對於 linear consecutive-k-out-of-n 和 linear consecutive-(r, s)-out-of-(m, n): F 系統最佳化的問題，目前已經有一些趨近的方法，這些方法都是以 reliability importance 為基礎；我們將使用遺傳演算法來求解 linear consecutive-k-out-of-n 和 linear consecutive-(r, s)-out-of-(m, n): F 系統最佳化的問題。
傳統的遺傳演算法在長時間之後，母體會有收斂到區域最佳解的情況，使得交配和突變的機制弱化。因此，我們在母體出現有可能收斂到區域最佳解時，對母體進行擾動，增加交配和突變的搜尋能力。

The linear consecutive-k-out-of-n and the linear consecutive-(r, s)-out-of-(m, n): F system both are enumerated minimum cuts easily. The optimization problem with linear consecutive-k-out-of-n: F system is discussed by many people. They proposed some heuristic methods for solving problem that are based on reliability importance. We propose the genetic algorithm for solving the optimization problem with linear consecutive-k-out-of-n: F system by introducing with reliability importance.
The population in a genetic algorithm may converge on a local optimal solution and the mechanisms of crossover and mutation are not very effective to search other solution when the algorithm runs long time. We propose a perturbation mechanism to let the population get diverge and solution search scope gets larger to escape the population converges on a local optimal solution.

Abstract i
List of Figures iv
List of Tables vi
List of Notations viii

Chapter 1 Introduction 1
1.1 Background and Motivation 1
1.2 Research Framework 6

Chapter 2 Literature Review 8
2.1 Consecutive-k-out-of-n: F System 8
2.1.1 Reliability Evaluation of Linear Consecutive-k-out-of-n: F System 8
2.1.2 The Optimal Assignment with Consecutive-k-out-of-n: F System 9
2.2 Reliability Importance 9
2.2.1 Birnbaum Importance and Invariant System 12
2.2.2 LK Heuristic 14
2.2.3 Pairwise Rearrangements 18
2.3 Genetic Algorithms 20

Chapter 3 Methodology 22
3.1 Problem Definition of One-dimensional System 22
3.2 Genetic Algorithms and Numerical Example (Programming Validation) 25
3.2.1 Chromosome Representation 26
3.2.2 Initial Population 26
3.2.3 Evaluation of Chromosome 29
3.2.4 Selection for crossover 31
3.2.5 Crossover 32
3.2.6 Selection for mutation 35
3.2.7 Mutation 36
3.2.8 Population Selection 38
3.3 Problem Definition of Two-dimensional System 42
3.3.1 Birnbaum Importance 45
3.3.2 Reliability Importance 47
3.4 Genetic Algorithms and Numerical Example (Programming Validation) 48
3.4.1 Chromosome Representation 49
3.4.2 Initial Population 50
3.4.3 Evaluation of Chromosome by Decomposition Approach 52
3.4.4 Evaluation of Chromosome by Crude Monte Carlo Simulation 54
3.4.5 Selection for crossover 56
3.4.6 Crossover 56
3.2.7 Selection for mutation 59
3.4.8 Mutation 59
3.4.9 Population Selection 61

Chapter 4 Computational Experiments and Analysis 63
4.1 Test Problem Generation and Solution Quality Measurement 63
4.2 Computational Experiments on One-dimensional System 65
4.2.1 Population Convergence and Perturbation 68
4.3 Computational Experiments on Two-dimensional System 72
4.3.1 The Gap between Crude Monte Carlo and Exact Evaluation 72
4.3.2 Result and Analysis 73

