# An adaptive maintenance policy for degrading units with an imperfect and a perfect inspection

Nicola Esposito, Antonio Piscopo, Bruno Castanier, Massimiliano Giorgio

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Nicola Esposito, Antonio Piscopo, Bruno Castanier, Massimiliano Giorgio. An adaptive maintenance policy for degrading units with an imperfect and a perfect inspection. European Journal of Operational Research, 2025, 10.1016/j.ejor.2025.10.014. hal-05446464

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# An adaptive maintenance policy for degrading units with an imperfect and a perfect inspection

Nicola Esposito<sup>a,b,✉</sup>, Antonio Piscopo<sup>a,b,c</sup>, Bruno Castanier<sup>a,✉</sup>, Massimiliano Giorgio<sup>c,*</sup>

## ARTICLE INFO

### Keywords:

- Stochastic processes

- Adaptive maintenance rule

- Imperfect and perfect inspections

- Perturbed gamma process with random effect

## ABSTRACT

This paper presents an adaptive condition-based maintenance policy for degrading units where at the inspection time an imperfect inspection is firstly carried out to decide whether to immediately replace the unit, to define its future replacement time, or to immediately carry out an additional, more costly, perfect inspection. When it is performed, the perfect inspection is used again to decide whether to immediately replace the unit or to define its future replacement time. The imperfect inspection is assumed to provide a degradation measurement affected by error. The perfect inspection is supposed to give an exact degradation measurement.

The distinguishing feature of the proposed policy is that the future replacement time is defined adaptively, unit-by-unit, based on the outcome of the inspections.

It is assumed that the inspections can be performed only once, at most, over the lifespan of the unit. The policy accounts for the joint presence of temporal variability, measurement error, and random effect. The temporal variability is modeled by using a gamma process.

Two comparative studies demonstrate the superiority of the proposed adaptive decision rule with respect to the classical control limit rule and show how its use allows to decide whether and which inspection is convenient to perform, depending on inspection costs and features of the degradation process.

## 1. Introduction

Most of the maintenance policies proposed in the literature for degrading units are developed by adopting either age-based maintenance (ABM) or condition-based maintenance (CBM) strategies.

Recent examples of applications of ABM and CBM are provided in Eryilmaz (2021), Cai et al. (2023), Zheng et al. (2023), Absi et al. (2024), Kivanç et al. (2024), Xu et al. (2024), and Andersen and Nielsen (2025). Comprehensive reviews of papers on ABM and CBM are provided in Ahmad and Kamaruddin (2012), Alaswad and Xiang (2017), and Arts et al. (2024). For a recent critical discussion of the effectiveness of preventive replacements in heterogeneous populations see Cha et al. (2025).

Overall, the use of CBM over ABM offers the (potential) twofold benefit of reducing the risk of failure and extending the operational life of units. Indeed, while ABM involves replacing units after an operational time that is determined a priori, CBM uses information on the degradation state of the units, gathered by performing ad hoc inspections over

the course of their operating life, to improve maintenance decisions.

However, carrying out the inspections needed for implementing CBM is not always easy and/or cheap (e.g., see Alaswad & Xiang, 2017). Performing an inspection may require logistic support, specialized equipment, skilled personnel, and long interruptions of operations, resulting in a cost that can reduce (and potentially outweigh) the benefits of CBM over ABM.

For this reason, in the last two decades, significant efforts have been devoted to the formulation of maintenance policies that rely on as few inspections as possible while still retaining the benefits of CBM (e.g., see Fauriat & Zio, 2020, Yuan et al., 2021, and Kim et al., 2022). Motivated by the intent of avoiding inspections which are expected to give rise to foregone decisions, Huynh et al. (2019), after remarking that most of the existing CBM policies allow replacements only at inspection times (e.g., see also Wang & Pham, 2013), proposed a policy that permits to schedule inspection and replacement times separately. Taking these concepts to the extreme, Crowder and Lawless (2007), Finkelstein et al. (2020), Cha et al. (2021), and Cha et al. (2022) proposed maintenance

<footnote><sup>a</sup> Université d'Angers/Laris, Angers, France</footnote>

<footnote><sup>b</sup> Scuola Superiore Meridionale, Napoli, Italy</footnote>

<footnote><sup>c</sup> Università di Napoli Federico II, P.le V. Tecchio 80, 80125 Napoli, Na, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: massimiliano.giorgio@unina.it (M. Giorgio).</footnote>
policies where a single inspection is used to decide whether to immediately replace the unit or to postpone its replacement to a predetermined future time.

A second issue with inspections is that, in practice, they are usually imperfect (e.g., see Lu et al., 2013). Nevertheless, most existing maintenance policies do not consider this eventuality (e.g., see Grall et al., 2002, Castanier et al., 2003, Castanier et al., 2004, Fouladirad & Grall, 2011, Huynh et al., 2011, Cholette et al., 2019, and Corset et al., 2024).

Motivated by this argument, Zhang et al. (2023), Wang et al. (2023), Li, Shi et al. (2023), Zhao et al. (2024), and Zhang et al. (2024) studied the impact of imperfect inspections on the diagnostic ability of CBM, focusing on the risk of incurring in a false negative (i.e., not detecting a failure that has occurred) or in a false positive (i.e., falsely diagnosing a failure that did not occur). For an updated historical survey on papers dealing with the impact of imperfect inspections in CBM see Ulansky and Raza (2024). Likewise, Kallen and van Noortwijk (2005), Esposito et al. (2021), and Deep et al. (2023) suggested CBM policies that formally account for the presence of measurement error.

Focusing on both cost and quality of inspections, Zequeira and Bérenguer (2006) suggested a maintenance policy that uses in a combined manner three types of inspections (partial, imperfect, and perfect). Similarly, Shen and Cui (2017) and Hao et al. (2020) proposed policies that use an inspection scheme where the first *m* inspections are (cheap) imperfect inspections and the following ones are (expensive) perfect inspections. Nguyen et al. (2019) proposed a dynamic scheme where the inspection quality is a design parameter.

Very recently, Esposito et al. (2023b) suggested a control limit maintenance policy that uses in a combined manner two inspection procedures that are used once at most over the lifetime of a unit at a single inspection epoch.

Still aiming to reduce the cost of inspections while retaining the advantages of CBM, Grall et al. (2002), Castanier et al. (2003), Castanier et al. (2004), Fauriat and Zio (2020), Cao (2021), Yao et al. (2024), and Hamzaoui et al. (2023) proposed maintenance policies where the time of the next inspection/replacement is planned adaptively, unit-by-unit, on the basis of the information available at the decision epoch. By following a similar logic, very recently, Zhuang et al. (2024) proposed a policy where candidate replacement times are determined by using remaining useful life predictions that are updated, in real-time, on the basis of several degradation measurements, along with some cost considerations.

Stimulated by this literature, in this paper, we propose an adaptive version of the CBM policy suggested in Esposito et al. (2023b). In fact, as in this latter paper, it is assumed that at the inspection epoch an imperfect inspection is performed first to decide whether to immediately replace the unit, to postpone its replacement, or to immediately perform a perfect inspection. In this latter case, the output of the perfect inspection is used again to decide whether to immediately replace the unit or to postpone its replacement. It is assumed that the two inspections can be carried out only once at most over the lifetime of the unit. The imperfect and perfect inspections are assumed to provide a measurement affected by error and an exact measurement of the degradation level, respectively.

The novelty of this new policy with respect to Esposito et al. (2023b) is that while in Esposito et al. (2023b) the postponement time is the same for all the units, here it is determined adaptively, unit-by-unit, at the inspection epoch, via a measurements-based decision rule.

The link between degradation and reliability is defined by assuming that a unit fails when its degradation level exceeds an assigned failure threshold. Failures are assumed to be not self-announcing (e.g., see Hu et al., 2020, Bautista et al., 2022, Bismut et al., 2022).

The model used to describe the evolution of the degradation over time is the same perturbed gamma process with random effect adopted in Esposito et al. (2023b). The temporal variability is modeled by using a classical gamma process (van Noortwijk, 2009, Alaswad & Xiang, 2017, and Chai et al., 2024). The random effect is modeled by treating the scale

parameter of the gamma process as a gamma random variable (Lawless & Crowder, 2004). The measurement error is modeled by using two alternative models, the inverse gamma-based model already used in Esposito et al. (2023b) and the Gaussian-based error model suggested in Pulcini (2016). Both the considered error models allow to describe cases where the measurement error depends in stochastic terms on the measured degradation level.

The reason for using this perturbed model is that it allows us to account for the joint presence of temporal variability, measurement error, and random effect, while easily modulating the relative contribution of these sources of uncertainty to the variance of the process. However, the degradation model does not constitute an original contribution of this paper. In fact, it is used only to simulate a reasonable diverse, realistic, set of possible experimental scenarios. In principle, the suggested decision rule can be used with any other perturbed degradation model.

The proposed maintenance policy is applied to a real-world inspired case of a buried pipeline. The optimal policy is determined by minimizing the long-run average maintenance cost rate.

The application includes two comparative studies. The first one shows the superiority of the proposed new adaptive decision rule with respect to the less flexible threshold limit rule used in Esposito et al. (2023b). The second study serves to demonstrate how, given an experimental scenario, the proposed approach, based on a cost-benefit analysis, allows to competently decide how many and which inspection/inspections is convenient to perform. This latter specific result also shows that, in some applications, focusing on maintenance models that rely on degradation measurements collected at a single inspection epoch does not represent a limitation, because there are cases where the information gathered by performing an inspection is not worth its cost.

The paper is structured as follows. Section 2 describes the adopted perturbed degradation model and introduces results that are used in the rest of the paper. Section 3 illustrates the proposed hybrid maintenance policy and the cost model. Section 4 is devoted to the formulation of the long-run average maintenance cost rate. Section 5 presents the results of the comparative studies. In particular, Section 5.1 discusses and motivates the choice of the functions used to implement the suggested adaptive decision rule, which constitutes one of the main contributions of the paper. Section 6 provides comments and conclusions. Finally, Appendices A and B give details about the derivation of the formulas used in the paper to compute the long-run average maintenance cost rate under all the considered maintenance policies.

## 2. The degradation process

The degradation process is modeled as in Esposito et al. (2023b). Namely, the actual (true) degradation process of the units {$W(t); t \ge 0$} is assumed to be a gamma process with random effect. The random effect (see Section 2.1) is described by treating the scale parameter $\Lambda$ of the gamma process as a random variable. The link existing between the noisy degradation measurement obtained by performing the imperfect inspection and the actual degradation state of the unit is described via the perturbed process {$Z(t); t \ge 0$}:

$$ \{Z(t) = W(t) + \varepsilon(t); t \ge 0\}, \quad (1) $$

where $Z(t)$ is the perturbed measurement obtained at time $t$ via the imperfect inspection, $W(t)$ is the actual degradation level of the unit at the same time, and $\varepsilon(t)$ is the measurement error.

It is assumed that, given $W(t_k)$, $\varepsilon(t_k)$ is independent of $\Lambda \forall k (k = 1, ..., n)$ and that for any $n > 1$, given $W(t_k)$, $\varepsilon(t_k)$ is conditionally independent both of $\varepsilon(t_j)$ and $W(t_j) \forall j \neq k (k, j = 1, ..., n)$. Obviously, based on this assumption, given $W(t_k)$, also $Z(t_k)$ is conditionally independent of $\Lambda \forall k (k = 1, ..., n)$ and of $Z(t_j)$ and $W(t_j) \forall j \neq k (k, j = 1, ..., n)$.
## 2.1. The actual degradation process

By following Lawless and Crowder (2004), the actual degradation process {$W(t)$; $t \ge 0$} is modeled by assuming that all the unit-specific degradation processes are gamma processes with the same age function $\eta(t)$ and scale parameter $\Lambda$ that varies randomly, from unit to unit, according to a gamma probability distribution. The main reason for using this model is that it allows to describe in a simple and effective manner populations where there are strong units (i.e., those characterized by high values of the unit-specific random parameter $\Lambda$) and weak units (i.e., those characterized by small values of the unit-specific random parameter $\Lambda$). Moreover, thanks to its simple structure, it also lets to easily modify/calibrate both the absolute and relative magnitude of temporal and unit-to-unit variability.

Specifically, the probability density function (pdf) of $\Lambda$ is expressed as:

$$f_{\Lambda}(\lambda) = \frac{c^d \lambda^{d-1}}{\Gamma(d)} e^{-c\lambda}, \quad c, d, \lambda > 0, \qquad (2)$$

and given $\Lambda = \lambda$, the conditional pdfs of $W(t)$ and $\Delta W(t, t+\Delta t) = W(t+\Delta t) - W(t)$ are expressed as:

$$f_{W(t)|\Lambda}(w|\lambda) = \frac{\lambda^{\eta(t)} \cdot w^{\eta(t)-1}}{\Gamma(\eta(t))} e^{-\lambda \cdot w}, \quad w > 0, \qquad (3)$$

and

$$f_{\Delta W(t, t+\Delta t)|\Lambda}(\delta|\lambda) = \frac{\lambda^{\Delta\eta(t, t+\Delta t)} \cdot \delta^{\Delta\eta(t, t+\Delta t)-1}}{\Gamma(\Delta\eta(t, t+\Delta t))} e^{-\lambda \cdot \delta}, \quad \delta > 0, \qquad (4)$$

respectively, where $\Delta W(t, t+\Delta t) = W(t+\Delta t) - W(t)$ is the degradation increment, $\Gamma(y) = \int_0^\infty u^{y-1} \cdot e^{-u} \cdot du$ is the complete gamma function, $\eta(t)$ is the age function, $\Delta\eta(t, t+\Delta t) = \eta(t+\Delta t) - \eta(t)$, and $c$ and $d$ are the scale and shape parameters of the gamma pdf in Eq. (2), respectively. The function $\eta(t)$ is monotonic increasing. It is assumed that $W(0) = \eta(0) = 0$.

Under this setting, the actual (marginal) degradation process {$W(t)$; $t \ge 0$} is Markovian (e.g., see Esposito et al., 2023a, and Giorgio & Pulcini, 2018). However, unlike the basic gamma process, it does not have stochastically independent increments.

