Imperfect Preventive Maintenance Optimization with Variable Age Reduction Factor and Independent Intervention Level

Abstract

Maintenance policies are crucial for ensuring the reliability, safety, and longevity of a system, as well as reducing the risk of accidents. Preventive maintenance (PM) is an effective strategy to keep equipment and systems in good working order by fixing potential issues before they cause downtime or safety hazards. However, optimizing the time intervals between PM activities is essential for minimizing the overall maintenance cost. This paper proposes an innovative approach that considers the intervention level of maintenance activities as an independent variable of PM times. The approach provides greater flexibility in creating maintenance plans, as it considers practical aspects that may impact maintenance activities beyond the time interval between PMs. The proposed approach uses a reliability model that incorporates imperfect preventive maintenance and a variable improvement factor based on age reduction. The improvement factor of each preventive maintenance activity (PMA) is defined based on the intervention level of the activity itself, which is determined by the number of tasks performed, execution time, and the number of items replaced in the maintenance plan. The proposed maintenance strategy determines not only the optimal times for PMAs and the respective intervention level but also the optimal number of maintenance activities that minimize the total maintenance cost along a fixed and user-defined planning horizon. The effectiveness and precision of the approach have been demonstrated through a series of numerical examples and a comprehensive case study involving three heat exchangers situated within the hydroelectric power plant.

1. Introduction

Maintenance policy is crucial for maintaining the reliability and safety of a system, as well as reducing the risk of accidents. It involves developing a strategic plan of action to avoid unexpected interruptions or, in case they happen, ensure that they are quickly addressed and that the system continues to function optimally with high availability, security, and durability. Proper maintenance not only reduces costs but also prolongs the useful life of the system and improves efficiency [1,2].

Preventive maintenance (PM), which involves performing routine maintenance activities on equipment or systems to prevent unexpected breakdowns or failures, is an excellent strategy to keep equipment and systems in good working order by fixing potential issues before they could cause downtime or safety hazards [3]. Optimizing the time intervals between preventive maintenance activities (PMAs) is, therefore, essential for minimizing the overall maintenance cost.

The total maintenance cost is calculated as the sum of the costs of corrective and preventive maintenance, divided by the forecast horizon. Longer time intervals between preventive maintenance generally result in lower corresponding costs. However, longer intervals between PM activities also increase the expected number of failures, which may require more corrective actions that increase the overall cost of the maintenance process. In general, the costs of corrective maintenance tend to be higher than those of preventive maintenance [4].

Preventive maintenance actions are associated with an intervention level that considers the number of tasks performed, execution time, and the number of items replaced in the maintenance plan [5]. The intervention level is usually considered dependent on the times between successive preventive maintenance actions, meaning that the longer the time interval between PMs, the greater the intervention level of the maintenance action, and the higher its cost. However, in practice, the number of tasks performed, execution time, and the number of items replaced in the maintenance plan may not depend solely on the time interval between PMs. Economic issues and contractual rules may also affect decision-making, while equipment usage regimes may vary significantly from the standard established in the manufacturer’s manuals.

This paper proposes an innovative approach that considers the intervention level of preventive maintenance activities as an independent variable of PM times. This approach provides greater flexibility in creating maintenance plans, as it considers practical aspects that may impact maintenance activities beyond the time interval between PMs. In practice, companies often operate under budgetary and time constraints, so an intervention that restores equipment to a state that is “good enough for safe and efficient operation” may be deemed sufficient, especially if complete restoration is significantly more costly or time-consuming. Additionally, the proposed approach uses a reliability model that incorporates imperfect preventive maintenance and a variable improvement factor based on age reduction. The improvement factor of each PMA is defined based on the intervention level of the activity itself, which is determined by the number of tasks performed, execution time, and the number of items replaced in the maintenance plan, according to [6].

The proposed maintenance strategy determines not only the optimal times for PMAs and the respective intervention level but also the optimal number of maintenance activities that minimize the total maintenance cost considering a user-defined planning horizon. To accomplish this task, a solution coding procedure for the genetic algorithm was developed to consider the times, intervention level, and number of maintenance activities simultaneously.