Chapter 5 Conclusion and Future 78

References 79
Appendix 83

[Birnb 1969] Birnbaum, Z. W. (1969). "On the importance of different components in a multi-component system." In P.R. Krishnaiah, (Ed.), Multivariate Analysis, Vol. 2, Academic Press, New York.
[Chian 1981] Chiang, D. T. and N., S. (1981). " Reliability of consecutive-k-out-of-n: F system." IEEE Transactions on Reliability 30: 87-89.
[Coit 1994] Coit, D. W. and S., Alice E. (1994). " Use of a genetic algorithm to optimize combinatorial reliability design problem." Proceedings of the Third IIE Research Conference, Atlanta: 467-472.
[Dengi 1997] Dengiz, B., Altiparmak, Fulya and Smith, Alice E. (1997). "Efficient optimization of all-terminal reliable networks using an evolutionary approach." IEEE Transactions on Reliability 46: 18-26.
[Derma 1982] Derman, C., Lieberman, G. J. and Ross, S. (1982). "On the consecutive-k-out-of-n: F system." IEEE Transactions on Reliability 31: 57-63.
[Fogel 1994] Fogel, D. (1994). "An Introduction to simulated evolutionary optimization." IEEE Transactions on Neural Networks 5: 3-14.
[Fu 1981] Fu, J. C. (1986). "Reliability of consecutive-k-out-of-n: F system with (k-1) step Markov dependence." IEEE Transactions on Reliability 35: 602-606.
[Gen 1993] Gen M., I., K. and Taguchi, T. (1993). "Reliability optimization problems: a novel genetic algorithm approach." Ashikaga Institute of Technology, Technical Report.
[Gen 1997] Gen, M. and Cheng, R. (1997). Genetic algorithms and engineering design, Ashikaga Institute of Technology Ashikaga, Japan.
[Griff 1986] Griffith, W. S. (1986). "On consecutive k-out-of-n failure systems and their generalizations.", A. P. Basu, (Ed.), Reliability and quality control, Elsevier (North-Holland) , New York.
[Holla 1975] Holland (1975). Adaptation in Natural and Artificial Systems, University of Michigan, Ann Arbor.
[Huang 2003] Huang, J. Z., MJ. and Fang, ZD. (2003). "Multi-state consecutive-k-out-of-n systems." IIE Transactions 35: 527-534.
[Hui 2005] Hui, K.-P., Bean, N. Kraetzl, M and Kroese (2005). "The Cross-Entropy method for network reliability estimation." Annals of Operations Research 134: 101-118.
[Hwang 1982] Hwang, F. K. (1982). "Fast solutions for consecutive-k-out-of-n: F system." IEEE Transactions on Reliability 31: 447-448.
[Hwang 1986] Hwang, F. K. and D., D. Z. (1986). " Optimal consecutive-2-out-of-n system." Mathematics of Operations Research 11: 187-191.
[Hwang 1989] Hwang, F. K. (1989). "Invariant permutations for consecutive-k-out-of-n: cycles." IEEE Transactions on Reliability 38: 65-67.
[Ishib 1994] Ishibuchi, H., Yamamoto, H., Murata, T. and Tanaka, H. (1994). "Genetic algorithms and neighborhood search algorithms for fuzzy flowsho scheduling problems." Fuzzy Sets and Systems 67: 81-100.
[Konto 1978a] Kontoleon, J. M. (1978). "Optimum allocation of components in a special 2-port network." IEEE Transactions on Reliability 27: 112-113.
[Konto 1978b] Kontoleon, J. M. (1978). "Analysis of a dynamic redundant system." IEEE Transactions on Reliability 27: 116-119.
[Konto 1980] Kontoleon, J. M. (1980). "Reliability determination of a r-successive-out-of-n: F system." IEEE Transactions on Reliability 29: 437.
[Koutr 1993] Koutras, M. V., Papadopoulos, George K. and Papastavridis, Stavros G. (1993). "Reliability of 2-dimensional consecutive-k-out-n: F systems." IEEE Transactions on Reliability 42: 658-661.
[Koutr 1994] Koutras, M. V., Papadopoulos, G.K. and Papastavridis, S.G. (1994). " Note: Pairwise rearrangements in reliability structures." Naval Research Logistics 41: 683-687.
