為延續企業的競爭優勢，業者亟需能符合成本效益、且能隨時保持高性能狀態的作業系統，因此適當的對系統進行預防維修與置換，將可有效的降低成本以及防止故障的發生。本論文將提出兩種廣義的不完美預防維修與年齡置換模型加以分析。
第一種是探討廣義的定期不完美預防維修模型，本模型將分三個方向加以討論。當系統失效或遭受衝擊時，系統的失效可分成兩種類型，而此時系統進行最小修理或維修，視嚴重情況而定。當經過多次維修週期後，系統即進行置換。第一個方向將考慮實際年齡函數與固定的失效型式機率。第二個方向將採用表現在失效率和有效年齡上的改進因子，且失效型式與衡擊的次數有關。第三個方向則同樣考慮改進因子，但失效型式則與系統的年齡有關。
另一種是探討在累積修理成本限制下的最佳年齡置換模型。系統遭受到衝擊係遵循非均勻卜瓦松過程，而系統一旦遭受衝擊將產生兩種失效類型。系統進行置換或最小修理，將決定於失效的嚴重程度、所累積修理成本與系統年齡等情況。本研究將考慮失效的型式與衝擊的次數有關。
針對上述四個模型策略，分別推導出相關的長期每單位成本函數，並探討使成本函數最小化的最佳解存在時之條件。本研究可作為類似系統或產品在訂定維修置換策略時的參考。

To gain and remain competitive advantage, manufacturers require a cost-effective system for maintaining its production machinery in peak operation condition. Preventive maintenances and replacements enable us to reduce the operating cost and prevent the occurrence of system failure. In this research we propose two categories of generalized imperfect preventive maintenance and age-replacement models.
The first category is a generalized periodic imperfect preventive maintenance model and is discussed in three directions. As failures or shocks occur, the system experiences two types of failures, and it is maintained or repaired minimally depending on the order of severity. After several maintained periods, the system is replaced. In the first direction, the virtual age function and constant probability of failure types are discussed. Second, the concept of improvement factors in the hazard-rate function and effective age is adopted and the shock number-dependent failure type is considered. Last, we still use improvement factors, whereas the age-dependent failure type is presented.
The second category is an optimal age-replacement model based on a cumulative repair-cost limit policy. A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As these shocks occur, the system experiences two types of failures, and it is replaced or repaired minimally depending on the order of severity, the accumulated repair cost, and age. In this study, the shock number-dependent failure type is considered.
For each of the above four models, the unique optimal solution which minimizes the long-run expected cost per unit of time is determined respectively. This research can be a reference for setting maintenance and replacement policies to analogous system and product.

[1] Abdel-Hameed, M. S., “An imperfect maintenance model with block replacement,” Applied Stochastic Models and Data Analysis, Vol.3, pp.63-72 (1987).
[2] Abdel-Hameed, M. S. and F. Proschan, “Non Stationary shock models,” Stochastic Processes and their Application, Vol.1, pp.383-404 (1973).
[3] Bai, D. S. and W. Y. Yun, “An age replacement policy with minimal repair cost limit,” IEEE Transactions on Reliability, Vol.R-35, pp.452-454 (1986).
[4] Barlow, R. E. and L. C. Hunter, “Optimal preventive maintenance policies,” Operations Research, Vol.8, pp.90-100 (1960).
[5] Beichelt, F., “A general preventive maintenance policy,” Mathematische Operations Forschung und Statistik, Vol.7, no.6, pp.927-932 (1976).
[6] Beichelt, F., “A replacement policy based on limits for repair cost rate,” IEEE Transactions on Reliability, Vol.R31, pp.401-403 (1982).
[7] Beichelt, F., “A unifying treatment of replacement policies with minimal repair,” Naval Research Logistics, Vol.40, pp.51-67 (1993).
[8] Beichelt, F., “A replacement policy based on limiting the cumulative maintenance cost,” International Journal of Quality and Reliability Management, Vol.18, pp.76-83 (2001).
[9] Beichelt, F. and K. Fisher, “On a basic equation of reliability theory,” Microelectrons and Reliability, Vol.19, pp.367-369 (1979).
[10] Berg, M., M. Bievenu, and R. Cléroux, “Age replacement policy with age-dependent minimal repair,” INFOR, Vol.24, pp.26-32 (1986).
[11] Berg, M. and R. Cléroux, “A marginal cost analysis for an age replacement policy with minimal repair,” INFOR, Vol.20, pp.258-263 (1982).