From Eqs. (2) and (3), $W(t)$ has marginal pdf:

$$f_{W(t)}(w) = \frac{c^d \cdot w^{\eta(t)-1}}{B(\eta(t), d) \cdot (w+c)^{\eta(t)+d}}, \quad w \ge 0, \qquad (5)$$

and marginal cumulative distribution function (cdf):

$$F_{W(t)}(w) = \mathbb{B}\left(\frac{w}{w+c}; \eta(t), d\right), \quad w \ge 0, \qquad (6)$$

where:

$$\mathbb{B}(z; \alpha, \beta) = \frac{\int_0^z u^{\alpha-1} \cdot (1-u)^{\beta-1} \cdot du}{B(\alpha, \beta)} \qquad (7)$$

is the regularized beta function and

$$B(\alpha, \beta) = \frac{\Gamma(\alpha) \cdot \Gamma(\beta)}{\Gamma(\alpha + \beta)} \qquad (8)$$

is the beta function.

The marginal mean of $W(t)$ (which exists for $d > 1$) is:

$$E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1}, \qquad (9)$$

and its marginal variance (which exists for $d > 2$) is:

$$V\{W(t)\} = \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)}. \qquad (10)$$

Given $W(t) = w$, the degradation increment $\Delta W(t, t+\Delta t)$ has conditional pdf:

$$f_{\Delta W(t, t+\Delta t)|W(t)}(\delta|w) = \frac{1}{B(\Delta\eta(t, t+\Delta t), \eta(t)+d)} \frac{(w+c)^{\eta(t)+d} \cdot \delta^{\Delta\eta(t, t+\Delta t)-1}}{(w+\delta+c)^{\eta(t+\Delta t)+d}}, \qquad (11)$$

and conditional cdf:

$$F_{\Delta W(t, t+\Delta t)|W(t)}(\delta|w) = \mathbb{B}\left(\frac{\delta}{\delta+w+c}; \Delta\eta(t, t+\Delta t), \eta(t)+d\right). \qquad (12)$$

## 2.2. The error model

For the measurement error $\varepsilon(t)$ we have considered two different modeling solutions. None of them is new.

As first option, by following Giorgio et al. (2019), we have assumed that, given $W(t) = w$, $\varepsilon(t)$ has conditional pdf:

$$f_{\varepsilon(t)|W(t)}(\varepsilon|w) = \frac{(\alpha(w))^{\beta(w)} \cdot (\varepsilon + w)^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{-\frac{\alpha(w)}{\varepsilon+w}}, \qquad (13)$$

where: $\beta(w) = \varphi \cdot w^{2-\nu} + 2$ and $\alpha(w) = [\beta(w) - 1] \cdot w$ with $\varphi > 0$ and $-\infty < \nu < \infty$.

As second option, by following Pulcini (2016), we have assumed that, given $W(t) = w$, $\varepsilon(t)$ has conditional pdf:

$$f_{\varepsilon(t)|W(t)}(\varepsilon|w) = \frac{1}{\sqrt{2 \cdot \pi \cdot \sigma^2(w)}} e^{-\frac{1}{2} \frac{\sigma^2(w)}{w}}, \quad -\infty < \varepsilon < \infty, \qquad (14)$$

where $\sigma^2(w) = w^\nu/\varphi$, $\varphi > 0$, and $-\infty < \nu < \infty$.

Both the error models in Eqs. (13) and (14) enable to describe experimental scenarios where the measurement error stochastically depends on the measured degradation level. The link between the variance of the error term and the measured degradation level is defined by using a power law function, which should be understood as a flexible empirical (i.e., data-driven) model. In fact, $\nu$ and $\varphi$ are unknown parameters that should be estimated from data.

From Eq. (1), under the error model in Eq. (13) the perturbed measurement $Z(t)$, given $W(t) = w$, has conditional pdf:

$$f_{Z(t)|W(t)}(z|w) = \frac{(\alpha(w))^{\beta(w)} \cdot z^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{-\frac{\alpha(w)}{z}}, \quad z \ge 0, \qquad (15)$$

and conditional cdf:

$$F_{Z(t)|W(t)}(z|w) = \frac{\gamma(\beta(w), \alpha(w)/z)}{\Gamma(\beta(w))}, \quad z \ge 0, \qquad (16)$$

where $\gamma(y, s) = \int_0^s u^{y-1} \cdot e^{-u} \cdot du$ is the lower incomplete gamma function.

Under the model in Eq. (14), given $W(t) = w$, the perturbed measurement $Z(t)$ has conditional pdf:

$$f_{Z(t)|W(t)}(z|w) = \frac{1}{\sqrt{2 \cdot \pi \cdot \sigma^2(w)}} e^{-\frac{1}{2} \frac{(z-w)^2}{\sigma^2(w)}}, \quad -\infty < z < \infty, \qquad (17)$$

and conditional cdf:

$$F_{Z(t)|W(t)}(z|w) = \Phi\left(\frac{z-w}{\sigma(w)}\right), \quad -\infty < z < \infty, \qquad (18)$$
where $\Phi(\cdot)$ is the cdf of the standard Gaussian random variable and $\sigma(w) = \sqrt{\sigma^2(w)}$ is the conditional standard deviation of $Z(t)$, given $W(t) = w$.

Under both the considered error models, given $W(t) = w$, the measurement error $\epsilon(t)$ has conditional mean $E\{\epsilon(t)|W(t) = w\} = 0$ and conditional variance $V\{\epsilon(t)|W(t) = w\} = w^\nu/\phi$.

Accordingly, given $W(t) = w$, the perturbed measurement $Z(t)$ has conditional mean:

$$E\{Z(t)|W(t) = w\} = E\{\epsilon(t)|W(t) = w\} + w = w, \quad (19)$$

and conditional variance:

$$V\{Z(t)|W(t) = w\} = V\{\epsilon(t)|W(t) = w\} = \frac{w^\nu}{\phi}. \quad (20)$$

The advantage of using the inverse gamma-based model in Eq. (13) is that, differently than the model in Eq. (14), it ensures that $Z(t)$ is non-negative, a constraint that could be necessary to impose due (for example) to technological/physical reasons. Indeed, in the case of the gamma process, using the inverse gamma-based error model gives also some mathematical advantages (see Giorgio et al. (2019) for details). On the other hand, the model in Eq. (14) could be preferred to the model in Eq. (13) because it includes as a special case the widely adopted error model which assumes that $\epsilon(t)$ is Gaussian distributed and independent of $\{W(t); t \ge 0\}$, a feature that could facilitate the comparison of the resulting perturbed model with other perturbed models suggested in the literature.

### 2.3. The marginal perturbed model

The marginal distribution of the perturbed measurement $Z(t)$ is not available in closed form. However, from Eq. (5), the marginal pdf and cdf of $Z(t)$ can be computed, numerically, by solving the following integrals:

$$\begin{aligned} f_{Z(t)}(z) &= \int_{0}^{\infty} f_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} f_{Z(t)|W(t)}(z|w) \cdot \frac{c^{d} \cdot w^{\eta(t)-1}}{B(\eta(t), d) \cdot (w + c)^{\eta(t)+d}} \cdot dw, \end{aligned} \quad (21)$$

$$\begin{aligned} F_{Z(t)}(z) &= \int_{0}^{\infty} F_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} F_{Z(t)|W(t)}(z|w) \cdot \frac{c^{d} \cdot w^{\eta(t)-1}}{B(\eta(t), d) \cdot (w + c)^{\eta(t)+d}} \cdot dw, \end{aligned} \quad (22)$$

where $f_{Z(t)|W(t)}(z|w)$ and $F_{Z(t)|W(t)}(z|w)$ are either those in Eqs. (15) and (16) or those in Eqs. (17) and (18), depending on the error model.

From Eqs. (9), (10), (19), and (20), by applying the laws of total expectation and total variance, the marginal mean in Eq. (23) and variance in Eq. (24) of $Z(t)$ can be readily obtained.

$$E\{Z(t)\} = E\{E\{Z(t)|W(t)\}\} = E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1}, \quad (23)$$

$$\begin{aligned} V\{Z(t)\} &= V\{E\{Z(t)|W(t)\}\} + E\{V\{Z(t)|W(t)\}\} \\ &= V\{W(t)\} + E\left\{\frac{(W(t))^\nu}{\phi}\right\} \\ &= \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)} + \frac{c^\nu \cdot \Gamma(d-\nu) \cdot \Gamma(\eta(t) + \nu)}{\Gamma(d) \cdot \Gamma(\eta(t))} \frac{1}{\phi}. \end{aligned} \quad (24)$$

The marginal mean exists for $d > 1$. The marginal variance exists for $d > 2$, $d - \nu > 0$, and $\eta(t) + \nu > 0$.

Although this feature does not impact on the results derived in the rest of this paper, it is also worth to remark that the perturbed process $\{Z(t); t \ge 0\}$ is not Markovian.

### 2.4. The useful life

A unit is assumed to fail when its actual degradation level exceeds a fixed failure threshold denoted by $w_M$. Thus, its useful life $X$ (i.e., its lifetime) can be defined as the first passage time of $\{W(t); t \ge 0\}$ to the failure threshold $w_M$:

$$X = \inf\{x : W(x) > w_M\} \quad (25)$$

From the conditional cdf in Eq. (12), denoting by $\tau$ a generic reference time (e.g., the inspection time) the conditional cdf of $X$ given $W(\tau) = w_\tau$, when $w_\tau \le w_M$, can be expressed as:

$$\begin{aligned} F_{X|W(\tau)}(x|w_\tau) &= P(X \le x | W(\tau) = w_\tau) = P(W(x) > w_M | W(\tau) = w_\tau) \\ &= \begin{cases} 0, & w_\tau \le w_M, x \le \tau \\ 1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, x), \eta(\tau) + d\right), & w_\tau \le w_M, x > \tau. \end{cases} \end{aligned} \quad (26)$$

It is worth to note that, the second equality in Eq. (26) holds because $\{W(t); t \ge 0\}$ is monotone increasing and hence the event $\{X \le x\}$ is equivalent to the event $\{W(x) > w_M\}$.

On the other hand, when $w_\tau > w_M$ the conditional cdf of $X$ given $W(\tau) = w_\tau$ can be expressed as:

$$\begin{aligned} F_{X|W(\tau)}(x|w_\tau) &= P(X \le x | W(\tau) = w_\tau) = P(W(x) > w_M | W(\tau) = w_\tau) \\ &= \begin{cases} 1 - \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right), & w_\tau > w_M, x \le \tau, \\ 1, & w_\tau > w_M, x > \tau. \end{cases} \end{aligned} \quad (27)$$

Details about the derivation of Eq. (27) are available in Esposito et al. (2023b).

For economy of notation, hereinafter, the cdfs in Eqs. (26) and (27) will be denoted by the same symbol. So, by $F_{X|W(\tau)}(x|w_\tau)$ we will intend the cdf in Eq. (26) when $w_\tau \le w_M$ and the one in Eq. (27) when $w_\tau > w_M$.

## 3. The maintenance policy and the cost model

In this section, we provide a detailed description of the proposed maintenance policy and of the related cost model.

### 3.1. The maintenance policy

The policy is formulated on the basis of the following assumptions:

* both a perfect and an imperfect inspection can be carried out;

* inspections are performed, at the same inspection time, only once over the lifespan of the unit;

* the imperfect inspection provides a perturbed measurement of the degradation level of the unit;

* the perfect inspection provides an exact measurement of the degradation level of the unit;

* inspections are non-destructive;

* units are considered failed when their degradation level passes an assigned failure threshold;
<table><caption>**Table 1**
Condition-based decision rule adopted at $\tau$.</caption><thead><tr><th colspan="3">Measurement at τ</th></tr><tr><th>Perturbed</th><th>Perfect</th><th>Decision</th></tr></thead><tbody><tr><td>$z_r > L_u$</td><td></td><td>Replacement at τ</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$w_r > D$</td><td>Replacement at τ</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$w_r \le D$</td><td>Replacement postponed to $\tau + \Delta\tau_w(w_r; \theta_w)$</td></tr><tr><td>$z_r \le L_l$</td><td></td><td>Replacement postponed to $\tau + \Delta\tau_z(z_r; \theta_z)$</td></tr></tbody></table>

* a failed unit could continue to operate, albeit with reduced performances/additional costs;

* failures are not self-announcing (e.g., see Hu et al., 2020, Bautista et al., 2022, and Bismut et al., 2022). That is, failures can be detected with certainty only by performing a perfect inspection;

* corrective and preventive replacements restore the unit to an “as good as new” state;

* the time between two successive replacements (i.e., the maintenance cycle) defines the cycle of a renewal process.

The policy consists in performing an imperfect inspection at a predetermined epoch $\tau$. The output $z_r$ of the imperfect inspection is used to decide whether to immediately replace the unit, to postpone its replacement to $\tau + \Delta\tau_z(z_r; \theta_z)$, or to perform, for an additional cost, a perfect inspection, which returns its exact degradation level $w_r$. In this latter case, based on $w_r$, it is decided again whether to immediately replace the unit or postpone its replacement to $\tau + \Delta\tau_w(w_r; \theta_w)$. The functions $\Delta\tau_z(z_r; \theta_z)$ and $\Delta\tau_w(w_r; \theta_w)$ are monotonic decreasing functions of $z_r$ and $w_r$, respectively. Depending on their shape, they can be indexed by a varying number of parameters, contained in the vectors $\theta_z$ and $\theta_w$.

The use of the functions $\Delta\tau_z(z_r; \theta_z)$ and $\Delta\tau_w(w_r; \theta_w)$ constitutes the main novelty with respect to Esposito et al. (2023b), where the postponement time is a predetermined design parameter. In fact, the policy suggested in Esposito et al. (2023b) can be obtained as a special case of this new policy by setting $\Delta\tau_z(z_r; \theta_z) = \Delta\tau_w(w_r; \theta_w) = \Delta\tau$.

