Abstract
In condition-based maintenance (CBM) planning, collected information from system condition monitoring is the basis of making decision about conducting the maintenance and repair activities. Recently, ample number of studies has been conducted in CBM field especially, in control-limit policy. In control-limit policy, using proportional Hazards model and results of monitoring system condition, one can estimate hazard rate function and its condition’s transition probability matrix. Then, considering replacement costs, optimal control-limit can be determined minimizing the average cost in the long run. The presented model considers repair policy and their implementation cost, and the assumptions of repair during interval inspection is ignored. Then, a model is presented to determine the optimal control-limit and the best repair policy, in which the average total cost per unit time in the long-run, is minimized. At the end, a numerical example is illustrated.
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Abbreviations
- \(T\) :
-
Random variable that represents the failure time
- \(S\) :
-
Condition space of \(\{Z(t)\}\) process
- \(Z(t)\) :
-
Failure diagnose variable in time \(t\)
- \(h(t, Z(t))\) :
-
Hazard rate function associated with time \(t\) and condition \(Z(t)\)
- \(\beta \) :
-
Shape parameter in Weibull distribution
- \(\eta \) :
-
Scale parameter in Weibull distribution
- \(\gamma \) :
-
Failure diagnose parameter represents the importance of \(Z(t)\) in hazard rate
- \(C\) :
-
Preventive replacement cost
- \(C+K\) :
-
Performing each replacement due to the failure
- \(P\) :
-
Transition probability matrix
- \(P_{ij}\) :
-
Probability of that the system condition in time \(t+\Delta \) be \(j\) provided that in time \(t\) be \(i\)and failure occurs after time \(t+\Delta \)
- \(P_{i-a_i r} (k)\) :
-
Transition probability matrix from \(i-a_i \) to\(r\) in the interval \(k\Delta \) and \((k+1)\Delta \) unit time difference and certain fixed repair policy
- \(a_i\) :
-
Kind of recommended repair that is proper for \(i\hbox {th}\) deterioration.
- \(A(i)\) :
-
Set of all recommended repair that is proper for \(i\hbox {th}\) deterioration
- \(X\) :
-
Set of all repair policies for the set of \(S\)
- \(x_k\) :
-
\(k\hbox {th}\) Repair policy which belongs to set of all recommended repair policy \(X\)
- \(x_0\) :
-
Initial repair policy
- \(C_j (i)\) :
-
The cost function of carrying out repair in the \(j\hbox {th}\) interval inspection provided that the system condition is equal to \(i\) and the repair activity is according to the repair policy \(x_k \)
- \(CP_j (i)\) :
-
The average repair cost from the first inspection until \(j\hbox {th}\) inspection provided that the system condition is equal to \(i\) and the repair activity is according to the repair policy \(x_k \)
- \(CP_{T_{d_{x_k}}}\) :
-
The average repair cost between first inspection and performing preventive replacement
- \(\hbox {CP}_\mathrm{T}\) :
-
The average repair cost from the first inspection until failure occurrence time
- \(d_{x_k}\) :
-
Control-limit for preventive replacement in repair policy \(x_k \)
- \(d_{x_k}^*\) :
-
Optimal control-limit for replacement and repair in repair policy \(x_k \)
- \(T_{d_{x_k}}\) :
-
Time of performing preventive replacement according to control-limit policy \(d_{x_k } \)
- \(t_i\) :
-
The first time that failure risk passes control-limit in which the system condition is \(\hbox {i}\)
- \(k_i\) :
-
Number of first inspection after \(t_i \)
- \(\varphi _{REP} \left( {d_{x_k}}\right) \) :
-
The average of replacement costs in unit time in long period
- \(\varphi _{rep} \left( {d_{x_k}}\right) \) :
-
The average of repair costs in unit time in long period
- \(\varphi \left( {{d_x}_k}\right) \) :
-
The average of all replacement and repair costs in unit time in long period
- \(Q\left( {{d_x}_k}\right) \) :
-
The probability of replacement due to the failure according to control-limit policy \(d_{x_k }\)
- \(W\left( {{d_x}_k}\right) \) :
-
The average time between two consecutive replacements (including preventive replacement and replacement due to the failure) according to control-limit policy \(d_{x_k } \)
- \(Q(j,i)\) :
-
The probability of replacement due to the failure provided that the system condition is \(i\) in the \(j\hbox {th}\) inspection
- \(W(j,i)\) :
-
The average remaining time to next replacement provided that the system condition is \(i\) in the \(j\hbox {th}\) inspection
- \(R(j,i,t)\) :
-
The conditional reliability function until time of \(j\Delta \,+\,t\) provided that the system condition does not experience repair or replacement in the \(j\hbox {th}\) inspection and in the inspection system condition is \(i\)
References
Amari, S. V., & McLaughlin, L. (2004). Optimal design of a condition-based maintenance model. In Reliability and Maintainability, 2004 Annual Symposium-RAMS (pp. 528–533). IEEE.