[12] Block, H. W., W. S. Borges and T. H. Savits, “Age dependent minimal repair,” Journal of Applied Probability, Vol.22, pp.370-385 (1985).
[13] Block, H. W., W. S. Borges and T. H. Savits, “A general age replacement model with minimal,” Naval Research Logistics, Vol.35, pp.365-372 (1988).
[14] Boland, P. J., “Periodic replacement when minimal repair costs vary with time,” Naval Research Logistics Quarterly, Vol.29, pp.541-546 (1982).
[15] Boland, P. J. and F. Proschan, “Periodic replacement with increasing minimal repair costs at failure,” Operations Research, Vol.30, pp.1183-1189 (1982).
[16] Boland, P. J. and F. Proschan, “Optimum replacement of a system subject to shocks,” Operations Research, Vol.31, pp.697-704 (1983).
[17] Brown, M. and F. Proschan, “Imperfect repair,” Journal of Applied Probability, Vol.20, pp.851-859 (1983).
[18] Chung, K. J. and S. D. Lin, “A simple procedure to compute the optimal repair cost under minimal repair,” International Journal of Quality and Reliability Management, Vol.13, pp.77-81 (1996).
[19] Cléroux, R., S. Dubuc and C. Tilquin, “The age replacement problem with minimal repair and random repair costs,” Operations Research, Vol.27, pp.1158-1167 (1979).
[20] Dohi, T., N. Kaio and O. Osaki, “A graphical method to repair-cost limit replacement policies with imperfect repair,” Mathematical and Computer Modeling, Vol.31, pp.99-106 (2000).
[21] Dohi, T., N. Kaio and O. Osaki, “A new graphical method to estimate the repair-time limit with incomplete repair and discounting,” Computers and Mathematics with Application, Vol.46, pp.999-1007 (2003).
[22] Drinkwater, R. W. and N. A. J. Hastings, “A economic replacement model,” Operational Research Quarterly, Vol.18, pp.121-138 (1967).
[23] Esary, J. D., A. W. Marshall and F. Proschan, “Shock models and wear process,” The Annals of Probability, Vol.1, pp.627-649 (1973).
[24] Hastings, N. A. J., “The repair limit replacement method,” Operational Research Quarterly, Vol.20, pp.337-349 (1969).
[25] Kapur, P. K., R. B. Garg and N. L. Butani, “Some replacement policies with minimal repairs and repair cost limit,” International Journal of System Science, Vol.20, pp.267-272 (1987).
[26] Kijima, M., “Some results for repairable systems with general repair,” Journal of Applied Probability, Vol.26, pp.89-102 (1989).
[27] Kijima, M., Morimura H., and Suzuki Y., “Periodical replacement problem without assuming minimal repair,” European Journal of Operational Research, Vol.69, pp.75-82 (1988).
[28] Levitin, G. and A. Lisnianski, “Optimization of imperfect preventive maintenance for multi-state systems,” Reliability Engineering and System Safety, Vol.67, pp.193-203 (2000).
[29] Liao, C. J. and W. J. Chen, “Single-machine scheduling with periodic maintenance and nonresumable jobs,” Computers and Operations Research, Vol.30, pp.1335-1347 (2003).
[30] Lie, C. H. and Y. H. Chun, “An algorithm for preventive maintenance policy,” IEEE Transactions on Reliability, Vol.35, pp.71-75 (1986).
[31] Lin, D., M. J. Zuo and R. C. M. Yam, “General sequential imperfect preventive maintenance models,” International Journal of Reliability, Quality and Safety Engineering, Vol.7, pp.253-266 (2000).
[32] Lotka, A. J., “A contribution to the theory of self-renewing aggregates with special reference to industrial replacement,” Annals of Mathematical Statistics, Vol.10, pp.1-25 (1939).
[33] Malik, M. A. K., “Reliable preventive maintenance scheduling,” AIIE Transactions, Vol.11, pp.221-228 (1979).
[34] Murthy, D. N. P. and D. G. Nguyen, “Optimal age-policy with imperfect preventive maintenance,” IEEE Transactions on Reliability, Vol.30, pp.80-81 (1981).
[35] Nakagawa, T., “Optimum policies when preventive maintenance is imperfect,” IEEE Transactions on Reliability, Vol.28, pp.331-332 (1979).