Table 1, where $L_l$, $L_u$, and $D$ are maintenance threshold levels, reports the condition-based rule adopted for decision-making. The rationale behind the formulation of the decision rule described in Table 1 is to resort to the more costly perfect inspection only when its use could significantly improve the decision made by using $z_r$ alone. We expect this situation to occur when $z_r$ takes an intermediate value (i.e., when $L_l < z_r \le L_u$). Whereas, when $z_r$ is either very low or very high, it is coherent to expect that knowing $w_r$ is less valuable.

The design parameters of the suggested policy are: $\tau$, $L_l$, $L_u$, $D$, $\theta_z$, and $\theta_w$. Their optimal value is determined by minimizing the long-run average maintenance cost rate. Hereinafter we will indicate by $\xi$ the vector of design parameters (i.e., $\xi = \{\tau, L_l, L_u, D, \theta_z, \theta_w\}$) and by $\xi^*$ the value that defines the optimal policy (i.e., $\xi^*$ is the value of $\xi$ that minimizes the long-run average maintenance cost rate). Note that $\theta_z$ and $\theta_w$ are themselves vectors, whose dimensions depend on the structure of the functions $\Delta\tau_z(z_r; \theta_z)$ and $\Delta\tau_w(w_r; \theta_w)$.

<table><caption>**Table 2**
Proposed policy: possible evolution of the maintenance cycle, with corresponding costs and lengths.</caption><thead><tr><th colspan="3">Measurement at τ</th></tr><tr><th>Perturbed</th><th>Perfect</th><th>Lifetime</th><th>C(z<sub>r</sub>, w<sub>r</sub>, X)</th><th>T(z<sub>r</sub>, w<sub>r</sub>)</th></tr></thead><tbody><tr><td>$z_r > L_u$</td><td></td><td>$X \le \tau$</td><td>$c_2 + c_1 + c_c + c_d(\tau - X)$</td><td>$\tau$</td></tr><tr><td>$z_r > L_u$</td><td></td><td>$X > \tau$</td><td>$c_2 + c_1 + c_p$</td><td>$\tau$</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$w_r > w_M$</td><td>$X \le \tau$</td><td>$c_2 + c_w + c_1 + c_c + c_d(\tau - X)$</td><td>$\tau$</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$D < w_r \le w_M$</td><td>$X > \tau$</td><td>$c_2 + c_w + c_1 + c_p$</td><td>$\tau$</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$w_r \le D$</td><td>$X \le \tau + \Delta\tau_w(w_r; \theta_w)$</td><td>$c_2 + c_w + 2c_1 + c_c + c_d(\tau + \Delta\tau_w(w_r; \theta_w) - X)$</td><td>$\tau + \Delta\tau_w(w_r; \theta_w)$</td></tr><tr><td>$L_l < z_r \le L_u$</td><td>$w_r \le D$</td><td>$X > \tau + \Delta\tau_w(w_r; \theta_w)$</td><td>$c_2 + c_w + 2c_1 + c_p$</td><td>$\tau + \Delta\tau_w(w_r; \theta_w)$</td></tr><tr><td>$z_r \le L_l$</td><td></td><td>$X \le \tau + \Delta\tau_z(z_r; \theta_z)$</td><td>$c_2 + 2c_1 + c_c + c_d(\tau + \Delta\tau_z(z_r; \theta_z) - X)$</td><td>$\tau + \Delta\tau_z(z_r; \theta_z)$</td></tr><tr><td>$z_r \le L_l$</td><td></td><td>$X > \tau + \Delta\tau_z(z_r; \theta_z)$</td><td>$c_2 + 2c_1 + c_p$</td><td>$\tau + \Delta\tau_z(z_r; \theta_z)$</td></tr></tbody></table>

Details about the functional forms used in this paper to model $\Delta\tau_z(z_r; \theta_z)$ and $\Delta\tau_w(w_r; \theta_w)$ as well as details about the strategy adopted to select them are provided in Section 5.1.

### 3.2. The cost model

The cost model considers the cost of a preventive replacement $c_p$, the cost of a corrective replacement $c_c$, ($c_c > c_p$), the cost of a perfect inspection $c_w$, the cost of an imperfect inspection $c_z$, a logistic cost $c_l$ (quantifying, for example, the cost of reaching the site where the unit is located) that is incurred every time a maintenance action is performed (i.e., either an inspection or a replacement), and a downtime cost, which accounts for cost caused by operating the unit in a failed state. This latter cost is computed by multiplying a fixed downtime cost rate $c_d$ by the downtime (i.e., the time elapsing between the failure occurrence and the replacement time).

Treating the logistic cost separately from inspection and replacement costs allows penalizing unrealistic maintenance policies which would lead to postpone the replacement for a very short time (i.e., those for which the economic benefit produced by postponing the replacement is smaller than the logistic cost).

Table 2, where $X$ is the lifetime, lists all the possible evolutions of the maintenance cycle, together with the corresponding cost and length, which are denoted by $C(z_r, w_r, X)$ and $T(z_r, w_r)$, respectively. In Table 2, coherently with the assumption that failures are not self-announcing, $X$ is always denoted with the capital letter, to intend that the lifetime of the unit should always be regarded as a random variable.

The first and second column of Table 2 report the outcome of the imperfect and perfect inspections, respectively. In the cases where the perfect inspection is not carried out in the second column there is an empty cell. It is useful to remark that, although $T(z_r, w_r)$ and $C(z_r, w_r, X)$ functionally depend on $\xi$, for economy of notation the adopted symbols do not highlight this dependency.

### 4. Formulation of the long-run average maintenance cost rate

By using the renewal/reward theorem (e.g., see Ross, 1983), the long-run average maintenance cost rate $C_\infty(\xi)$ can be formulated as:

$$C_\infty(\xi) = \frac{E\{C(Z(\tau), W(\tau), X)\}}{E\{T(Z(\tau), W(\tau))\}} \qquad (28)$$

where expectations have to be taken with respect to $Z(\tau)$, $W(\tau)$, and $X$.

The results reported in this section are obtained by assuming that the degradation process is the one described in Section 2. However, analogous formulas can be derived for other similar continuous-state continuous-time perturbed degradation processes (e.g., Wiener-based and inverse Gaussian-based perturbed processes).

The numerator of Eq. (28) can be expressed as in Eq. (29), where $f_{Z(t)|W(t)}(z|w)$ and $F_{Z(t)|W(t)}(z|w)$ are either those in Eqs. (15) and (16) or those in Eqs. (17) and (18), depending on the adopted error model. The
denominator can be expressed as in Eq. (30). The integrals in Eqs. (29) and (30) do not allow for a closed form solution. However, they can be easily computed numerically. Details about the derivation of Eqs. (29) and (30) are provided in Appendix A.

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= c_L + c_z + c_p + (c_c - c_p + c_d \cdot \tau) \cdot \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right. \nonumber \\
&\qquad \left. + c_w \cdot \int_0^\infty \left[F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)\right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \right. \nonumber \\
&\qquad \left. + c_L \cdot \int_0^\infty \left[F_{Z(\tau)|W(\tau)}(L_l|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)\right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \right. \nonumber \\
&\qquad \left. + c_d \cdot \int_0^\infty \Delta \tau_z (z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dz_\tau \cdot dw_\tau \right. \nonumber \\
&\qquad \left. + c_d \cdot \int_0^\infty \Delta \tau_w (w_\tau; \theta_w) \cdot \left[F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)\right] \times \right. \nonumber \\
&\qquad \left. \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \right. \nonumber \\
&\qquad \left. + (c_c - c_p) \cdot \int_0^D \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_w(w_\tau; \theta_w)), \eta(\tau) + d\right) \right] \times \right. \nonumber \\
&\qquad \left. \left[F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)\right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \right. \nonumber \\
&\qquad \left. + (c_c - c_p) \cdot \int_0^\infty \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_z(z_\tau; \theta_z)), \eta(\tau) + d\right) \right] \times \right. \nonumber \\
&\qquad \left. \times f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dz_\tau \cdot dw_\tau \right. \nonumber \\
&\qquad \left. - c_d \cdot \int_0^\infty \int_0^{L_1} \int_0^{\tau + \Delta\tau_z(z_\tau, \theta_z)} \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \times \right. \nonumber \\
&\qquad \left. \times f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dx \cdot dz_\tau \cdot dw_\tau \right. \nonumber \\
&\qquad \left. - c_d \cdot \int_0^\infty \int_0^\tau \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \eta(x, \tau)\right) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dx \cdot dw_\tau, \right. \tag{29}
\end{align}
$$

$$
\begin{align}
E\{T(Z(\tau), W(\tau))\} &= \tau + \int_0^\infty \int_0^{L_1} \Delta \tau_z(z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \times \nonumber \\
&\quad \times \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dz_\tau \cdot dw_\tau + \int_0^D \Delta \tau_w(w_\tau; \theta_w) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) \nonumber \\
&\quad - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau. \tag{30}
\end{align}
$$

5. Comparative studies

In this section, we report the results of two comparative studies.

These studies are performed by modeling $\Delta\tau_z(z_\tau; \theta_z)$ and $\Delta\tau_w(w_\tau; \theta_w)$ as in Eqs. (31) and (32), respectively:

$$
\Delta \tau_z (z_\tau; \theta_z) = \Delta \tau^z + \theta_1^z \cdot \left( 1 - \mathbb{B} \left( \frac{z_\tau}{L_1}; \theta_2^z, \theta_3^z \right) \right), \quad (31)
$$

$$
\Delta \tau_w (w_\tau; \theta_w) = \Delta \tau^w + \theta_1^w \cdot \left( 1 - \mathbb{B} \left( \frac{w_\tau}{D}; \theta_2^w, \theta_3^w \right) \right), \quad (32)
$$

where $\theta_z = \{\Delta\tau^z, \theta_1^z, \theta_2^z, \theta_3^z\}$ and $\theta_w = \{\Delta\tau^w, \theta_1^w, \theta_2^w, \theta_3^w\}$. The choice of these functions is discussed and motivated in Section 5.1.

The first study is developed to show that the suggested adaptive policy, say $\mathcal{P}_0$, significantly outperforms the less flexible threshold limit rule suggested in Esposito et al. (2023b), say $\mathcal{P}_1$. The second study demonstrates how, depending on the considered experimental scenario, the proposed approach allows to competently decide how many and which inspection/inspections is convenient to carry out.

To achieve this second objective, we have compared the proposed policy with three alternative ones:

i) an adaptive policy that uses the imperfect inspection only, say $\mathcal{P}_2$,

ii) an adaptive policy that uses the perfect inspection only, say $\mathcal{P}_3$, and

iii) a pure age-based policy that does not use any inspection, say $\mathcal{P}_4$.

All these competing policies are defined specializing the policies $\mathcal{P}_0$. Namely, the policy $\mathcal{P}_2$ is defined by setting $L_l = L_u$, the policy $\mathcal{P}_3$ is defined by setting $L_l = 0$, $L_u = +\infty$, and $c_z = 0$, and the policy $\mathcal{P}_4$ is defined by setting $L_l = L_u = 0$ and $c_z = c_w = 0$.

It is worth to note that, by using the functions Eqs. (31) and (32), the policy $\mathcal{P}_0$ has 12 design parameters (i.e., $\tau$, $L_l$, $L_u$, $D$, $\Delta\tau^z$, $\theta_1^z$, $\theta_2^z$, $\theta_3^z$, $\Delta\tau^w$, $\theta_1^w$, $\theta_2^w$, and $\theta_3^w$). Accordingly, the policy $\mathcal{P}_2$ has 6 design parameters (i.e., $\tau$, $L_l = L_u = L$, $\Delta\tau^z$, $\theta_1^z$, $\theta_2^z$, and $\theta_3^z$), the policy $\mathcal{P}_3$ has 6 design parameters (i.e., $\tau$, $D$, $\Delta\tau^w$, $\theta_1^w$, $\theta_2^w$, and $\theta_3^w$), while the policy $\mathcal{P}_4$ has 1 design parameter (i.e., $\tau$).

Incidentally, it is worth to remark that, also the non-adaptive policy $\mathcal{P}_1$ (i.e., the one suggested in Esposito et al., 2023b) can be understood as a special case of the policy $\mathcal{P}_0$, since it can be obtained from $\mathcal{P}_0$ by setting $\Delta\tau_w(w_\tau; \theta_w) = \Delta\tau_z(z_\tau; \theta_z) = \Delta\tau$. Indeed, consistently, the policy $\mathcal{P}_1$ has 5 design parameters (i.e., $\tau$, $L_l$, $L_u$, $D$, and $\Delta\tau$).

Both studies are developed by applying the considered policies to a real-world inspired case of a buried pipeline whose failures are caused by excess of corrosion. The motivation for focusing on this application is that it allows to convincingly justify both the assumptions that (perfect and/or imperfect) inspections can be performed only at one inspection epoch during the operating life of the pipeline and that failures are not self-announcing. In fact, in the case of pipelines that are buried underground or under the sea, as inspections are difficult, very expensive, and cause downtimes and service disruptions, even the cost of only one (either perfect or imperfect) inspection could counterweigh the advantages of CBM over ABM. Moreover, as the excess of corrosion typically causes small leaks, most failures can only be detected by performing a perfect ad hoc inspection.

The comparative studies are developed considering different experimental scenarios, defined by adopting different setups both for the cost model and for the degradation model. Specifically, the scenarios are defined considering two different values (referred to as high and low) of $c_w$ and $c_z$, two different couples of values of $c_c$ and $c_d$, two different couples of values of $c$ and $d$, and two different values of $\varphi$. The parameters $c$ and $d$ modulate the magnitude of the unit-to-unit heterogeneity. In fact, the two couples of values used for $c$ and $d$ define scenarios where the random effect has the same mean (i.e., $c/d$) and two different values
(referred to as high and low) of the variance (i.e., $c/d^2$). The parameter $\varphi$ modulates the magnitude of the variance of the measurement error (which decreases with $\varphi$). Combining the resulting models we have defined 32 different (i.e., $2^3 \cdot 2^2 = 8.4 = 32$) scenarios.

