本論文的主要目的是建立一不完美維護模式，結合在虛擬年齡與利用率下之最佳維護策略，而使得總成本最小。
本論文以Liu et al. (1995)的概念為基礎，除了虛擬年齡外，系統的利用率也會影響系統的失敗率，其中利用率是一個隨機過程且具有馬可夫性質，所以是一馬可夫鏈。在此假設下，我們考慮在一有限時間內運用動態規劃法求得最佳的維護區間，並且依據不同的利用率探討維護區間的變化。

The main objective of this thesis is to construct an imperfect maintenance model, which combines virtual age and utilization, for determining the optimal policy to minimize the total cost.
The concept of our virtual age function is based on Liu, Makis and Jardine (1995). However, we add a variable, utilization rate here which is different from the original concept. The utilization rate is a stochastic process with Markov property, so the stochastic process is a Markov chain. The restoration degree and utilization of the system are both assumed to influence the failure intensity following a virtual age process. In this situation, we finally use dynamic programming to find the optimal overhaul intervals with finite planning horizon, and show how it varies with alternation of utilization.

[1] Anderson J. A. and A. Senthilselvan (1980), “Smooth Estimates for the Hazard Function,” Journal of the Royal Statistical Society. Series B (Methodological),42 (3),322-327
[2] Bandopadhyaya, A. (1994), “An Estimation of the Hazard Rate of Firms Under Chapter 11 Protection,” The Review of Economics and Statistics 76 (2), 346-350
[3] Chun, Y. H. (1992), “Optimal number of periodic preventive maintenance operations under warranty,” Reliability Engineering and System Safety 37 (3), 223–225.
[4] Cox, D. R. (1972), “Regression models and life-tables,” Journal of the Royal Statistical Society. Series B (Methodological), 34 (2), 187-220.
[5] Kijima, M. (1989), “Some Results for Repairable Systems with General Repair,” Journal of Applied Probability, 26, 89-102.
[6] Lie, C.H. and Y. H. Chun. (1986), “An algorithm for preventive maintenance policy,” IEEE Transactions on Reliability R- 35/l, 71-75.
[7] Liu, X.G., V. Makis and A.K.S. Jardine (1995), “A Replacement Model with Overhauls and Repairs,” Naval Research Logistics, 42, 1063-1079.
[8] Nakagawa, T. (1984), “Optimal policy of continuous and discrete replacement with minimal repair at failure,” Naval Research Logistics Quarterly 31 (4), 543–550.
[9] Nakagawa, T. (1979), “Optimum Policies when Preventive Maintenance is Imperfect,” IEEE Transactions on Reliability R- 28/4,331-332.
[10] Nakagawa, T. (1981a), “A summary of periodic replacement with minimal repair at failure,” Journal of the Operations Research Society of Japan, 24, 213–228.
[11] Nakagawa, T. (1981b), “Modified periodic replacement with minimal repair at failure,” IEEE Transactions on Reliability R 30, 165–168.
[12] Nakagawa, T. (1986), “Periodic and Sequential Maintenance Policies,” Journal of Applied Probability, 23, 536-542.
[13] Nakagawa, T. (1988), “Sequential Imperfect Preventive Maintenance Policies,” IEEE Trans Reliability, 37, 295–8.
[14] Pham, H. and H. Wang (1996), “Imperfect Maintenance,” European Journal of Operational Research, 94, 425-438.
[15] Pham, H. and H. Wang (1999), “Some maintenance models and availability with imperfect maintenance in production systems,” Annals of Operations Research 91, 305–318.
[16] Prentice, R. L. (1973), “Exponential survivals with censoring and explanatory variables,” Biomnetrika, 60(2), 279-288
[17] Sheu, S., W. S. Griffith and T. Nakagawa, (1995), “Extended optimal replacement model with random minimal repair costs,” European Journal of Operational Research 85, 636–649.
[18] Sheu, S., C. Kuo and T. Nakagawa, (1993), “Extended optimal age replacement policy with minimal repair,” RAIRO: Recherche Operationnelle 27 (3), 337–351.
[19] Wang, H. (2002), “A Survey of Maintenance Policies of Deteriorating Systems,” European Journal of Operational Research”, 139, 469–489.