# A Hybrid Maintenance Policy for a Deteriorating Unit in the Presence of Three Forms of Variability

Nicola Esposito, Agostino Mele, Bruno Castanier, Massimiliano Giorgio

► To cite this version:

Nicola Esposito, Agostino Mele, Bruno Castanier, Massimiliano Giorgio. A Hybrid Maintenance Policy for a Deteriorating Unit in the Presence of Three Forms of Variability. Reliability Engineering and System Safety, 2023, pp.109320. 10.1016/j.ress.2023.109320. hal-04095371

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https://univ-angers.hal.science/hal-04095371

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# A Hybrid Maintenance Policy for a Deteriorating Unit in the Presence of Three Forms of Variability

Nicola Esposito, Agostino Mele, Bruno Castanier, Massimiliano Giorgio

► To cite this version:

Nicola Esposito, Agostino Mele, Bruno Castanier, Massimiliano Giorgio. A Hybrid Maintenance Policy for a Deteriorating Unit in the Presence of Three Forms of Variability. Reliability Engineering and System Safety, 2023, pp.109320. 10.1016/j.ress.2023.109320. hal-04095371

HAL Id: hal-04095371

https://univ-angers.hal.science/hal-04095371v1

Submitted on 23 Jan 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# A hybrid maintenance policy for a deteriorating unit in the presence of three forms of variability

Nicola Esposito<sup>a, b</sup>, Agostino Mele<sup>c, d</sup>, Bruno Castanier<sup>a</sup>, Massimiliano GIORGIO<sup>b, *</sup>

<figure><img src="image_3.png" /></figure>

Check for
updates

## ARTICLE INFO

### Keywords:

- Gamma degradation process

- Measurement error

- Random effect

- Long-run average maintenance cost rate

- Condition-based decision rule

## ABSTRACT

In this paper, a hybrid condition-/age-based maintenance policy is proposed for deteriorating units whose observed degradation paths are affected by temporal variability, unit to unit variability, and measurement error. The hidden degradation process is described by using a gamma model with random effect. The measurement error is modeled as a non-Gaussian random variable. Units are assumed to fail when their degradation level exceeds a given threshold.

The suggested maintenance policy consists in using the information obtained at a predetermined inspection time to decide whether to immediately replace the unit or to postpone its replacement to a successive pre-determined time. The inspection involves measuring the degradation level of the unit. It is supposed that an imperfect procedure is firstly adopted, which provides a perturbed measurement of the true degradation level. However, it is also supposed that a more expensive perfect measurement procedure can be optionally used, depending on the result provided by the imperfect one. The proposed maintenance policy is applied to a real-world inspired case of a pipeline. The performances of the policy are measured in terms of the long-run average maintenance cost rate. Obtained results demonstrate the affordability and utility of the proposed approach.

## 1. Introduction

The aim of this paper is to propose a new hybrid condition-/age-based maintenance policy for deteriorating units whose observed degradation paths are affected by temporal variability, unit to unit variability, and measurement error that uses in a combined manner an imperfect (low cost) measurement procedure and a perfect measurement procedure. The units are assumed to fail when their degradation level exceeds an assigned failure threshold. Maintenance strategies of units subjected to this kind of failure are customarily developed by adopting either an age-based or a condition-based approach (e.g., see Ahmad and Kamaruddin [1] and Alaswad and Xiang [2]). These strategies aim at optimizing some objective function, generally the long-run average maintenance cost rate (e.g., see Wang and Pham [3], Gertsbakh [4], Finkelstein et al. [5], and Cha et al. [6]).

As a rule of thumb, condition-based maintenance (CBM) strategies tend to give advantages with respect to age-based maintenance (ABM)

ones, allowing to reduce the occurrence of failures and to use the units for a larger portion of their useful life (see de Jonge et al. [7] for an extensive discussion about the relative benefits of CBM with respect to ABM).

In this paper, we focus on cases where the degradation phenomenon is described by gamma process-based models. The performance of CBM in the case of gamma deteriorating units has been largely investigated in the literature (e.g., see van Noortwijk [8] and Alaswad and Xiang [2]). A potential limitation of many existing works (e.g., see Grall et al. [9], Castanier et al. [10], Castanier et al. [11], Fouladirad and Grall [12], Huynh et al. [13], Cholette et al. [14]) is that they assume that inspections provide the exact value of the measured degradation level. Unfortunately, especially when the degradation level is measured through in-service nondestructive inspection methods (e.g., see Lu et al. [15]), the measurements are often affected by errors that can undermine the performances of CBM strategies. A simple CBM policy for gamma deteriorating units that accounts for the presence of measurement error

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

<footnote><sup>b</sup> Dipartimento di Ingegneria Industriale, Università di Napoli Federico II, P.le V. Tecchio 80, Napoli 80125, Italia.</footnote>

<footnote><sup>c</sup> Università degli studi della Campania “Luigi Vanvitelli”, Aversa, Italia.</footnote>

<footnote><sup>d</sup> Kineton, Napoli, Italia.</footnote>

<footnote>* Corresponding.</footnote>

<footnote>E-mail address: massimiliano.giorgio@unina.it (M. GIORGIO).</footnote>
<figure><img src="image_2.png" /></figure>

# A hybrid maintenance policy for a deteriorating unit in the presence of three forms of variability

Nicola Esposito<sup>a, b</sup>, Agostino Mele<sup>c, d</sup>, Bruno Castanier<sup>a</sup>, Massimiliano GIORGIO<sup>b, *</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Gamma degradation process

Measurement error

Random effect

Long-run average maintenance cost rate

Condition-based decision rule

## ABSTRACT

In this paper, a hybrid condition-/age-based maintenance policy is proposed for deteriorating units whose observed degradation paths are affected by temporal variability, unit to unit variability, and measurement error. The hidden degradation process is described by using a gamma model with random effect. The measurement error is modeled as a non-Gaussian random variable. Units are assumed to fail when their degradation level exceeds a given threshold.

The suggested maintenance policy consists in using the information obtained at a predetermined inspection time to decide whether to immediately replace the unit or to postpone its replacement to a successive pre-determined time. The inspection involves measuring the degradation level of the unit. It is supposed that an imperfect procedure is firstly adopted, which provides a perturbed measurement of the true degradation level. However, it is also supposed that a more expensive perfect measurement procedure can be optionally used, depending on the result provided by the imperfect one. The proposed maintenance policy is applied to a real-world inspired case of a pipeline. The performances of the policy are measured in terms of the long-run average maintenance cost rate. Obtained results demonstrate the affordability and utility of the proposed approach.

## 1. Introduction

The aim of this paper is to propose a new hybrid condition-/age-based maintenance policy for deteriorating units whose observed degradation paths are affected by temporal variability, unit to unit variability, and measurement error that uses in a combined manner an imperfect (low cost) measurement procedure and a perfect measurement procedure. The units are assumed to fail when their degradation level exceeds an assigned failure threshold. Maintenance strategies of units subjected to this kind of failure are customarily developed by adopting either an age-based or a condition-based approach (e.g., see Ahmad and Kamaruddin [1] and Alaswad and Xiang [2]). These strategies aim at optimizing some objective function, generally the long-run average maintenance cost rate (e.g., see Wang and Pham [3], Gertsbakh [4], Finkelstein et al. [5], and Cha et al. [6]).

As a rule of thumb, condition-based maintenance (CBM) strategies tend to give advantages with respect to age-based maintenance (ABM)

ones, allowing to reduce the occurrence of failures and to use the units for a larger portion of their useful life (see de Jonge et al. [7] for an extensive discussion about the relative benefits of CBM with respect to ABM).

In this paper, we focus on cases where the degradation phenomenon is described by gamma process-based models. The performance of CBM in the case of gamma deteriorating units has been largely investigated in the literature (e.g., see van Noortwijk [8] and Alaswad and Xiang [2]). A potential limitation of many existing works (e.g., see Grall et al. [9], Castanier et al. [10], Castanier et al. [11], Fouladirad and Grall [12], Huynh et al. [13], Cholette et al. [14]) is that they assume that inspections provide the exact value of the measured degradation level. Unfortunately, especially when the degradation level is measured through in-service nondestructive inspection methods (e.g., see Lu et al. [15]), the measurements are often affected by errors that can undermine the performances of CBM strategies. A simple CBM policy for gamma deteriorating units that accounts for the presence of measurement error

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

<footnote><sup>b</sup> Dipartimento di Ingegneria Industriale, Università di Napoli Federico II, P.le V. Tecchio 80, Napoli 80125, Italia.</footnote>

<footnote><sup>c</sup> Università degli studi della Campania “Luigi Vanvitelli”, Aversa, Italia.</footnote>

<footnote><sup>d</sup> Kineton, Napoli, Italia.</footnote>

<footnote>* Corresponding.</footnote>

<footnote>E-mail address: massimiliano.giorgio@unina.it (M. GIORGIO).</footnote>
is suggested in Kallen and van Noortwijk [16]. More recently, Nguyen et al. [17] proposed a CBM policy where the quality of the inspection procedure is included in the maintenance model as a design parameter. Similarly, Hao et al. [18] developed a gamma process-based CBM strategy where measurements are performed considering the possibility of using in a combined manner a perfect and an imperfect inspection procedure.

Another potential limitation of many existing CBM policies is that they assume that preventive replacements are linked to inspections, in the sense that the formers can be made only at inspection epochs (e.g., see Wang and Pham [19] and Huynh et al. [20]). However, given that inspection costs can be very high, it is reasonable to presume that maintenance policies that allow to avoid inspections that are expected to give rise to foregone decisions could provide economic advantages (e.g., see Fauriat and Zio [21], Yuan et al. [22], and Kim et al. [23]). With this idea in mind, Huynh et al. [20] proposed a new maintenance policy where repairs/replacements and inspections are decided and scheduled separately, showing that this strategy can give benefits in terms of long-run average maintenance cost rate. In this paper, the observed degradation phenomenon is described by using a standard homogeneous gamma process, which can represent a potential limitation for its use in practical settings. However, Huynh et al. [20] are neither the first nor the only ones to propose a maintenance policy that exploits this idea. In fact, a maintenance scheme where a degradation measurement is used to decide whether to perform an immediate replacement or to plan a future replacement time was already developed in Crowder and Lawless [24]. In this latter paper, for illustrative purposes, the authors discuss in detail the case where the unit is subjected to a single inspection. Nonetheless, they also envisage the possibility of extending the policy by making a second inspection at the future scheduled replacement time, instead of performing an automatic replacement. Very recently, this general idea has been further addressed in Finkelstein et al. [25], Cha et al. [26], and Cha et al. [27], who have developed policies based on the stronger assumptions that only one inspection is allowed over a maintenance cycle (i.e., between two consecutive repairs/replacements) and that the future replacement time is a design parameter whose value is defined a priori, regardless of the measured degradation level. This latter maintenance policy is especially well suited in the case of units whose degradation state cannot be continuously monitored, such as (e.g., see Alaswad and Xiang [2]) buried pipelines in oil and gas industries, reformer tubes in an ammonia plant, or drilling rigs. In fact, the state of these units can be checked only through expensive (ad hoc) inspections, which usually entail the interruption of operation or the shutdown of a plant.

In Crowder and Lawless [24], the degradation process of the considered units is modeled by using the gamma process with random effect firstly presented in Lawless and Crowder [28]. In Finkelstein et al. [25], the degradation process is modeled by using a non-homogeneous gamma process. In Cha et al. [26] it is described by using a transformed gamma process (Giorgio et al. [29]), whereas in Cha et al. [27] it is considered the case of a mixed population composed of two different (i.e., strong and weak) gamma degrading items.

Inspired by the maintenance scheme proposed in Finkelstein et al. [25], this paper suggests a hybrid condition-/age-based maintenance policy, where the information on the degradation level of the unit, obtained through a single inspection performed at a predetermined time, is used to decide whether to immediately replace the unit or to postpone its replacement to a successive predetermined time.

The main novelty with respect to Finkelstein et al. [25], Cha et al. [26], and Cha et al. [27], is that here the maintenance policy is developed by considering the joint presence of temporal variability, unit-to-unit variability (random effect), and measurement error. In fact, the observed degradation phenomenon is modeled by using the perturbed gamma process with random effect proposed in Esposito et al. [30], where the (hidden) true degradation phenomena is the gamma process with random effect firstly suggested in Lawless and Crowder [28] and the measurement error is a non-Gaussian random variable that

depends in stochastic sense on the measured degradation level.

Another relevant difference with respect to the mentioned works is that, here, failures are assumed to be not self-announcing (Hu et al. [31], Bautista et al. [32], Bismut et al. [33]). This type of failures are frequently encountered in practice because failure thresholds are often defined on the basis of a contractual clause or a regulatory constraint that includes a safety factor. Thus, exceeding the threshold does entail a failure as such. As a direct consequence of this assumption, in this paper it is supposed that failures can be detected only through inspections and that measurements affected by error do not allow to assess with certainty whether a unit is failed or not. Indeed, more specifically, it is assumed that failed units can continue to operate, albeit with reduced performance and/or additional cost, without giving evidence that their degradation level has already passed the threshold. Anyhow, the lifetime of a unit is assumed to coincide with the first passage time of its actual (hidden) degradation process to the specified failure threshold.

Finally, another distinguish feature of the suggested maintenance policy is that, alike Hao et al. [18], it uses in a combined manner two measurement procedures. Specifically, the proposed strategy consists in exploiting the information provided by an imperfect (low quality, perturbed) measurement procedure to decide whether to immediately replace the unit, to postpone its replacement to a future predefined replacement time, or to resort to a second (more costly) perfect measurement procedure, which reveals the exact degradation level of the unit. In the latter case, the decision about whether to replace the unit is finally made on the basis of the exact degradation measurement.

The key idea of the policy is to use the information acquired through the inspection to timely identify and immediately replace the units that are already failed or close to fail, while postponing to the successive predetermined time the replacement of those whose degradation level is safely below the failure threshold. We believe that, especially in the presence of random effect, where the inspection can allow to distinguish weak units from strong ones (see also Cha et al. [27]), the suggested hybrid maintenance policy could outperform the standard condition-based and age-based ones by allowing for a globally more convenient trade-off between inspection, corrective maintenance, and preventive maintenance costs. Obviously, the presence of measurement error and the possibility of adopting two different measurement procedures complicate the decision-making. In fact, while using only the imperfect measurement procedure saves maintenance costs, paying for the perfect one allows for better-informed decision-making. The ratio-nale of the suggested decision rule is to pay the (optional) extra cost only when it is believed that knowing the exact degradation level of the unit could affect the final decision.

In order to demonstrate the utility and feasibility of the proposed maintenance policy, we have applied it to a real-world inspired case of a buried pipeline. In the application, the performances of the suggested policy are evaluated and compared to those of three alternative simpler policies obtained as special cases from the suggested one, by assuming that only one (either the imperfect or the perfect one, respectively) or none of the considered inspections is made. The comparison is made in terms of long-run average maintenance cost rate. Obtained results demonstrate the affordability and the utility of the proposed maintenance policy.

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, the cost model, and the formulation of the long-run average maintenance cost rate. Section 4 presents the application example. Section 5 provides some conclusions.

## 2. The considered perturbed gamma degradation process with random effect

A perturbed degradation process {$Z(t); t \ge 0$} is a model that is adopted when the actual degradation process {$W(t); t \ge 0$} cannot be
is suggested in Kallen and van Noortwijk [16]. More recently, Nguyen et al. [17] proposed a CBM policy where the quality of the inspection procedure is included in the maintenance model as a design parameter. Similarly, Hao et al. [18] developed a gamma process-based CBM strategy where measurements are performed considering the possibility of using in a combined manner a perfect and an imperfect inspection procedure.

Another potential limitation of many existing CBM policies is that they assume that preventive replacements are linked to inspections, in the sense that the formers can be made only at inspection epochs (e.g., see Wang and Pham [19] and Huynh et al. [20]). However, given that inspection costs can be very high, it is reasonable to presume that maintenance policies that allow to avoid inspections that are expected to give rise to foregone decisions could provide economic advantages (e.g., see Fauriat and Zio [21], Yuan et al. [22], and Kim et al. [23]). With this idea in mind, Huynh et al. [20] proposed a new maintenance policy where repairs/replacements and inspections are decided and scheduled separately, showing that this strategy can give benefits in terms of long-run average maintenance cost rate. In this paper, the observed degradation phenomenon is described by using a standard homogeneous gamma process, which can represent a potential limitation for its use in practical settings. However, Huynh et al. [20] are neither the first nor the only ones to propose a maintenance policy that exploits this idea. In fact, a maintenance scheme where a degradation measurement is used to decide whether to perform an immediate replacement or to plan a future replacement time was already developed in Crowder and Lawless [24]. In this latter paper, for illustrative purposes, the authors discuss in detail the case where the unit is subjected to a single inspection. Nonetheless, they also envisage the possibility of extending the policy by making a second inspection at the future scheduled replacement time, instead of performing an automatic replacement. Very recently, this general idea has been further addressed in Finkelstein et al. [25], Cha et al. [26], and Cha et al. [27], who have developed policies based on the stronger assumptions that only one inspection is allowed over a maintenance cycle (i.e., between two consecutive repairs/replacements) and that the future replacement time is a design parameter whose value is defined a priori, regardless of the measured degradation level. This latter maintenance policy is especially well suited in the case of units whose degradation state cannot be continuously monitored, such as (e.g., see Alaswad and Xiang [2]) buried pipelines in oil and gas industries, reformer tubes in an ammonia plant, or drilling rigs. In fact, the state of these units can be checked only through expensive (ad hoc) inspections, which usually entail the interruption of operation or the shutdown of a plant.

In Crowder and Lawless [24], the degradation process of the considered units is modeled by using the gamma process with random effect firstly presented in Lawless and Crowder [28]. In Finkelstein et al. [25], the degradation process is modeled by using a non-homogeneous gamma process. In Cha et al. [26] it is described by using a transformed gamma process (Giorgio et al. [29]), whereas in Cha et al. [27] it is considered the case of a mixed population composed of two different (i.e., strong and weak) gamma degrading items.