The age function $\eta(t)$ of the gamma process is modeled by using the linear function $a \cdot t$, which has proved to be appropriate in the case of corrosion (e.g., see Ellingwood & Mori, 1993, Li, Yang et al., 2023). The parameter $a$ is always set to the value 0.17, which is close to the one suggested in Mahmoodian and Alani (2014). The parameter $\nu$ of the error term is always set to 1. The values of the coefficients of the cost model are inspired by those found in Dey (2004). Table 3 lists the 8 setups considered for the cost model (denoted by letters ranging from A to H) and Table 4 lists the 4 setups considered for the degradation model (denoted by numbers ranging from 1 to 4).

The 32 scenarios are thus identified by a letter and by a subscripted number. For example, the scenario $A_2$ is obtained by setting the coefficients of the cost model to the values given in the first row of Table 3 and the parameters of the degradation model to the values given in the second row of Table 4. Similarly, the scenario $F_3$ is obtained by using values given in the sixth row of Table 3 and the third row of Table 4.

Hereinafter, the expression “scenarios Y” (where $Y = A, B, ..., H$) will be used to collectively refer to scenarios $Y_1, Y_2, Y_3$, and $Y_4$ (i.e., all the scenarios defined by using the cost model Y), while the expression “scenarios i” (where $i = 1, 2, 3, 4$) will be used to collectively refer to scenarios $A_i, B_i, ..., H_i$ (i.e., all scenarios that are defined by using the degradation model i).

Scenario by scenario, the comparison of the performances of the policy $P_0$ and $P_k$ ($k = 1, 2, 3, 4$) is done by using the percent relative long-run average maintenance cost rate difference, $PRD(k)$:

$$PRD(k) = \frac{^{k}C_{\infty}(\cdot) - {}^{0}C_{\infty}(\cdot)}{{}^{0}C_{\infty}(\cdot)} \cdot 100. \quad (33)$$

where ${}^{k}C_{\infty}(\cdot)$ denotes the (optimal) long-run average maintenance cost rate obtained by adopting the policy $P_k$ ($k = 0, ..., 4$). Hence, based on its definition, the higher $PRD(k)$ the better is $P_0$ with respect to $P_k$.

A pipeline is considered failed when its degradation level exceeds the value $w_M = 35$ mm.

Details about the computation of the long-run average maintenance cost rate ${}^{k}C_{\infty}(\cdot)$ are provided in Appendix B.

## 5.1. Modeling of the postponement time functions

Choosing appropriate forms for the functions $\Delta\tau_z(z_\tau; \theta_z)$ and $\Delta\tau_w(w_\tau; \theta_w)$ is crucial to guarantee the effectiveness of the proposed policy. We discuss this task in this section because our idea is that this choice should be understood as application-dependent. Nevertheless, we are confident that the functions we suggest in this section, thanks to their extreme flexibility, could be suitable in many different experimental situations.

As general requirements, we want $\Delta\tau_z(z_\tau; \theta_z)$ and $\Delta\tau_w(w_\tau; \theta_w)$ to be non-negative, monotonic decreasing with $z_\tau$ and $w_\tau$, respectively, and

<caption>**Table 3**</caption>

Values of the coefficients of the cost model used to define the 32 experimental scenarios (costs are expressed in monetary units).

<table><caption>**Table 4**</caption><thead><tr><th></th><th>c<sub>l</sub></th><th>c<sub>p</sub></th><th>c<sub>c</sub></th><th>c<sub>d</sub></th><th>c<sub>z</sub></th><th>c<sub>w</sub></th></tr></thead><tbody><tr><td>A</td><td></td><td></td><td></td><td></td><td>0.1</td><td>0.8</td></tr><tr><td>B</td><td></td><td></td><td></td><td></td><td></td><td>0.4</td></tr><tr><td>C</td><td></td><td>6</td><td>1</td><td></td><td></td><td>0.8</td></tr><tr><td>D</td><td></td><td></td><td></td><td></td><td>0.3</td><td>0.4</td></tr><tr><td>E</td><td>0.2</td><td>2</td><td></td><td></td><td></td><td>0.8</td></tr><tr><td>F</td><td></td><td></td><td></td><td></td><td>0.1</td><td>0.4</td></tr><tr><td>G</td><td></td><td></td><td>3</td><td>0.05</td><td></td><td>0.8</td></tr><tr><td>H</td><td></td><td></td><td></td><td></td><td>0.3</td><td>0.4</td></tr></tbody></table>

Values of the parameters of the degradation model used to define the 32 experimental scenarios.

<table><thead><tr><th></th><th>ν</th><th>a</th><th>c</th><th>d</th><th>φ</th></tr></thead><tbody><tr><td>1</td><td></td><td></td><td></td><td></td><td>0.1</td></tr><tr><td>2</td><td></td><td></td><td>5</td><td>2.125</td><td>1</td></tr><tr><td>3</td><td>1</td><td>0.17</td><td></td><td></td><td>0.1</td></tr><tr><td>4</td><td></td><td></td><td>15</td><td>6.375</td><td>1</td></tr></tbody></table>

bounded above. Obviously, the list of potential candidates is endless.

Thus, to select the functional forms adopted in this paper we have conducted an extended preliminary analysis. In particular, for the sole purpose of containing the computational burden, and only for this specific task, we have developed the complete preliminary analysis considering the special case $P_3$ of the proposed policy $P_0$ that (as the ones suggested in Esposito et al., 2022 and Esposito et al., 2024) adopts only the perfect inspection procedure. Accordingly, this preliminary study has provided direct indications for the choice of the function $\Delta\tau_w(w_\tau; \theta_w)$, which we have assumed suitable also for $\Delta\tau_z(z_\tau; \theta_z)$.

As possible candidates for modeling $\Delta\tau_w(w_\tau; \theta_w)$ we have considered the functions given in Eqs. (34)-(38), which have been obtained by modifying the Pareto, the Weibull, the loglogistic, and the beta complementary cdf (ccdf). Indeed, although $\Delta\tau_w(w_\tau; \theta_w)$ does not have a conceptual link with a ccdf, this approach, thanks to the inherent features of the ccdf, allows to easily formulate simple, yet flexible, functions that satisfy all the above-mentioned requirements. In addition, for maximum flexibility, as a further option, we have also considered the step function in Eq. (39), which, at the cost of a large number of parameters, can approximate any conceivable decreasing function. For simplicity, hereinafter, the functional forms in Eqs. (34)-(39), will be referred to by using the notations $f_P(\cdot)$, $f_W(\cdot)$, $f_L(\cdot)$, $f_B(\cdot)$, $f_{DB}(\cdot)$, and $f_S(\cdot)$ respectively.

$$f_P(w_\tau; \theta_w = \{\Delta\tau, \theta_1, \theta_2\}) = \Delta\tau_w(w_\tau; \theta_w) = \Delta\tau + \theta_1 \cdot (D - w_\tau)^{\theta_2}, \quad w_\tau \le D \quad (34)$$

$$ \begin{aligned} f_W(w_\tau; \theta_w = \{\Delta\tau, \theta_1, \theta_2, \theta_3\}) &= \Delta\tau_w(w_\tau; \theta_w) \\ &= \Delta\tau + \theta_1 \cdot \left(1 - e^{-\left(\frac{D-w_\tau}{\theta_2}\right)^{\theta_3}}\right), \quad w_\tau \le D \end{aligned} \quad (35) $$

$$f_L(w_\tau; \theta_w = \{\Delta\tau, \theta_1, \theta_2, \theta_3\}) = \Delta\tau_w(w_\tau; \theta_w) = \Delta\tau + \frac{\theta_1}{1 + \left(\frac{\theta_2}{D-w_\tau}\right)^{\theta_3}}, \quad w_\tau \le D \quad (36)$$

$$ \begin{aligned} f_B(w_\tau; \theta_w = \{\Delta\tau, \theta_1, \theta_2, \theta_3\}) &= \Delta\tau_w(w_\tau; \theta_w) \\ &= \Delta\tau + \theta_1 \cdot \left(1 - \mathbb{B}\left(\frac{w_\tau}{D}; \theta_2, \theta_3\right)\right), \quad w_\tau \le D \end{aligned} \quad (37) $$

$$ \begin{aligned} f_{DB}(w_\tau; \theta_w = \{\Delta\tau, \theta_1, \theta_2, \theta_3, \theta_4, \theta_5, \theta_6\}) &= \Delta\tau_w(w_\tau; \theta_w) \\ &= \Delta\tau + \theta_1 \cdot \left(1 - \mathbb{B}\left(\frac{w_\tau}{D}; \theta_2, \theta_3\right)\right) + \theta_4 \cdot \left(1 - \mathbb{B}\left(\frac{w_\tau}{D}; \theta_5, \theta_6\right)\right), \quad w_\tau \le D \end{aligned} \quad (38) $$

$$ f_S(w_\tau; \theta_w = \{\Delta\tau_1, \Delta\tau_2, \dots, \Delta\tau_k, \theta_1, \theta_2, \dots, \theta_{k-1}\}) = \Delta\tau_w(w_\tau; \theta_w) = \begin{cases} \Delta\tau_1 & w_\tau \le \theta_1 \\ \Delta\tau_2 & \theta_1 < w_\tau \le \theta_2 \\ \vdots & \vdots \\ \Delta\tau_k & \theta_{k-1} < w_\tau \le D \end{cases} \quad (39) $$

Note that the function $f_{DB}(\cdot)$ in Eq. (38) includes as a special case the function $f_B(\cdot)$ in Eq. (37) (i.e., $f_{DB}(\cdot)$ reduces to $f_B(\cdot)$ when $\theta_4 = 0$).

The step function $f_S(\cdot)$ in Eq. (39) was already adopted in Esposito
et al. (2022) and Esposito et al. (2024). To use this latter function, in addition to the parameter vector $\theta_w$, it is also necessary to determine the value of the parameter $k$, which consists in the number of steps. As argued in Esposito et al. (2022) and Esposito et al. (2024), choosing the value of $k$ involves a trade-off between performance of the policy and computational burden. For the comparative analysis, we have set $k = 20$, which based on the results obtained in Esposito et al. (2022) and Esposito et al. (2024), is deemed appropriate for this task.

We have optimized the considered policy by adopting for $\Delta\tau_w(w_r; \theta_w)$ all the mentioned alternative functions under all scenarios defined in Tables 3 and 4. In particular, given that here we focus on the policy that uses only the perfect inspection, the scenarios that differ in terms of the values of $c_z$ and/or $\varphi$ are considered only once.

The results reported in Table 5 show that, although the considered competing functions yield very similar values of the optimal long-run average maintenance cost rate, under all the setups the best policies are always obtained by using the functions $f_B(\cdot)$ and $f_{DB}(\cdot)$, which give long-run average maintenance cost rates that coincide with each other up to the third significant figure (at least). In fact, under all the considered scenarios, $f_{DB}(\cdot)$ factually reduces to its special case $f_B(\cdot)$.

Just to give a visual example of the difference between the various functions, in Fig. 1 we report the shapes of $f_P(\cdot)$, $f_W(\cdot)$, $f_L(\cdot)$, $f_B(\cdot)$, $f_{DB}(\cdot)$, and $f_S(\cdot)$ which minimize the average long-run maintenance cost rate under the setup $A_1$. Only for this graphical evaluation, to facilitate the comparison, $\tau$ was set to the value that minimizes the cost under the function $f_B(\cdot)$. Thus, values of the long-run average cost rate obtained under the functions $f_P(\cdot)$, $f_W(\cdot)$, $f_L(\cdot)$, $f_{DB}(\cdot)$, and $f_S(\cdot)$ should be intended as constrained optima. The corresponding (constrained) optimal long-run average maintenance cost rates are reported in Table 6. The symbol $C_\infty^e(\cdot)$ in the second row of the table indicates that the value reported in that row are constrained optima.

Not all shapes are clearly visible in Fig. 1. Indeed, $f_W(\cdot)$, $f_P(\cdot)$, and $f_L(\cdot)$ are almost perfectly superimposed with each other and $f_B(\cdot)$ practically overlaps $f_{DB}(\cdot)$. This situation is also reflected by the optimal cost reported in Table 6, where the optimal cost obtained under $f_W(\cdot)$, $f_P(\cdot)$, and $f_L(\cdot)$ are coincident (at least up to the third significant figure), and so are $f_B(\cdot)$ and $f_{DB}(\cdot)$.

Based on this preliminary analysis, considered that the results obtained by using the functions $f_B(\cdot)$ and $f_{DB}(\cdot)$ coincide and that $f_B(\cdot)$ has three parameters less than $f_{DB}(\cdot)$, for the subsequent analyses, we have decided to model $\Delta\tau_w(w_r; \theta_w)$ and $\Delta\tau_z(z_r; \theta_z)$ as in Eqs. (31) and (32), by using the function $f_B(\cdot)$.

Fig. 2 gives some graphical examples of the various shapes the function in Eq. (37) can assume by changing the values of the parameters $\theta_2$ and $\theta_3$. In this figure, $f_B(\cdot)$ is represented by using normalized scales on both axes (i.e., it is $D = 1$, $\Delta\tau = 0$, $\theta_1 = 1$, and both $w_r$ and $\Delta\tau$ range between 0 and 1). However, by choosing suitable values of $D$, $\Delta\tau$, and $\theta_1$, the functions depicted in the figure can be rescaled and translated to cover any desired finite range.

<caption>**Table 5**</caption>

Average long-run maintenance cost rate obtained by using the alternative functional forms of $\Delta\tau_w(w_r; \theta_w)$ indicated in the first row. Values are multiplied by 100 for the sake of readability.

