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 \(i\)
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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