To provide context and identify gaps in the current literature, this paper begins with a review of related works in Section 2. Section 3 describes the proposed method, while Section 4 details its application in a case study and the corresponding results. Finally, the conclusions appear in Section 5, followed by the bibliographical references that supported the research.

2. Imperfect Preventive Maintenance

A repairable system can be defined as “a system that, after failing to perform an activity, can be restored to proper functioning satisfactorily by some method” [7]. As defined in [8], a repairable system can be fixed by replacing some of its components, rather than restoring the entire system after a failure.

The research conducted by [5,9] reports five categories of maintenance activities, which are:

• Perfect maintenance: the maintenance operation restores the system to a status “as good as new”. The maintenance operation restores the system such that its operational performance and integrity are indistinguishable from when it was brand new. This maintenance effectively resets the reliability curve, bringing it back to the initial high reliability associated with a new system. All accumulated wear and tear or degradation is completely reversed.
• Minimal maintenance: maintenance operation that restores the system to a state of “as bad as old”. The maintenance operation addresses only immediate faults or damages, but the cumulative wear, age, and minor degradations persist, rendering the system’s condition nearly unchanged post-maintenance. There’s little to no improvement in the reliability after the maintenance. The curve remains near its current level, which is closer to the system’s end-of-life reliability.
• Imperfect maintenance: maintenance operation that restores the system to a state of “worse than new, but better than old”. The maintenance operation rejuvenates the system, but not to its original state. While certain wear and degradations are mitigated, the system does not return to its brand-new condition. The reliability improves post-maintenance but does not reach the peak reliability of a new system. The curve is lifted but stays between its original high (new system) and its current degraded state.
• More than perfect maintenance: the maintenance operation returns the system to a state of “better than new”. The maintenance operation not only restores the system but also enhances its performance, durability, and other attributes, making it surpass its original specifications. Post-maintenance, the reliability exceeds that of the original system. The curve is elevated beyond the initial reliability, indicating enhanced performance and longevity due to maintenance upgrades or innovations.
• Destructive maintenance: maintenance operation that restores the system to a state of “worse than old”. Instead of rejuvenating or restoring the system, the maintenance operation inadvertently introduces flaws, damages, or reduces the system’s overall integrity. The reliability deteriorates post-maintenance, moving the curve downwards, closer to complete system failure. This is a non-ideal outcome, often resulting from erroneous maintenance procedures or unforeseen complications.

According to [9], the fourth and fifth categories have little presence in reality; almost all correspond to specific events in the life of the system. In the first category, the most commonly used model that precisely fits the described scenario is the perfect renewal process (PRP) as mentioned by [4,10]. In the second category, a non-homogeneous Poisson process (NHPP) [4,10] is used, among others. In the third category, several failure process models, including variants of generalized geometric processes [11], are used.

Recent works on reliability and imperfect preventive maintenance models have introduced new approaches where artificial intelligence (AI) algorithms are used together with statistical models to assist in optimizing maintenance policies. Many of these works focus on finding possible solutions for specific problems identified in certain industrial sectors. In this context, it is observed that the use of evolutionary and metaheuristic algorithms is widespread.

In [12], optimization techniques using evolutionary algorithms for complex system maintenance costs are presented. The aim is to minimize maintenance costs while maintaining system reliability. The algorithms determine the optimal times for maintenance activities on system components and the work compares the performance of different methods to find the global optimum. A case study is presented using data from an Algerian power distributor with a mathematical model for reliability constraints. The algorithms generate different solutions that are evaluated based on performance criteria. The results indicate that reliability constraints are crucial for obtaining viable and efficient solutions, and the study highlights that evolutionary algorithms are an effective tool for solving maintenance problems in complex systems.

A mathematical model to optimize product availability with imperfect preventive maintenance was developed by Wang et al. [13]. The model maximizes availability by optimizing product replacement time and ideal preventive maintenance intervals. The researchers compared numerical and evolutionary algorithms to find the global optimum and reported that the evolutionary algorithm was faster but slightly less precise than the numerical algorithm. The study demonstrates the applicability of the evolutionary algorithm for this problem. Future work includes constructing a model that considers warranty and availability costs, predictive risk of failure, and equipment utilization rate using operational research algorithms and different distributions, such as normal and Weibull distributions.