<table><thead><tr><th></th><th>f<sub>P</sub>(·)</th><th>f<sub>W</sub>(·)</th><th>f<sub>L</sub>(·)</th><th>f<sub>B</sub>(·)</th><th>f<sub>DB</sub>(·)</th><th>f<sub>S</sub>(·)</th></tr></thead><tbody><tr><td>A<sub>1</sub></td><td>6.80</td><td>6.76</td><td>6.76</td><td>6.75</td><td>6.75</td><td>6.83</td></tr><tr><td>A<sub>3</sub></td><td>5.92</td><td>5.92</td><td>5.92</td><td>5.92</td><td>5.92</td><td>5.93</td></tr><tr><td>B<sub>1</sub></td><td>6.26</td><td>6.22</td><td>6.22</td><td>6.21</td><td>6.21</td><td>6.29</td></tr><tr><td>B<sub>3</sub></td><td>5.34</td><td>5.34</td><td>5.34</td><td>5.34</td><td>5.34</td><td>5.36</td></tr><tr><td>E<sub>1</sub></td><td>4.55</td><td>4.61</td><td>4.55</td><td>4.55</td><td>4.55</td><td>4.67</td></tr><tr><td>E<sub>3</sub></td><td>4.44</td><td>4.48</td><td>4.44</td><td>4.44</td><td>4.44</td><td>4.47</td></tr><tr><td>F<sub>1</sub></td><td>4.16</td><td>4.24</td><td>4.16</td><td>4.16</td><td>4.16</td><td>4.18</td></tr><tr><td>F<sub>3</sub></td><td>4.01</td><td>4.06</td><td>4.01</td><td>4.01</td><td>4.01</td><td>4.03</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Shapes of the function $f_P(\cdot)$, $f_W(\cdot)$, $f_L(\cdot)$, $f_B(\cdot)$, $f_{DB}(\cdot)$, and $f_S(\cdot)$ that minimize the long-run average maintenance cost rate in the case of scenario $A_1$.</figcaption></figure>

<caption>**Table 6**</caption>

Values of the long-run average maintenance cost rate obtained by adopting the functions represented in Fig. 1 in the case of setup $A_1$. Values are multiplied by 100 for the sake of readability.

<table><thead><tr><th></th><th>f<sub>P</sub>(·)</th><th>f<sub>W</sub>(·)</th><th>f<sub>L</sub>(·)</th><th>f<sub>B</sub>(·)</th><th>f<sub>DB</sub>(·)</th><th>f<sub>S</sub>(·)</th></tr></thead><tbody><tr><td>C<sub>∞</sub>(·)</td><td></td><td></td><td></td><td>6.75</td><td></td><td></td></tr><tr><td>C<sub>∞</sub>(·)</td><td>6.76</td><td>6.76</td><td>6.76</td><td>6.75</td><td>6.83</td><td></td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Behavior of $f_B(\cdot)$ for various values of parameters $\theta_2$ and $\theta_3$. All the curves are obtained by setting $D = 1$, $\Delta\tau = 0$, and $\theta_1 = 1$.</figcaption></figure>

## 5.2. Comparative study 1

This first comparative study aims to measure and highlight the economic benefit that could be obtained by adopting the proposed new adaptive policy $P_0$ in place of the less flexible threshold limit policy $P_1$ suggested in Esposito et al. (2023b).

The decision rule is described in Table 1. As we already remarked, the non-adaptive policy $P_1$ can be obtained from the policy $P_0$ by setting $\Delta\tau_w(w_r; \theta_w) = \Delta\tau_z(z_r; \theta_z) = \Delta\tau$. This means that the only distinguishing element between these two policies is that, when it is
<caption>**Table 7**</caption>

Optimal values $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$ of the long-run average maintenance cost rates $^0C_\infty(\cdot)$ and $^1C_\infty(\cdot)$ and corresponding values of the index PRD(1), obtained under each scenario, by using the policies $\mathcal{P}_0$ and $\mathcal{P}_1$. The error term is modeled as in Eq. (13). Costs reported in the table are multiplied by 100 for visualization purposes.

<table><caption>**Table 8**</caption><thead><tr><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>1</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(1)</th><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>1</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(1)</th></tr></thead><tbody><tr><td>A<sub>1</sub></td><td>6.53</td><td>6.85</td><td>4.86</td><td>E<sub>1</sub></td><td>4.17</td><td>4.21</td><td>0.87</td></tr><tr><td>A<sub>2</sub></td><td>6.08</td><td>6.56</td><td>7.88</td><td>E<sub>2</sub></td><td>3.98</td><td>4.1</td><td>3.04</td></tr><tr><td>A<sub>3</sub></td><td>5.49</td><td>5.65</td><td>2.97</td><td>E<sub>3</sub></td><td>3.94</td><td>3.95</td><td>0.4</td></tr><tr><td>A<sub>4</sub></td><td>5.12</td><td>5.38</td><td>5.03</td><td>E<sub>4</sub></td><td>3.81</td><td>3.86</td><td>1.29</td></tr><tr><td>B<sub>1</sub></td><td>6.25</td><td>6.71</td><td>7.42</td><td>F<sub>1</sub></td><td>4.14</td><td>4.19</td><td>1.3</td></tr><tr><td>B<sub>2</sub></td><td>6.08</td><td>6.53</td><td>7.40</td><td>F<sub>2</sub></td><td>3.98</td><td>4.10</td><td>3.04</td></tr><tr><td>B<sub>3</sub></td><td>5.39</td><td>5.59</td><td>3.71</td><td>F<sub>3</sub></td><td>3.94</td><td>3.95</td><td>0.4</td></tr><tr><td>B<sub>4</sub></td><td>5.12</td><td>5.38</td><td>4.97</td><td>F<sub>4</sub></td><td>3.79</td><td>3.86</td><td>1.75</td></tr><tr><td>C<sub>1</sub></td><td>6.83</td><td>7.15</td><td>4.61</td><td>G<sub>1</sub></td><td>4.35</td><td>4.38</td><td>0.64</td></tr><tr><td>C<sub>2</sub></td><td>6.37</td><td>6.86</td><td>7.65</td><td>G<sub>2</sub></td><td>4.21</td><td>4.28</td><td>1.63</td></tr><tr><td>C<sub>3</sub></td><td>5.82</td><td>5.99</td><td>2.84</td><td>G<sub>3</sub></td><td>4.15</td><td>4.16</td><td>0.29</td></tr><tr><td>C<sub>4</sub></td><td>5.43</td><td>5.70</td><td>4.95</td><td>G<sub>4</sub></td><td>4.03</td><td>4.07</td><td>1.07</td></tr><tr><td>D<sub>1</sub></td><td>6.52</td><td>7.01</td><td>7.43</td><td>H<sub>1</sub></td><td>4.35</td><td>4.37</td><td>0.38</td></tr><tr><td>D<sub>2</sub></td><td>6.37</td><td>6.83</td><td>7.15</td><td>H<sub>2</sub></td><td>4.21</td><td>4.28</td><td>1.63</td></tr><tr><td>D<sub>3</sub></td><td>5.69</td><td>5.91</td><td>3.82</td><td>H<sub>3</sub></td><td>4.15</td><td>4.16</td><td>0.29</td></tr><tr><td>D<sub>4</sub></td><td>5.43</td><td>5.70</td><td>4.85</td><td>H<sub>4</sub></td><td>4.01</td><td>4.07</td><td>1.46</td></tr></tbody></table>

Optimal values $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$ of the long-run average maintenance cost rates $^0C_\infty(\cdot)$ and $^1C_\infty(\cdot)$ and corresponding values of the index PRD(1), obtained under each scenario, by using the policies $\mathcal{P}_0$ and $\mathcal{P}_1$. The error term is modeled as in Eq. (14). Costs reported in the table are multiplied by 100 for visualization purposes.

<table><thead><tr><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>1</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(1)</th><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>1</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(1)</th></tr></thead><tbody><tr><td>A<sub>1</sub></td><td>6.71</td><td>7.14</td><td>6.38</td><td>E<sub>1</sub></td><td>4.24</td><td>4.26</td><td>0.31</td></tr><tr><td>A<sub>2</sub></td><td>6.19</td><td>6.57</td><td>6.27</td><td>E<sub>2</sub></td><td>4.03</td><td>4.10</td><td>1.90</td></tr><tr><td>A<sub>3</sub></td><td>5.83</td><td>5.95</td><td>2.01</td><td>E<sub>3</sub></td><td>4.02</td><td>4.02</td><td>0</td></tr><tr><td>A<sub>4</sub></td><td>5.20</td><td>5.41</td><td>4.05</td><td>E<sub>4</sub></td><td>3.82</td><td>3.86</td><td>1.13</td></tr><tr><td>B<sub>1</sub></td><td>6.40</td><td>6.87</td><td>7.32</td><td>F<sub>1</sub></td><td>4.18</td><td>4.24</td><td>1.39</td></tr><tr><td>B<sub>2</sub></td><td>6.18</td><td>6.54</td><td>5.85</td><td>F<sub>2</sub></td><td>4.03</td><td>4.10</td><td>1.88</td></tr><tr><td>B<sub>3</sub></td><td>5.48</td><td>5.76</td><td>5.04</td><td>F<sub>3</sub></td><td>3.99</td><td>4.02</td><td>0.65</td></tr><tr><td>B<sub>4</sub></td><td>5.20</td><td>5.41</td><td>4.05</td><td>F<sub>4</sub></td><td>3.82</td><td>3.86</td><td>1.11</td></tr><tr><td>C<sub>1</sub></td><td>6.98</td><td>7.41</td><td>6.23</td><td>G<sub>1</sub></td><td>4.41</td><td>4.42</td><td>0.25</td></tr><tr><td>C<sub>2</sub></td><td>6.48</td><td>6.87</td><td>6.12</td><td>G<sub>2</sub></td><td>4.22</td><td>4.28</td><td>1.55</td></tr><tr><td>C<sub>3</sub></td><td>6.12</td><td>6.27</td><td>2.42</td><td>G<sub>3</sub></td><td>4.22</td><td>4.22</td><td>0.01</td></tr><tr><td>C<sub>4</sub></td><td>5.51</td><td>5.73</td><td>4.01</td><td>G<sub>4</sub></td><td>4.04</td><td>4.08</td><td>0.95</td></tr><tr><td>D<sub>1</sub></td><td>6.67</td><td>7.15</td><td>7.16</td><td>H<sub>1</sub></td><td>4.36</td><td>4.40</td><td>1.01</td></tr><tr><td>D<sub>2</sub></td><td>6.45</td><td>6.84</td><td>5.94</td><td>H<sub>2</sub></td><td>4.22</td><td>4.28</td><td>1.55</td></tr><tr><td>D<sub>3</sub></td><td>5.78</td><td>6.07</td><td>5.00</td><td>H<sub>3</sub></td><td>4.22</td><td>4.22</td><td>0</td></tr><tr><td>D<sub>4</sub></td><td>5.51</td><td>5.72</td><td>3.81</td><td>H<sub>4</sub></td><td>4.04</td><td>4.08</td><td>0.94</td></tr></tbody></table>

decided to postpone the replacement, while under the policy $\mathcal{P}_1$ the replacement is always postponed to the same predetermined time $\tau + \Delta\tau$, under the policy $\mathcal{P}_0$ the postponement time is determined adaptively, unit-by-unit, according to the outputs of the imperfect and perfect inspections, via the functions in Eqs. (31) and (32), respectively.

Tables 7 and 8 report the values $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$ of the optimal long-run average maintenance cost rates $^0C_\infty(\cdot)$ and $^1C_\infty(\cdot)$ obtained under the policy $\mathcal{P}_0$ and $\mathcal{P}_1$, respectively, for each scenario defined in Tables 3 and 4. More specifically, Table 7 reports the results obtained by using the inverse gamma-based error term in Eq. (13) and Table 8 reports the corresponding results obtained by using the Gaussian-based error term in Eq. (14).

The optimal values $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$ are determined by setting, scenario by scenario, the design parameters of the policies $\mathcal{P}_0$ and $\mathcal{P}_1$ to the values that minimize the long-run average maintenance cost rate.

For economy of space, Tables 7 and 8 do not report the (optimal) values of the design parameters used to obtain $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$.

The results reported in Tables 7 and 8 give evidence that, in the majority of considered experimental scenarios, the policy $\mathcal{P}_0$ gives significant economic benefits over its special case $\mathcal{P}_1$. In fact, all the values of the index PRD(1) reported in Tables 7 and 8 are non-negative, ranging from 0.29 (scenario $G_3$ and $H_3$) to 7.88 (scenario $A_2$) in the cases where the error term is modeled as in Eq. (13) (i.e., Table 7) and from 0.01 (scenarios $E_3$ and $H_3$) to 7.32 (scenario $B_1$), in the cases where the error term is modeled as in Eq. (14) (i.e., Table 8).

In particular, obtained results indicate that the superiority of the policy $\mathcal{P}_0$ over $\mathcal{P}_1$ increases as the unit-to-unit heterogeneity increases. In fact, the index PRD(1) increases passing from the scenarios 3 and 4 (where the heterogeneity is low) to the scenarios 1 and 2 (where the heterogeneity is high). Conversely, the presence of measurement error mitigates this superiority. Indeed, as shown in Table 7, the index PRD(1) decreases passing from the scenarios 2 and 4 (where the variance of the measurement error is small) to the scenarios 1 and 3 (where the variance of the measurement error is high).

Regarding costs, obtained results also show that, as was natural to expect, advantages obtained by using the adaptive policy $\mathcal{P}_0$ diminish as the coefficients $c_c$ and $c_d$, which determine the cost caused by a failure, diminish (that is, moving from setups A, B, C, and D to setups E, F, G, and H). In fact, when $c_c$ and $c_d$ decrease the index PRD(1) decreases.

