A condition-based maintenance policy for a production system under excessive environmental degradation

Abstract

In this paper a condition-based maintenance model is proposed for a single-unit system of production of goods and services. The system is subject to random deterioration which impacts not only the product quality but also the environment. We assume that the environment degrades whenever a specific level of system deterioration is reached. The proposed maintenance model aims to assess the degradation in such a way to reduce the deterioration of the environment. To control this deterioration, inspections are performed and after which the system is preventively replaced or left as it is. Preventive replacement occurs whenever the level of the system degradation reaches a specific level threshold. The objective is to determine optimal inspection dates which minimize the average total cost per unit of time in the infinite horizon. Cost function is composed of inspection and maintenance costs in addition to a penalty cost due to environmental deterioration. The maintenance optimization model is formally derived. On the basis of Nelder–Mead method, inspection dates as optimal solutions are computed. A numerical example is provided to illustrate the proposed maintenance model.

A Appendix

A.1 Proof of Proposition 2

Let us recall that inspections are carried out until the degradation level exceeds the given level $L$. It follows that the number $N$ of inspections during a cycle is a geometric random variable whose expectation is such that:

$$\begin{aligned} \displaystyle E(N)=\sum _{i=1}^{\infty }i ~Pr\left\{ N=i\right\} , \end{aligned}$$

where $Pr\left\{ N=i\right\}$ is the probability of performing exactly $i$ inspection actions during a cycle. To calculate such a probability, let us note that:

$$\begin{aligned} \displaystyle Pr \left\{ N< i+1\right\} =Pr \left\{ N< i\right\} +Pr \left\{ N=i\right\} , \end{aligned}$$

which implies that:

$$\begin{aligned} \displaystyle Pr\left\{ N=i\right\} =Pr \left\{ N< i+1\right\} -Pr \left\{ N< i\right\} . \end{aligned}$$

The event $^{\prime \prime }N< i^{\prime \prime }$ is equivalent to the fact that the cycle ends with a corrective maintenance, i.e. the production system fails before the $i^{th}$ inspection date $\theta _i$. Under the assumption that the production system fails only after exceeding the threshold level of environmental degradation, the event $^{\prime \prime }N< i^{\prime \prime }$ is equivalent to $^{\prime \prime }T+X< \theta _i^{\prime \prime }$. Consequently, for all $i$ we have:

$$\begin{aligned} \displaystyle Pr \left\{ N< i\right\} =Pr \left\{ T+X \le \theta _{i} \right\} . \end{aligned}$$

Since random variable $X$ is assumed to be the residual lifetime of the production system after exceeding the alarm threshold, it follows that:

$$\begin{aligned} \displaystyle Pr \left\{ T+X \le \theta _{i} \right\} =\int \limits _{0}^{\theta _{i}} G(\theta _{i} -\tau )~f(\tau )d\tau . \end{aligned}$$

As a result, we have:

$$\begin{aligned}&\displaystyle Pr\left\{ N=i\right\} =\int \limits _{0}^{\theta _{i+1} }G(\theta _{i+1} -\tau ) ~f(\tau )d\tau \nonumber \\&\quad -\int \limits _{0}^{\theta _{i} }G(\theta _{i} -\tau ) ~f(\tau )d\tau . \end{aligned}$$

This ends the proof.

A.2 Proof of Proposition 3

Let us assume that the degradation level exceeds the threshold value $L$ at $T=\tau$, and $\theta _{i-1} < T \le \theta _{i}$. It follows that duration $T_d$ of excessive environmental degradation in a cycle could be written as:

$$\begin{aligned} T_d=\left\{ \begin{array}{lll} X&~~~~&if ~~X \le \theta _{i} +H-\tau \\ \theta _i+H-\tau&~~~~&otherwise \end{array} \right. \end{aligned}$$

In other words, $T_d = min(X, \theta _i+H-\tau )$. The average time $E(T_d)$ of excessive environmental degradation in a cycle can be expressed as:

$$\begin{aligned} \displaystyle E(T_d )&= \sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } E\left(T_d\left|_{\tau < T\le \tau +d\tau } \right.\right) ~f(\tau )d\tau \\&= \sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } E\left(min(X, \theta _i+H-\tau )\left|_{\tau < T\le \tau +d\tau } \right.\right) ~f(\tau )d\tau \\&= \sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } \left(\int \limits _{0}^{\theta _{i}+H-\tau } \left[1 - G(x)\right]dx \right) ~f(\tau )d\tau \end{aligned}$$

This ends the proof.

A.3 Proof of Proposition 4

To prove Proposition 4, let us note that the cycle time represented by the random variable $T_c$ can be written as:

$$\begin{aligned} T_c = T + T_d. \end{aligned}$$

It follows that the average cycle time $E(T_c)$ is:

$$\begin{aligned} E(T_c) = E(T) + E(T_d). \end{aligned}$$

By exploiting the result of Proposition 3, the average cycle time $E(T_c)$ can be written as:

$$\begin{aligned} \displaystyle E(T_c )&= \sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } \tau ~f(\tau )d\tau \nonumber \\&\quad +\sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } \left(\int \limits _{0}^{\theta _{i}+H-\tau } \left[1 - G(x)\right]dx \right) ~f(\tau )d\tau \\&= \sum _{i=1}^{\infty }\int \limits _{\theta _{i-1} }^{\theta _{i} } \left(\tau +\int \limits _{0}^{\theta _{i}+H-\tau } \left[1 - G(x)\right]dx \right) ~f(\tau )d\tau . \end{aligned}$$

This ends the proof.