Gholizadeh et al. [14] proposed a mathematical optimization model to schedule a flow-shop system in the disposable appliance manufacturing industry while considering a preventive maintenance policy (PMP). Real data from Pilka Plast Haraz Polymer Company in Iran were used to develop the model, which aims to minimize production and maintenance costs and considers uncertainties associated with the processes. The model uses operation times and system availability to optimize the processing sequence. Evolutionary algorithms were also used to solve the proposed model, indicating that such methods are effective in obtaining the optimal global solution that minimizes the total maintenance costs. The authors suggest future work on rigorous problem-solving techniques, introducing the development of new products, and implementing quality control (QC) in the mathematical model.

Raghav et al. [15] proposed a statistical model to estimate system availability and optimize the times for preventive maintenance to minimize maintenance costs and maximize system availability. The proposed approach was applied in a power generation plant and evaluated using multi-objective optimization algorithms, such as particle swarm optimization (PSO) and the genetic algorithm (GA). The results showed that these algorithms can significantly reduce maintenance costs compared to existing preventive maintenance policies. Future work could involve using real data to determine the distribution and parameters, such as component failure detection rates, resource utilization rates, and factors related to component quality, that can be considered.

In the research conducted by Suzuki and Itu [16], the authors propose a maintenance scheduling model based on metaheuristic algorithms, considering reliability constraints, to minimize the total number of maintenance activities throughout the lifetime of a nuclear power plant. The proposed model was applied to find the best maintenance plans, the global optimum, for components that constitute a water injection function. Comparing the results obtained with data from an existing method, it was possible to verify that the proposed model produced a superior maintenance schedule, in which the total number of activities was minimized, and the system’s unreliability was restricted within a predefined upper limit. In future work, the researchers mention that they intend to consider the non-linearity of component failure rates and the difference in cost of each type of maintenance to optimize the schedule under more realistic conditions.

It is worth mentioning that alternative optimization methods, including exact approaches, could be employed to address this problem. For instance, the utilization of the Nelder–Mead method, which is commonly employed in solving nonlinear optimization problems, as discussed in Lin and Pham’s study [17], did not yield superior outcomes compared to metaheuristic methods in this specific application. Moreover, analytical techniques that assume a constant probability of failure over time cannot be utilized for problems that involve non-homogeneous counting processes, similar to the examples examined in this research [18].

Moreover, unlike the works above and those found in the literature, the proposed approach uses a reliability model that considers imperfect preventive maintenance and a variable improvement factor based on age reduction. So, the improvement factor of each preventive maintenance is defined according to the intervention level of the maintenance action, considering a fixed and pre-defined planning horizon. This approach differs from the usual methods in two main aspects, namely:

• The intervention level of maintenance activities is considered independent of the time between consecutive preventive maintenance, in contrast to the common approach in which the intervention level is treated as a time-dependent variable. In general, the longer the time between maintenance actions, the greater the intervention level of the action. The proposed approach eliminates this restriction.
• The proposed maintenance strategy determines not only the optimal times for PMAs and their respective intervention levels but also defines the optimal number of maintenance activities that minimize the overall maintenance cost, considering a user-defined planning horizon. Coding and decoding approaches to the genetic algorithm solutions were developed to simultaneously consider the times, intervention levels, and the number of maintenance activities.

3. The Proposed Approach

The methodological procedure adopted in this work can be divided into two steps, namely:

• Adjustment of the reliability model from the actual data of the system(s) under observation, with the possibility of considering more than one similar system.
• Imperfect preventive maintenance optimization to determine the optimal times for PMAs, their respective optimal intervention levels, and the optimal number of maintenance activities that minimize the overall maintenance cost.

The intervention levels of preventive maintenance actions are expressed in percentages, indicating that maintenance can be performed with different severities. The higher the level of intervention, the greater the impact of maintenance on the system condition, characterizing the approach with a variable improvement factor. In practice, the definitions of different levels of preventive maintenance intervention are closely related to the system characteristics and should be determined by the operator. These definitions may consider different conditions and activities of maintenance plans related to the number of tasks performed or items replaced in preventive action [6].