Inspired by the maintenance scheme proposed in Finkelstein et al. [25], this paper suggests a hybrid condition-/age-based maintenance policy, where the information on the degradation level of the unit, obtained through a single inspection performed at a predetermined time, is used to decide whether to immediately replace the unit or to postpone its replacement to a successive predetermined time.

The main novelty with respect to Finkelstein et al. [25], Cha et al. [26], and Cha et al. [27], is that here the maintenance policy is developed by considering the joint presence of temporal variability, unit-to-unit variability (random effect), and measurement error. In fact, the observed degradation phenomenon is modeled by using the perturbed gamma process with random effect proposed in Esposito et al. [30], where the (hidden) true degradation phenomena is the gamma process with random effect firstly suggested in Lawless and Crowder [28] and the measurement error is a non-Gaussian random variable that

depends in stochastic sense on the measured degradation level.

Another relevant difference with respect to the mentioned works is that, here, failures are assumed to be not self-announcing (Hu et al. [31], Bautista et al. [32], Bismut et al. [33]). This type of failures are frequently encountered in practice because failure thresholds are often defined on the basis of a contractual clause or a regulatory constraint that includes a safety factor. Thus, exceeding the threshold does entail a failure as such. As a direct consequence of this assumption, in this paper it is supposed that failures can be detected only through inspections and that measurements affected by error do not allow to assess with certainty whether a unit is failed or not. Indeed, more specifically, it is assumed that failed units can continue to operate, albeit with reduced performance and/or additional cost, without giving evidence that their degradation level has already passed the threshold. Anyhow, the lifetime of a unit is assumed to coincide with the first passage time of its actual (hidden) degradation process to the specified failure threshold.

Finally, another distinguish feature of the suggested maintenance policy is that, alike Hao et al. [18], it uses in a combined manner two measurement procedures. Specifically, the proposed strategy consists in exploiting the information provided by an imperfect (low quality, perturbed) measurement procedure to decide whether to immediately replace the unit, to postpone its replacement to a future predefined replacement time, or to resort to a second (more costly) perfect measurement procedure, which reveals the exact degradation level of the unit. In the latter case, the decision about whether to replace the unit is finally made on the basis of the exact degradation measurement.

The key idea of the policy is to use the information acquired through the inspection to timely identify and immediately replace the units that are already failed or close to fail, while postponing to the successive predetermined time the replacement of those whose degradation level is safely below the failure threshold. We believe that, especially in the presence of random effect, where the inspection can allow to distinguish weak units from strong ones (see also Cha et al. [27]), the suggested hybrid maintenance policy could outperform the standard condition-based and age-based ones by allowing for a globally more convenient trade-off between inspection, corrective maintenance, and preventive maintenance costs. Obviously, the presence of measurement error and the possibility of adopting two different measurement procedures complicate the decision-making. In fact, while using only the imperfect measurement procedure saves maintenance costs, paying for the perfect one allows for better-informed decision-making. The ratio-nale of the suggested decision rule is to pay the (optional) extra cost only when it is believed that knowing the exact degradation level of the unit could affect the final decision.

In order to demonstrate the utility and feasibility of the proposed maintenance policy, we have applied it to a real-world inspired case of a buried pipeline. In the application, the performances of the suggested policy are evaluated and compared to those of three alternative simpler policies obtained as special cases from the suggested one, by assuming that only one (either the imperfect or the perfect one, respectively) or none of the considered inspections is made. The comparison is made in terms of long-run average maintenance cost rate. Obtained results demonstrate the affordability and the utility of the proposed maintenance policy.

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, the cost model, and the formulation of the long-run average maintenance cost rate. Section 4 presents the application example. Section 5 provides some conclusions.

## 2. The considered perturbed gamma degradation process with random effect

A perturbed degradation process {$Z(t); t \ge 0$} is a model that is adopted when the actual degradation process {$W(t); t \ge 0$} cannot be
observed due to the presence of a perturbing term $\epsilon(t)$, here intended as a measurement error. To fully define this kind of process it is necessary to give a stochastic description of the hidden degradation process, of the measurement error, and of their reciprocal relationship.

By following Esposito et al. [30], the perturbed gamma process with random effect adopted in this paper is defined by assuming that:

(i) the perturbed measurement $Z(t)$ can be expressed as:

$$Z(t) = W(t) + \epsilon(t) \quad (1)$$

where $W(t)$ is the actual (measured) degradation level of the unit at the time $t$ and $\epsilon(t)$ is the associated measurement error;

(ii) the hidden process $\{W(t); t \ge 0\}$ is the gamma process with random effect firstly suggested in Lawless and Crowder [28]. Under this model, the degradation process of the units pertaining to the (heterogeneous) population of interest is described by using a gamma process that has an assigned age function and scale parameter that varies randomly from unit to unit according to a gamma probability distribution. More specifically, the conditional pdf of the degradation increment $\Delta W(t, t + \Delta t) = W(t + \Delta t) - W(t)$, given the value $\lambda$ of the random scale parameter $\Lambda$, is expressed as:

$$f_{\Delta W(t, t+\Delta t)|\Lambda}(\delta|\lambda) = \frac{\lambda^{\Delta\eta(t, t+\Delta t)} \cdot \delta^{\Delta\eta(t, t+\Delta t)-1}}{\Gamma(\Delta\eta(t, t+\Delta t))} \cdot e^{-\lambda \cdot \delta}, \quad \delta \ge 0 \quad (2)$$

and the scale parameter $\Lambda$ is assumed to have the gamma pdf:

$$f_{\Lambda}(\lambda) = \frac{e^{d} \cdot \lambda^{d-1}}{\Gamma(d)} \cdot e^{-c \cdot \lambda}, \quad \lambda > 0, \quad (3)$$

where $\Gamma(y) = \int_{0}^{\infty} u^{y-1} \cdot e^{-u} \cdot du$ denotes the complete gamma function, $\eta(t)$ is a monotonic increasing function usually referred to as the age function, $\eta(t, t + \Delta t) = \eta(t + \Delta t) - \eta(t)$, $d > 0$ is the shape parameter of the pdf (3), and $c > 0$ is the scale parameter. It is assumed that $W(0) = 0$ and that $\eta(0) = 0$;

(iii) the measurement error $\epsilon(t)$, given $W(t) = w$, has the conditional probability density function (pdf):

$$f_{\epsilon(t)|W(t)}(\epsilon|w) = \frac{[\alpha(w)]^{\beta(w)} \cdot (\epsilon + w)^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{\frac{a(w)}{\epsilon+w}}, \quad -w \le \epsilon < \infty \quad (4)$$

where:

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

Hence, given $W(t) = w$, the measurement error $\epsilon(t)$ has null conditional mean:

$$E\{\epsilon(t)|W(t) = w\} = \frac{\alpha(w)}{\beta(w) - 1} - w = 0 \quad (5)$$

and conditional variance that is a power function of $w$:

$$V\{\epsilon(t)|W(t) = w\} = \frac{[\alpha(w)]^2}{[\beta(w) - 1]^2 \cdot [\beta(w) - 2]} = \frac{w^\nu}{\varphi}. \quad (6)$$

The pdf (4) consents to model simply and effectively the case where the measurement error depends, in a stochastic sense, on the measured degradation level $W(t)$ (see also Pulcini [34]) and allows the perturbed measurement to be surely non negative, a result that is not guaranteed by the classical assumption that the measurement error is Gaussian distributed, especially when measurement uncertainty is large, a situation that often occurs

when degradation measurements are obtained through low cost indirect measurement procedures. Further motivations for the use of the pdf (4) are provided in Giorgio et al. [35].

(iv) for any $n > 1$, any $i = 1, ..., n$, and any set of times $t_1, ..., t_n$, the measurement error $\epsilon(t_i)$ given $W(t_i) = w_i$ is conditionally independent both on $\epsilon(t_1), ..., \epsilon(t_{i-1}), \epsilon(t_{i+1}), ..., \epsilon(t_n)$ and $W(t_1), ..., W(t_{i-1}), W(t_{i+1}), ..., W(t_n)$. Consequently, it also results that $Z(t_i)$ given $W(t_i) = w_i$ is conditionally independent both on $Z(t_1), ..., Z(t_{i-1}), Z(t_{i+1}), ..., Z(t_n)$ and $W(t_1), ..., W(t_{i-1}), W(t_{i+1}), ..., W(t_n)$.

From (4), the perturbed measurement $Z(t) = W(t) + \epsilon(t)$, given $W(t) = w$, is an inverse gamma random variable with conditional pdf:

$$f_{Z(t)|W(t)}(z|w) = \frac{[\alpha(w)]^{\beta(w)} \cdot z^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{-\frac{\alpha(w)}{z}}, \quad z \ge 0, \quad (7)$$

conditional cumulative distribution function (cdf):

$$F_{Z(t)|W(t)}(z|w) = \frac{\gamma(\beta(w), \alpha(w)/z)}{\Gamma(\beta(w))}, \quad z \ge 0, \quad (8)$$

conditional mean:

$$E\{Z(t)|W(t) = w\} = E\{\epsilon(t)|W(t) = w\} + w = w, \quad (9)$$

and conditional variance:

$$V\{Z(t)|W(t) = w\} = V\{Z(t)|W(t) = w\} = \frac{w^\nu}{\varphi}, \quad (10)$$

where $\gamma(y, s) = \int_{0}^{s} u^{y-1} \cdot e^{-u} \cdot du$ is the lower incomplete gamma function.

Moreover, from (2) and (3), it results that the hidden degradation process $W(t)$ has marginal pdf:

$$f_{W(t)}(w) = \frac{c^d \cdot w^{\eta(t)-1}}{B(\eta(t), d) \cdot (w + c)^{\eta(t)+d}}, \quad w \ge 0, \quad (11)$$

marginal cdf:

$$F_{W(t)}(w) = \mathbb{B}\left(\frac{w}{w+c}; \eta(t), d\right), \quad w \ge 0, \quad (12)$$

mean (that exists for $d > 1$):

$$E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1}, \quad (13)$$

and variance (that exists for $d > 2$):

$$V\{W(t)\} = \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)}, \quad (14)$$

where:

$$\mathbb{B}(z; \alpha, \beta) = \frac{\int_{0}^{z} u^{\alpha-1} \cdot (1-u)^{\beta-1} \cdot du}{B(\alpha, \beta)}, \quad (15)$$

is the regularized beta function and

$$B(\alpha, \beta) = \frac{\Gamma(\alpha) \cdot \Gamma(\beta)}{\Gamma(\alpha + \beta)}, \quad (16)$$

is the beta function.

More specifically, it is possible to prove (e.g., see Esposito et al. [30]) that under the considered setting the hidden (marginal) degradation process $\{W(t); t \ge 0\}$, is Markovian and that the degradation increment $\Delta W(t, t + \Delta t)$, given $W(t) = w$, has conditional pdf:
observed due to the presence of a perturbing term $\epsilon(t)$, here intended as a measurement error. To fully define this kind of process it is necessary to give a stochastic description of the hidden degradation process, of the measurement error, and of their reciprocal relationship.

By following Esposito et al. [30], the perturbed gamma process with random effect adopted in this paper is defined by assuming that:

(i) the perturbed measurement $Z(t)$ can be expressed as:

$$Z(t) = W(t) + \epsilon(t) \quad (1)$$

where $W(t)$ is the actual (measured) degradation level of the unit at the time $t$ and $\epsilon(t)$ is the associated measurement error;

(ii) the hidden process $\{W(t); t \ge 0\}$ is the gamma process with random effect firstly suggested in Lawless and Crowder [28]. Under this model, the degradation process of the units pertaining to the (heterogeneous) population of interest is described by using a gamma process that has an assigned age function and scale parameter that varies randomly from unit to unit according to a gamma probability distribution. More specifically, the conditional pdf of the degradation increment $\Delta W(t, t + \Delta t) = W(t + \Delta t) - W(t)$, given the value $\lambda$ of the random scale parameter $\Lambda$, is expressed as:

$$f_{\Delta W(t, t+\Delta t)|\Lambda}(\delta|\lambda) = \frac{\lambda^{\Delta\eta(t, t+\Delta t)} \cdot \delta^{\Delta\eta(t, t+\Delta t)-1}}{\Gamma(\Delta\eta(t, t+\Delta t))} \cdot e^{-\lambda \cdot \delta}, \quad \delta \ge 0 \quad (2)$$

and the scale parameter $\Lambda$ is assumed to have the gamma pdf:

$$f_{\Lambda}(\lambda) = \frac{e^{d} \cdot \lambda^{d-1}}{\Gamma(d)} \cdot e^{-c \cdot \lambda}, \quad \lambda > 0, \quad (3)$$

where $\Gamma(y) = \int_{0}^{\infty} u^{y-1} \cdot e^{-u} \cdot du$ denotes the complete gamma function, $\eta(t)$ is a monotonic increasing function usually referred to as the age function, $\eta(t, t + \Delta t) = \eta(t + \Delta t) - \eta(t)$, $d > 0$ is the shape parameter of the pdf (3), and $c > 0$ is the scale parameter. It is assumed that $W(0) = 0$ and that $\eta(0) = 0$;

(iii) the measurement error $\epsilon(t)$, given $W(t) = w$, has the conditional probability density function (pdf):

$$f_{\epsilon(t)|W(t)}(\epsilon|w) = \frac{[\alpha(w)]^{\beta(w)} \cdot (\epsilon + w)^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{\frac{\alpha(w)}{\epsilon+w}}, \quad -w \le \epsilon < \infty \quad (4)$$

where:

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

Hence, given $W(t) = w$, the measurement error $\epsilon(t)$ has null conditional mean:

$$E\{\epsilon(t)|W(t) = w\} = \frac{\alpha(w)}{\beta(w) - 1} - w = 0 \quad (5)$$

and conditional variance that is a power function of $w$:

$$V\{\epsilon(t)|W(t) = w\} = \frac{[\alpha(w)]^2}{[\beta(w) - 1]^2 \cdot [\beta(w) - 2]} = \frac{w^\nu}{\varphi}. \quad (6)$$

The pdf (4) consents to model simply and effectively the case where the measurement error depends, in a stochastic sense, on the measured degradation level $W(t)$ (see also Pulcini [34]) and allows the perturbed measurement to be surely non negative, a result that is not guaranteed by the classical assumption that the measurement error is Gaussian distributed, especially when measurement uncertainty is large, a situation that often occurs

when degradation measurements are obtained through low cost indirect measurement procedures. Further motivations for the use of the pdf (4) are provided in Giorgio et al. [35].

(iv) for any $n > 1$, any $i = 1, ..., n$, and any set of times $t_1, ..., t_n$, the measurement error $\epsilon(t_i)$ given $W(t_i) = w_i$ is conditionally independent both on $\epsilon(t_1), ..., \epsilon(t_{i-1}), \epsilon(t_{i+1}), ..., \epsilon(t_n)$ and $W(t_1), ..., W(t_{i-1}), W(t_{i+1}), ..., W(t_n)$. Consequently, it also results that $Z(t_i)$ given $W(t_i) = w_i$ is conditionally independent both on $Z(t_1), ..., Z(t_{i-1}), Z(t_{i+1}), ..., Z(t_n)$ and $W(t_1), ..., W(t_{i-1}), W(t_{i+1}), ..., W(t_n)$.

From (4), the perturbed measurement $Z(t) = W(t) + \epsilon(t)$, given $W(t) = w$, is an inverse gamma random variable with conditional pdf:

$$f_{Z(t)|W(t)}(z|w) = \frac{[\alpha(w)]^{\beta(w)} \cdot z^{-\beta(w)-1}}{\Gamma(\beta(w))} \cdot e^{-\frac{\alpha(w)}{z}}, \quad z \ge 0, \quad (7)$$

conditional cumulative distribution function (cdf):

$$F_{Z(t)|W(t)}(z|w) = \frac{\gamma(\beta(w), \alpha(w)/z)}{\Gamma(\beta(w))}, \quad z \ge 0, \quad (8)$$

conditional mean:

$$E\{Z(t)|W(t) = w\} = E\{\epsilon(t)|W(t) = w\} + w = w, \quad (9)$$

and conditional variance:

$$V\{Z(t)|W(t) = w\} = V\{Z(t)|W(t) = w\} = \frac{w^\nu}{\varphi}, \quad (10)$$

where $\gamma(y, s) = \int_{0}^{s} u^{y-1} \cdot e^{-u} \cdot du$ is the lower incomplete gamma function.

Moreover, from (2) and (3), it results that the hidden degradation process $W(t)$ has marginal pdf:

$$f_{W(t)}(w) = \frac{c^d \cdot w^{\eta(t)-1}}{B(\eta(t), d) \cdot (w + c)^{\eta(t)+d}}, \quad w \ge 0, \quad (11)$$

marginal cdf:

$$F_{W(t)}(w) = \mathbb{B}\left(\frac{w}{w+c}; \eta(t), d\right), \quad w \ge 0, \quad (12)$$

mean (that exists for $d > 1$):

$$E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1}, \quad (13)$$

and variance (that exists for $d > 2$):

$$V\{W(t)\} = \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)}, \quad (14)$$

where:

$$\mathbb{B}(z; \alpha, \beta) = \frac{\int_{0}^{z} u^{\alpha-1} \cdot (1-u)^{\beta-1} \cdot du}{B(\alpha, \beta)}, \quad (15)$$

is the regularized beta function and

$$B(\alpha, \beta) = \frac{\Gamma(\alpha) \cdot \Gamma(\beta)}{\Gamma(\alpha + \beta)}, \quad (16)$$

is the beta function.