To better clarify how the policy $\mathcal{P}_0$ attains the discussed improvement, in Figs. 3 and 4 we provide a graphical representation of the decision rules the policies $\mathcal{P}_0$ and $\mathcal{P}_1$ use at $\tau$ to decide whether to postpone or not the replacement and to define the postponement time. The error term is defined as in Eq. (13). The figures refer to the scenario $D_3$. For the sake of simplifying the comparison (and only in the case of this comparison), under the non-adaptive policy $\mathcal{P}_1$ we have set the decision parameter $\tau$ to the optimal value obtained for the same

<figure><img src="image_5.png" /><figcaption>Fig. 3. Perturbed measurement-based decision rule used under the policies $\mathcal{P}_0$ (panel A) and $\mathcal{P}_1$ (panel B) in the case of scenario $D_3$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Perfect measurement-based decision rule used under the policies $P_0$ (panel A) and $P_1$ (panel B) in the case of the scenario $D_3$.</figcaption></figure>

<caption>Table 9</caption>

Optimal value $^0C_\infty^*(\cdot)$ and (constrained) optimal value $^1C_\infty^*(\cdot)$ of $^0C_\infty^*(\cdot)$ and $^1C_\infty^*(\cdot)$ and corresponding values of the index PRD(1), obtained under the scenario $D_3$. The error term is modeled as in Eq. (13). Costs reported in the table are multiplied by 100 for visualization purposes.

<table><thead><tr><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>1</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(1)</th></tr></thead><tbody><tr><td>D<sub>3</sub></td><td>5.69</td><td>5.92</td><td>3.95</td></tr></tbody></table>

parameter under the policy $P_0$. In Table 9 the optimal value of $^1C_\infty^*(\cdot)$ obtained under this constraint is indicated as $^1C_\infty^*(\cdot)$. The corresponding unconstrained optimum (i.e., the one obtained under the same scenario $D_3$) $^1C_\infty^*(\cdot)$ is reported in Table 7. As $^1C_\infty^*(\cdot)$ (5.91) differs from $^1C_\infty^*(\cdot)$ (5.92), also the values of PRD(1) reported in the same tables do not coincide.

Figs. 3 and 4 show that, compared to the policy $P_0$ which, unit-by-unit, can define adaptive (i.e., measurement-dependent) values of $\Delta\tau_z(z_r; \theta_z)$ and $\Delta\tau_w(w_r; \theta_w)$ the policy $P_1$ uses a predetermined value $\Delta\tau$ which ends up determining postponement times lower than optimal for strong units and greater than optimal for weak units.

All in all, based on the discussed results, it is possible to affirm that the superiority of the policy $P_0$ over the policy $P_1$ is due to its enhanced ability to use the outcomes of the inspections for planning better-informed postponement times. Indeed, as remarked in Esposito et al. (2023b), in the case of the considered gamma process with random effect, this ability strongly depends on the possibility of quickly distinguishing strong units (i.e., those for which the unit-specific parameter $\Lambda$ takes a value that is above the mean) from weak units (i.e., those for which the unit-specific parameter $\Lambda$ is below the mean). In fact, the driving idea of both the CBM policies $P_0$ and $P_1$ is to perform early replacements of the weak units and to delay the replacement of strong ones. Obviously, the advantages obtained by gathering information

<caption>Table 10</caption>

Optimal values of $^0C_\infty^*(\cdot)$, $^2C_\infty^*(\cdot)$, $^3C_\infty^*(\cdot)$, and $^4C_\infty^*(\cdot)$ and corresponding PRD computed, under each scenario, by using the policies $P_0$, $P_2$, $P_3$, and $P_4$. The error term is modeled as in Eq. (13). Costs reported in the table are multiplied by 100 for visualization purposes.

<table><thead><tr><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>2</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(2)</th><th><sup>3</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(3)</th><th><sup>4</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(4)</th></tr></thead><tbody><tr><td>A<sub>1</sub></td><td>6.53</td><td>6.53</td><td>0</td><td>6.75</td><td>3.31</td><td>7.63</td><td>16.85</td></tr><tr><td>A<sub>2</sub></td><td>6.08</td><td>6.09</td><td>0.01</td><td>6.75</td><td>11.0</td><td>7.72</td><td>27.0</td></tr><tr><td>A<sub>3</sub></td><td>5.49</td><td>5.49</td><td>0</td><td>5.92</td><td>7.73</td><td>5.96</td><td>8.48</td></tr><tr><td>A<sub>4</sub></td><td>5.12</td><td>5.12</td><td>0</td><td>5.92</td><td>15.5</td><td>5.99</td><td>16.9</td></tr><tr><td>B<sub>1</sub></td><td>6.25</td><td>6.53</td><td>4.54</td><td>6.21</td><td>-0.65</td><td>7.63</td><td>22.2</td></tr><tr><td>B<sub>2</sub></td><td>6.08</td><td>6.09</td><td>0.01</td><td>6.21</td><td>1.89</td><td>7.72</td><td>26.7</td></tr><tr><td>B<sub>3</sub></td><td>5.39</td><td>5.49</td><td>1.92</td><td>5.34</td><td>-0.91</td><td>5.96</td><td>10.6</td></tr><tr><td>B<sub>4</sub></td><td>5.12</td><td>5.12</td><td>0</td><td>5.34</td><td>4.24</td><td>5.99</td><td>16.9</td></tr><tr><td>C<sub>1</sub></td><td>6.83</td><td>6.84</td><td>0.17</td><td>6.75</td><td>-1.22</td><td>7.63</td><td>11.7</td></tr><tr><td>C<sub>2</sub></td><td>6.37</td><td>6.37</td><td>0</td><td>6.75</td><td>5.9</td><td>7.72</td><td>21.2</td></tr><tr><td>C<sub>3</sub></td><td>5.82</td><td>5.82</td><td>0</td><td>5.92</td><td>1.59</td><td>5.96</td><td>2.30</td></tr><tr><td>C<sub>4</sub></td><td>5.43</td><td>5.43</td><td>0</td><td>5.92</td><td>8.87</td><td>5.99</td><td>10.2</td></tr><tr><td>D<sub>1</sub></td><td>6.52</td><td>6.84</td><td>4.9</td><td>6.21</td><td>-4.84</td><td>7.63</td><td>17.0</td></tr><tr><td>D<sub>2</sub></td><td>6.37</td><td>6.37</td><td>0</td><td>6.21</td><td>-2.59</td><td>7.72</td><td>21.2</td></tr><tr><td>D<sub>3</sub></td><td>5.69</td><td>5.82</td><td>2.32</td><td>5.34</td><td>-6.17</td><td>5.96</td><td>4.68</td></tr><tr><td>D<sub>4</sub></td><td>5.43</td><td>5.43</td><td>0</td><td>5.34</td><td>-1.74</td><td>5.99</td><td>10.2</td></tr><tr><td>E<sub>1</sub></td><td>4.17</td><td>4.17</td><td>0</td><td>4.59</td><td>10.16</td><td>4.28</td><td>2.70</td></tr><tr><td>E<sub>2</sub></td><td>3.98</td><td>4.02</td><td>1.06</td><td>4.59</td><td>15.4</td><td>4.29</td><td>7.86</td></tr><tr><td>E<sub>3</sub></td><td>3.94</td><td>3.95</td><td>0.36</td><td>4.44</td><td>12.69</td><td>3.94</td><td>0.01</td></tr><tr><td>E<sub>4</sub></td><td>3.81</td><td>3.81</td><td>0</td><td>4.44</td><td>16.5</td><td>3.94</td><td>3.44</td></tr><tr><td>F<sub>1</sub></td><td>4.14</td><td>4.17</td><td>0.73</td><td>4.23</td><td>2.09</td><td>4.28</td><td>3.47</td></tr><tr><td>F<sub>2</sub></td><td>3.98</td><td>4.02</td><td>1.03</td><td>4.23</td><td>6.17</td><td>4.29</td><td>7.86</td></tr><tr><td>F<sub>3</sub></td><td>3.94</td><td>3.95</td><td>0.35</td><td>4.01</td><td>1.77</td><td>3.94</td><td>0</td></tr><tr><td>F<sub>4</sub></td><td>3.79</td><td>3.81</td><td>0.05</td><td>4.01</td><td>5.70</td><td>3.94</td><td>3.91</td></tr><tr><td>G<sub>1</sub></td><td>4.35</td><td>4.35</td><td>0</td><td>4.59</td><td>5.62</td><td>4.28</td><td>-1.54</td></tr><tr><td>G<sub>2</sub></td><td>4.21</td><td>4.22</td><td>0.02</td><td>4.59</td><td>9.00</td><td>4.29</td><td>1.86</td></tr><tr><td>G<sub>3</sub></td><td>4.15</td><td>4.15</td><td>0</td><td>4.44</td><td>6.91</td><td>3.94</td><td>-5.12</td></tr><tr><td>G<sub>4</sub></td><td>4.03</td><td>4.03</td><td>0.02</td><td>4.44</td><td>10.05</td><td>3.94</td><td>-2.28</td></tr><tr><td>H<sub>1</sub></td><td>4.35</td><td>4.35</td><td>0</td><td>4.23</td><td>-2.87</td><td>4.28</td><td>-1.54</td></tr><tr><td>H<sub>2</sub></td><td>4.21</td><td>4.22</td><td>0.02</td><td>4.23</td><td>0.23</td><td>4.29</td><td>1.83</td></tr><tr><td>H<sub>3</sub></td><td>4.15</td><td>4.15</td><td>0</td><td>4.01</td><td>-3.44</td><td>3.94</td><td>-5.13</td></tr><tr><td>H<sub>4</sub></td><td>4.01</td><td>4.03</td><td>0.41</td><td>4.01</td><td>0</td><td>3.94</td><td>-1.90</td></tr></tbody></table>

about $\Lambda$ increase as the variance of $\Lambda$ increases. On the other hand, by using the same arguments, it is also obvious that the utility of the information gathered by performing the imperfect inspection decreases as the variance of the measurement error increases (i.e., as $\varphi$ decreases).

### 5.3. Comparative study 2

In this second comparative study, we compare the performances of the policy $P_0$ with those of its special cases $P_2$, $P_3$, and $P_4$. The main aim of this study is to show how, by using the policy $P_0$ and its special cases $P_2$, $P_3$, and $P_4$, it is possible to decide, scenario by scenario, whether it is appropriate to perform inspections and what kind of inspections (i.e., imperfect and/or perfect) is more convenient to use. In fact, obtained results prove that, depending on the scenario, the policy which provides the smallest (optimal) long-run average maintenance cost rate could be any of the four considered policies.

In addition, this study also serves to show that, in experimental situations where the inspection costs are very high, the best policy could be even the pure age-based policy $P_4$. This result demonstrates that in the considered experimental situation, focusing on maintenance models that rely on (either imperfect and/or perfect) degradation measurements gathered at a single inspection epoch could effectively solve the trade-off
between model performance and model complexity.

The study has been conducted by applying the policies $P_2$, $P_3$, and $P_4$ in each of the 32 experimental scenarios we already considered to develop the comparative study 1.

Table 10 reports the optimal values $^0C_\infty^*(\cdot)$, $^3C_\infty^*(\cdot)$, and $^4C_\infty^*(\cdot)$ of the long-run average maintenance cost rates $^2C_\infty(\cdot)$, $^3C_\infty(\cdot)$, and $^4C_\infty(\cdot)$ and the corresponding values of the index PRD(2), PRD(3), and PRD(4), obtained under each scenario, by using the policies $P_2$, $P_3$, and $P_4$. For the convenience of readers, Table 10 also reports the values of $^0C_\infty^*(\cdot)$ we already reported in Table 7.

All the results reported in Table 10 have been obtained by using the error term in Eq. (13). For economy of space, Table 10 does not report the optimal values of the design parameters of the considered policies.

As a preliminary comment, from Table 10 we note that, although the policies $P_3$ and $P_4$ can be intended as special cases of the policy $P_0$, the indexes PRD(3) and PRD(4) can take negative values. This is possible because to let $P_0$ exactly reduce to $P_3$ and $P_4$ it is necessary to set $c_z = 0$, and $c_z = c_w = 0$, respectively. In fact, while under the policy $P_3$ accessing the information provided by the perfect measurement incurs the cost $c_w$, under the policy $P_0$ it incurs the cost $c_w + c_z$, since the imperfect information must always be performed first. Therefore, in experimental scenarios where the policy $P_0$ uses very frequently the perfect inspection it could result more convenient to adopt the policy $P_3$.

In the case two policies exhibit equal performances we will assume that the most effective is the simplest (i.e., the one that uses fewer inspections).

The results reported in Table 10 give evidence that:

* under the scenarios A, B, C, and D, where the corrective cost $c_c$ and the downtime cost rate $c_d$ are set to the high values (i.e., when the costs caused by failures are high), regardless of the values of c, d, or $\varphi$, the proposed policy $P_0$ greatly outperforms the pure age-based policy $P_4$. Actually, under the same scenarios, the policy $P_4$ is outperformed also by the CBM policies $P_2$ and $P_3$ (which use only the imperfect and perfect inspection, respectively). This result confirms that (e.g., see also Esposito et al., 2023b), the main advantage of CBM over ABM is that CBM allows exploiting the information gathered by performing inspections to prolong the useful lifetime of the units while mitigating the risk of failure. This ability is further boosted under the scenarios where the unit-to-unit heterogeneity is high (i.e., those where the variance $c/d^2$ is large) and the variance of the measurement error is small (i.e., those where $\varphi$ is set to the high value). In fact, for example, the value of the index PRD(4) decreases passing from the scenario $C_1$ to $C_3$ that differ only in the variance of the random effect (which is high in $C_1$ and low in $C_3$) and increases passing from the scenario $C_1$ to $C_2$ which differ only in the variance of the measurement error (which is high in $C_1$ and low in $C_2$);

* as the costs induced by failures decrease, the advantages of using CBM policies with respect to the ABM policy $P_4$ decrease. In fact, as $c_c$ and $c_d$ decrease (i.e., moving from $A_i$ to $E_i$, $B_i$ to $F_i$, $C_i$ to $G_i$, and $D_i$ to $H_i$, $\forall i = 1, 2, 3, 4$) advantages provided by the CBM policies $P_0$, $P_2$, and $P_3$ with respect to the ABM policy $P_4$ decrease. Indeed, by comparing the values obtained for the index PRD(2), PRD(3), and

<caption>Table 11</caption>

Optimal values of $^0C_\infty(\cdot)$, $^2C_\infty(\cdot)$, $^3C_\infty(\cdot)$, and $^4C_\infty(\cdot)$ and corresponding PRD computed, for each scenario, by using the policies $P_0$, $P_2$, $P_3$, and $P_4$. The error term is modeled as in Eq. (14). Costs reported in the table are multiplied by 100 for visualization purposes.