In the following subsections, these steps are described in detail, highlighting the necessary calculations and the methods and tools used in each of them.

3.1. Adjustment of the Reliability Model

The reliability models addressed in this work consider the study of repair systems in which the failure rate is described by a non-homogeneous Poisson process (NHPP). This means that the time intervals between the failures are not independent, and the failure rate is described by a fault intensity function that varies with time, in general, in an increasing way. Therefore, the first step of the reliability model adjustment is the definition of an intensity function parameterization that best fits systems failure data.

Here, we apply the traditional Power Law proposed by Crow [19], which is one of the main ways to characterize the failure intensity function in Equation (1),

$$u\left(t\right)=\frac{1}{{\lambda }^{\beta }}\cdot \beta \cdot t^{\left(\beta -1\right)}$$

(1)

in which $\lambda ,$ and $\beta$ are the model parameters that must be estimated, with $\left(\lambda >0,\beta >0\right)$, where $\lambda$ is the parameter of scale and can be interpreted as the time during which exactly a failure is expected to occur, and $\beta$ is the parameter of shape and represents the variability of the expected number of faults compared to time.

The choice of the power law to describe the failure intensity function is due to its great adjustment power to the most diverse formats by the variation of its parameters, as can be seen in Figure 1.

The estimation of the parameters $\lambda$ and $\beta$ also results in the function of an average failure number $\mu \left(t\right)$, which describes the accumulated number of expected failures to time $t$, according to Equation (2).

$$\mu \left(t\right)={\int }_0^{t}u\left(t\right)dt=\frac{1}{{\lambda }^{\beta }}\cdot t^{\beta }$$

(2)

3.1.1. Imperfect Preventive Maintenance Model

The failure intensity function describes the failure rate as a function of time but does not consider the imperfection of preventive maintenance actions. In fact, the function defined in Equation (1) assumes that the failure intensity does not change after a preventive maintenance action and that the systems return to the state of “as good as new” after this action (i.e., perfect PM).

However, in practice, preventive maintenance improves system performance for an intermediate condition, reducing failure intensity to a value between a state somewhere between “as good as new” and “as bad as old” after preventive activities [20]. Thus, considering the imperfect maintenance, the failure intensity function is rewritten by applying the improvement factor $a$ after each preventive maintenance action in the instant $T_j$, according to Equation (3).

$$u\left(t\right)=\left\{\begin{array}{cc}{u}_0\left(t\right),& {\mathrm{T}}_0\le t<{\mathrm{T}}_1\\ u_0\left(t-a{\mathrm{T}}_1\right),& {\mathrm{T}}_1\le t<{\mathrm{T}}_2\\ u_0\left(t-a{\mathrm{T}}_2\right),& {\mathrm{T}}_2\le t<{\mathrm{T}}_3\\ ⋮& ⋮\\ u_0\left(t-a{\mathrm{T}}_{\mathrm{j}-1}\right),& {\mathrm{T}}_{\mathrm{j}-1}\le t<{\mathrm{T}}_{\mathrm{j}}\end{array}\right.$$

(3)

Therefore, in addition to estimating the parameters $\lambda$ and $\beta$, it is necessary to estimate the improvement factor that represents the influence of maintenance action on system failure intensity. In this paper, the improvement factor is defined as a function of the intervention level $s$, as $a\left(s\right)$, resulting in new expressions for the fault intensity function $u\left(t\right)$ and the average number of failures $\mu \left(t\right)$, according to Equations (4) and (5), respectively:

$$u\left(t\right)=\frac{1}{{\lambda }^{\beta }}·\beta ·(t-a(s_j)·T_{j-1}{)}^{(\beta -1)}, T_{j-1}\le t<T_j$$

(4)

$$\begin{gathered} \mu \left(t\right)=\sum _{j=1}^{c-1}\left\{\left[\frac{1}{{\lambda }^{\beta }}·{(T_j-a(s}_j)T_{j-1}{)}^{\beta }-\frac{1}{{\lambda }^{\beta }}·{\left(\right(1-a(s}_j))T_{j-1})\right]\right\} \\ +\left[\frac{1}{{\lambda }^{\beta }}·{(t-a(s}_{c-1})T_{c-1}{)}^{\beta }-\frac{1}{{\lambda }^{\beta }}·{\left(\right(1-a(s}_{c-1}))T_{c-1}{)}^{\beta }\right] \end{gathered}$$