More specifically, it is possible to prove (e.g., see Esposito et al. [30]) that under the considered setting the hidden (marginal) degradation process $\{W(t); t \ge 0\}$, is Markovian and that the degradation increment $\Delta W(t, t + \Delta t)$, given $W(t) = w$, has conditional pdf:
$$f_{\Delta W(t,t+\Delta t)|W(t)}(\delta|w) = \frac{1}{B(\Delta\eta(t, t+\Delta t), \eta(t)+d)} \frac{(w+c)^{\eta(t)+d} \cdot \delta^{\Delta\eta(t,t+\Delta t)-1}}{(w+\delta+c)^{\eta(t+\Delta t)+d}} \quad (17)$$

and conditional cdf:

$$F_{\Delta W(t,t+\Delta t)|W(t)}(\delta|w) = \mathbb{E}\left(\frac{\delta}{\delta+w+c}; \Delta\eta(t, t+\Delta t), \eta(t)+d\right). \quad (18)$$

In addition, the marginal distribution of the perturbed measurement $Z(t)$ is not available in closed form. However, from (7), (8), and (11), the marginal pdf and cdf of $Z(t)$ can be easily calculated by numerically solving the integrals:

$$\begin{aligned} f_{Z(t)}(z) &= \int_{0}^{\infty} f_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} \frac{[\alpha(w)]^{\beta(w)} \cdot z^{-\beta(w)-1} \cdot e^{\frac{a(w)}{c}} \cdot c^{d} \cdot w^{\eta(t)-1}}{\Gamma(\beta(w)) \cdot \mathbb{B}(\eta(t), d) \cdot (w+c)^{\eta(t)+d}} \cdot dw \end{aligned} \quad (19)$$

and

$$\begin{aligned} F_{X|W(\tau)}(x|w_{\tau}) &= P(X \le x | W(\tau) = w_{\tau}) = P(W(x) > w_{M} | W(\tau) = w_{\tau}) \\ &= \begin{cases} 0, & w_{\tau} \le w_{M}, x \le \tau \\ 1 - \mathbb{E}\left(\frac{w_{M}}{w_{\tau}+c}; \eta(\tau, x), \Delta\eta(\tau, x), \eta(\tau)+d\right), & w_{\tau} \le w_{M}, x > \tau. \end{cases} \end{aligned} \quad (24)$$

$$\begin{aligned} V\{Z(t)\} &= V\{E\{Z(t)|W(t)\}\} + E\{V\{Z(t)|W(t)\}\} = V\{W(t)\} + E\left\{\frac{(W(t))^{\nu}}{\varphi}\right\} \\ &= \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)} + \frac{c^\nu \cdot \Gamma(d-\nu) \cdot \Gamma(\eta(t) + \nu)}{\Gamma(d) \cdot \Gamma(\eta(t))} \cdot \frac{1}{\varphi} \end{aligned} \quad (22)$$

As already mentioned, we assume that a unit fails when its actual (hidden) degradation level exceeds an assigned failure threshold, hereinafter indicated by $w_M$. Thus, given that the hidden process $\{W(t); t \ge 0\}$ is monotone increasing, the useful life $X$ of the unit (i.e., its time to failure):

$$X = \inf\{x : W(x) > w_M\} \quad (23)$$

can be defined as the first and sole passage time of $\{W(t); t \ge 0\}$ to the failure threshold $w_M$.

Further results which involve the lifetime $X$ are derived here. From the conditional cdf (18), denoting by $\tau$ a generic reference time (that will be intended as the inspection time in the following sections), it is possible to readily obtain the conditional cdf of $X$ given $W(\tau) = w_\tau$ in the cases where $w_\tau \le w_M$. In fact, this cdf can be expressed as:

$$\begin{aligned} F_{X|W(\tau)}(x|w_{\tau}) &= P(X \le x | W(\tau) = w_{\tau}) \\ &= P(W(x) > w_{M} | W(\tau) = w_{\tau}) = \begin{cases} 1 - \mathbb{E}\left(\frac{w_{M}}{w_{\tau}}; \eta(x), \Delta\eta(x, \tau)\right), & w_{\tau} > w_{M}, x \le \tau, \\ 1, & w_{\tau} > w_{M}, x > \tau. \end{cases} \end{aligned} \quad (25)$$

where the first equality can be explained by observing that, since the process $\{W(t); t \ge 0\}$ is monotone increasing, the event $\{X \le x\}$ is equivalent to the event $\{W(x) > w_M\}$.

Differently, in the case where $w_\tau > w_M$ the conditional cdf of $X$ given $W(\tau) = w_\tau$ can be expressed as in (25):

$$\begin{aligned} F_{Z(t)}(z) &= \int_{0}^{\infty} F_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} \frac{\gamma(\beta(w), \alpha(w)/z) \cdot c^d \cdot w^{\eta(t)-1}}{\Gamma(\beta(w)) \cdot \mathbb{B}(\eta(t), d) \cdot (w+c)^{\eta(t)+d}} \cdot dw. \end{aligned} \quad (20)$$

In fact, being:

$$\begin{aligned} P(W(x) > w_M | W(\tau) = w_\tau) &= 1 - F_{W(x)|W(\tau)}(w_M | w_\tau) \\ &= 1 - \int_{0}^{w_M} f_{W(x)|W(\tau)}(w | w_\tau) \cdot dw \\ &= 1 - \int_{0}^{w_M} \frac{f_{W(\tau)|W(x)}(w_\tau | w) \cdot f_{W(x)}(w)}{f_{W(\tau)}(w_\tau)} \cdot dw \\ &= 1 - \int_{0}^{w_M} \frac{f_{\Delta W(x,\tau)|W(x)}(w_\tau - w | w) \cdot f_{W(x)}(w)}{f_{W(\tau)}(w_\tau)} \cdot dw \end{aligned} \quad (26)$$

From (9), (10), (13), and (14), by using the laws of total expectation and total variance, it is easy to verify that the marginal mean of $Z(t)$ (that exists for $d > 1$) results in:

$$E\{Z(t)\} = E\{E\{Z(t)|W(t)\}\} = E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1} \quad (21)$$

and that its marginal variance (that exists for $d > \nu$) is equal to:

from (11), (17), and (15) we have:
$$f_{\Delta W(t,t+\Delta t)|W(t)}(\delta|w) = \frac{1}{B(\Delta\eta(t, t+\Delta t), \eta(t)+d)} \frac{(w+c)^{\eta(t)+d} \cdot \delta^{\Delta\eta(t,t+\Delta t)-1}}{(w+\delta+c)^{\eta(t+\Delta t)+d}} \quad (17)$$

and conditional cdf:

$$F_{\Delta W(t,t+\Delta t)|W(t)}(\delta|w) = \mathbb{E}\left(\frac{\delta}{\delta+w+c}; \Delta\eta(t, t+\Delta t), \eta(t)+d\right). \quad (18)$$

In addition, the marginal distribution of the perturbed measurement $Z(t)$ is not available in closed form. However, from (7), (8), and (11), the marginal pdf and cdf of $Z(t)$ can be easily calculated by numerically solving the integrals:

$$\begin{aligned} f_{Z(t)}(z) &= \int_{0}^{\infty} f_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} \frac{[\alpha(w)]^{\beta(w)} \cdot z^{-\beta(w)-1} \cdot e^{-\frac{\alpha(w)}{c}} \cdot c^{d} \cdot w^{\eta(t)-1}}{\Gamma(\beta(w)) \cdot \mathbb{B}(\eta(t), d) \cdot (w+c)^{\eta(t)+d}} \cdot dw \end{aligned} \quad (19)$$

and

$$\begin{aligned} F_{X|W(\tau)}(x|w_{\tau}) &= P(X \le x | W(\tau) = w_{\tau}) = P(W(x) > w_{M} | W(\tau) = w_{\tau}) \\ &= \begin{cases} 0, & w_{\tau} \le w_{M}, x \le \tau \\ 1 - \mathbb{E}\left(\frac{w_{M}}{w_{\tau}+c}; \eta(\tau, x), \Delta\eta(\tau, x), \eta(\tau)+d\right), & w_{\tau} \le w_{M}, x > \tau. \end{cases} \end{aligned} \quad (24)$$

$$\begin{aligned} V\{Z(t)\} &= V\{E\{Z(t)|W(t)\}\} + E\{V\{Z(t)|W(t)\}\} = V\{W(t)\} + E\left\{\frac{(W(t))^{\nu}}{\varphi}\right\} \\ &= \frac{c^2 \cdot \eta(t) \cdot [\eta(t) + d - 1]}{(d-1)^2 \cdot (d-2)} + \frac{c^\nu \cdot \Gamma(d-\nu) \cdot \Gamma(\eta(t) + \nu)}{\Gamma(d) \cdot \Gamma(\eta(t))} \cdot \frac{1}{\varphi} \end{aligned} \quad (22)$$

As already mentioned, we assume that a unit fails when its actual (hidden) degradation level exceeds an assigned failure threshold, hereinafter indicated by $w_M$. Thus, given that the hidden process $\{W(t); t \ge 0\}$ is monotone increasing, the useful life $X$ of the unit (i.e., its time to failure):

$$X = \inf\{x : W(x) > w_M\} \quad (23)$$

can be defined as the first and sole passage time of $\{W(t); t \ge 0\}$ to the failure threshold $w_M$.

Further results which involve the lifetime $X$ are derived here. From the conditional cdf (18), denoting by $\tau$ a generic reference time (that will be intended as the inspection time in the following sections), it is possible to readily obtain the conditional cdf of $X$ given $W(\tau) = w_\tau$ in the cases where $w_\tau \le w_M$. In fact, this cdf can be expressed as:

$$\begin{aligned} F_{X|W(\tau)}(x|w_{\tau}) &= P(X \le x | W(\tau) = w_{\tau}) \\ &= P(W(x) > w_{M} | W(\tau) = w_{\tau}) = \begin{cases} 1 - \mathbb{E}\left(\frac{w_{M}}{w_{\tau}}; \eta(x), \Delta\eta(x, \tau)\right), & w_{\tau} > w_{M}, x \le \tau, \\ 1, & w_{\tau} > w_{M}, x > \tau. \end{cases} \end{aligned} \quad (25)$$

where the first equality can be explained by observing that, since the process $\{W(t); t \ge 0\}$ is monotone increasing, the event $\{X \le x\}$ is equivalent to the event $\{W(x) > w_M\}$.

Differently, in the case where $w_\tau > w_M$ the conditional cdf of $X$ given $W(\tau) = w_\tau$ can be expressed as in (25):

$$\begin{aligned} F_{Z(t)}(z) &= \int_{0}^{\infty} F_{Z(t)|W(t)}(z|w) \cdot f_{W(t)}(w) \cdot dw \\ &= \int_{0}^{\infty} \frac{\gamma(\beta(w), \alpha(w)/z) \cdot c^d \cdot w^{\eta(t)-1}}{\Gamma(\beta(w)) \cdot \mathbb{B}(\eta(t), d) \cdot (w+c)^{\eta(t)+d}} \cdot dw. \end{aligned} \quad (20)$$

From (9), (10), (13), and (14), by using the laws of total expectation and total variance, it is easy to verify that the marginal mean of $Z(t)$ (that exists for $d > 1$) results in:

$$E\{Z(t)\} = E\{E\{Z(t)|W(t)\}\} = E\{W(t)\} = \frac{c \cdot \eta(t)}{d-1} \quad (21)$$

and that its marginal variance (that exists for $d > \nu$) is equal to:

In fact, being:

$$\begin{aligned} P(W(x) > w_M | W(\tau) = w_\tau) &= 1 - F_{W(x)|W(\tau)}(w_M | w_\tau) \\ &= 1 - \int_{0}^{w_M} f_{W(x)|W(\tau)}(w | w_\tau) \cdot dw \\ &= 1 - \int_{0}^{w_M} \frac{f_{W(\tau)|W(x)}(w_\tau | w) \cdot f_{W(x)}(w)}{f_{W(\tau)}(w_\tau)} \cdot dw \\ &= 1 - \int_{0}^{w_M} \frac{f_{\Delta W(x,\tau)|W(x)}(w_\tau - w | w) \cdot f_{W(x)}(w)}{f_{W(\tau)}(w_\tau)} \cdot dw \end{aligned} \quad (26)$$

from (11), (17), and (15) we have:
$$
\begin{align*}
F_{W(x)|W(\tau)}(w_M|w_\tau) &= 1 - \int_0^{w_M} \frac{(w+\tau)^{\eta(x)+d} \cdot (w_\tau-w)^{\Delta\eta(x)-1}}{B(\Delta\eta(x,\tau), \eta(x)+d) \cdot (w_\tau+c)^{\eta(\tau)+d} \cdot B(\eta(x),d) \cdot (w+\tau+c)^{\eta(x)+d}} \cdot \frac{e^{d \cdot w_\tau^{\eta(x)-1}}}{B(\eta(\tau),d) \cdot (w_\tau+c)^{\eta(\tau)+d}} \cdot \mathrm{d}w \\
&= 1 - \frac{B(\eta(\tau), d)}{B(\Delta\eta(x, \tau), \eta(x) + d) \cdot B(\eta(x), d)} \int_0^{w_M} \frac{(w_\tau - w)^{\Delta\eta(x,\tau)-1} \cdot w_\tau^{\eta(x)-1}}{w_\tau^{\eta(\tau)-1}} \cdot \mathrm{d}w \\
&= 1 - \frac{1}{B(\eta(x), \Delta\eta(x, \tau))} \int_0^{w_M} \left(\frac{w}{w_\tau}\right)^{\eta(x)-1} \cdot \left(1 - \frac{w}{w_\tau}\right)^{\Delta\eta(x,\tau)-1} \cdot \frac{1}{w_\tau} \cdot \mathrm{d}w \\
&= 1 - \frac{1}{B(\eta(x), \Delta\eta(x, \tau))} \int_0^{w_M/w_\tau} y^{\eta(x)-1} \cdot (1-y)^{\Delta\eta(x,\tau)-1} \cdot \mathrm{d}y \\
&= 1 - \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \tag{27}
\end{align*}
$$

that coincides with the cdf (25).

The cdf (25) is linked to the concept of a bridge process. A bridge is a stochastic process that describes the evolution of the state variable over a finite interval conditionally to the values it assumes at the endpoints (e.g., see Hoyle et al. [36]). It is interesting to note that the cdf (25) obtained under the considered gamma process with random effect coincides with the analog conditional cdf obtained in the case of the basic gamma bridge process: that is, in the absence of random effect (e.g., see Khale et al. [37]).

Note that, for the sake of economy of notation, the cdfs in (24) and (25) have been indicated by using the same symbol. However, as it is explicitly specified in the text, the (24) should be used if and only if $w_\tau \le w_M$ and the (25) if and only if $w_\tau > w_M$.

From (25), when $w_\tau > w_M$, the conditional mean of X, given $W(\tau) = w_\tau$, can be expressed as:

$$
\begin{align}
E\{X|W(\tau) = w_\tau\} &= \int_0^\tau x \cdot f_{X|W(\tau)}(x|w_\tau) \cdot \mathrm{d}x \nonumber \\
&= \left[-x \cdot (1 - F_{X|W(\tau)}(x|w_\tau))\right]_0^\tau + \int_0^\tau (1 - F_{X|W(\tau)}(x|w_\tau)) \cdot \mathrm{d}x \tag{28} \\
&= \int_0^\tau \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \cdot \mathrm{d}x. \nonumber
\end{align}
$$

Finally, from (24), when $w_\tau \le w_M$, given $W(\tau) = w_\tau$, the conditional mean of the variable $g(X)$ defined by the following transformation:

$$
g(X) = \begin{cases} \tau + \Delta\tau - X, & X \le \tau + \Delta\tau \\ 0, & X > \tau + \Delta\tau \end{cases} \tag{29}
$$

can be computed as:

$$
E\{g(X)|W(\tau) = w_\tau\} = \int_{\tau}^{\tau + \Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot \mathrm{d}x
$$

<table><caption>**Table 1**
Condition-based rule.</caption><thead><tr><th colspan="2">Measurement at τ</th><th rowspan="2">Decision</th></tr><tr><th>Perturbed</th><th>Perfect</th></tr></thead><tbody><tr><td>w<sub>τ</sub> &gt; L<sub>l</sub></td><td></td><td>Replacement at τ</td></tr><tr><td>L<sub>l</sub> &lt; w<sub>τ</sub> &lt; L<sub>u</sub></td><td>w<sub>τ</sub> &gt; L<sub>W</sub></td><td>Replacement at τ</td></tr><tr><td>L<sub>l</sub> &lt; w<sub>τ</sub> &lt; L<sub>u</sub></td><td>w<sub>τ</sub> &le; L<sub>W</sub></td><td>Replacement at τ + Δτ</td></tr><tr><td>w<sub>τ</sub> &le; L<sub>l</sub></td><td></td><td>Replacement at τ + Δτ</td></tr></tbody></table>

$$
\begin{align}
&= \left[ -(\tau + \Delta\tau - x) \cdot (1 - F_{X|W(\tau)}(x|w_\tau)) \right]_{\tau}^{\tau+\Delta\tau} - \int_{\tau}^{\tau+\Delta\tau} (1 - F_{X|W(\tau)}(x|w_\tau)) \cdot \mathrm{d}x \nonumber \\
&= \Delta\tau - \int_{\tau}^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot \mathrm{d}x. \tag{30}
\end{align}
$$

It is worth to remark that, under the considered setting, the marginal process $\{W(t); t \ge 0\}$ is age and state dependent, a feature that the process preserves also when (as it is assumed in Section 5) the temporal variability of the unit-specific paths is described by homogeneous gamma processes (e.g., see Giorgio and Pulcini [38]). The dependence on the state is generated by the presence in the population of units whose degradation levels evolve over time at different speeds (i.e., the future degradation growth of units whose degradation level is high is predicted to be high because these units are suspected to be weak). In fact, it is exactly the existence (and the understanding of the nature) of this dependency that allows to distinguish the weak units from the strong ones. Note that the presence of age and state dependency also implies that, differently than in the case of failure threshold models constructed by using the basic homogenous gamma process (e.g., Grall et al. [9], Castanier et al. [10], Castanier et al. [11], and Huynh et al. [20]), where the probability that a failure occurs in a future time interval depends only on the gap existing between the current degradation level of the unit and the failure threshold, under the considered gamma process with random effect the same probability directly depends both on the threshold, on the current degradation level of the unit, and on its current age.

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.

### 3. The maintenance policy, the cost model, and formulation of the long-run average maintenance cost rate

We consider a heterogeneous population of degrading units where the observed degradation process can be described by the perturbed process with random effect presented in Section 2. Maintenance activities refer to a generic unit randomly selected from the considered population. The proposed policy is developed based on the following assumptions:

* only one inspection can be performed during the lifetime of the unit;

* inspections are instantaneous and non-destructive;

* failures are not self-announcing;

* the unit can continue to operate even in a failed state;

* operating a failed unit causes additional costs due to reduced performance;

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

* times between successive replacements define the cycles of a renewal process.