<table><thead><tr><th></th><th><sup>0</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th><sup>2</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(2)</th><th><sup>3</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(3)</th><th><sup>4</sup>C<sub>∞</sub><sup>*</sup>(·)</th><th>PRD(4)</th></tr></thead><tbody><tr><td>A<sub>1</sub></td><td>6.71</td><td>7.42</td><td>10.6</td><td>6.75</td><td>0.51</td><td>7.63</td><td>13.7</td></tr><tr><td>A<sub>2</sub></td><td>6.19</td><td>6.23</td><td>0.75</td><td>6.75</td><td>9.07</td><td>7.72</td><td>24.8</td></tr><tr><td>A<sub>3</sub></td><td>5.83</td><td>6.04</td><td>3.60</td><td>5.92</td><td>1.50</td><td>5.96</td><td>2.21</td></tr><tr><td>A<sub>4</sub></td><td>5.20</td><td>5.21</td><td>0.22</td><td>5.92</td><td>13.9</td><td>5.99</td><td>15.3</td></tr><tr><td>B<sub>1</sub></td><td>6.40</td><td>7.42</td><td>15.9</td><td>6.21</td><td>-3.07</td><td>7.63</td><td>19.2</td></tr><tr><td>B<sub>2</sub></td><td>6.18</td><td>6.23</td><td>0.90</td><td>6.21</td><td>0.48</td><td>7.72</td><td>25.0</td></tr><tr><td>B<sub>3</sub></td><td>5.48</td><td>6.04</td><td>10.1</td><td>5.34</td><td>-2.61</td><td>5.96</td><td>8.66</td></tr><tr><td>B<sub>4</sub></td><td>5.20</td><td>5.21</td><td>0.21</td><td>5.34</td><td>2.77</td><td>5.99</td><td>15.3</td></tr><tr><td>C<sub>1</sub></td><td>6.98</td><td>7.69</td><td>10.3</td><td>6.75</td><td>-3.32</td><td>7.63</td><td>9.35</td></tr><tr><td>C<sub>2</sub></td><td>6.48</td><td>6.51</td><td>0.44</td><td>6.75</td><td>4.18</td><td>7.72</td><td>19.2</td></tr><tr><td>C<sub>3</sub></td><td>6.12</td><td>6.36</td><td>3.90</td><td>5.92</td><td>-3.42</td><td>5.96</td><td>-2.74</td></tr><tr><td>C<sub>4</sub></td><td>5.51</td><td>5.51</td><td>0.09</td><td>5.92</td><td>7.41</td><td>5.99</td><td>8.73</td></tr><tr><td>D<sub>1</sub></td><td>6.67</td><td>7.69</td><td>15.3</td><td>6.21</td><td>-7.01</td><td>7.63</td><td>14.34</td></tr><tr><td>D<sub>2</sub></td><td>6.45</td><td>6.51</td><td>0.82</td><td>6.21</td><td>-3.81</td><td>7.72</td><td>19.6</td></tr><tr><td>D<sub>3</sub></td><td>5.78</td><td>6.36</td><td>10.2</td><td>5.34</td><td>-7.58</td><td>5.96</td><td>3.12</td></tr><tr><td>D<sub>4</sub></td><td>5.51</td><td>5.51</td><td>0.09</td><td>5.34</td><td>-3.06</td><td>5.99</td><td>8.73</td></tr><tr><td>E<sub>1</sub></td><td>4.24</td><td>4.29</td><td>1.04</td><td>4.59</td><td>8.23</td><td>4.28</td><td>0.90</td></tr><tr><td>E<sub>2</sub></td><td>4.03</td><td>4.03</td><td>0.06</td><td>4.59</td><td>14.1</td><td>4.29</td><td>6.64</td></tr><tr><td>E<sub>3</sub></td><td>4.02</td><td>4.04</td><td>0.44</td><td>4.44</td><td>10.3</td><td>3.94</td><td>-2.09</td></tr><tr><td>E<sub>4</sub></td><td>3.82</td><td>3.82</td><td>0</td><td>4.44</td><td>16.1</td><td>3.94</td><td>3.12</td></tr><tr><td>F<sub>1</sub></td><td>4.18</td><td>4.29</td><td>2.59</td><td>4.23</td><td>1.10</td><td>4.28</td><td>2.47</td></tr><tr><td>F<sub>2</sub></td><td>4.03</td><td>4.03</td><td>0.05</td><td>4.23</td><td>4.95</td><td>4.29</td><td>6.63</td></tr><tr><td>F<sub>3</sub></td><td>3.99</td><td>4.04</td><td>1.08</td><td>4.01</td><td>0.28</td><td>3.94</td><td>-1.47</td></tr><tr><td>F<sub>4</sub></td><td>3.82</td><td>3.82</td><td>0</td><td>4.01</td><td>4.87</td><td>3.94</td><td>3.10</td></tr><tr><td>G<sub>1</sub></td><td>4.41</td><td>4.44</td><td>0.78</td><td>4.59</td><td>4.26</td><td>4.28</td><td>-2.81</td></tr><tr><td>G<sub>2</sub></td><td>4.22</td><td>4.22</td><td>0.04</td><td>4.59</td><td>8.91</td><td>4.29</td><td>1.77</td></tr><tr><td>G<sub>3</sub></td><td>4.22</td><td>4.23</td><td>0.26</td><td>4.44</td><td>5.12</td><td>3.94</td><td>-6.71</td></tr><tr><td>G<sub>4</sub></td><td>4.04</td><td>4.04</td><td>0</td><td>4.44</td><td>9.79</td><td>3.94</td><td>-2.51</td></tr><tr><td>H<sub>1</sub></td><td>4.36</td><td>4.44</td><td>1.88</td><td>4.23</td><td>-3.07</td><td>4.28</td><td>-1.75</td></tr><tr><td>H<sub>2</sub></td><td>4.22</td><td>4.22</td><td>0.04</td><td>4.23</td><td>0.17</td><td>4.29</td><td>1.77</td></tr><tr><td>H<sub>3</sub></td><td>4.22</td><td>4.23</td><td>0.27</td><td>4.01</td><td>-5.05</td><td>3.94</td><td>-6.70</td></tr><tr><td>H<sub>4</sub></td><td>4.04</td><td>4.04</td><td>0.00</td><td>4.01</td><td>-0.85</td><td>3.94</td><td>-2.52</td></tr></tbody></table>

PRD(4), it also results that under the scenarios $F_3$, $G_1$, $G_3$, $G_4$, $H_3$, and $H_4$, the ABM policy $P_4$ even outperforms the CBM policy $P_0$, $P_2$, and $P_3$, being equivalent to $P_0$ under the scenario $F_3$. Obtained results also show (see scenarios $E_1$, $E_2$, $E_3$, $E_4$, $G_1$, $G_2$, and $H_1$) that the convenience of making inspections depends on the adopted decision rule. For example, in the scenario $F_4$, the ABM policy $P_4$ outperforms the CBM policy $P_3$, while it is worse than the CBM policies $P_0$ and $P_2$;

* advantages gained by performing the perfect inspection decrease when its cost $c_w$ increases. This effect can be observed by comparing the results obtained under the scenarios which differ only in the cost $c_w$ of the perfect inspection (i.e., $A_i$ vs $B_i$, $C_i$ vs $D_i$, $E_i$ vs $F_i$, and $G_i$ vs $H_i$, $\forall i = 1, 2, 3, 4$).

In fact, for example, the value of the index PRD(2) decreases passing from the scenario $D_1$, where the cost $c_w$ of the perfect inspection is low, to the corresponding scenario $C_1$, where $c_w$ is high. Actually, values obtained for the index PRD(2), PRD(3), and PRD(4), show that under the scenarios $A_1$, $A_3$, $A_4$, $C_2$, $C_3$, $C_4$, $E_1$, and $E_4$,
where $c_w$ is high, the policy $\mathcal{P}_2$ is the one that provides the best results, being factually equivalent to the more complex policy $\mathcal{P}_0$;

* advantages gained by performing the imperfect inspection decrease as its cost $c_z$ increases. This effect can be observed by comparing the results obtained under the scenarios which differ only in the cost $c_z$ of the imperfect inspection (i.e., $A_i$ vs $C_i$, $B_i$ vs $D_i$, $E_i$ vs $G_i$, and $F_i$ vs $H_i$, $\forall i = 1, 2, 3, 4$). In fact, for example, the index PRD(3) which serves to compare the performance of policy $\mathcal{P}_0$ with those of the policy $\mathcal{P}_3$ (which uses only the perfect inspection) decreases passing from the scenario $B_1$, where $c_z$ is low, to the corresponding scenario $D_1$, where $c_z$ is high. Actually, the values obtained for the index PRD(2), PRD(3), and PRD(4), show that under the scenarios $C_1, D_1, D_2, D_3, D_4$, and $H_1$, where $c_z$ is high, the policy $\mathcal{P}_3$ is the one that provides the best results;

* the advantages provided by the policy $\mathcal{P}_0$ with respect to the policy $\mathcal{P}_2$, which uses only the imperfect inspection, increase as the variance of the random effect increases. This effect can be observed by comparing the results obtained under the scenarios which differ only in the amount of heterogeneity (i.e., 3 vs 1 and 4 vs 2). In fact, for example, the values of the index PRD(2) increase moving from the scenarios 3 and 4 where the variance of the random effect is low to the scenarios 1 and 2, respectively, where the variance of the random effect is high;

* the advantages provided by the policy $\mathcal{P}_0$ with respect to the policy $\mathcal{P}_3$, which uses only the perfect inspection, increase as the variance of the measurement error decreases. This effect can be observed by comparing the results obtained under the scenarios which differ only in the value of the variance of the measurement error (i.e., 1 vs 2 and 3 vs 4). In fact, for example, the value of the index PRD(3) increases moving from the scenarios 1 and 3, where the variance of the measurement error is high, to the scenarios 2 to 4, respectively, where the variance of the measurement error is low.

More in general, the results reported in Table 10 show that, depending on the combined effects of variance of the random effect, variance of the measurement error, inspection costs, and costs induced by failures, any of the considered competing maintenance policies can be preferred to the other ones. In fact:

* the policy $\mathcal{P}_0$ is the best one under the scenarios $A_2, B_2, E_2, E_3, F_1, F_2, F_4, G_2$, and $H_2$;

* the policy $\mathcal{P}_2$ is the best one under the scenarios $A_1, A_3, A_4, B_4, C_2, C_3, C_4, E_1$, and $E_4$;

* the policy $\mathcal{P}_3$ is the best one under the scenarios $B_1, B_3, C_1, D_1, D_2, D_3, D_4$, and $H_1$;

* the policy $\mathcal{P}_4$ is the best one under the scenarios $F_3, G_1, G_3, G_4, H_3$, and $H_4$.

We have repeated the same comparative study by using the error term in Eq. (14). The results we obtained are reported in Table 11.

As a first general comment on the results reported in Table 11, we see that almost all the considerations we made about the results reported in Table 10 still apply also for the results reported in Table 11. Nonetheless, from Table 11 it also seems that the long-run average maintenance cost rates obtained by using the Gaussian error term are generally higher than those obtained by using the inverse gamma one, despite the mean and the variance of the considered error terms coinciding. Indeed, it also

seems that said difference tends to decrease as the variance of the error term decreases (i.e., passing from the scenarios 1 and 3 to the scenarios 2 and 4).

The study we performed does not allow to provide a definitive explanation for this effect. However, we think that it could depend on the fact that under the Gaussian error term the perturbed measurement is allowed to take negative values, whereas under the inverse gamma term it is constrained to be non-negative. This implies that when the variance of the error term is high (in particular when the error term can take values that are comparable to the measured degradation level) the shapes of the conditional pdfs of the error terms in Eqs. (13) and (14) tend to differ one from the other more than when the variance is small.

Finally, in general, it also seems that under the Gaussian error term the imperfect inspection is less valuable than under the inverse gamma one, especially when the magnitude of the measurement error is high.

## 6. Conclusions

In this paper, we have proposed a new adaptive maintenance policy for gradually degrading units which operate in the simultaneous presence of unit-to-unit variability and measurement error. The policy consists in performing, at a predefined epoch, an imperfect inspection, which provides a perturbed measurement of the degradation level of the unit. Based on the outcome of this measurement, by using a condition-based rule, it is decided whether to immediately replace the unit, to postpone its replacement to a later time, or to perform (for an additional cost) a perfect inspection which returns the actual degradation level of the unit. When this latter measurement is performed, its outcome is then used to decide whether to immediately replace the unit or to postpone its replacement to a future time. The postponement time is defined at the inspection time, unit-by-unit, by using an adaptive rule, on the basis of the available degradation measurements. Postponed replacements are carried out without performing other inspections, regardless of the degradation state of the units. Each replacement restores the unit to an as good as new state. The optimal policy is determined by minimizing the long-run average maintenance cost rate.

The actual degradation process is described by using a gamma model with random effect. The measurement error is modeled by using two different solutions, consisting in an inverse gamma- and a Gaussian-based error model. Both solutions can describe cases where the measurement error depends in stochastic terms on the measured degradation level. Failures are defined by the first passage time of the actual degradation process to a fixed threshold and are supposed to be not self-announcing.

The utility and effectiveness of the proposed policy have been demonstrated via two comparative studies where the maintained unit is a buried pipeline. The first comparative study has shown that the proposed adaptive policy outperforms a similar policy, recently suggested in the literature, which uses a (less flexible) threshold limit-based decision rule. The second comparative study has demonstrated how, depending on the considered experimental scenario, the proposed approach allows to competently decide how many and which inspection/inspections is convenient to carry out. Incidentally, obtained results also show that performing inspections (more than one or even only one) is not always convenient and that convenience depends both on inspection/maintenance costs and on the effectiveness of the adopted inspection rule.
We consider useful emphasizing that the main objective of the paper was to demonstrate the advantages that the proposed adaptive decision rule can provide over the standard threshold limit rule. We focused on a policy that uses the decision rule only once because this maintenance scheme facilitates the comparison. We know that this assumption could be considered too restrictive. For example, we recognize that it would be interesting to extend the proposed maintenance policy by assuming that the information gathered at the inspection time can be used not only to plan the replacement time (as it is now) but also to plan additional inspections, if deemed cost-effective. Nevertheless, we decided not to deal with this extension in this paper since it would imply a huge increase of the complexity of the resulting maintenance model. This is because, differently than the majority of the literature related to maintenance of gamma degrading units (where the reference degradation process is a homogenous gamma process), the degradation model we consider has increments that are neither independent nor stationary and the

perturbed process is even non-Markovian.

