可靠度是影響系統正常運作的重要因素，為提高系統的可靠度，冗餘配置問題 (redundancy allocation problem, RAP) 漸漸成為系統在初始規劃、設計及控制階段中重要的工具。近幾年，單層次的冗餘配置問題已相繼發展成多層次冗餘配置問題 (multi-level redundancy allocation problem, MRAP) 與複合多層次冗餘配置問題 (multiple MRAP, MMRAP) 以解決現實生活中更複雜的多層次系統。

然而，上述所提及的兩個多層次問題在設計與建構模型時仍有其侷限，導致運用於現實中存有諸多限制，因而失去普遍性與一般性。為突破問題求解上的侷限，本研究建構了新的多層次冗餘配置問題模型：一般化多層次冗餘配置問題 (general multi-level redundancy allocation problem, GMRAP)，此模型的設計可更廣泛地適用於多種關鍵的系統，如製造系統、電腦伺服器系統及軟體系統。GMRAP不僅是個NP-hard問題，還是個非線性整數規劃與階層最佳化問題，較傳統的RAP, MRAP和MMRAP複雜，容易陷入區域解。為此，本研究提出一個新穎的演算法：模組搜尋式簡化群體演算法 (simplified swarm optimization with modular search, SSO-MS)，此演算法是以SSO為核心透過模組式的尋優方式來處理GMRAP，這也是首次將SSO演算法運用於具有階層 (hierarchy) 結構的冗餘配置問題上。

透過實作與驗證，將SSO-MS演算法與基因演算法 (genetic algorithm, GA) 以及粒子群演算法 (particle swarm optimization, PSO) 進行比較，結果顯示本研究所提出的SSO-MS演算法在三種演算法中具有相當的競爭力與優異的成效。同時，本研究以實例演示並驗證GMRAP，結果顯示其可行性與一般性優於MRAP與MMRAP。

In recent decades, the redundancy allocation problem (RAP) is becoming an increasingly important tool in the initial stages of planning, designing, and controlling of systems. Moreover, the multi-level redundancy allocation problem (MRAP) and multiple multi-level redundancy allocation problem (MMRAP) are extensions derived from the RAP for practical modeling of real-life problems.

However, the two problems mentioned above still have restrictions that may be the best resolution in some special cases, but not in general. Therefore, this paper formulates a new kind of the MRAP called the general multi-level redundancy allocation problem (GMRAP) so as to break the restrictions and generalize previous problems. GMRAP designs are widespread in many critical systems, such as manufacturing systems, computing systems and software systems. GMRAP is not only NP-hard, but a nonlinear integer optimization problem with hierarchy. The complexity of GMRAP is much larger than the traditional RAP, MRAP and MMRAP. Furthermore, a novel algorithm called simplified swarm optimization with modular search (SSO-MS) is proposed to solve the GMRAP in this paper. To the best of our knowledge, this is the first attempt to use SSO for the hierarchal RAP instead of genetic algorithms.

Finally, the result obtained by SSO-MS has been compared with those obtained from genetic algorithms and particle swarm optimization algorithms. Computational results show that the proposed SSO-MS is very competitive and demonstrate the effectiveness and the practical viability of this approach and model.

CHINESE ABSTRACT I
ABSTRACT II
LIST OF FIGURES V
LIST OF TABLES VI
LIST OF ACRONYMS AND NOTATIONS VII
CHAPTER 1 INTRODUCTION 1
1.1 BACKGROUND AND MOTIVATION 1
1.2 RESEARCH FRAMEWORK 3
CHAPTER 2 LITERATURE REVIEW 5
2.1 REDUNDANCY ALLOCATION PROBLEM (RAP) 5
2.2 MULTI-LEVEL REDUNDANCY ALLOCATION PROBLEM (MRAP) 7
2.3 MULTIPLE MULTI-LEVEL REDUNDANCY ALLOCATION PROBLEM(MMRAP) 11
2.4 PARTICLE SWARM OPTIMIZATION (PSO) 12
2.5 SIMPLIFIED SWARM OPTIMIZATION (SSO) 13
2.6 SUMMARY AND CONCLUSION 14
CHAPTER 3 MODEL FORMULATION 16
3.1 PROBLEM STATEMENT 16
3.2 THE MATHEMATICAL MODEL 18
3.3 OBJECTIVE FUNCTION 20
3.4 SUMMARY AND CONCLUSION 22
CHAPTER 4 PROPOSED SSO-MS ALGORITHM 23
4.1 SOLUTION REPRESENTATION 23
4.2 CODING PROCEDURES 24
4.3 INITIALIZATION FOR UPPER BOUND 26
4.4 INITIAL POPULATION 29
4.5 UPDATE MECHANISM 31
4.6 PENALTY FUNCTION 34
CHAPTER 5 EXPERIMENTAL RESULTS AND ANALYSIS 36
5.1 THE EXPERIMENT STATEMENT 36
5.2 COMPARISONS OF METHODS UNDER DIFFERENT COST CONSTRAINTS 38
5.3 VERIFICATION AND VALIDATION OF MODELS 44
CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 47
REFERENCES 49
APPENDIX: SOLUTIONS FOR THE NUMERICAL EXAMPLE 57

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