The suggested policy consists in using the information obtained by performing an ad hoc inspection at a certain epoch $\tau$ to decide whether to replace the unit immediately or to postpone its replacement to a future time $\tau + \Delta\tau$, according to a condition-based rule. In case of postponement, no other inspections will be performed and the unit will be replaced at $\tau + \Delta\tau$ regardless of its state. The inspection is assumed to consist in measuring the degradation level of the unit. It is supposed that the measurement is firstly performed by using an imperfect procedure that provides a perturbed measurement (i.e., a measurement affected by error). Nevertheless, it is also supposed that, at the same inspection time, it is possible to resort to a different perfect measurement procedure that allows to observe the exact value of the degradation state of the unit. The
$$
\begin{align*}
F_{W(x)|W(\tau)}(w_M|w_\tau) &= 1 - \int_0^{w_M} \frac{(w+\tau)^{\eta(x)+d} \cdot (w_\tau-w)^{\Delta\eta(x)-1}}{B(\Delta\eta(x,\tau), \eta(x)+d) \cdot (w_\tau+c)^{\eta(\tau)+d} \cdot B(\eta(x),d) \cdot (w+\tau+c)^{\eta(\tau)+d}} \cdot \frac{e^{d \cdot w_\tau^{\eta(x)-1}}}{e^{d \cdot w_\tau^{\eta(\tau)-1}} \cdot B(\eta(\tau),d) \cdot (w_\tau+c)^{\eta(\tau)+d}} \cdot dw \\
&= 1 - \frac{B(\eta(\tau), d)}{B(\Delta\eta(x, \tau), \eta(x) + d) \cdot B(\eta(x), d)} \int_0^{w_M} \frac{(w_\tau - w)^{\Delta\eta(x, \tau)-1} \cdot w_\tau^{\eta(x)-1}}{w_\tau^{\eta(\tau)-1}} \cdot dw \\
&= 1 - \frac{1}{B(\eta(x), \Delta\eta(x, \tau))} \int_0^{w_M} \left(\frac{w}{w_\tau}\right)^{\eta(x)-1} \cdot \left(1 - \frac{w}{w_\tau}\right)^{\Delta\eta(x, \tau)-1} \cdot \frac{1}{w_\tau} \cdot dw \\
&= 1 - \frac{1}{B(\eta(x), \Delta\eta(x, \tau))} \int_0^{w_M/w_\tau} y^{\eta(x)-1} \cdot (1-y)^{\Delta\eta(x, \tau)-1} \cdot dy \\
&= 1 - \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \tag{27}
\end{align*}
$$

that coincides with the cdf (25).

The cdf (25) is linked to the concept of a bridge process. A bridge is a stochastic process that describes the evolution of the state variable over a finite interval conditionally to the values it assumes at the endpoints (e.g., see Hoyle et al. [36]). It is interesting to note that the cdf (25) obtained under the considered gamma process with random effect coincides with the analog conditional cdf obtained in the case of the basic gamma bridge process: that is, in the absence of random effect (e.g., see Khale et al. [37]).

Note that, for the sake of economy of notation, the cdfs in (24) and (25) have been indicated by using the same symbol. However, as it is explicitly specified in the text, the (24) should be used if and only if $w_\tau \le w_M$ and the (25) if and only if $w_\tau > w_M$.

From (25), when $w_\tau > w_M$, the conditional mean of $X$, given $W(\tau) = w_\tau$, can be expressed as:

$$
\begin{align}
E\{X|W(\tau) = w_\tau\} &= \int_0^\tau x \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \nonumber \\
&= [-x \cdot (1 - F_{X|W(\tau)}(x|w_\tau))]_0^\tau + \int_0^\tau (1 - F_{X|W(\tau)}(x|w_\tau)) \cdot dx \tag{28} \\
&= \int_0^\tau \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \cdot dx. \nonumber
\end{align}
$$

Finally, from (24), when $w_\tau \le w_M$, given $W(\tau) = w_\tau$, the conditional mean of the variable $g(X)$ defined by the following transformation:

$$
g(X) = \begin{cases} \tau + \Delta\tau - X, & X \le \tau + \Delta\tau \\ 0, & X > \tau + \Delta\tau \end{cases} \tag{29}
$$

can be computed as:

$$
E\{g(X)|W(\tau) = w_\tau\} = \int_{\tau}^{\tau+\Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx
$$

<table><caption>**Table 1**
Condition-based rule.</caption>
  <thead>
    <tr>
      <th colspan="2">Measurement at τ</th>
      <th rowspan="2">Decision</th>
    </tr>
    <tr>
      <th>Perturbed</th>
      <th>Perfect</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>z<sub>τ</sub> &gt; L<sub>l</sub></td>
      <td></td>
      <td>Replacement at τ</td>
    </tr>
    <tr>
      <td>L<sub>l</sub> &lt; z<sub>τ</sub> &le; L<sub>u</sub></td>
      <td>w<sub>τ</sub> &gt; L<sub>W</sub></td>
      <td>Replacement at τ</td>
    </tr>
    <tr>
      <td>L<sub>l</sub> &lt; z<sub>τ</sub> &le; L<sub>u</sub></td>
      <td>w<sub>τ</sub> &le; L<sub>W</sub></td>
      <td>Replacement at τ + Δτ</td>
    </tr>
    <tr>
      <td>z<sub>τ</sub> &le; L<sub>l</sub></td>
      <td></td>
      <td>Replacement at τ + Δτ</td>
    </tr>
  </tbody>
</table>

$$
\begin{align}
&= \left[ -(\tau + \Delta\tau - x) \cdot (1 - F_{X|W(\tau)}(x|w_\tau)) \right]_{\tau}^{\tau+\Delta\tau} - \int_{\tau}^{\tau+\Delta\tau} (1 - F_{X|W(\tau)}(x|w_\tau)) \cdot dx \nonumber \\
&= \Delta\tau - \int_{\tau}^{\tau+\Delta\tau} \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot dx. \tag{30}
\end{align}
$$

It is worth to remark that, under the considered setting, the marginal process $\{W(t); t \ge 0\}$ is age and state dependent, a feature that the process preserves also when (as it is assumed in Section 5) the temporal variability of the unit-specific paths is described by homogeneous gamma processes (e.g., see Giorgio and Pulcini [38]). The dependence on the state is generated by the presence in the population of units whose degradation levels evolve over time at different speeds (i.e., the future degradation growth of units whose degradation level is high is predicted to be high because these units are suspected to be weak). In fact, it is exactly the existence (and the understanding of the nature) of this dependency that allows to distinguish the weak units from the strong ones. Note that the presence of age and state dependency also implies that, differently than in the case of failure threshold models constructed by using the basic homogenous gamma process (e.g., Grall et al. [9], Castanier et al. [10], Castanier et al. [11], and Huynh et al. [20]), where the probability that a failure occurs in a future time interval depends only on the gap existing between the current degradation level of the unit and the failure threshold, under the considered gamma process with random effect the same probability directly depends both on the threshold, on the current degradation level of the unit, and on its current age.

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.

### 3. The maintenance policy, the cost model, and formulation of the long-run average maintenance cost rate

We consider a heterogeneous population of degrading units where the observed degradation process can be described by the perturbed process with random effect presented in Section 2. Maintenance activities refer to a generic unit randomly selected from the considered population. The proposed policy is developed based on the following assumptions:

* only one inspection can be performed during the lifetime of the unit;

* inspections are instantaneous and non-destructive;

* failures are not self-announcing;

* the unit can continue to operate even in a failed state;

* operating a failed unit causes additional costs due to reduced performance;

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

* times between successive replacements define the cycles of a renewal process.

The suggested policy consists in using the information obtained by performing an ad hoc inspection at a certain epoch $\tau$ to decide whether to replace the unit immediately or to postpone its replacement to a future time $\tau + \Delta\tau$, according to a condition-based rule. In case of postponement, no other inspections will be performed and the unit will be replaced at $\tau + \Delta\tau$ regardless of its state. The inspection is assumed to consist in measuring the degradation level of the unit. It is supposed that the measurement is firstly performed by using an imperfect procedure that provides a perturbed measurement (i.e., a measurement affected by error). Nevertheless, it is also supposed that, at the same inspection time, it is possible to resort to a different perfect measurement procedure that allows to observe the exact value of the degradation state of the unit. The
<table><caption>**Table 2**
Proposed policy: maintenance actions.</caption><thead><tr><th colspan="3">Measurement at τ</th><th rowspan="2">State at τ</th><th rowspan="2">Lifetime</th><th rowspan="2">Maintenance action</th><th rowspan="2">Cycle length T(z<sub>τ</sub>, w<sub>τ</sub>)</th></tr><tr><th>Perturbed</th><th>Perfect</th><th></th></tr></thead><tbody><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ</td><td>τ</td></tr><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>Preventive at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>L<sub>W</sub> < w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>Preventive at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>τ < X ≤ τ + Δτ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>X > τ + Δτ</td><td>Preventive at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td></td><td>X > τ + Δτ</td><td>Preventive at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>τ < X ≤ τ + Δτ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr></tbody></table>

<table><caption>**Table 3**
Maintenance costs.</caption><thead><tr><th colspan="3">Measurement at τ</th><th rowspan="2">State at τ</th><th rowspan="2">X</th><th rowspan="2">Maintenance cost C(z<sub>τ</sub>, w<sub>τ</sub>, X)</th></tr><tr><th>Perturbed</th><th>Perfect</th><th></th></tr></thead><tbody><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ - X)</td></tr><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>p</sub></td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ - X)</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>L<sub>W</sub> < w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>p</sub></td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>τ < X ≤ τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>X > τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>p</sub></td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td></td><td>X > τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>p</sub></td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>τ < X ≤ τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr></tbody></table>

use of the perfect measurement procedure is optional and has an extra cost and the decision about whether to use it or not is made on the basis of the value obtained by performing the perturbed measurement. In particular, here it is assumed that the perfect measurement is made when $L_l < z_τ \le L_u$.

Considered that failures are not self-announcing (i.e., given that failed units do not reveal their state), it is assumed that replacements are executed only at the predetermined inspection time $τ$ or at the replacement time $τ + Δτ$.

The condition-based rule adopted to make the decision is described in Table 1, where $z_τ$ indicates the observed value of the perturbed degradation measurement $Z(τ)$ obtained at the inspection time $τ$, $w_τ$ indicates the true value of the measured degradation level $W(τ)$, and $L_l$, $L_u$, and $L_W$ are (maintenance) threshold levels.

Table 2 summarizes all possible experimental scenarios and reports for each of them the maintenance action to be taken and the resulting length of a maintenance cycle, denoted by $T(z_τ, w_τ)$. In this table, $w_{τ+Δτ}$ indicates the true value of the degradation level $W(τ + Δτ)$ at $τ + Δτ$.

The advantage of using the second measurement procedure is that it allows to know the exact degradation level of the unit at $τ$. This information, for example, could allow to mitigate the risk of postponing the replacement of units that are already failed or close to fail as well as to avoid the premature replacement of units whose true degradation state is safely below the threshold.

The parameters $τ$, $Δτ$, $L_l$, $L_u$, and $L_W$ should be intended as design parameters of the considered policy. Hereinafter, the vector of design parameters will be denoted by $\xi = \{τ, Δτ, L_l, L_u, L_W\}$. The value of $\xi$ should be set a priori with the aim of maximizing/minimizing a selected objective function. In this paper, $\xi$ is set with the aim of minimizing the long-run average maintenance cost rate. The value of $\xi$ that defines the optimal policy is indicated as $\xi^*$.

The cost model is formulated as shown in Table 3 in respect to the different maintenance scenarios presented in Table 2. In Table 3, $c_c$ is the

cost of a corrective replacement, $c_p$ is the cost of a preventive replacement, $c_z$ is the cost of the imperfect inspection, $c_w$ is the cost of the perfect (optional) inspection, $c_l$ is the logistic cost (incurred each time a maintenance action is taken), and $c_d$ is the downtime cost rate per unit of time caused by operating the unit in a failed state. The total downtime cost in a maintenance cycle is computed as the product of the downtime and this fixed cost rate. The downtime is assumed to coincide with the time elapsing from the failure occurrence to the time at which the unit is replaced. Finally, $C(z_τ, w_τ, X)$ is the maintenance cost per cycle formulated, scenario by scenario, for given values of the arguments $z_τ, w_τ$, and $X$.

It is worth to emphasize that, in line with the assumption that failures are not self-announcing, in Table 3 $X$ is always denoted with the capital letter, even when $z_τ$ and/or $w_τ$ are assumed to be known, to intend that (even in these special cases) the lifetime of the unit should be regarded as a random variable.

It is finally useful to remark that, although $T(z_τ, w_τ)$ and $C(z_τ, w_τ, X)$ functionally depend on $\xi = \{τ, Δτ, L_l, L_u, L_W\}$, for economy of notation the adopted symbols do not highlight this dependency.

Finally, by using the renewal/reward theorem (e.g., see Ross [39]), the long-run average maintenance cost rate $C_∞(\xi)$ can be formulated as:

$$C_{\infty}(\xi) = \frac{E\{C(Z(\tau), W(\tau), X)\}}{E\{T(Z(\tau), W(\tau))\}} \quad (31)$$

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

The expectations at the numerator and denominator of (31) do not allow for closed form expressions. However, they can be computed numerically by using the (32) and (33), respectively. Deriving (32)-(33) is rather cumbersome, although conceptually simple. Details about their derivation are provided in Appendix A.
<table><caption>**Table 2**
Proposed policy: maintenance actions.</caption><thead><tr><th colspan="3">Measurement at τ</th><th rowspan="2">State at τ</th><th rowspan="2">Lifetime</th><th rowspan="2">Maintenance action</th><th rowspan="2">Cycle length T(z<sub>τ</sub>, w<sub>τ</sub>)</th></tr><tr><th>Perturbed</th><th>Perfect</th><th></th></tr></thead><tbody><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ</td><td>τ</td></tr><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>Preventive at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>L<sub>W</sub> < w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>Preventive at τ</td><td>τ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>τ < X ≤ τ + Δτ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>X > τ + Δτ</td><td>Preventive at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td></td><td>X > τ + Δτ</td><td>Preventive at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>τ < X ≤ τ + Δτ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>Corrective at τ + Δτ</td><td>τ + Δτ</td></tr></tbody></table>

<table><caption>**Table 3**
Maintenance costs.</caption><thead><tr><th colspan="3">Measurement at τ</th><th rowspan="2">State at τ</th><th rowspan="2">X</th><th rowspan="2">Maintenance cost C(z<sub>τ</sub>, w<sub>τ</sub>, X)</th></tr><tr><th>Perturbed</th><th>Perfect</th><th></th></tr></thead><tbody><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ - X)</td></tr><tr><td>z<sub>τ</sub> > L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>p</sub></td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ - X)</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>L<sub>W</sub> < w<sub>τ</sub> ≤ w<sub>M</sub></td><td>X ≤ τ</td><td>c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>p</sub></td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>τ < X ≤ τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr><tr><td>L<sub>l</sub> < z<sub>τ</sub> ≤ L<sub>u</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ L<sub>W</sub></td><td>X > τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>w</sub> + c<sub>p</sub></td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td></td><td>X > τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>p</sub></td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> ≤ w<sub>M</sub></td><td>τ < X ≤ τ + Δτ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr><tr><td>z<sub>τ</sub> ≤ L<sub>l</sub></td><td></td><td></td><td>w<sub>τ</sub> > w<sub>M</sub></td><td>X ≤ τ</td><td>2·c<sub>l</sub> + c<sub>z</sub> + c<sub>c</sub> + c<sub>d</sub>·(τ + Δτ - X)</td></tr></tbody></table>

use of the perfect measurement procedure is optional and has an extra cost and the decision about whether to use it or not is made on the basis of the value obtained by performing the perturbed measurement. In particular, here it is assumed that the perfect measurement is made when $L_l < z_τ \le L_u$.

Considered that failures are not self-announcing (i.e., given that failed units do not reveal their state), it is assumed that replacements are executed only at the predetermined inspection time $τ$ or at the replacement time $τ + Δτ$.

The condition-based rule adopted to make the decision is described in Table 1, where $z_τ$ indicates the observed value of the perturbed degradation measurement $Z(τ)$ obtained at the inspection time $τ$, $w_τ$ indicates the true value of the measured degradation level $W(τ)$, and $L_l$, $L_u$, and $L_W$ are (maintenance) threshold levels.

Table 2 summarizes all possible experimental scenarios and reports for each of them the maintenance action to be taken and the resulting length of a maintenance cycle, denoted by $T(z_τ, w_τ)$. In this table, $w_{τ+Δτ}$ indicates the true value of the degradation level $W(τ + Δτ)$ at $τ + Δτ$.

The advantage of using the second measurement procedure is that it allows to know the exact degradation level of the unit at $τ$. This information, for example, could allow to mitigate the risk of postponing the replacement of units that are already failed or close to fail as well as to avoid the premature replacement of units whose true degradation state is safely below the threshold.

The parameters $τ$, $Δτ$, $L_l$, $L_u$, and $L_W$ should be intended as design parameters of the considered policy. Hereinafter, the vector of design parameters will be denoted by $\xi = \{τ, Δτ, L_l, L_u, L_W\}$. The value of $\xi$ should be set a priori with the aim of maximizing/minimizing a selected objective function. In this paper, $\xi$ is set with the aim of minimizing the long-run average maintenance cost rate. The value of $\xi$ that defines the optimal policy is indicated as $\xi^*$.

The cost model is formulated as shown in Table 3 in respect to the different maintenance scenarios presented in Table 2. In Table 3, $c_c$ is the

cost of a corrective replacement, $c_p$ is the cost of a preventive replacement, $c_z$ is the cost of the imperfect inspection, $c_w$ is the cost of the perfect (optional) inspection, $c_l$ is the logistic cost (incurred each time a maintenance action is taken), and $c_d$ is the downtime cost rate per unit of time caused by operating the unit in a failed state. The total downtime cost in a maintenance cycle is computed as the product of the downtime and this fixed cost rate. The downtime is assumed to coincide with the time elapsing from the failure occurrence to the time at which the unit is replaced. Finally, $C(z_τ, w_τ, X)$ is the maintenance cost per cycle formulated, scenario by scenario, for given values of the arguments $z_τ, w_τ$, and $X$.