[36] Nakagawa, T., “A summary of imperfect preventive maintenance policies with minimal repair,” RAIRO Operations Research, Vol.14, pp.249-255 (1980).
[37] Nakagawa, T., “Periodic and sequential preventive maintenance policies,” Journal of Applied Probability, Vol.23, pp.536-542 (1986).
[38] Nakagawa, T., “Sequential imperfect preventive maintenance policies,” IEEE Transactions on Reliability, Vol.37, pp.295-298 (1988).
[39] Nakagawa, T. and K. Yasui, “Periodic-replacement models with threshold levels,” IEEE Transactions on Reliability, Vol.40, pp.395-397 (1991).
[40] Nakagawa, T. and M. Kowada, “Analysis of a system with minimal repair and its application to replacement policy,” European Journal of Operational Research, Vol.12, pp.176-182 (1983).
[41] Nakagawa, T. and S. Osaki, “The optimum repair limit replacement policies,” Operational Research Quarterly, Vol.25, pp.311-317 (1974).
[42] Nguyen, D. G. and D. N. P. Murthy, “A note on the repair limit replacement policy,” Journal of the Operational Research Society, Vol.31, pp.1103-1104 (1980).
[43] Nguyen, D. G. and D. N. P. Murthy, “Optimal preventive maintenance policies for repairable systems,” Operations Research, Vol.29, pp.1181-1194 (1981).
[44] Park, K. S., “Cost limit replacement under minimal repair,” Microelectronics and Reliability, Vol.23, pp.347-349 (1979).
[45] Park, K. S., “Pseudo dynamic cost limits replacement model under minimal repair,” Microelectronics and Reliability, Vol.25, pp.573-579 (1985).
[46] Pham, H. and H. Wang, “Imperfect maintenance,” European Journal of Operational Research, Vol.94, pp.425-438 (1996).
[47] Puri, P. S. and H. Singh, “Optimum replacement of a system subject to shocks: A mathematical lemma,” Operations Research, Vol.34, pp.782-789 (1986).
[48] Ross, S.M., “Applied probability models with optimization applications,” Holden-Day: San Francisco (1970).
[49] Savage, I. R., “Cycling,” Naval Research Logistics, Vol.3, pp.163-175 (1956).
[50] Savits, T. H., “Some multivariate distributions period from a non-fatal shock model,” Journal of Applied Probability, Vol.25, pp.383-390 (1988).
[51] Seo, J. H. and D. S. Bai, “An optimal maintenance policy for a system under periodic overhaul,” Mathematical and Computer Modelling, Vol.39, pp.373-380 (2004).
[52] Sheu, S. H., “A generalized block replacement policy with minimal repair and general random repair costs for a multi-unit system,” Journal of the Operational Research Society, Vol. 42, pp. 331-341 (1991).
[53] Sheu, S. H., “Optimal block replacement policy with multiple choice at failure,” Journal of Applied Probability, Vol.29, pp.129-141 (1992).
[54] Sheu, S. H., “A modified block replacement policy with two variables general and random minimal repair cost,” Journal of Applied Probability, Vol.33, pp.557-572 (1996).
[55] Sheu, S. H., “A generalized age and block replacement of a system subject to shocks,” European Journal of Operational Research, Vol.108, pp.345-362 (1998).
[56] Sheu, S. H. and T. H. Chang, “Generalized sequential preventive maintenance policy of a system subject to shocks,” International Journal of Systems Science, Vol.33, pp.267-276 (2002).
[57] Sheu, S. H. and W. S. Griffith, “Optimal number of minimal repairs before replacement of a system subject to shocks,” Naval Research Logistics Quarterly, Vol.43, pp.319-333 (1996).
[58] Sheu, S. H., W. S. Griffith and T. Nakagawa, “Extended optimal replacement model with random minimal repair costs,” European Journal of Operational Research, Vol.85, pp.636-649 (1995).
[59] Sheu, S. H. and C. T. Liou, “A generalized sequential preventive maintenance policy for repairable system with general random repair costs,” International Journal of Systems Science, Vol.26, pp.681-690 (1995).
[60] Tilquin, C. and R. Cléroux, “Periodic replacement with minimal repair at failure and general cost function,” Journal of Statistical Computing and Simulation, Vol.4, pp.63-67 (1975).
[61] Weiss, G., “On the theory of replacement of machinery with a random failure time,” Naval Research Logistics, Vol.3, pp.279-293 (1956).