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研究生:王佩淳
研究生(外文):Pei-Chun Wang
論文名稱:運用確定性全域最佳化方法求解工程與管理問題
論文名稱(外文):A Deterministic Global Optimization Approach for Engineering and Management Problems
指導教授:蔡榮發教授
口試委員:郭人介教授張錦特教授邱志洲教授翁頌舜教授
口試日期:2012-06-20
學位類別:博士
校院名稱:國立臺北科技大學
系所名稱:工商管理研究所
學門:商業及管理學門
學類:其他商業及管理學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:英文
論文頁數:76
中文關鍵詞:全域最佳化凸化線性化管理科學
外文關鍵詞:Global optimizationConvexificationLinearizationManagement science
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確定性的全域最佳化演算法普遍應用在各種工程與管理的問題,運用最佳化演算法在管理方面可量化並結構化獲得最佳決策,而在工程最佳化方面要求絕對的精確度,因此全域最佳解成為求解的重點。然而實際的工程與管理問題所建構的數學模型大部份皆為非凸規劃(nonconvex programming)模型,無法以簡單的最佳化演算法求得全域最佳解。若能將非凸規劃轉換為凸規劃(convex programming),便能以簡單的確定性最佳化演算法求得全域最佳解。因此,許多凸化(convexification)方法皆探討如何將非凸函數有效率的轉換為凸函數以確保獲得全域最佳解。然而為確保求解品質,模型中的非凸函數通常需要運用不同的凸化方法來轉換,此外,由函數轉換而產生的非線性(nonlinear)限制式需透過線性化(linearization)來達成獲得全域最佳解的目的。因此,本研究運用全域最佳化演算法求解不同的工程與管理問題以獲得全域最佳解,透過整合運用不同的凸化方法,同時將模型中包括正(posynomial)與非正(signomial)的非凸函數轉換為凸函數,接著,利用逐段線性化(piecewise linearization)方法將非線性限制式線性化,使模型成為一凸規劃模型達成確保獲取全域最佳解的目的。本研究主要具有以下優點,首先,全域最佳化演算法在工程與管理的應用方面能夠獲得滿足各條件下的精確解答以及最佳決策。第二是相較於啟發式演算法,本研究所探討的最佳化演算法可保證其解為全域最佳解。第三,本研究所探討的最佳化演算法能夠以最少的限制式與二元變數進行轉換,相較於其他確定性演算法為最有效率之演算法。第四,本研究運用多重解演算法獲取最佳的替代策略,提昇最佳化管理應用方面的決策彈性,使企業具備快速應變的能力。

Deterministic global optimization has been applied in many applications of engineering and management area. Global optimization enables management problems to obtain an optimal decision, and applications to engineering problems usually require very precise solution. Thus the global optimal solution of the applications plays critical roles. These real-world optimization problems lead to nonconvex problems. In most of the previously study on engineering and management problems involving nonconvexties, the emphasis has not been on global optimization since it is difficult to globally optimize such models. However, most nonconvex problems cannot be dealt with by conventional optimization algorithms to guarantee global optimality. Therefore, deterministic optimization approaches have been developed for convexifying the nonconvex function to obtain globally optimal solutions. This dissertation utilizes deterministic optimization approach to find the global optimum of various engineering and management problems. The presented deterministic optimization approach transforms a nonconvex program into a convex program by convexification and linearization techniques and is thus guaranteed to reach a global optimum. The advantages of this study are summarized as follows. First, deterministic global optimization applications to engineering and management problems can obtain precise solution and optimal decision. Second, the obtained solution is guaranteed to reach a global optimum thus better than heuristic algorithms. Third, compared with other deterministic methods, the presented method utilizes less additional binary variables and constraints to reformulate the problem. Then the computational efficiency can be improved greatly. Fourth, the presented approach finds alternative optimal solutions to enhance the flexibility of the decision making.

摘 要 i
ABSTRACT ii
誌 謝 iii
CONTENTS iv
LIST OF FIGURES vi
LIST OF TABLES vii
Chapter 1 Introduction 1
1.1 Research Background 1
1.2 Research Motivation and Purpose 3
1.3 Framework of the Dissertation 4
Chapter 2 Deterministic Global Optimization Approaches 7
2.1 Convexity 7
2.1.1 Convex Sets 7
2.1.2 Convex Functions 8
2.2 Convex Underestimation Techniques 11
2.3 Convexification Techniques 14
2.4 Linearization Techniques 16
2.5 Multiple Solutions 18
Chapter 3 Application to Rectangular Packing Problems 20
3.1 Introduction of Rectangular Packing Problem 20
3.2 Problem Formulation 22
3.3 Reformulation of Rectangular Packing Problems 24
3.4 Numerical Examples 27
3.5 Summary 31
Chapter 4 Application to Engineering Design Problems 33
4.1 Introduction of Engineering Design Problems 34
4.2 Convexification and Linearization Techniques for MINLP 35
4.3 Engineering Design Problems 36
4.4 Summary 45
Chapter 5 Application to Cooperative Alliance Problems 46
5.1 Introduction 47
5.2 Optimal Expansion of Incorporating Multilevel Competence sets 51
5.3 Multiple Solutions of Cooperative Alliance Problem 55
5.4 Numerical example of cooperative alliance 55
5.5 Summary 62
Chapter 6 Discussion and Conclusions 63
REFERENCES 66

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