It is worth to emphasize that, in line with the assumption that failures are not self-announcing, in Table 3 $X$ is always denoted with the capital letter, even when $z_τ$ and/or $w_τ$ are assumed to be known, to intend that (even in these special cases) the lifetime of the unit should be regarded as a random variable.

It is finally useful to remark that, although $T(z_τ, w_τ)$ and $C(z_τ, w_τ, X)$ functionally depend on $\xi = \{τ, Δτ, L_l, L_u, L_W\}$, for economy of notation the adopted symbols do not highlight this dependency.

Finally, by using the renewal/reward theorem (e.g., see Ross [39]), the long-run average maintenance cost rate $C_∞(\xi)$ can be formulated as:

$$C_{\infty}(\xi) = \frac{E\{C(Z(\tau), W(\tau), X)\}}{E\{T(Z(\tau), W(\tau))\}} \quad (31)$$

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

The expectations at the numerator and denominator of (31) do not allow for closed form expressions. However, they can be computed numerically by using the (32) and (33), respectively. Deriving (32)-(33) is rather cumbersome, although conceptually simple. Details about their derivation 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) \left[ 1 - \mathbb{E} \left( \frac{w_M}{w_M + c} ; \eta(\tau), d \right) \right] \nonumber \\
& + c_w \cdot \int_0^\infty \left[ \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \nonumber \\
& + (c_l + c_d \cdot \Delta\tau) \cdot \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \nonumber \\
& + (c_c - c_p) \cdot \int_0^{L_w} \left[ 1 - \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c} ; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \tag{32} \\
& + (c_c - c_p) \cdot \int_0^{L_w} \left[ 1 - \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c} ; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x \nonumber \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c} ; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x \nonumber \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c} ; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x \nonumber \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{E} \left( \frac{w_M - w_\tau}{w_M + c} ; \eta(x), \Delta\eta(x, \tau) \right) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x, \nonumber
\end{align}
$$

$$
\begin{align}
E\{T(Z(\tau), W(\tau))\} ={}& \tau + \Delta\tau \cdot \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \nonumber \\
& + \Delta\tau \cdot \int_0^{L_w} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u) - \gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau. \tag{33}
\end{align}
$$

4. Application example

To demonstrate its effectiveness, we have applied the proposed policy to a real-world inspired case of a corroding pipeline. Pipelines are often buried underground, offshore, or in general in places that are difficult to access, so inspections are usually very costly and failures (e.g., the considered soft failure consisting in excess of corrosion) and/or small leaks are difficult to detect unless specific inspections are performed.

The application has been developed by considering eight realistic experimental scenarios, which have been defined by adopting the setups described in Table 4. The cost parameters reported in the table have been inspired by Dey [40]. Without loss of generality, in this example, it

<table><caption>**Table 4**
Setups used to evaluate the performance of the proposed policy.</caption>
  <thead>
    <tr>
      <th></th>
      <th>c<sub>l</sub></th>
      <th>c<sub>z</sub></th>
      <th>c<sub>w</sub></th>
      <th>c<sub>p</sub></th>
      <th>c<sub>c</sub></th>
      <th>c<sub>d</sub></th>
      <th>&phi; [mm<sup>-1</sup>]</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Setup A<sub>1</sub></td>
      <td>0.2</td>
      <td>0.2</td>
      <td>1</td>
      <td>2</td>
      <td>6</td>
      <td>0.6</td>
      <td>0.01</td>
    </tr>
    <tr>
      <td>Setup A<sub>2</sub></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td>0.005</td>
    </tr>
    <tr>
      <td>Setup B<sub>1</sub></td>
      <td>0.2</td>
      <td>0.2</td>
      <td>0.8</td>
      <td>2</td>
      <td>6</td>
      <td>0.6</td>
      <td>0.01</td>
    </tr>
    <tr>
      <td>Setup B<sub>2</sub></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td>0.005</td>
    </tr>
    <tr>
      <td>Setup C<sub>1</sub></td>
      <td>0.2</td>
      <td>0.2</td>
      <td>0.4</td>
      <td>2</td>
      <td>6</td>
      <td>0.6</td>
      <td>0.01</td>
    </tr>
    <tr>
      <td>Setup C<sub>2</sub></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td>0.005</td>
    </tr>
    <tr>
      <td>Setup D<sub>1</sub></td>
      <td>0.2</td>
      <td>0.8</td>
      <td>1.2</td>
      <td>2</td>
      <td>6</td>
      <td>0.1</td>
      <td>0.01</td>
    </tr>
    <tr>
      <td>Setup D<sub>2</sub></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td>0.005</td>
    </tr>
  </tbody>
</table>

is assumed that the unit-specific gamma processes are homogeneous with age function $\eta(t) = \alpha \cdot t$, an assumption that was found to be appropriate for modeling the effect of corrosion (e.g., see Ellingwood and Mori [41]). However, other modeling solutions can be adopted to describe degradation phenomena of different kinds (e.g., aging, reinforcement corrosion, etc.), without serious additional difficulties.

The parameter $\varphi$ has been calibrated to define two groups of experimental situations where the measurement errors have different magnitude. Specifically, subscript 1 identifies the scenarios where the measurement errors are moderate (in this case it is $\varphi = 0.01$) while subscript 2 identifies those where the measurement errors are severe (in this case it is $\varphi = 0.005$). The setups, such as $\mathcal{A}_1$ and $\mathcal{A}_2$, that are indicated by the same letter with a different subscript only differ in the magnitude of the measurement errors. The logistic cost $c_l$ is always equal to 0.2. The cost of the preventive replacement $c_p$ is always equal to 2. The cost of the corrective replacement $c_c$ is always equal to 6. The values of the other cost parameters vary according to the letter (i.e., $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, or $\mathcal{D}$) used to identify the setup. The units are considered failed when their degradation level exceeds the value $w_M = 35$ mm.

The parameters of the hidden degradation process are set to the values reported in Table 5, which have been selected by using as rough reference those reported in Mahmoodian and Alani [42]. Without loss of generality, the value of the parameter $\nu$ of the pdf (4) is always set equal to 1.

The results obtained by applying the suggested policy, hereinafter indicated by $\mathcal{P}_0$, are compared to those obtained by adopting three alternative policies, indicated by $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$. The policies $\mathcal{P}_1$ and $\mathcal{P}_2$ are defined by assuming that, at the inspection time, only the perturbed and only the perfect measurement procedure can be adopted,

<table><caption>**Table 5**
Values of the parameters of the degradation process.</caption>
  <thead>
    <tr>
      <th>a [years]</th>
      <th>c [mm]</th>
      <th>d</th>
      <th>ν</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0.17</td>
      <td>5</td>
      <td>2.125</td>
      <td>1</td>
    </tr>
  </tbody>
</table>
$$
\begin{equation}
\begin{aligned}
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{E}\left(\frac{w_M}{w_M + c}; \eta(\tau), d\right) \right] \\
& + c_w \cdot \int_0^\infty \left[ \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(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 \\
& + (c_l + c_d \cdot \Delta\tau) \cdot \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(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 \\
& + (c_c - c_p) \cdot \int_0^L \left[ 1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right) \right] \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(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 \\
& + (c_c - c_p) \cdot \int_0^{L_w} \left[ 1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right) \right] \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_0^L \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(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 \cdot dx \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x), \Delta\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 dw_\tau \cdot dx,
\end{aligned}
\tag{32}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
E\{T(Z(\tau), W(\tau))\} ={}& \tau + \Delta\tau \cdot \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \\
& + \Delta\tau \cdot \int_0^{L_w} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u) - \gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau.
\end{aligned}
\tag{33}
\end{equation}
$$

4. Application example

To demonstrate its effectiveness, we have applied the proposed policy to a real-world inspired case of a corroding pipeline. Pipelines are often buried underground, offshore, or in general in places that are difficult to access, so inspections are usually very costly and failures (e.g., the considered soft failure consisting in excess of corrosion) and/or small leaks are difficult to detect unless specific inspections are performed.

The application has been developed by considering eight realistic experimental scenarios, which have been defined by adopting the setups described in Table 4. The cost parameters reported in the table have been inspired by Dey [40]. Without loss of generality, in this example, it

<table><caption>**Table 4**
Setups used to evaluate the performance of the proposed policy.</caption><thead><tr><th></th><th>c<sub>l</sub></th><th>c<sub>z</sub></th><th>c<sub>w</sub></th><th>c<sub>p</sub></th><th>c<sub>c</sub></th><th>c<sub>d</sub></th><th>&phi; [mm<sup>-1</sup>]</th></tr></thead><tbody><tr><td>Setup A<sub>1</sub></td><td>0.2</td><td>0.2</td><td>1</td><td>2</td><td>6</td><td>0.6</td><td>0.01</td></tr><tr><td>Setup A<sub>2</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.005</td></tr><tr><td>Setup B<sub>1</sub></td><td>0.2</td><td>0.2</td><td>0.8</td><td>2</td><td>6</td><td>0.6</td><td>0.01</td></tr><tr><td>Setup B<sub>2</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.005</td></tr><tr><td>Setup C<sub>1</sub></td><td>0.2</td><td>0.2</td><td>0.4</td><td>2</td><td>6</td><td>0.6</td><td>0.01</td></tr><tr><td>Setup C<sub>2</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.005</td></tr><tr><td>Setup D<sub>1</sub></td><td>0.2</td><td>0.8</td><td>1.2</td><td>2</td><td>6</td><td>0.1</td><td>0.01</td></tr><tr><td>Setup D<sub>2</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.005</td></tr></tbody></table>

is assumed that the unit-specific gamma processes are homogeneous with age function $\eta(t) = \alpha \cdot t$, an assumption that was found to be appropriate for modeling the effect of corrosion (e.g., see Ellingwood and Mori [41]). However, other modeling solutions can be adopted to describe degradation phenomena of different kinds (e.g., aging, reinforcement corrosion, etc.), without serious additional difficulties.

The parameter $\varphi$ has been calibrated to define two groups of experimental situations where the measurement errors have different magnitude. Specifically, subscript 1 identifies the scenarios where the measurement errors are moderate (in this case it is $\varphi = 0.01$) while subscript 2 identifies those where the measurement errors are severe (in this case it is $\varphi = 0.005$). The setups, such as $\mathcal{A}_1$ and $\mathcal{A}_2$, that are indicated by the same letter with a different subscript only differ in the magnitude of the measurement errors. The logistic cost $c_l$ is always equal to 0.2. The cost of the preventive replacement $c_p$ is always equal to 2. The cost of the corrective replacement $c_c$ is always equal to 6. The values of the other cost parameters vary according to the letter (i.e., $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, or $\mathcal{D}$) used to identify the setup. The units are considered failed when their degradation level exceeds the value $w_M = 35$ mm.

The parameters of the hidden degradation process are set to the values reported in Table 5, which have been selected by using as rough reference those reported in Mahmoodian and Alani [42]. Without loss of generality, the value of the parameter $\nu$ of the pdf (4) is always set equal to 1.

The results obtained by applying the suggested policy, hereinafter indicated by $\mathcal{P}_0$, are compared to those obtained by adopting three alternative policies, indicated by $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$. The policies $\mathcal{P}_1$ and $\mathcal{P}_2$ are defined by assuming that, at the inspection time, only the perturbed and only the perfect measurement procedure can be adopted,

<table><caption>**Table 5**
Values of the parameters of the degradation process.</caption><thead><tr><th>a [years]</th><th>c [mm]</th><th>d</th><th>&nu;</th></tr></thead><tbody><tr><td>0.17</td><td>5</td><td>2.125</td><td>1</td></tr></tbody></table>
<table><caption>**Table 6**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{A}_1$ and $\mathcal{A}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup A<sub>1</sub></th><th colspan="6">Setup A<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.1005</td><td></td><td>22.2</td><td>43.9</td><td>3.76</td><td>7.81</td><td>10.3</td><td>0.1009</td><td></td><td>22.2</td><td>44.2</td><td>3.55</td><td>7.72</td><td>10.3</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>1.62%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>1.83%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.1062</td><td>5.58%</td><td>24.6</td><td>48.7</td><td></td><td></td><td>10.5</td><td>0.1062</td><td>5.20%</td><td>24.6</td><td>48.7</td><td></td><td></td><td>10.5</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>26.9%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>26.4%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 7**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{B}_1$ and $\mathcal{B}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup B<sub>1</sub></th><th colspan="6">Setup B<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0993</td><td></td><td>22.4</td><td>44.6</td><td>3.41</td><td>9.42</td><td>10.2</td><td>0.0996</td><td></td><td>22.4</td><td>44.8</td><td>3.23</td><td>9.35</td><td>10.2</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>2.94%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>3.22%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.1020</td><td>2.77%</td><td>24.0</td><td>47.9</td><td></td><td></td><td>10.3</td><td>0.1020</td><td>2.47%</td><td>24.0</td><td>47.9</td><td></td><td></td><td>10.3</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>28.5%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>28.2%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 8**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of the setups $\mathcal{C}_1$ and $\mathcal{C}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup C<sub>1</sub></th><th colspan="6">Setup C<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0954</td><td></td><td>22.5</td><td>45.1</td><td>2.68</td><td>16.85</td><td>10.0</td><td>0.0956</td><td></td><td>22.5</td><td>45.2</td><td>2.55</td><td>17</td><td>9.96</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>7.09%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>7.54%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.0934</td><td>-2.06%</td><td>22.8</td><td>46.0</td><td></td><td></td><td>9.83</td><td>0.0934</td><td>-2.19%</td><td>22.8</td><td>46.0</td><td></td><td></td><td>9.83</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>33.7%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>33.5%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 9**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{D}_1$ and $\mathcal{D}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup D<sub>1</sub></th><th colspan="6">Setup D<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0809</td><td></td><td>47.5</td><td>93.5</td><td>7.73</td><td>10.57</td><td>15.9</td><td>0.0812</td><td></td><td>48.0</td><td>94.3</td><td>7.33</td><td>10.3</td><td>16.0</td></tr><tr><td>P<sub>1</sub></td><td>0.0810</td><td>0.20%</td><td>46.6</td><td>90.7</td><td>9.15</td><td>9.15</td><td></td><td>0.0813</td><td>0.23%</td><td>47.0</td><td>90.9</td><td>8.86</td><td>8.86</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.0792</td><td>-2.06%</td><td>44.5</td><td>92.1</td><td></td><td></td><td>15.2</td><td>0.0792</td><td>-2.40%</td><td>44.5</td><td>92.1</td><td></td><td></td><td>15.2</td></tr><tr><td>P<sub>3</sub></td><td>0.0772</td><td>-4.55%</td><td>75.7</td><td></td><td></td><td></td><td></td><td>0.0772</td><td>-4.87%</td><td>75.7</td><td></td><td></td><td></td><td></td></tr></tbody></table>

respectively. The policy $\mathcal{P}_3$ is a classical (pure) age-based maintenance policy, which assumes that the unit is replaced at $\tau$, regardless of its state, without performing any inspection. It is worth to remark that the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ can be obtained as special cases from the policy $\mathcal{P}_0$ by imposing the following constraints:

* $\mathcal{P}_1$ (which uses only the imperfect measurement procedure) is obtained from $P_0$ by setting $L_l = L_u$;

* $\mathcal{P}_2$ (which uses only the perfect measurement procedure) is obtained from $P_0$ by setting $L_l = 0, L_u = +\infty$, and $c_z = 0$;

* $\mathcal{P}_3$ (which does not use any measurement procedure) is obtained from $P_0$ by setting $L_l = L_u = 0$ and $c_z = 0$.

All the policies have been applied to determine, under each setup, the optimal set of design parameters and the corresponding long-run average maintenance cost rate.