Banjevic, D., Jardine, A. K. S., Makis, V., & Ennis, M. (2001). A control-limit policy and software for condition-based maintenance optimization. INFOR-OTTAWA-, 39(1), 32–50.
Banjevic, D., & Jardine, A. K. S. (2006). Calculation of reliability function and remaining useful life for a Markov failure time process. IMA Journal of Management Mathematics, 17(2), 115–130.
Chen, D., & Trivedi, K. S. (2002). Closed-form analytical results for condition-based maintenance. Reliability Engineering & System Safety, 76(1), 43–51.
Chen, D., & Trivedi, K. S. (2005). Optimization for condition-based maintenance with semi-Markov decision process. Reliability Engineering & System Safety, 90(1), 25–29.
Cox, D. D. R., & Oakes, D. (1984). Analysis of survival data (Vol. 21). Boca Raton: CRC Press.
Hosseini, M. M., Kerr, R. M., & Randall, R. B. (2000). An inspection model with minimal and major maintenance for a system with deterioration and Poisson failures. IEEE Transactions on Reliability, 49(1), 88–98.
Jardine, A. K. S., Banjevic, D., & Makis, V. (1997). Optimal replacement policy and the structure of software for condition-based maintenance. Journal of Quality in Maintenance Engineering, 3(2), 109–119.
Jardine, A. K. S., Makis, V., Banjevic, D., Braticevic, D., & Ennis, M. (1998). A decision optimization model for condition-based maintenance. Journal of Quality in Maintenance Engineering, 4(2), 115–121.
Jardine, A. K. S., Joseph, T., & Banjevic, D. (1999). Optimizing condition-based maintenance decisions for equipment subject to vibration monitoring. Journal of Quality in Maintenance Engineering, 5(3), 192–202.
Jardine, A. K., Lin, D., & Banjevic, D. (2006). A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical systems and signal processing, 20(7), 1483–1510.
Khatab, A. (2013). Hybrid hazard rate model for imperfect preventive maintenance of systems subject to random deterioration. Journal of Intelligent Manufacturing, 1–8.
Kim, M. J., & Makis, V. (2009). Optimal maintenance policy for a multi-state deteriorating system with two types of failures under general repair. Computers & Industrial Engineering, 57(1), 298–303.
Love, C. E., Zhang, Z. G., Zitron, M. A., & Guo, R. (2000). A discrete semi-Markov decision model to determine the optimal repair/replacement policy under general repairs. European Journal of Operational Research, 125(2), 398–409.
Makis, V., & Jardine, A. K. (1992). Optimal replacement in the proportional hazards model. Infor, 30(1), 172–183.
Makis, V., Jiang, X., & Cheng, K. (2000). Optimal preventive replacement under minimal repair and random repair cost. Mathematics of operations research, 25(1), 141–156.
Moustafa, M. S., Maksoud, E. Y., & Sadek, S. (2004). Optimal major and minimal maintenance policies for deteriorating systems. Reliability Engineering & System Safety, 83(3), 363–368.
Tian, Z., Lin, D., & Wu, B. (2012). Condition based maintenance optimization considering multiple objectives. Journal of Intelligent Manufacturing, 23(2), 333–340.
Vlok, P. J., Wnek, M., & Zygmunt, M. (2004). Utilising statistical residual life estimates of bearings to quantify the influence of preventive maintenance actions. Mechanical systems and signal processing, 18(4), 833–847.
Wang, W. (2003). Modelling condition monitoring intervals: A hybrid of simulation and analytical approaches. Journal of the Operational Research Society, 54(3), 273–282.
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Mousavi, S.M., Shams, H. & Ahmadi, S. Simultaneous optimization of repair and control-limit policy in condition-based maintenance. J Intell Manuf 28, 245–254 (2017). https://doi.org/10.1007/s10845-014-0974-8
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DOI: https://doi.org/10.1007/s10845-014-0974-8