CRediT authorship contribution statement

**Nicola Esposito:** Writing – review & editing, Writing – original draft, Software, Methodology, Formal analysis, Conceptualization.
**Antonio Piscopo:** Writing – review & editing, Software, Methodology, Conceptualization. **Bruno Castanier:** Writing – review & editing, Supervision, Methodology, Conceptualization. **Massimiliano Giorgio:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Formal analysis, Conceptualization.

Declaration of competing interest

The author(s) declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix A

Eq. (29) can be derived starting as follows:

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= \nonumber \\
&= \int_0^\infty \int_0^\infty \int_0^\infty C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&= \int_0^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&= \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \tag{A.1} \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\tau C(z_\tau, w_\tau, x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber
\end{align}
$$

Hence, from the Eq. (A.1), replacing $C(z_\tau, w_\tau, x)$ (integral by integral) with the appropriate expression specified in Table 2, the Eq. (A.2) is readily obtained:
$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} = & \int_0^{w_M} \int_{L_u}^{\infty} \int_{\tau}^{\infty} (c_l + c_z + c_p) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_{w_M}^{\infty} \int_{L_u}^{\infty} \int_{0}^{\tau} [c_l + c_z + c_c + c_d \cdot (\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_0^{D} \int_{L_l}^{L_u} \int_{\tau+\Delta\tau_w(w_\tau, \theta_w)}^{\infty} [2 \cdot c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau_w(w_\tau; \theta_w) - x)] f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_{w_M}^{w_M} \int_{L_l}^{L_u} \int_{\tau}^{\infty} (c_l + c_z + c_w + c_p) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \tag{A.2} \\
& + \int_0^{\infty} \int_{L_l}^{\infty} \int_{0}^{\tau} [c_l + c_z + c_w + c_c + c_d \cdot (\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_{w_M}^{w_M} \int_{L_l}^{L_u} \int_{0}^{\tau+\Delta\tau_z(z_\tau, \theta_z)} [2 \cdot c_l + c_z + c_c + c_d \cdot (\tau + \Delta\tau_z(z_\tau; \theta_z) - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_0^{\infty} \int_{L_l}^{\infty} \int_{0}^{\tau+\Delta\tau_z(z_\tau, \theta_z)} (2 \cdot c_l + c_z + c_p) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + \int_{w_M}^{w_M} \int_{0}^{\infty} \int_{0}^{\tau} [2 \cdot c_l + c_z + c_c + c_d \cdot (\tau + \Delta\tau_z(z_\tau; \theta_z) - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \nonumber
\end{align}
$$

where:

$$
\int_0^{w_M} \int_{L_u}^{\infty} \int_{\tau}^{\infty} (c_l + c_z + c_p) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau = \int_0^{w_M} \int_{L_u}^{\infty} (c_l + c_z + c_p) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau
$$

and:

$$
\int_D^{w_M} \int_{L_l}^{L_u} \int_{\tau}^{\infty} (c_l + c_z + c_w + c_p) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau = \int_D^{w_M} \int_{L_l}^{L_u} (c_l + c_z + c_w + c_p) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau.
$$

Eq. (A.2) can be then rewritten as:

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \nonumber \\
& + c_l \cdot F_{Z(\tau)}(L_l) + c_l \cdot \int_0^{L_l} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& + (c_c - c_p) \cdot \int_0^{w_M} F_{X|W(\tau)}(\tau + \Delta\tau_w(w_\tau; \theta_w))|w_\tau| \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& + (c_c - c_p) \cdot \int_0^{w_M} \int_0^{L_l} F_{X|W(\tau)}(\tau + \Delta\tau_z(z_\tau; \theta_z)|w_\tau) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \nonumber \\
& + c_d \cdot \int_0^{w_M} \int_0^{L_l} \int_{\tau}^{\tau+\Delta\tau_z(z_\tau; \theta_z)} \left[ \int_{\tau}^{\tau+\Delta\tau_z(z_\tau; \theta_z)} (\tau + \Delta\tau_z(z_\tau; \theta_z) - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \right] \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \tag{A.3} \\
& + c_d \cdot \int_0^{w_M} \int_0^{L_l} \int_{\tau}^{\tau+\Delta\tau_w(w_\tau; \theta_w)} \left[ \int_{\tau}^{\tau+\Delta\tau_w(w_\tau; \theta_w)} (\tau + \Delta\tau_w(w_\tau; \theta_w) - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \right] \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& - c_d \cdot \int_0^{w_M} \int_0^{\infty} \left[ \int_{0}^{\tau} x \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \right] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber
\end{align}
$$

where, integrating by parts, and using the cdfs in Eqs (26) and (27), the integrals in the last three lines of (A.3) that involves the useful life X can be reformulated as it follows:
$$ \int_{\tau}^{\tau+Y} (\tau+Y-x) f_{X|W(\tau)}(x|w_{\tau}) \cdot dx = - (\tau+Y-x) \cdot (1-F_{X|W(\tau)}(x|w_{\tau})) \Big|_{\tau}^{\tau+Y} - \int_{\tau}^{\tau+Y} (1-F_{X|W(\tau)}(x|w_{\tau})) \cdot dx = Y - \int_{\tau}^{\tau+Y} \mathbb{B} \left( \frac{w_M - w_{\tau}}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot dx, $$

with Y equal either to $\Delta\tau_w(w_\tau; \theta_w)$ or to $\Delta\tau_z(z_\tau; \theta_z)$, and

$$ \int_{0}^{\tau} x \cdot f_{X|W(\tau)}(x|w_{\tau}) \cdot dx = [-x \cdot (1 - F_{X|W(\tau)}(x|w_{\tau}))]_{0}^{\tau} + \int_{0}^{\tau} (1 - F_{X|W(\tau)}(x|w_{\tau})) \cdot dx = \int_{0}^{\tau} \mathbb{B} \left( \frac{w_M}{w_{\tau}}, \eta(x), \Delta\eta(x, \tau) \right) \cdot dx. $$

By exploiting also these results Eq. (A.3) can be rearranged as:

$$ \begin{aligned} E\{C(Z(\tau), W(\tau), X)\} = & c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \\ & + c_l \cdot F_{Z(\tau)}(L_l) + c_d \cdot \int_{0}^{\infty} \int_{0}^{\tau} \Delta\tau_z(z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \\ & + (c_c - c_p) \cdot \int_{0}^{\infty} \int_{0}^{\tau} \Delta\tau_z(z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \\ & + (c_c - c_p) \cdot \int_{0}^{\infty} \int_{0}^{\tau} \int_{0}^{L_l} \int_{0}^{\tau+\Delta\tau_z(z_\tau; \theta_z)} \mathbb{B} \left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_w(w_\tau; \theta_w)), \eta(\tau) + d \right) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \\ & + (c_c - c_p) \cdot \int_{0}^{\infty} \int_{0}^{\tau} \int_{0}^{L_l} \int_{0}^{\tau+\Delta\tau_z(z_\tau; \theta_z)} \mathbb{B} \left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_z(z_\tau; \theta_z)), \eta(\tau) + d \right) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \\ & + c_d \cdot \int_{0}^{\infty} \int_{0}^{\tau} \Delta\tau_w(w_\tau; \theta_w) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \\ & - c_d \cdot \int_{0}^{\infty} \int_{0}^{L_l} \int_{0}^{\tau+\Delta\tau_z(z_\tau; \theta_z)} \mathbb{B} \left( \frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dz_\tau \cdot dw_\tau \\ & - c_d \cdot \int_{0}^{\infty} \int_{0}^{\tau+\Delta\tau_w(w_\tau; \theta_w)} \mathbb{B} \left( \frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dw_\tau \\ & - c_d \cdot \int_{0}^{\infty} \int_{0}^{\tau} \int_{0}^{L_l} \int_{0}^{\tau+\Delta\tau_w(w_\tau; \theta_w)} \mathbb{B} \left( \frac{w_M}{w_\tau}; \eta(x), \eta(x, \tau) \right) \cdot f_{W(\tau)}(w_\tau) \cdot dx \cdot dw_\tau. \end{aligned} \quad (A.4) $$

Eq. (A.4) reduces to Eq. (29) once $F_{W(\tau)}(\cdot)$ and $f_{W(\tau)}(\cdot)$ are replaced with the expressions given in Eqs. (5) and (6), respectively.

Eq. (30) can be more easily derived by following the scheme given in Table 2. In fact, more directly, it is:

$$ \begin{aligned} E\{T(Z(\tau), W(\tau))\} &= \tau \cdot [1 - F_{Z(\tau)}(L_u)] + \tau \cdot \int_{D}^{\infty} \int_{L_l}^{L_u} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau \\ &\quad + \int_{0}^{\infty} \int_{0}^{L_l} (\tau + \Delta\tau_z(z_\tau; \theta_z)) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau + \int_{0}^{\infty} \int_{L_l}^{L_u} (\tau + \Delta\tau_w(w_\tau; \theta_w)) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \\ &= \tau + \int_{0}^{\infty} \int_{0}^{L_l} \Delta\tau_z(z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau + \int_{0}^{\infty} \int_{L_l}^{L_u} \Delta\tau_w(w_\tau; \theta_w) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dz_\tau \cdot dw_\tau. \end{aligned} \quad (A.5) $$

which reduces to Eq. (30) once $f_{W(\tau)}(\cdot)$ is replaced with the expression given in Eq. (5).

## Appendix B

Under the policy $\mathcal{P}_k(k = 1, 2, 3, 4)$, the long-run average maintenance cost rate is computed as:

$$ \mathcal{P}_k = \frac{E\{C(Z(\tau), W(\tau), X)\}}{E\{T(Z(\tau), W(\tau))\}} $$

where $E\{C(Z(\tau), W(\tau), X)\}$ and $E\{T(Z(\tau), W(\tau))\}$ are obtained as special cases from Eqs. (29) and (30), respectively, by imposing the constraint relative to policy $\mathcal{P}_k$.

Thus, in the case of policy $\mathcal{P}_2$, by imposing the constraint $L_l = L_u$ the Eq. (29) reduces to:
$$
\begin{align}
E\{^2C(Z(\tau), W(\tau), X)\} &= c_l + c_z + c_p + (c_c - c_p + c_d \cdot \tau) \cdot \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M + c}; \eta(\tau), d\right) \right] \nonumber \\
&\quad + c_l \cdot \int_0^\infty \int_0^\infty f_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \nonumber \\
&\quad + c_d \cdot \int_0^\infty \int_0^\infty \Delta\tau_z(z_\tau; \theta_z) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}z_\tau \cdot \mathrm{d}w_\tau \tag{B.1} \\
&\quad + (c_c - c_p) \cdot \int_0^\infty \int_0^\infty \left(1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_z(z_\tau; \theta_z)), \eta(\tau) + d\right)\right) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}z_\tau \cdot \mathrm{d}w_\tau \nonumber \\
&\quad - c_d \cdot \int_0^\infty \int_0^\infty \int_0^\infty \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}x \cdot \mathrm{d}z_\tau \cdot \mathrm{d}w_\tau \nonumber \\
&\quad - c_d \cdot \int_{w_M}^\infty \int_0^\tau \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \eta(x, \tau)\right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}x \cdot \mathrm{d}w_\tau, \nonumber
\end{align}
$$

and the Eq. (30) reduces to:

$$
E\{^2T(Z(\tau), W(\tau))\} = \tau + \int_0^\infty \int_0^\infty \Delta\tau_z(z_\tau; \theta_z) \cdot f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}z_\tau \cdot \mathrm{d}w_\tau \quad (B.2)
$$

Similarly, in the case of policy $\mathcal{P}_3$, by imposing $L_l = 0, L_u = +\infty$, and $c_z = 0$, it is:

$$
\begin{align}
E\{^3C(Z(\tau), W(\tau), X)\} &= c_l + c_p + c_w + (c_c - c_p + c_d \cdot \tau) \cdot \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M + c}; \eta(\tau), d\right) \right] \nonumber \\
&\quad + c_l \cdot \int_0^D \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau + c_d \cdot \int_0^D \Delta\tau_w(w_\tau; \theta_w) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \nonumber \\
&\quad + (c_c - c_p) \cdot \int_0^D \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau_w(w_\tau; \theta_w)), \eta(\tau) + d\right) \right] \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \tag{B.3} \\
&\quad - c_d \cdot \int_0^D \int_\tau^\tau \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}x \cdot \mathrm{d}w_\tau \nonumber \\
&\quad - c_d \cdot \int_{w_M}^\infty \int_0^\tau \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \eta(x, \tau)\right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}x \cdot \mathrm{d}w_\tau, \nonumber
\end{align}
$$

$$
E\{^3T(Z(\tau), W(\tau))\} = \tau + \int_0^D \Delta\tau_w(w_\tau; \theta_w) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau. \quad (B.4)
$$

Finally, in the case of policy $\mathcal{P}_4$, by imposing $L_l = L_u = 0$, and $c_z = 0$, it results:

$$
E\{^4C(Z(\tau), W(\tau), X)\} = c_l + c_p + (c_c - c_p + c_d \cdot \tau) \cdot \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M + c}; \eta(\tau), d\right) \right] - c_d \cdot \int_{w_M}^{\infty} \int_0^{\tau} \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}x \cdot \mathrm{d}w_\tau, \quad (B.5)
$$

$$
E\{^4T(Z(\tau), W(\tau))\} = \tau \qquad (B.6)
$$

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