Setup by setup, to compare the performances of the policy $\mathcal{P}_0$ with those of the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ the percent relative difference PRD(k) ($k = 1, 2, 3$) has been evaluated:

$$PRD(k) = \frac{C_{\infty}(k\xi^*) - C_{\infty}(0\xi^*)}{C_{\infty}(0\xi^*)} \cdot 100. \quad (34)$$

where $C_{\infty}(\cdot)$ denotes the long-run average maintenance cost rate obtained by adopting the policy $\mathcal{P}_k$ ($k = 0, \dots, 3$) and $k\xi^*$ indicates the
<table><caption>**Table 6**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{A}_1$ and $\mathcal{A}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup A<sub>1</sub></th><th colspan="6">Setup A<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.1005</td><td></td><td>22.2</td><td>43.9</td><td>3.76</td><td>7.81</td><td>10.3</td><td>0.1009</td><td></td><td>22.2</td><td>44.2</td><td>3.55</td><td>7.72</td><td>10.3</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>1.62%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>1.83%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.1062</td><td>5.58%</td><td>24.6</td><td>48.7</td><td></td><td></td><td>10.5</td><td>0.1062</td><td>5.20%</td><td>24.6</td><td>48.7</td><td></td><td></td><td>10.5</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>26.9%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>26.4%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 7**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{B}_1$ and $\mathcal{B}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup B<sub>1</sub></th><th colspan="6">Setup B<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0993</td><td></td><td>22.4</td><td>44.6</td><td>3.41</td><td>9.42</td><td>10.2</td><td>0.0996</td><td></td><td>22.4</td><td>44.8</td><td>3.23</td><td>9.35</td><td>10.2</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>2.94%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>3.22%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.1020</td><td>2.77%</td><td>24.0</td><td>47.9</td><td></td><td></td><td>10.3</td><td>0.1020</td><td>2.47%</td><td>24.0</td><td>47.9</td><td></td><td></td><td>10.3</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>28.5%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>28.2%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 8**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of the setups $\mathcal{C}_1$ and $\mathcal{C}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup C<sub>1</sub></th><th colspan="6">Setup C<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0954</td><td></td><td>22.5</td><td>45.1</td><td>2.68</td><td>16.85</td><td>10.0</td><td>0.0956</td><td></td><td>22.5</td><td>45.2</td><td>2.55</td><td>17</td><td>9.96</td></tr><tr><td>P<sub>1</sub></td><td>0.1022</td><td>7.09%</td><td>21.0</td><td>39.6</td><td>5.02</td><td>5.02</td><td></td><td>0.1027</td><td>7.54%</td><td>20.9</td><td>39.7</td><td>4.81</td><td>4.81</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.0934</td><td>-2.06%</td><td>22.8</td><td>46.0</td><td></td><td></td><td>9.83</td><td>0.0934</td><td>-2.19%</td><td>22.8</td><td>46.0</td><td></td><td></td><td>9.83</td></tr><tr><td>P<sub>3</sub></td><td>0.1276</td><td>33.7%</td><td>32.0</td><td></td><td></td><td></td><td></td><td>0.1276</td><td>33.5%</td><td>32.0</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 9**
Optimal values of the design parameters, of the long-run average maintenance cost rate, and of PRD(·) under all the considered policies in the case of setups $\mathcal{D}_1$ and $\mathcal{D}_2$.</caption><thead><tr><th rowspan="2"></th><th colspan="6">Setup D<sub>1</sub></th><th colspan="6">Setup D<sub>2</sub></th></tr><tr><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th><th><sup>k</sup>C<sub>∞</sub>(<sup>k</sup>ξ<sup>*</sup>)</th><th>PRD(k)</th><th><sup>k</sup>τ<sup>*</sup></th><th><sup>k</sup>Δτ<sup>*</sup></th><th><sup>k</sup>L<sub>l</sub><sup>*</sup></th><th><sup>k</sup>L<sub>u</sub><sup>*</sup></th><th><sup>k</sup>L<sub>W</sub><sup>*</sup></th></tr></thead><tbody><tr><td>P<sub>0</sub></td><td>0.0809</td><td></td><td>47.5</td><td>93.5</td><td>7.73</td><td>10.57</td><td>15.9</td><td>0.0812</td><td></td><td>48.0</td><td>94.3</td><td>7.33</td><td>10.3</td><td>16.0</td></tr><tr><td>P<sub>1</sub></td><td>0.0810</td><td>0.20%</td><td>46.6</td><td>90.7</td><td>9.15</td><td>9.15</td><td></td><td>0.0813</td><td>0.23%</td><td>47.0</td><td>90.9</td><td>8.86</td><td>8.86</td><td></td></tr><tr><td>P<sub>2</sub></td><td>0.0792</td><td>-2.06%</td><td>44.5</td><td>92.1</td><td></td><td></td><td>15.2</td><td>0.0792</td><td>-2.40%</td><td>44.5</td><td>92.1</td><td></td><td></td><td>15.2</td></tr><tr><td>P<sub>3</sub></td><td>0.0772</td><td>-4.55%</td><td>75.7</td><td></td><td></td><td></td><td></td><td>0.0772</td><td>-4.87%</td><td>75.7</td><td></td><td></td><td></td><td></td></tr></tbody></table>

respectively. The policy $\mathcal{P}_3$ is a classical (pure) age-based maintenance policy, which assumes that the unit is replaced at $\tau$, regardless of its state, without performing any inspection. It is worth to remark that the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ can be obtained as special cases from the policy $\mathcal{P}_0$ by imposing the following constraints:

* $\mathcal{P}_1$ (which uses only the imperfect measurement procedure) is obtained from $P_0$ by setting $L_l = L_u$;

* $\mathcal{P}_2$ (which uses only the perfect measurement procedure) is obtained from $P_0$ by setting $L_l = 0, L_u = +\infty$, and $c_z = 0$;

* $\mathcal{P}_3$ (which does not use any measurement procedure) is obtained from $P_0$ by setting $L_l = L_u = 0$ and $c_z = 0$.

All the policies have been applied to determine, under each setup, the optimal set of design parameters and the corresponding long-run average maintenance cost rate.

Setup by setup, to compare the performances of the policy $\mathcal{P}_0$ with those of the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ the percent relative difference PRD(k) ($k = 1, 2, 3$) has been evaluated:

$$PRD(k) = \frac{C_{\infty}(k\xi^*) - C_{\infty}(0\xi^*)}{C_{\infty}(0\xi^*)} \cdot 100. \quad (34)$$

where $C_{\infty}(\cdot)$ denotes the long-run average maintenance cost rate obtained by adopting the policy $\mathcal{P}_k$ ($k = 0, \dots, 3$) and $k\xi^*$ indicates the
corresponding optimal set of design parameters, being $0^\xi = \{0^\tau, 0^{\Delta\tau}, 0^{L_1}, 0^{L_u}, 0^{L_W}\}$, $1^\xi = \{1^\tau, 1^{\Delta\tau}, 1^{L_1} = 1^{L_u}\}$, $2^\xi = \{2^\tau, 2^{\Delta\tau}, 2^{L_W}\}$, and $3^\xi = \{3^\tau\}$. Thus, $0^{C_\infty}(\cdot)$ is the long-run average maintenance cost rate given in (31). The expressions of $^kC_\infty(\cdot)$ under the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ are provided in Appendix B.

Table 6 reports the optimal long-run average maintenance cost rate and the corresponding optimal values of the design parameters obtained under all the considered policies in the case of the setups $\mathcal{A}_1$ and $\mathcal{A}_2$. Tables 7, 8, and 9 report the same quantities in the cases of the setups $\mathcal{B}_1$ and $\mathcal{B}_2$, $\mathcal{C}_1$ and $\mathcal{C}_2$, and $\mathcal{D}_1$ and $\mathcal{D}_2$, respectively. Obviously, the results obtained under the policies $\mathcal{P}_2$ and $\mathcal{P}_3$, which do not use the imperfect measurement procedure, are not affected by the magnitude of the measurement errors. Thus, these results do not change passing from the left to the right side of the mentioned tables. For similar reasons, the results obtained under the policy $\mathcal{P}_1$ do not depend on $c_w$, those obtained under the policy $\mathcal{P}_2$ do not depend on $c_z$, and those obtained under the policy $\mathcal{P}_3$ depend neither on $c_w$ nor on $c_z$.

Table 6 shows that, under the setups $\mathcal{A}_1$ and $\mathcal{A}_2$, the policy which yields the lowest long-run average maintenance cost rate (i.e., the best policy) is $\mathcal{P}_0$. It also shows that, under these setups, $\mathcal{P}_1$ outperforms $\mathcal{P}_2$, which in turn performs better than $\mathcal{P}_3$. In fact, for example, under the setup $\mathcal{A}_1$ the use of the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$, in place of the policy $\mathcal{P}_0$, leads to percent increases of the long-run average maintenance cost rate equal to 1.62%, 5.58%, and 26.9%, respectively.

The setups $\mathcal{B}_1$ and $\mathcal{B}_2$ differ from $\mathcal{A}_1$ and $\mathcal{A}_2$ only for the value of $c_w$, which is 0.8 instead of 1. Table 7 shows that under the setups $\mathcal{B}_1$ and $\mathcal{B}_2$ the policies $\mathcal{P}_0$ and $\mathcal{P}_3$ are still the best and the worst one, respectively. On the other hand, due to the reduced value of $c_w$, which incentivizes the use of the perfect inspection, in these cases $\mathcal{P}_2$ outperforms $\mathcal{P}_1$. In fact, for example, in the case of the setup $\mathcal{B}_1$, using the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ instead of $\mathcal{P}_0$ produces a percent increase of the long-run average maintenance cost rate equal to 2.94%, 2.77%, and 28.5%, respectively.

The setups $\mathcal{C}_1$ and $\mathcal{C}_2$ differ from the setups $\mathcal{A}_1$, $\mathcal{B}_1$ and $\mathcal{A}_2$, $\mathcal{B}_2$ in the value of $c_w$ only, which is smaller than in the other mentioned cases. These setups reflect an experimental scenario in which the perfect measurement procedure is only slightly more expensive than the imperfect one. Table 8 shows that, under $\mathcal{C}_1$ and $\mathcal{C}_2$, the best policy is $\mathcal{P}_2$, which uses the perfect measurement procedure only. The same table also shows that $\mathcal{P}_0$, which allows for the optional use of the perfect measurement procedure, performs better than $\mathcal{P}_1$. As in the cases of the setups $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{B}_1$, and $\mathcal{B}_2$, the policies $\mathcal{P}_0$, $\mathcal{P}_1$, and $\mathcal{P}_2$ outperform the (pure) age-based policy $\mathcal{P}_3$. In fact, for example, under the setup $\mathcal{C}_1$ the use of $\mathcal{P}_2$ in place of $\mathcal{P}_0$ leads to a percent reduction of the long-run average maintenance cost rate equal to 2.06%, while $\mathcal{P}_1$ and $\mathcal{P}_3$ provide long-run average maintenance cost rates that are 7.09% and 33.7% higher than the one provided by $\mathcal{P}_0$.

With respect to the other setups, $\mathcal{D}_1$ and $\mathcal{D}_2$ depict scenarios characterized by higher inspection costs $c_w$ and $c_z$ and smaller downtime cost rate $c_d$. Table 9 shows that, under these setups, the best policy is $\mathcal{P}_3$, followed by $\mathcal{P}_2$ and $\mathcal{P}_0$, which performs slightly better than $\mathcal{P}_1$.

The results obtained under these latter setups show that, when the inspection costs are high with respect to other maintenance costs, performing inspections does not necessarily give economic advantages in the long run.

Tables 6-9 also show that the magnitude of the measurement error (within the considered range) does not significantly influence the discussed results. Hence, the comments provided above apply both when measurements are moderately and severely perturbed. In fact, the value of $\varphi$ only influences the long-run average maintenance cost rate under the policies $\mathcal{P}_0$ and $\mathcal{P}_1$, which in general slightly increases as $\varphi$ decreases. Consequently, also the percent relative difference $\text{PRD}(\cdot)$ slightly changes passing from the case where the error is moderate to the one when it is severe.

The severity of the measurement error can be expressed in terms of the

ratio $V\{Z(t)\}/V\{W(t)\}$. Unfortunately, both the variance of $W(t)$ and $Z(t)$ (see (14) and (22)), as well as $V\{Z(t)\}/V\{W(t)\}$ depend on $t$. Hence, it is not possible to quantify the ratio $V\{Z(t)\}/V\{W(t)\}$ by a unique number. However, it can be useful to note that, for example, in the case of the setup $\mathcal{A}_1$ at $0^{\tau^*} = 22.2$ it is $V\{Z(0^{\tau^*})\}/V\{W(0^{\tau^*})\} = 1.57$ while under the setup $\mathcal{A}_2$ at $0^{\tau^*} = 22.2$ it is $V\{Z(0^{\tau^*})\}/V\{W(0^{\tau^*})\} = 2.15$. Similarly, under the setup $\mathcal{B}_1$ at $0^{\tau^*} = 47.5$ it is $V\{Z(0^{\tau^*})\}/V\{W(0^{\tau^*})\} = 1.30$ while under the setup $\mathcal{B}_2$ at $0^{\tau^*} = 48.0$ it is $V\{Z(0^{\tau^*})\}/V\{W(0^{\tau^*})\} = 1.61$.

To deepen and better understand how the results provided by the considered policies change with the setup, it is useful to investigate how the long-run average maintenance cost rate $^kC_\infty(\cdot)$ splits into the components $^kC_c(\cdot)$, $^kC_d(\cdot)$, $^kC_\infty(\cdot)$, $^kC_p(\cdot)$, $^kC_w(\cdot)$, and $^kC_\infty(\cdot)$ that denote the contributions to $^kC_\infty(\cdot)$ of corrective replacement, downtime, logistic, preventive replacement, perfect inspection, and perturbed inspection costs, respectively. The expressions of these cost rates are given in Appendix C. Obviously, it is $^kC_\infty(\cdot) = ^kC_c(\cdot) + ^kC_d(\cdot) + ^kC_p(\cdot) + ^kC_w(\cdot) + ^kC_\infty(\cdot)$.

Figs. 1 and 2 show how in the case of setups $\mathcal{A}_1$ and $\mathcal{D}_1$, respectively, the (minimum) long-run average maintenance cost rate splits into the aforementioned contributions.

The setup $\mathcal{D}_1$ differs from the setup $\mathcal{A}_1$ for the values of $c_w$, $c_z$, and $c_d$, which pass from 0.8, 0.2, and 0.6 under setup $\mathcal{A}_1$ to 1.2, 0.8, and 0.1 under setup $\mathcal{B}_1$, respectively.

From Fig. 1, it is apparent that in the case of the setup $\mathcal{A}_1$ the main

<figure><img src="image_4.png" /><figcaption>Fig. 1. Values of $^kC_\infty(\cdot)$, $^kC_d(\cdot)$, $^kC_\infty(\cdot)$, $^kC_p(\cdot)$, $^kC_w(\cdot)$, and $^kC_\infty(\cdot)$ under each of the considered policies in the case of the setup $\mathcal{A}_1$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Values of $^kC_\infty(\cdot)$, $^kC_d(\cdot)$, $^kC_\infty(\cdot)$, $^kC_p(\cdot)$, $^kC_w(\cdot)$, and $^kC_\infty(\cdot)$ under each of the considered policies in the case of the setup $\mathcal{B}_1$.</figcaption></figure>
corresponding optimal set of design parameters, being $0\xi = \{0\tau, 0\Delta\tau, 0L_1, 0L_u, 0L_W\}$, $1\xi = \{1\tau, 1\Delta\tau, 1L_1 = 1L_u\}$, $2\xi = \{2\tau, 2\Delta\tau, 2L_W\}$, and $3\xi = \{3\tau\}$. Thus, $0C_\infty(\cdot)$ is the long-run average maintenance cost rate given in (31). The expressions of $kC_\infty(\cdot)$ under the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ are provided in Appendix B.

Table 6 reports the optimal long-run average maintenance cost rate and the corresponding optimal values of the design parameters obtained under all the considered policies in the case of the setups $\mathcal{A}_1$ and $\mathcal{A}_2$. Tables 7, 8, and 9 report the same quantities in the cases of the setups $\mathcal{B}_1$ and $\mathcal{B}_2$, $\mathcal{C}_1$ and $\mathcal{C}_2$, and $\mathcal{D}_1$ and $\mathcal{D}_2$, respectively. Obviously, the results obtained under the policies $\mathcal{P}_2$ and $\mathcal{P}_3$, which do not use the imperfect measurement procedure, are not affected by the magnitude of the measurement errors. Thus, these results do not change passing from the left to the right side of the mentioned tables. For similar reasons, the results obtained under the policy $\mathcal{P}_1$ do not depend on $c_w$, those obtained under the policy $\mathcal{P}_2$ do not depend on $c_z$, and those obtained under the policy $\mathcal{P}_3$ depend neither on $c_w$ nor on $c_z$.

Table 6 shows that, under the setups $\mathcal{A}_1$ and $\mathcal{A}_2$, the policy which yields the lowest long-run average maintenance cost rate (i.e., the best policy) is $\mathcal{P}_0$. It also shows that, under these setups, $\mathcal{P}_1$ outperforms $\mathcal{P}_2$, which in turn performs better than $\mathcal{P}_3$. In fact, for example, under the setup $\mathcal{A}_1$ the use of the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$, in place of the policy $\mathcal{P}_0$, leads to percent increases of the long-run average maintenance cost rate equal to 1.62%, 5.58%, and 26.9%, respectively.

The setups $\mathcal{B}_1$ and $\mathcal{B}_2$ differ from $\mathcal{A}_1$ and $\mathcal{A}_2$ only for the value of $c_w$, which is 0.8 instead of 1. Table 7 shows that under the setups $\mathcal{B}_1$ and $\mathcal{B}_2$ the policies $\mathcal{P}_0$ and $\mathcal{P}_3$ are still the best and the worst one, respectively. On the other hand, due to the reduced value of $c_w$, which incentivizes the use of the perfect inspection, in these cases $\mathcal{P}_2$ outperforms $\mathcal{P}_1$. In fact, for example, in the case of the setup $\mathcal{B}_1$, using the policies $\mathcal{P}_1$, $\mathcal{P}_2$, and $\mathcal{P}_3$ instead of $\mathcal{P}_0$ produces a percent increase of the long-run average maintenance cost rate equal to 2.94%, 2.77%, and 28.5%, respectively.

The setups $\mathcal{C}_1$ and $\mathcal{C}_2$ differ from the setups $\mathcal{A}_1$, $\mathcal{B}_1$ and $\mathcal{A}_2$, $\mathcal{B}_2$ in the value of $c_w$ only, which is smaller than in the other mentioned cases. These setups reflect an experimental scenario in which the perfect measurement procedure is only slightly more expensive than the imperfect one. Table 8 shows that, under $\mathcal{C}_1$ and $\mathcal{C}_2$, the best policy is $\mathcal{P}_2$, which uses the perfect measurement procedure only. The same table also shows that $\mathcal{P}_0$, which allows for the optional use of the perfect measurement procedure, performs better than $\mathcal{P}_1$. As in the cases of the setups $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{B}_1$, and $\mathcal{B}_2$, the policies $\mathcal{P}_0$, $\mathcal{P}_1$, and $\mathcal{P}_2$ outperform the (pure) age-based policy $\mathcal{P}_3$. In fact, for example, under the setup $\mathcal{C}_1$ the use of $\mathcal{P}_2$ in place of $\mathcal{P}_0$ leads to a percent reduction of the long-run average maintenance cost rate equal to 2.06%, while $\mathcal{P}_1$ and $\mathcal{P}_3$ provide long-run average maintenance cost rates that are 7.09% and 33.7% higher than the one provided by $\mathcal{P}_0$.

With respect to the other setups, $\mathcal{D}_1$ and $\mathcal{D}_2$ depict scenarios characterized by higher inspection costs $c_w$ and $c_z$ and smaller downtime cost rate $c_d$. Table 9 shows that, under these setups, the best policy is $\mathcal{P}_3$, followed by $\mathcal{P}_2$ and $\mathcal{P}_0$, which performs slightly better than $\mathcal{P}_1$.

The results obtained under these latter setups show that, when the inspection costs are high with respect to other maintenance costs, performing inspections does not necessarily give economic advantages in the long run.

