated from a case of Master Production Scheduling (MPS), namely optimization in permutation problems (in short, permutation optimization (PO)) and optimization in fuzzy linear programs (in short, fuzzy linear optimization (FLO)). Both optimization problems are of significance in application and in theorem. As optimization approaches, this study is meant to investigate the model structures and algorithms for PO, and for FLO. Both developments are outlined below:
(1) Permutation property has been recognized as a common but challenging feature in combinatorial problems. We express a general form of PO, which is capable of presenting the structures and complexity of various permutation problems. Because of their complexity, recent research has turned to genetic algorithms for solving such problems. Although genetic algorithms have been proven to facilitate the entire space search, they lack in fine-tuning capability for obtaining the global optimum. Therefore, in this study a hybrid genetic algorithm is developed by integrating both evolutional and neighborhood searches for PO. On the analysis of such hybridization, the pros and cons compensation between genetic algorithm and neighborhood search are particularly addressed.
(2) In real-world applications, certain kinds of uncertainty are not stochastic. For intrinsic uncertainty, the concept of fuzzy sets was suggested. With these fuzzy input data that are presented by subjective membership functions, fuzzy linear programming is to enhance the capability of linear programs by individual’s perception. Although a number of researches have focused on the development of optimization in fuzzy linear programs, how to explicitly present one’s preference has never been addressed, neither the overall tolerance, and solution procedure. In this study, we developed an FLO model based on preference approach, which is capable of incorporating one’s optimistic or pessimistic attitude as well as admitting tolerances of all coefficients. Because of its complex with a non-linear model, the time consumed in finding a compromise solution is also a core issue for the development of FLO. However, by investigating its inherited linear character, we elaborate a basis-based algorithm for solving the proposed FLO problem.
The results of this study have shown to be effectively applicable for the case of MPS. Experimental results of the MPS problem indicate that the hybrid genetic algorithm outperforms the other tested methods, in particular for larger scaled problems. Moreover, this implementation verifies the need of applying our proposed preference approach for the fuzzy optimization, and signifies the superiority of the proposed basis-based algorithm comparing to the Dinkelbach-type-2 algorithm and the bisection procedure.
中文摘要 I
Abstract II
Acknowledgements IV
Table of Contents V
List of Tables VIII
List of Figures IX
List of Notations X
List of Abbreviations XII
Chapter 1. Introduction 1
1.1 Motivation 1
1.2 Objectives 2
1.3 Methodology 5
1.4 Framework and Organization 6
Chapter 2. Permutation Property and Uncertainty in a Case of
Master Production Scheduling 9
2.1 Preliminary 9
2.2 Problem Statement and Literature Review 11
2.2.1 The Role of Master Production Scheduling 11
2.2.2 Literature Review 12
2.2.3 A Case of MPS 14
2.3 Modeling of the Considered MPS Problem 18
2.3.1 Production Environment 18
2.3.2 The Proposed MPS Model (MS0) 19
2.3.3 Properties of Model (MS0) 22
2.3.4 Decomposition of Model (MS0) 25
2.4 Solution Approaches for Model (MS0) with Extraction and Resolution 27
2.4.1 Mixed-binary Programming Approach 27
2.4.2 Pattern-based Heuristic Approach 28
2.4.3 Extracting Sub-model (MS1) as a Permutation Problem 33
2.4.4 Resolving Sub-model (MS2) as a Fuzzy Linear Program 34
2.5 Summary 35
Chapter 3. Optimization in Permutation Problems based on
a Hybrid Genetic Algorithm 38
3.1 Preliminary 38
3.2 Problem Statement and Solution Review 40
3.2.1 Modeling of Permutation Optimization Problems as Model (PO) 40
3.2.2 Review of Solution Approaches to Model (PO) 43
3.2.3 Resolution of Solution Approaches to Model (PO) 45
3.3 A Survey of NS and GA for Model (PO) 46
3.3.1 Foundation of Neighborhood Search 46
3.3.2 Foundation of Genetic Algorithm 50
3.3.3 Foundation of a Hybrid GA 56
3.4 The Proposed Hybrid GA 57
3.4.1 Solution Framework 57
3.4.2 Functional Features of the NS 59
3.4.3 Functional Features of the GA 60
3.4.4 Measure of Performance and Termination Condition of the Hybrid GA 61
3.4.5 Summary of the Hybrid GA with Illustrations 62
3.5 Simulation Results 65
3.5.1 Evaluation and Comparative Studies of Efficiency 66
3.5.2 Evaluation and Comparative Studies of Effectiveness 70
3.5.3 Discussion of Performance Measure 72
3.5.4 Discussion of the Number of Evaluated Solutions 74
3.5.5 Application to the Quadratic Assignment Problem 76
3.6 Summary 78
Chapter 4. Optimization in Fuzzy Linear Programs based on
Preference Approach 80
4.1 Preliminary 80
4.2 Problem Statement and Solution Review 82
4.2.1 The Basic Concept of Fuzzy Programming Approach 82
4.2.2 Modeling of Symmetric Fuzzy Linear Programs as Model (FLO) 84
4.2.3 Review of Solution Approaches to Model (FLO) 85
4.2.4 Resolution of Solution Approaches to Model (FLO) 88
4.3 Preference Presentation in a Fuzzy Linear Inequality 90
4.3.1 Target Hyperplanes of a Fuzzy Linear Inequality 90
4.3.2 Validation of the Extremes of Coefficients 91
4.3.3 Membership Functions based on Target Hyperplanes 93
4.4 Preference Approach to Fuzzy Linear Optimization 96
4.4.1 Preference Constraint for the Objective with Target Levels 96
4.4.2 Representation of Model (FLO) with the Proposed Membership Function 98
4.4.3 Existing Solution Approaches to the Auxiliary Model 101
4.5 The Proposed Solution Procedure for the Auxiliary Model 104
4.5.1 Problem Reformulation and Karush-Kuhn-Tucker Condition 104
4.5.2 Basic Solutions and Optimality 106
4.5.3 The Proposed Solution Procedure 108
4.6 The Proposed Fuzzy Linear Optimization Approach with a Numerical Illustration 110
4.7 Numerical Results of the Application Case 117
4.7.1 The Efficiency of our Solution Procedure 118
4.7.2 The Applicability of our FLO Approach 119
4.8 Summary 120
Chapter 5. Conclusions and Future Research 123
5.1 Summary of the Results 123
5.2 Conclusions with Further Research 125
References 128
Appendix 137
A. Solution Procedures of the Proposed Hybrid GA 137
B. An Introduction of Fuzzy Sets and Fuzzy Numbers 139
C. Collections of the Proofs Related to Chapter 4 140
D. Procedures of both Bisection and Dinkelbach-type-2 Algorithms 143
E. The Fuzzy Model corresponding to Model (MS2) 145
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