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

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研究生：	吳光耀 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 Optimization in Permutation Problems and Fuzzy Linear Programs with Applications to a Case of Master Production Scheduling
指導教授：	王小璠 Hsiao-Fan Wang
口試委員:
學位類別：	博士 Doctor
系所名稱：	工學院 - 工業工程與工程管理學系 Department of Industrial Engineering and Engineering Management
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 approach, permutation problem, fuzzy linear programming, genetic algorithm, preference modeling, generalized linear fractional programming, master production scheduling
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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  Motivation    1
2  Objectives    2
3  Methodology    5
4  Framework and Organization    6
Chapter 2. Permutation Property and Uncertainty in a Case of
Master Production Scheduling    9
1  Preliminary    9
2  Problem Statement and Literature Review    11
2.1  The Role of Master Production Scheduling    11
2.2  Literature Review    12
2.3  A Case of MPS    14
3  Modeling of the Considered MPS Problem    18
3.1  Production Environment    18
3.2  The Proposed MPS Model (MS0)    19
3.3  Properties of Model (MS0)    22
3.4  Decomposition of Model (MS0)    25
4  Solution Approaches for Model (MS0) with Extraction and Resolution    27
4.1  Mixed-binary Programming Approach    27
4.2  Pattern-based Heuristic Approach    28
4.3  Extracting Sub-model (MS1) as a Permutation Problem    33
4.4  Resolving Sub-model (MS2) as a Fuzzy Linear Program    34
5  Summary    35
Chapter 3. Optimization in Permutation Problems based on
a Hybrid Genetic Algorithm    38
1  Preliminary    38
2  Problem Statement and Solution Review    40
2.1  Modeling of Permutation Optimization Problems as Model (PO)    40
2.2  Review of Solution Approaches to Model (PO)    43
2.3  Resolution of Solution Approaches to Model (PO)    45
3  A Survey of NS and GA for Model (PO)    46
3.1  Foundation of Neighborhood Search    46
3.2  Foundation of Genetic Algorithm    50
3.3  Foundation of a Hybrid GA    56
4  The Proposed Hybrid GA    57
4.1  Solution Framework    57
4.2  Functional Features of the NS    59
4.3  Functional Features of the GA    60
4.4  Measure of Performance and Termination Condition of the Hybrid GA    61
4.5  Summary of the Hybrid GA with Illustrations    62
5  Simulation Results    65
5.1  Evaluation and Comparative Studies of Efficiency    66
5.2  Evaluation and Comparative Studies of Effectiveness    70
5.3  Discussion of Performance Measure    72
5.4  Discussion of the Number of Evaluated Solutions    74
5.5  Application to the Quadratic Assignment Problem    76
6  Summary    78
Chapter 4. Optimization in Fuzzy Linear Programs based on
Preference Approach    80
1  Preliminary    80
2  Problem Statement and Solution Review    82
2.1  The Basic Concept of Fuzzy Programming Approach    82
2.2  Modeling of Symmetric Fuzzy Linear Programs as Model (FLO)    84
2.3  Review of Solution Approaches to Model (FLO)    85
2.4  Resolution of Solution Approaches to Model (FLO)    88
3  Preference Presentation in a Fuzzy Linear Inequality    90
3.1  Target Hyperplanes of a Fuzzy Linear Inequality    90
3.2  Validation of the Extremes of Coefficients    91
3.3  Membership Functions based on Target Hyperplanes    93
4  Preference Approach to Fuzzy Linear Optimization    96
4.1  Preference Constraint for the Objective with Target Levels    96
4.2  Representation of Model (FLO) with the Proposed Membership Function    98
4.3  Existing Solution Approaches to the Auxiliary Model    101
5  The Proposed Solution Procedure for the Auxiliary Model    104
5.1  Problem Reformulation and Karush-Kuhn-Tucker Condition    104
5.2  Basic Solutions and Optimality    106
5.3  The Proposed Solution Procedure    108
6  The Proposed Fuzzy Linear Optimization Approach with a Numerical Illustration    110
7  Numerical Results of the Application Case    117
7.1  The Efficiency of our Solution Procedure    118
7.2  The Applicability of our FLO Approach    119
8  Summary    120
Chapter 5. Conclusions and Future Research    123
1  Summary of the Results    123
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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