Tables 6-9 also show that the magnitude of the measurement error (within the considered range) does not significantly influence the discussed results. Hence, the comments provided above apply both when measurements are moderately and severely perturbed. In fact, the value of $\varphi$ only influences the long-run average maintenance cost rate under the policies $\mathcal{P}_0$ and $\mathcal{P}_1$, which in general slightly increases as $\varphi$ decreases. Consequently, also the percent relative difference $\text{PRD}(\cdot)$ slightly changes passing from the case where the error is moderate to the one when it is severe.

The severity of the measurement error can be expressed in terms of the

ratio $V\{Z(t)\}/V\{W(t)\}$. Unfortunately, both the variance of $W(t)$ and $Z(t)$ (see (14) and (22)), as well as $V\{Z(t)\}/V\{W(t)\}$ depend on $t$. Hence, it is not possible to quantify the ratio $V\{Z(t)\}/V\{W(t)\}$ by a unique number. However, it can be useful to note that, for example, in the case of the setup $\mathcal{A}_1$ at $0\tau^* = 22.2$ it is $V\{Z(0\tau^*)\}/V\{W(0\tau^*)\} = 1.57$ while under the setup $\mathcal{A}_2$ at $0\tau^* = 22.2$ it is $V\{Z(0\tau^*)\}/V\{W(0\tau^*)\} = 2.15$. Similarly, under the setup $\mathcal{B}_1$ at $0\tau^* = 47.5$ it is $V\{Z(0\tau^*)\}/V\{W(0\tau^*)\} = 1.30$ while under the setup $\mathcal{B}_2$ at $0\tau^* = 48.0$ it is $V\{Z(0\tau^*)\}/V\{W(0\tau^*)\} = 1.61$.

To deepen and better understand how the results provided by the considered policies change with the setup, it is useful to investigate how the long-run average maintenance cost rate $kC_\infty(\cdot)$ splits into the components $k_cC_\infty(\cdot)$, $k_dC_\infty(\cdot)$, $k_lC_\infty(\cdot)$, $k_pC_\infty(\cdot)$, $k_wC_\infty(\cdot)$, and $k_zC_\infty(\cdot)$ that denote the contributions to $kC_\infty(\cdot)$ of corrective replacement, downtime, logistic, preventive replacement, perfect inspection, and perturbed inspection costs, respectively. The expressions of these cost rates are given in Appendix C. Obviously, it is $kC_\infty(\cdot) = k_cC_\infty(\cdot) + k_dC_\infty(\cdot) + k_lC_\infty(\cdot) + k_pC_\infty(\cdot) + k_wC_\infty(\cdot) + k_zC_\infty(\cdot)$.

Figs. 1 and 2 show how in the case of setups $\mathcal{A}_1$ and $\mathcal{D}_1$, respectively, the (minimum) long-run average maintenance cost rate splits into the aforementioned contributions.

The setup $\mathcal{D}_1$ differs from the setup $\mathcal{A}_1$ for the values of $c_w$, $c_z$, and $c_d$, which pass from 0.8, 0.2, and 0.6 under setup $\mathcal{A}_1$ to 1.2, 0.8, and 0.1 under setup $\mathcal{D}_1$, respectively.

From Fig. 1, it is apparent that in the case of the setup $\mathcal{A}_1$ the main

<figure><img src="image_4.png" /><figcaption>Fig. 1. Values of $k_cC_\infty(\cdot)$, $k_dC_\infty(\cdot)$, $k_lC_\infty(\cdot)$, $k_pC_\infty(\cdot)$, $k_wC_\infty(\cdot)$, and $k_zC_\infty(\cdot)$ under each of the considered policies in the case of the setup $\mathcal{A}_1$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Values of $k_cC_\infty(\cdot)$, $k_dC_\infty(\cdot)$, $k_lC_\infty(\cdot)$, $k_pC_\infty(\cdot)$, $k_wC_\infty(\cdot)$, and $k_zC_\infty(\cdot)$ under each of the considered policies in the case of the setup $\mathcal{D}_1$.</figcaption></figure>
contribution, under all the policies, is $^k_pC_\infty(\cdot)$, followed by $^k_dC_\infty(\cdot)$ and $^k_cC_\infty(\cdot)$. This shows that, under this setup, the maintenance cycle ends preferentially with a preventive replacement, hence units are generally replaced when they are not yet failed. In fact, replacing failed units implies not only a higher (compared to the preventive one) corrective replacement cost but also a downtime cost that, depending on the setup, can be quite hefty.

Conversely, in the case of the setup $\mathcal{S}_1$, where the downtime cost rate $c_d$ is lower than under the other setups, the consequences of failures are milder. Hence, the setup $\mathcal{S}_1$, compared to the setup $\mathcal{A}_1$, leads all of the considered policies to incentivize more corrective replacements, by increasing the probability of failure in the maintenance cycle. This situation is clearly shown in Fig. 2, where it is apparent that (under all the considered policies) the dominant contribution is now $^k_cC_\infty(\cdot)$. Tables 6 and 9 show that this result is primarily accomplished by lengthening the maintenance cycle. In fact, from these tables, it is immediate to verify that the values of $^k\tau^*$ and $^k\Delta\tau^*$ obtained under the setup $\mathcal{S}_1$ are all higher than the corresponding values obtained under the setup $\mathcal{A}_1$.

## 5. Conclusions

In this paper, a new maintenance policy has been suggested for degrading units whose observed degradation paths can be modeled by a perturbed gamma process with random effect. The hidden degradation process is described by using a gamma model with random effect, while the measurement error is modeled as a non-Gaussian random variable that stochastically depends on the measured degradation level. Failures are assumed to occur when the hidden degradation level firstly passes a fixed threshold and are supposed to be not self-announcing.

The proposed policy consists in performing, at a predefined epoch, an inspection aimed at measuring the degradation level of the unit. The measurement is initially performed by using an imperfect procedure, that provides a perturbed measurement of the true degradation level. Based on the outcome of this first measurement procedure, a condition-based rule is then used to decide whether to immediately replace the unit, to postpone its replacement to a (also predefined) later time, or to use second perfect measurement procedure, which has an extra cost. In this latter case, based on the outcome of this second measurement procedure it is finally decided whether to immediately replace the unit or to postpone its replacement to a predefined future time. In case the replacement is postponed, no further inspections will be performed and the unit will be replaced regardless of its degradation state. Each replacement is assumed to restore the unit to an “as good as new” state. The performances of the maintenance policy are evaluated in terms of long-run average maintenance cost rate.

## Appendix A

Eq. (32) can be derived starting as follows:

An application example of the proposed approach to a real-world inspired case of a buried pipeline has been developed. The performances of the suggested policy are compared with those of three alternative policies obtained by assuming that only one (either the perturbed one or the perfect one, respectively) or none of the considered measurement procedures is adopted. The comparative study has been developed by considering eight setups that differ in the values of some of the cost parameters and/or in the magnitude of the measurement error. The performances of the considered policies have been evaluated and compared in terms of long-run average maintenance cost rate. Obtained results demonstrate the affordability and utility of the suggested approach.

## Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research activity was supported by Università di Napoli Federico II in the frame of the international agreement between Dipartimento di Ingegneria Industriale and Polytech Angers (codice identificativo 000011-ALTRI-2021-M-GIORGIO_001_001, prot. 40,162 del 20/4/2021), by the Université Franco Italien within the frame of the chapitre 2 of Vinci project (subvention N °C2-221), and by the WISE project of the Région Pays de la Loire (France).

## CRediT authorship contribution statement

**Nicola Esposito:** Conceptualization, Formal analysis, Methodology, Software, Writing – original draft. **Agostino Mele:** Conceptualization, Formal analysis. **Bruno Castanier:** Conceptualization, Formal analysis, Methodology, Supervision, Funding acquisition, Writing – review & editing. **Massimiliano GIORGIO:** Conceptualization, Formal analysis, Methodology, Supervision, Funding acquisition, Writing – original draft, Writing – review & editing.

## 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.

## Data availability

The data that supports the findings of this study are available within the article.
contribution, under all the policies, is $^k_pC_\infty(\cdot)$, followed by $^k_dC_\infty(\cdot)$ and $^k_cC_\infty(\cdot)$. This shows that, under this setup, the maintenance cycle ends preferentially with a preventive replacement, hence units are generally replaced when they are not yet failed. In fact, replacing failed units implies not only a higher (compared to the preventive one) corrective replacement cost but also a downtime cost that, depending on the setup, can be quite hefty.

Conversely, in the case of the setup $\mathcal{S}_1$, where the downtime cost rate $c_d$ is lower than under the other setups, the consequences of failures are milder. Hence, the setup $\mathcal{S}_1$, compared to the setup $\mathcal{A}_1$, leads all of the considered policies to incentivize more corrective replacements, by increasing the probability of failure in the maintenance cycle. This situation is clearly shown in Fig. 2, where it is apparent that (under all the considered policies) the dominant contribution is now $^k_cC_\infty(\cdot)$. Tables 6 and 9 show that this result is primarily accomplished by lengthening the maintenance cycle. In fact, from these tables, it is immediate to verify that the values of $^k\tau^*$ and $^k\Delta\tau^*$ obtained under the setup $\mathcal{S}_1$ are all higher than the corresponding values obtained under the setup $\mathcal{A}_1$.

## 5. Conclusions

In this paper, a new maintenance policy has been suggested for degrading units whose observed degradation paths can be modeled by a perturbed gamma process with random effect. The hidden degradation process is described by using a gamma model with random effect, while the measurement error is modeled as a non-Gaussian random variable that stochastically depends on the measured degradation level. Failures are assumed to occur when the hidden degradation level firstly passes a fixed threshold and are supposed to be not self-announcing.

The proposed policy consists in performing, at a predefined epoch, an inspection aimed at measuring the degradation level of the unit. The measurement is initially performed by using an imperfect procedure, that provides a perturbed measurement of the true degradation level. Based on the outcome of this first measurement procedure, a condition-based rule is then used to decide whether to immediately replace the unit, to postpone its replacement to a (also predefined) later time, or to use second perfect measurement procedure, which has an extra cost. In this latter case, based on the outcome of this second measurement procedure it is finally decided whether to immediately replace the unit or to postpone its replacement to a predefined future time. In case the replacement is postponed, no further inspections will be performed and the unit will be replaced regardless of its degradation state. Each replacement is assumed to restore the unit to an “as good as new” state. The performances of the maintenance policy are evaluated in terms of long-run average maintenance cost rate.

## Appendix A

Eq. (32) can be derived starting as follows:

An application example of the proposed approach to a real-world inspired case of a buried pipeline has been developed. The performances of the suggested policy are compared with those of three alternative policies obtained by assuming that only one (either the perturbed one or the perfect one, respectively) or none of the considered measurement procedures is adopted. The comparative study has been developed by considering eight setups that differ in the values of some of the cost parameters and/or in the magnitude of the measurement error. The performances of the considered policies have been evaluated and compared in terms of long-run average maintenance cost rate. Obtained results demonstrate the affordability and utility of the suggested approach.

## Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research activity was supported by Università di Napoli Federico II in the frame of the international agreement between Dipartimento di Ingegneria Industriale and Polytech Angers (codice identificativo 000011-ALTRI-2021-M-GIORGIO_001_001, prot. 40,162 del 20/4/2021), by the Université Franco Italien within the frame of the chapitre 2 of Vinci project (subvention N °C2-221), and by the WISE project of the Région Pays de la Loire (France).

## CRediT authorship contribution statement

**Nicola Esposito:** Conceptualization, Formal analysis, Methodology, Software, Writing – original draft. **Agostino Mele:** Conceptualization, Formal analysis. **Bruno Castanier:** Conceptualization, Formal analysis, Methodology, Supervision, Funding acquisition, Writing – review & editing. **Massimiliano GIORGIO:** Conceptualization, Formal analysis, Methodology, Supervision, Funding acquisition, Writing – original draft, Writing – review & editing.

## 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.

## Data availability

The data that supports the findings of this study are available within the article.
$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= \int_0^\infty \int_0^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&= \int_0^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \tag{A.1} \\
&\quad + \int_{L_W}^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_0^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber
\end{align}
$$

Hence, from the (A.1), replacing $C(z_\tau, w_\tau, x)$ (integral by integral) with the appropriate expression specified in Table 3, the (A.2) is readily obtained:

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= \int_0^\infty \int_0^\infty \int_0^\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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty [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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty [2 \cdot c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_u}^\infty \int_0^\infty (2 \cdot 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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_i}^\infty \int_0^\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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty [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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty [2 \cdot c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_i}^\infty \int_0^\infty (2 \cdot 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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_0^\infty \int_0^\infty [c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - X)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \tag{A.2}
\end{align}
$$

where:

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

and

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

The (A.2) can be then rewritten as:
$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= \int_0^\infty \int_0^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&= \int_0^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \tag{A.1} \\
&\quad + \int_{L_W}^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_0^\infty \int_0^\infty C(z_\tau, w_\tau, x) f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber
\end{align}
$$

Hence, from the (A.1), replacing $C(z_\tau, w_\tau, x)$ (integral by integral) with the appropriate expression specified in Table 3, the (A.2) is readily obtained:

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} &= \int_0^\infty \int_0^\infty \int_0^\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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty [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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty [2 \cdot c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_u}^\infty \int_0^\infty (2 \cdot 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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{L_W}^\infty \int_{L_i}^\infty \int_0^\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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty [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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty [2 \cdot c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - x)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_{w_M}^\infty \int_{L_u}^\infty \int_0^\infty (2 \cdot 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 dw_\tau \cdot dz_\tau \cdot dx \nonumber \\
&\quad + \int_0^\infty \int_{L_i}^\infty \int_0^\infty [c_l + c_z + c_w + c_c + c_d \cdot (\tau + \Delta\tau - X)] f_{X|W(\tau)}(x|w_\tau) f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \cdot dx \tag{A.2}
\end{align}
$$

where:

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

and

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

The (A.2) can be then rewritten as:
$$
\begin{equation}
\begin{aligned}
E\{C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \\
& + c_l \cdot F_{Z(\tau)}(L_l) + c_d \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau + (c_c - c_p) \cdot \int_0^{L_W} \int_\tau^{\tau+\Delta\tau} f_{X|W(\tau)}(x|w_\tau) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \\
& + (c_c - c_p) \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} f_{X|W(\tau)}(x|w_\tau) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx + c_d \cdot \Delta\tau \cdot \int_{W_M}^{\infty} F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \\
& + c_d \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \\
& + c_d \cdot \int_0^{L_W} \left[ \int_\tau^{\tau+\Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \right] \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau - c_d \cdot \int_{W_M}^{\infty} \int_0^\tau x \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau.
\end{aligned}
\tag{A.3}
\end{equation}
$$

Hence, from (A.3), by exploiting the results given in (24), (28), and (30), the following expression is obtained:

$$
\begin{equation}
\begin{aligned}
E\{C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \\
& + (c_l + c_d \cdot \Delta\tau) \cdot F_{Z(\tau)}(L_l) + c_d \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \\
& + (c_c - c_p) \cdot \int_0^{L_W} \left[1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right)\right] \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \\
& + (c_c - c_p) \cdot \int_0^{W_M} \left[1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right)\right] \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \\
& + c_d \cdot \Delta\tau \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau - c_d \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_0^{L_W} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_{W_M}^{\infty} \int_0^\tau \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx.
\end{aligned}
\tag{A.4}
\end{equation}
$$

It is not hard to verify that the (A.4) reduces to the (32) once $F_{W(\tau)}(\cdot)$, $F_{Z(\tau)}(\cdot)$, $F_{Z(\tau)|W(\tau)}(\cdot|\cdot)$ and $f_{W(\tau)}(\cdot)$ are replaced with the expressions given in (12), (20), (8), and (11), respectively.

The (33) can be more easily derived by following the scheme given in Table 2. In fact, more directly, it is:

$$
\begin{align}
E\{T(Z(\tau), W(\tau))\} &= \tau \cdot [1 - F_{Z(\tau)}(L_u)] + \tau \cdot \int_{L_u}^{\infty} \int_{L_l}^{L_W} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau + (\tau + \Delta\tau) \cdot F_{Z(\tau)}(L_l) + (\tau + \Delta\tau) \cdot \int_0^{L_W} \int_{L_l}^{L_u} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \nonumber \\
&= \tau + \Delta\tau \cdot F_{Z(\tau)}(L_l) + \Delta\tau \cdot \int_0^{L_W} \int_{L_l}^{L_u} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \tag{A.5}
\end{align}
$$

which reduces to (33) once $F_{Z(\tau)}(\cdot)$, $F_{Z(\tau)|W(\tau)}(\cdot|\cdot)$, and $f_{W(\tau)}(\cdot)$ are replaced with the expressions given in (20), (8), and (11), respectively.

## Appendix B

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

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

where $E\{ {}^k C(Z(\tau), W(\tau), X) \}$ and $E\{ {}^k T(Z(\tau), W(\tau)) \}$ are obtained as special cases from (32) and (33), respectively, by imposing the constraint relative to policy $\mathcal{P}_k$.

Thus, in the case of policy $\mathcal{P}_1$, by imposing the constraint $L_l = L_u$ the (32) simplifies to:
$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \nonumber \\
& + c_l \cdot F_{Z(\tau)}(L_l) + c_d \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau + (c_c - c_p) \cdot \int_0^{L_W} \int_\tau^{\tau+\Delta\tau} f_{X|W(\tau)}(x|w_\tau) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \nonumber \\
& + (c_c - c_p) \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} f_{X|W(\tau)}(x|w_\tau) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx + c_d \cdot \Delta\tau \cdot \int_{W_M}^{\infty} F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& + c_d \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \nonumber \\
& + c_d \cdot \int_0^{L_W} \left[ \int_\tau^{\tau+\Delta\tau} (\tau + \Delta\tau - x) \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \right] \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau - c_d \cdot \int_{W_M}^{\infty} \int_0^\tau x \cdot f_{X|W(\tau)}(x|w_\tau) \cdot dx \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau. \tag{A.3}
\end{align}
$$

Hence, from (A.3), by exploiting the results given in (24), (28), and (30), the following expression is obtained:

$$
\begin{align}
E\{C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + c_w \cdot [F_{Z(\tau)}(L_u) - F_{Z(\tau)}(L_l)] + (c_c - c_p + c_d \cdot \tau) \cdot [1 - F_{W(\tau)}(w_M)] \nonumber \\
& + (c_l + c_d \cdot \Delta\tau) \cdot F_{Z(\tau)}(L_l) + c_d \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& + (c_c - c_p) \cdot \int_0^{L_W} \left[1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right)\right] \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \nonumber \\
& + (c_c - c_p) \cdot \int_0^{W_M} \left[1 - \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right)\right] \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \tag{A.4} \\
& + c_d \cdot \Delta\tau \cdot \int_0^{L_W} [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau - c_d \cdot \int_0^{W_M} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot F_{Z(\tau)|W(\tau)}(L_l|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \nonumber \\
& - c_d \cdot \int_0^{L_W} \int_\tau^{\tau+\Delta\tau} \mathbb{E}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot [F_{Z(\tau)|W(\tau)}(L_u|w_\tau) - F_{Z(\tau)|W(\tau)}(L_l|w_\tau)] \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx \nonumber \\
& - c_d \cdot \int_{W_M}^{\infty} \int_0^\tau \mathbb{E}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau)\right) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dx. \nonumber
\end{align}
$$

It is not hard to verify that the (A.4) reduces to the (32) once $F_{W(\tau)}(\cdot)$, $F_{Z(\tau)}(\cdot)$, $F_{Z(\tau)|W(\tau)}(\cdot|\cdot)$ and $f_{W(\tau)}(\cdot)$ are replaced with the expressions given in (12), (20), (8), and (11), respectively.

