Reliability and maintenance modeling for a production system by means of point process observations

Abstract

This paper develops a reward model for the optimization of preventive maintenance for a complex production system functioning in any one of k unobservable operating states. The changes of the states are driven by a non-homogeneous Markov (NHM) process X(t) with known characteristics. The system fails according to a point process whose intensity is modulated by the unobservable state. Failures are rectified through minimal repairs (MRs) whose costs are associated with age and the state process X(t). The modeling approach also allows both the revenue stream and the preventive maintenance cost to be characterized by the state process X(t). The paper first formulates the reward model depending on the unobservable state process estimated through the filtering theorem argument by projection on the observed history including failure point process observations. The estimation of the state process allows failure prediction and maximizing revenue stream implemented through scheduling periodic overhauls. A case study is provided to illustrate the proposed method.

Appendix A. Proof of Proposition 1

One can note that the cumulative process $\Lambda (t)$ is arisen by the stochastic integration of the random function

$$\begin{aligned} \phi _t=\phi (t,X(t))=\sum _{d=0}^{k-1}\underbrace{\left( \phi (t,d)-\phi (t,d-1)\right) }_{\Delta \phi (t,d)}\textbf{1}(X(t)-d) \end{aligned}$$

with respect to the counting process N(t). In other words,

$$\begin{aligned} \begin{aligned} \int _0^t\phi _sdN(s)&=\int _0^t\sum _{d=0}^{k-1}\left( \phi (s,d)-\phi (s,d-1)\right) \\&\qquad \times \textbf{1}(X(s)-d)dN(s) \end{aligned} \end{aligned}$$

(A1)

or, based on the relation

$$\begin{aligned} \Delta Y_t=\textbf{1}(X(s)-d)\Delta N(s), \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned} \int _0^t\phi _sdN(s)&=\int _0^t\sum _{d=0}^{k-1}\Delta \phi (s,d)dY_s\\&=\sum _{i=1}^{N(t)}\left[ \sum _{d=0}^{k-1}\Delta \phi \left( T_i,d\right) \textbf{1}\left( X(T_i)-d\right) \right] \\&=\sum _{i=1}^{N(t)}\phi \left( T_i,X(T_i)\right) =\Lambda (t). \end{aligned} \end{aligned}$$

(A2)

From (A2) it follows that

$$\begin{aligned} d\Lambda (t)=\sum _{d=0}^{k-1}\Delta \phi (t,d)dY_t=\sum _{d=0}^{k-1}\Delta \phi (t,d)\textbf{1}(X(t)-d)dN(t). \end{aligned}$$

Since $\Delta \phi (t,d)$ is deterministic and $\textbf{1}(X(t)-d)$ is $\mathcal {F}_t$-measurable we have

$$\begin{aligned} \begin{aligned} \lambda _c(t)dt&=\mathbb {E}\left[ d\Lambda (t)\mid \mathcal {F}_t\right] \\&=\sum _{d=0}^{k-1}\Delta \phi (t,d)\textbf{1}(X(t)-d)\mathbb {E}\left[ dN(t)\mid \mathcal {F}_t\right] \\&=\sum _{d=0}^{k-1}\Delta \phi (t,d)\textbf{1}(X(t)-d)\lambda _{X(t)}dt\\&=\sum _{d=0}^{k-1}\left( \lambda _d\Delta \phi (t,d)+\Delta \lambda _d\phi (t,d-1)\right) \times \textbf{1}(X(t)-d)dt \end{aligned} \end{aligned}$$

(A3)

or,

$$\begin{aligned} \lambda _c(t)=\sum _{d=0}^{k-1}\left( \lambda _d\Delta \phi (t,d)+\Delta \lambda _d\phi (t,d-1)\right) \times \textbf{1}(X(t)-d). \end{aligned}$$

That means the cumulative process $\Lambda (t)$ admits the $\mathcal {F}_t$-intensity $\lambda _c(t)$:

$$\begin{aligned} \begin{aligned} \lambda _c(t)&=\sum _{d=0}^{k-1}\left( \lambda _d\left( \phi (t,d)-\phi (t,d-1)\right) +\left( \lambda _d-\lambda _{d-1}\right) \phi (t,d-1)\right) \\&\qquad \times \textbf{1}(X(t)-d). \end{aligned} \end{aligned}$$

(A4)