可靠度是評估系統運作績效的重要指標，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、設計及控制階段中的重要工具。近幾年，由於市場對於資訊量的需求愈來越大，多元狀態 (multi-state systems, MSSs) 的冗餘配置問題開始被廣泛的討論與應用。相比於傳統的二元狀態 (binary-state systems, BSSs) 冗餘配置問題，多態冗餘配置問題更能符合存在於現實生活中那些複雜且多元的系統。
然而，上述所提及的多態冗餘配置問題在設計與建構模型時為了計算上的簡化，仍舊限制組成元件的狀態數量，導致與現實情況還是有些許出入，為此失去普遍性與一般性。本研究為使此問題能夠更貼近真實情況，建構了新的多態冗餘配置問題：可維修式多態冗餘配置問題 (repairable redundancy allocation problem in multi-state systems, RRAP in MSSs)。此模型的設計能更貼近實況且更廣泛的被運用於多種常見且重要的系統，如電力系統、傳輸系統以及電腦伺服器系統等。可維修式多態冗餘配置問題相比於傳統的二態冗餘配置問題在計算方面是較為複雜的，傳統的可靠度運算方法並無法輕易地運用在此問題上。因此本研究使用了一般生成函數 (universal generating function, UGF)，藉由代數多項式的運算方法來求得系統所有的狀態與其發生的機率，然後再進一步計算系統的可靠度。除此之外，本問題不僅是個NP-hard問題，還是個非線性整數規劃的最佳化問題，容易陷入區域解。因此本研究採用一個簡單卻又強悍的演算法：簡化群體演算法 (simplified swarm optimization, SSO)。最後透過實作與驗證，將SSO演算法與基因演算法 (GA) 進行比較，結果顯示SSO具有強大的競爭力與優異的成效。

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 redundancy allocation problem in multi-state systems (RAP in MSSs) is the extension derived from the traditional redundancy allocation problem in binary-state systems (RAP in BSSs) for practical modeling in real life.
However, RAP in MSSs still has some restrictions that components have only two performances: perfect functionality and complete failure. Therefore, this paper formulates a new kind of RAP called repairable redundancy allocation problem in multi-state systems (RRAP in MSSs) so as to break the restrictions and generalize the previous problem. RRAP in MSSs designs are more closed to reality and widespread in many critical systems, such as power systems, transportation systems, and computing systems. Because of this, the computation of MSSs is more complicated than the BSSs’, and it is difficult to use the traditional technique to calculate system reliability. At this point, this research uses the universal generating function (UGF) to find out the whole system states and the corresponding probabilities with algebraic procedure, and then the system reliability is able to calculate. Moreover, RRAP in MSSs is not only an NP-hard problem, but also a nonlinear integer optimization problem. So this paper applies a novel algorithm called simplified swarm optimization (SSO) which is simple but powerful. Finally, the results obtained by SSO have been compared with that obtained from genetic algorithm (GA). Computational results show that the SSO is very competitive and effective in this problem.

CHINESE ABSTRACT I
ABSTRACT II
CONTENTS III
LIST OF FIGURES V
LIST OF TABLES VI
LIST OF ACRONYM AND NOTATIONS VII
CHAPTER 1 INTRODUCTION 1
1.1 BACKGROUND AND MOTIVATION 1
1.2 RESEARCH FRAMEWORK 3
CHAPTER 2 LITERATURE REVIEW 4
2.1 REDUNDANCY ALLOCATION PROBLEM (RAP) 4
2.2 REDUNDANCY ALLOCATION PROBLEM IN MULTI-STATE SYSTEMS 6
2.3 REPAIRABLE COMPONENT 8
2.4 UNIVERSAL GENERATING FUNCTION (UGF) 9
2.5 SIMPLIFIED SWARM OPTIMIZATION (SSO) 10
2.6 SUMMARY AND CONCLUSION 12
CHAPTER 3 MODEL FORMULATION 14
3.1 PROBLEM STATEMENT 14
3.2 THE PROBLEM ASSUMPTION 18
3.3 STRUCTURE REPRESENTATION 19
3.4 THE MATHEMATICAL MODEL 22
3.5 SUMMARY AND CONCLUSION 23
CHAPTER 4 RESEARCH METHOD 24
4.1 UNIVERSAL GENERATING FUNCTION 24
4.2 SIMPLIFIED SWARM OPTIMIZATION 29
4.3 SUMMARY AND CONCLUSION 32
CHAPTER 5 EXPERIMENTAL RESULTS AND ANALYSIS 33
5.1 THE EXPERIMENT STATEMENT 33
5.2 THE EXPERIMENT RESULTS 37
CHAPTER 6 CONCLUSIONS AND FUTURE REARCH 44
REFERENCES 46

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