The (33) can be more easily derived by following the scheme given in Table 2. In fact, more directly, it is:

$$
\begin{align}
E\{T(Z(\tau), W(\tau))\} &= \tau \cdot [1 - F_{Z(\tau)}(L_u)] + \tau \cdot \int_{L_u}^{\infty} \int_{L_l}^{L_W} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau + (\tau + \Delta\tau) \cdot F_{Z(\tau)}(L_l) + (\tau + \Delta\tau) \cdot \int_0^{L_W} \int_{L_l}^{L_u} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \nonumber \\
&= \tau + \Delta\tau \cdot F_{Z(\tau)}(L_l) + \Delta\tau \cdot \int_0^{L_W} \int_{L_l}^{L_u} f_{Z(\tau)|W(\tau)}(z_\tau|w_\tau) \cdot f_{W(\tau)}(w_\tau) \cdot dw_\tau \cdot dz_\tau \tag{A.5}
\end{align}
$$

which reduces to (33) once $F_{Z(\tau)}(\cdot)$, $F_{Z(\tau)|W(\tau)}(\cdot|\cdot)$, and $f_{W(\tau)}(\cdot)$ are replaced with the expressions given in (20), (8), and (11), respectively.

## Appendix B

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

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

where $E\{ {}^k C(Z(\tau), W(\tau), X) \}$ and $E\{ {}^k T(Z(\tau), W(\tau)) \}$ are obtained as special cases from (32) and (33), respectively, by imposing the constraint relative to policy $\mathcal{P}_k$.

Thus, in the case of policy $\mathcal{P}_1$, by imposing the constraint $L_l = L_u$ the (32) simplifies to:
$$
\begin{equation}
\begin{aligned}
E\{^1C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] + (c_l + c_d \cdot \Delta\tau) \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
& + (c_c - c_p) \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M+c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right) \right] \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x \\
& - c_d \cdot \int_0^{w_M} \int_{\tau}^{\tau+\Delta\tau} \mathbb{B}\left(\frac{w_M - w_\tau}{w_M+c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x \\
& - c_d \cdot \int_{w_M}^{\infty} \int_{\tau}^{\tau} \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
\tag{B.1}
\end{equation}
$$

and the (33) reduces to:

$$
E\{^1T(Z(\tau), W(\tau))\} = \tau + \Delta\tau \cdot \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau. \quad (B.2)
$$

Similarly, in the case of policy $\mathcal{P}_2$, by imposing $L_l = 0, L_u = +\infty$, and $c_z = 0$, it is:

$$
\begin{equation}
\begin{aligned}
E\{^2C(Z(\tau), W(\tau), X)\} ={}& c_l + c_p + c_w + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] + (c_l + c_d \cdot \Delta\tau) \int_0^{L_w} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
& + (c_c - c_p) \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M+c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d\right) \right] \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
& - c_d \cdot \int_0^{L_w} \int_{\tau}^{\tau+\Delta\tau} \mathbb{B}\left(\frac{w_M - w_\tau}{w_M+c}; \eta(x) - \eta(\tau), \eta(\tau) + d\right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x \\
& - c_d \cdot \int_{w_M}^{\infty} \int_{\tau}^{\tau} \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
\tag{B.3}
\end{equation}
$$

and

$$
E\{^2T(Z(\tau), W(\tau))\} = \tau + \Delta\tau \cdot \int_0^{L_w} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau. \quad (B.4)
$$

Finally, in the case of policy $\mathcal{P}_3$, by imposing $L_l = L_u = 0$, and $c_z = 0$, it results:

$$
E\{^3C(Z(\tau), W(\tau), X)\} = c_l + c_p + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] - c_d \cdot \int_{w_M}^{\infty} \int_{\tau}^{\tau} \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x, \quad (B.5)
$$

and

$$
E\{^3T(Z(\tau), W(\tau))\} = \tau. \tag{B.6}
$$

Appendix C

The long-run average maintenance cost rate $^kC_\infty(\cdot)$ can be intended as the sum of the contributions of corrective replacement costs, preventive replacement costs, perturbed inspection costs, perfect inspection costs, logistic costs, and downtime costs. These contributions can be expressed as:

$$
^k C_{\infty}(\cdot) = \frac{^k C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}, \quad ^d C_{\infty}(\cdot) = \frac{^d C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}, \quad \dots, \quad ^z C_{\infty}(\cdot) = \frac{^z C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}
$$

where $E\{^1T(Z(\tau), W(\tau))\}$ is the expected length of a maintenance cycle, given in (33), and the numerators, from (32), can be expressed as:
$$
\begin{equation}
\begin{aligned}
E\{^1C(Z(\tau), W(\tau), X)\} ={}& c_l + c_z + c_p + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] + (c_l + c_d \cdot \Delta\tau) \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \\
& + (c_c - c_p) \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_0^{w_M} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_{w_M}^{\infty} \int_0^\tau \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx,
\end{aligned}
\tag{B.1}
\end{equation}
$$

and the (33) reduces to:

$$
E\{^1T(Z(\tau), W(\tau))\} = \tau + \Delta\tau \cdot \int_0^\infty \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau. \quad (B.2)
$$

Similarly, in the case of policy $\mathcal{P}_2$, by imposing $L_l = 0, L_u = +\infty$, and $c_z = 0$, it is:

$$
\begin{equation}
\begin{aligned}
E\{^2C(Z(\tau), W(\tau), X)\} ={}& c_l + c_p + c_w + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] + (c_l + c_d \cdot \Delta\tau) \int_0^{L_w} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \\
& + (c_c - c_p) \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \\
& - c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left(\frac{w_M - w_\tau}{w_M + c}; \eta(x) - \eta(\tau), \eta(\tau) + d \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx \\
& - c_d \cdot \int_{w_M}^{\infty} \int_0^\tau \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx,
\end{aligned}
\tag{B.3}
\end{equation}
$$

and

$$
E\{^2T(Z(\tau), W(\tau))\} = \tau + \Delta\tau \cdot \int_0^{L_w} \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau. \quad (B.4)
$$

Finally, in the case of policy $\mathcal{P}_3$, by imposing $L_l = L_u = 0$, and $c_z = 0$, it results:

$$
E\{^3C(Z(\tau), W(\tau), X)\} = c_l + c_p + (c_c - c_p + c_d \cdot \tau) \left[ 1 - \mathbb{B}\left(\frac{w_M}{w_M+c}; \eta(\tau), d\right) \right] - c_d \cdot \int_{w_M}^{\infty} \int_0^{\tau} \mathbb{B}\left(\frac{w_M}{w_\tau}; \eta(x), \Delta\eta(x, \tau) \right) \cdot \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{B(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot dw_\tau \cdot dx, \quad (B.5)
$$

and

$$
E\{^3T(Z(\tau), W(\tau))\} = \tau. \tag{B.6}
$$

Appendix C

The long-run average maintenance cost rate $^kC_\infty(\cdot)$ can be intended as the sum of the contributions of corrective replacement costs, preventive replacement costs, perturbed inspection costs, perfect inspection costs, logistic costs, and downtime costs. These contributions can be expressed as:

$$
^k C_{\infty}(\cdot) = \frac{^k C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}, \quad ^d C_{\infty}(\cdot) = \frac{^d C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}, \quad \dots, \quad ^z C_{\infty}(\cdot) = \frac{^z C(\cdot)}{E\{^1T(Z(\tau), W(\tau))\}}
$$

where $E\{^1T(Z(\tau), W(\tau))\}$ is the expected length of a maintenance cycle, given in (33), and the numerators, from (32), can be expressed as:
$$ 
\begin{aligned}
{}^k_c C(\cdot) &= c_c \cdot \left[ 1 - \mathbb{B}\left( \frac{w_M}{w_M + c}; \eta(\tau), d \right) \right] + c_c \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \times \\
&\quad \times \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \quad (C.1) \\
&\quad + c_c \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_d C(\cdot) &= c_d \cdot \tau \cdot \left[ 1 - \mathbb{B}\left( \frac{w_M}{w_M + c}; \eta(\tau), d \right) \right] + c_d \cdot \Delta\tau \cdot \int_0^{\infty} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \\
&\quad + c_d \cdot \Delta\tau \cdot \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \\
&\quad - c_d \cdot \int_0^{w_M} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \eta(\tau) - \eta(\tau), \eta(\tau) + d \right) \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x \quad (C.2)
\end{aligned}
$$

$$ 
\begin{aligned}
& -c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \eta(\tau) - \eta(\tau), \eta(\tau) + d \right) \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x \\
& -c_d \cdot \int_{w_M}^{\infty} \int_0^{\tau} \mathbb{B}\left( \frac{w_M}{w_\tau}; \eta(\tau), \Delta\eta(x, \tau) \right) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_p C(\cdot) &= c_p - c_p' \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \\
&\quad - c_p' \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_l C(\cdot) &= c_l \cdot \left[ 1 + \int_0^{\infty} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau + \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau \right],
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_w C(\cdot) &= c_w \cdot \int_0^{\infty} \left[ \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \cdot \mathrm{d}w_\tau,
\end{aligned}
$$

and

$$ 
\begin{aligned}
{}^k_z C(\cdot) &= c_z.
\end{aligned}
$$

As already remarked in Section 4, under $P_1$ it is $L_l = L_u$ (under this setting it results ${}^1_w C(\cdot) = 0$), under $P_2$ it is $L_l = 0, L_u = \infty$, and $c_z = 0$ (under these settings it is ${}^2_w C(\cdot) = 0$), and under $P_3$ it is $L_l = L_u = 0$ and $c_z = c_w = 0$ (under these settings it results ${}^3_w C(\cdot) = {}^2_w C(\cdot) = 0$).

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$$ 
\begin{aligned}
{}^k_c C(\cdot) &= c_c \cdot \left[ 1 - \mathbb{B}\left( \frac{w_M}{w_M + c}; \eta(\tau), d \right) \right] + c_c \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \times \\
&\quad \times \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \quad \text{(C.1)} \\
&\quad + c_c \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_d C(\cdot) &= c_d \cdot \tau \cdot \left[ 1 - \mathbb{B}\left( \frac{w_M}{w_M + c}; \eta(\tau), d \right) \right] + c_d \cdot \Delta\tau \cdot \int_0^{\infty} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
&\quad + c_d \cdot \Delta\tau \cdot \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
&\quad - c_d \cdot \int_0^{w_M} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \eta(\tau) - \eta(\tau), \eta(\tau) + d \right) \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x \quad \text{(C.2)}
\end{aligned}
$$

$$ 
\begin{aligned}
&-c_d \cdot \int_0^{L_w} \int_\tau^{\tau+\Delta\tau} \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \eta(\tau) - \eta(\tau), \eta(\tau) + d \right) \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x \\
&-c_d \cdot \int_{w_M}^{\infty} \int_0^{\tau} \mathbb{B}\left( \frac{w_M}{w_\tau}; \eta(\tau), \Delta\eta(x, \tau) \right) \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_p C(\cdot) &= c_p - c_p' \cdot \int_0^{L_w} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \\
&\quad - c_p' \cdot \int_0^{w_M} \left[ 1 - \mathbb{B}\left( \frac{w_M - w_\tau}{w_M + c}; \Delta\eta(\tau, \tau + \Delta\tau), \eta(\tau) + d \right) \right] \cdot \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \cdot \mathrm{d}x,
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_l C(\cdot) &= c_l' \left[ 1 + \int_0^{\infty} \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l) \cdot c^d \cdot w_\tau^{\eta(\tau)-1}}{\Gamma(\beta(w_\tau)) \cdot \mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau + \int_0^{L_w} \left[ \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau \right],
\end{aligned}
$$

$$ 
\begin{aligned}
{}^k_w C(\cdot) &= c_w' \int_0^{\infty} \left[ \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_u)}{\Gamma(\beta(w_\tau))} - \frac{\Gamma(\beta(w_\tau), \alpha(w_\tau)/L_l)}{\Gamma(\beta(w_\tau))} \right] \frac{c^d \cdot w_\tau^{\eta(\tau)-1}}{\mathbf{B}(\eta(\tau), d) \cdot (w_\tau + c)^{\eta(\tau)+d}} \mathrm{d}w_\tau,
\end{aligned}
$$

and

$$ 
\begin{aligned}
{}^k_z C(\cdot) &= c_z.
\end{aligned}
$$

As already remarked in Section 4, under $P_1$ it is $L_l = L_u$ (under this setting it results ${}^1_w C(\cdot) = 0$), under $P_2$ it is $L_l = 0, L_u = \infty$, and $c_z = 0$ (under these settings it is ${}^2_w C(\cdot) = 0$), and under $P_3$ it is $L_l = L_u = 0$ and $c_z = c_w = 0$ (under these settings it results ${}^3_w C(\cdot) = {}^2_w C(\cdot) = 0$).

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[18] Hao S, Yang J, Bérenguer C. Condition-based maintenance with imperfect inspections for continuous degradation processes. Appl Math Modell 2020;86: 311-34.

[19] Wang Y, Pham H. Maintenance modeling and policies, in: stochastic reliability and maintenance modeling. Springer; 2013. p. 141-58.

[20] Huynh KT, Grall A, Berenguer C. A parametric predictive maintenance decision making framework considering improved system health prognosis precision. IEEE Trans Reliab 2019;68(1):375-96.

[21] Fauriat W, Zio E. Optimization of an aperiodic sequential inspection and condition-based maintenance policy driven by value of information. Reliab Eng Syst Saf 2020;204:107133.

[22] Yuan XX, Higo E, Pandey MD. Estimation of the value of an inspection and maintenance program: a Bayesian gamma process model. Reliab Eng Syst Saf 2021; 216:107912.

[23] Kim S, Choi JH, Kim NH. Inspection schedule for prognostics with uncertainty management. Reliab Eng Syst Saf 2022;222:108391.

[24] Crowder M, Lawless J. On a scheme for predictive maintenance. Eur J Oper Res 2007;176(3):1713-22.

[25] Finkelstein M, Cha JH, Levitin G. On a new age-replacement policy for items with observed stochastic degradation. Qual Reliab Eng Int 2020;36(3):1132-43.

[26] Cha JH, Finkelstein M, Levitin G. Age replacement policy for items described by stochastic degradation with dependent increments. J Manag Math 2021;33(2): 273-87.

[27] Cha JH, Finkelstein M, Levitin G. Replacement policy for heterogeneous items subject to gamma degradation processes. Methodol Comput Appl Probab 2022;24: 1323-40.

[28] Lawless J, Crowder M. Covariates and random effects in a gamma process model with application to degradation and failure. Lifetime Data Anal 2004;10(3): 213-27.

[29] Giorgio M, Guida M, Pulcini G. A new class of Markovian processes for deteriorating units with state dependent increments and covariates. IEEE Trans Reliab 2015;64(2):562-78.

[30] Esposito N, Mele A, Castanier B, Giorgio M. A new gamma degradation process with random effect and state-dependent measurement error. Proc Inst Mech Eng O J Risk Reliab 2022. https://doi.org/10.1177/1748006X211067299.

[31] Hu J, Shen J, Shen L. Opportunistic maintenance for two-component series systems subject to dependent degradation and shock. Reliab Eng Syst Saf 2020;201: 106995.

[32] Bautista L, Castro IT, Landesa L. Condition-based maintenance for a system subject to multiple degradation processes with stochastic arrival intensity. Eur J Oper Res 2022;302(2):560-74.

[33] Bismut E, Pandey MD, Straub D. Reliability-based inspection and maintenance planning of a nuclear feeder piping system. Reliab Eng Syst Saf 2022;224:108521.

[34] Pulcini G. A perturbed gamma process with statistically dependent measurement errors. Reliab Eng Syst Saf 2018;152:296-306.

[35] Giorgio M, Mele A, Pulcini G. A perturbed gamma degradation process with degradation dependent non-Gaussian measurement errors. Appl Stoch Models Bus Ind 2019;35(2):198-210.

[36] Hoyle E, Hughston LP, Macrina A. Lévy random bridges and the modelling of financial information. Stoch Process their Appl 2011;121(4):856-84.

[37] Kahle W, Mercier S, Paroissin C. Degradation process in reliability. John Wiley & Sons; 2016.

[38] Giorgio M, Pulcini G. A new state-dependent degradation process and related model misidentification problems. Eur J Oper Res 2018;267:1027-38.

[39] Ross SM. Stochastic processes. New York: John Wiley; 1983.

[40] Dey PK. Decision support system for inspection and maintenance: a case study of oil pipelines. IEEE Trans Eng Manag 2004;51(1):47-56.

[41] Ellingwood BR, Mori Y. Probabilistic methods for condition assessment and life prediction of concrete structures in nuclear power plants. Nucl Eng Des 1993;142: 155166.

[42] Mahmoodian M, Alani A. Modeling deterioration in concrete pipes as a stochastic gamma process for time-dependent reliability analysis. J Pipeline Syst Eng Pract 2014;5(1):0413008.