[Kuo 1990] Kuo, W. and Z., M. (1990). " Design and performance analysis of consecutive-k-out-of-n structure." Naval Research Logistics 37: 203-230.
[Kuo 2000] Kuo, W. and P., V. Rajendra (2000). "An annotated overview of system-reliability optimization." IEEE Transactions on Reliability 49: 176-187.
[Kuo 2002] Kuo, W. and Lin, Fen-Hui (2002). "Reliability importance and invariant optimal allocation." Journal of Heuristics 8: 155-171.
[Kuo 2003] Kuo, W. and Z., Ming J. (2003). Optimal reliability modeling: principles and applications, John Wiley & Sons, Inc., Hoboken, New Jersey.
[Lashe 1990] Lasher, W. C. and S., Anthony A. (1990). "2-Dimensional consecutive-k-out-of-n: F models." IEEE Transactions on Reliability 39: 382-385.
[Levit 2003] Levitin, G. (2003). "Element availability importance in generalized k-out-of-r-from-n systems." IIE Transactions 35: 1125-1131.
[Lin 1984] Lin, G. D. and C., M. T. (1984). "Economical design of large consecutive-k-out-of-n: F systems." IEEE Transactions on Reliability 33: 411-413.
[Lin 1999] Lin, F.-H. and K., Way (1999). "Structure importance of consecutive-k-out-of-n systems." Operations Research Letters 25: 101-107.
[Malon 1984] Malon, D. M. (1984). " Optimal consecutive-k-out-of-n: F component sequencing." IEEE Transactions on Reliability 33: 414-418.
[Malon 1985] Malon, D. M. (1985). "Optimal consecutive-k-out-of-n: F component sequencing." IEEE Transactions on Reliability 34: 46-49.
[Miyak 1995] Miyakawa, M. and Y. H. (1995). "Reliability of a linear connected-(r, s)-out-of-(m, n)-F lattice system." IEEE Transactions on Reliability 44: 333-336.
[Papas 1987] Papastavridis, S. G. (1987). "The most important component in a consecutive-k-out-of-n: F system." IEEE Transactions on Reliability 36: 266-268.
[Pslla 1995] Psllakis, Z. M. (1995). " A simulation algorithm for computing failure probability of a consecutive-k-out-of-r-from-n: F system." IEEE Transactions on Reliability 44: 523-531.
[Sfaki 1991] Sfakianakis, M. E. and P., Stavros G. (1991). "Optimal-arrangement and importance of the components in a consecutive-k-out-of-r-from-n: F system." IEEE Transactions on Reliability 40: 277-279.
[Smith 1996] Smith, A. E. and C., David W. (1996). "Reliability optimization of series-parallel systems using a genetic algorithm." IEEE Transactions on Reliability 45: 254-260.
[Smith 1997] Smith, D. L. D. and A. E. (1997). "Heuristic optimization of network design considering all-terminal reliability." Proceedings of the 1997 Annual reliability and Maintainability Symposium, Philadelphia PA: 194-199.
[Yamam 2005a] Yamamoto, H. and A., T. (2005). "Evaluating methods for the reliability of a large 2-dimensional rectangular k-within-consecutive-(r, s)-out-of-(m, n): F system." Naval Research Logistics 52: 243-252.
[Yamam 2005b] Yamamoto, H. and M., M. (2005). "A recursive algorithm for the reliability of a circular connected-(r, s)-out-of-(m, n)-F lattice system." Computers and Industrial Engineering 48: 22-34.
[Yokot 1995] Yokota, T., Gen, M. and Ida, K. (1995). "System reliability of optimization problems with several failure modes by genetic algorithm." Japanese Journal of Fuzzy Theory and Systems 7: 117-185.
[Zuo 2000] Zuo, M. J. L., Daming and Wu, Yanhong (2000). " Reliability evaluation of combined k-out-of-n:F, consecutive-k-out-of-n:F, and linear connected-(r, s)-out-of-(m, n) F system structures." IEEE Transactions on Reliability 49: 99-104.