(5)

3.1.2. Improvement Factor Parameterization

The preventive maintenance action performed at time $T_j$ with intervention level $s_j$ promotes a reduction in system failure intensity for all maintenance cycles. To represent the improvement factor, an exponential function that expresses the factor value regarding the intervention level of the maintenance is used, defined as Equation (6):

$$a\left(s_j\right)=1-EXP\left(-s_j·\theta \right), j=1,...,c$$

(6)

So, the adjustment of the reliability model is performed by obtaining the value of the parameters $\theta ,\lambda ,$ and $\beta ,$ as defined above. Here, the traditional maximum likelihood estimation method has been adopted, as described in the following subsection.

3.1.3. Maximum Likelihood Estimation Method

The estimation of the parameters $\lambda$, $\beta ,$ and $\theta$ was performed to solve a problem of maximizing the likelihood function, defined according to Equation (7):

$$\begin{gathered} L\left(\lambda ,\beta ,\theta \right)=\prod _{k=1}^m\left\{\prod _{j=1}^{c_k}\left[\prod _{i=1}^{n_{kj}}\frac{1}{{\lambda }^{\beta }}·\beta ·{(t_{k,j,i}-a(s}_{k,j-1})T_{k,j-1}{)}^{\beta -1}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \right.·EXP\left.\left[-\sum _{j=1}^{c_k}{\left(\frac{1}{{\lambda }^{\beta }}·{(T_{k,j}-a(s}_{k,j-1})T_{k,j-1}{)}^{\beta }-\frac{1}{{\lambda }^{\beta }} \\ ·{{\left(\right(1-a(s}_{k,j-1}))T_{k,j-1})}^{\beta }\right)}^{ }\right]\right\} \end{gathered}$$

(7)

which is defined considering $m$ repairable systems, indicated by $k=1,\dots ,m$ and subject to $c_k$ cycles of preventive maintenance performed at times $T_{k,j}$, with $k=1,\dots ,m$ and $j=1,\dots ,{ c}_k$. Each of the PM actions is associated with an intervention level, which is known and denoted by $s_{k,j}$. In each maintenance cycle $j$, $n_{k,j}$ failures are observed for the kth system, which occurs at times $t_{k,j,i}$. In Equations (4) and (5) the values of $T_{k,0}$ and $s_{k,0}$ are equal to zero and coincide with the beginning of systems observation.

3.2. Maintenance Optimization

The optimization problem with the proposed model is aimed at finding optimal values for the following variables: the number of maintenance actions, the times to perform these maintenance activities, and their severities that minimize the overall maintenance cost. This second phase of optimization was performed using the genetic algorithm, following the steps illustrated in Figure 2.

3.2.1. Representation and Decoding of Candidate Solutions

The proposed optimization approach uses a real representation for candidate solutions, which considers a variable number of maintenance actions, with their respective severities, in a user-defined time horizon of $N$ months. The approach determines the number of PM activities ($c$) and the times and intervention level for these activities, $T_j\mathrm{e}{s}_j,\mathrm{f}\mathrm{o}\mathrm{r}j=1,\dots ,c$.

Numerically, each solution is represented as a vector of $N$ real numbers, denoted as $X=\left\{x_0,{\cdots ,x}_{N-1}\right\}$. Each element $x_i$ of a solution $X$ is decomposed into an integer part $⌊x_i⌋$ and a fractional part ${\{x}_i\},$ such as ${\{x}_i\}=x_i-⌊x_i⌋$.

The occurrence of a preventive maintenance activity in a given month is defined by the value of $⌊x_i⌋$. Each element $x_i$ of a candidate solution is defined in the interval $\left(0;2\left|s\right|-1\right)$, in which $\left|s\right|$ is the number of intervention levels considered. Thus, the solution is decoded by calculating the rest of the division of $⌊x_i⌋$ by 2, $mod(⌊x_i⌋,2)$, indicating the occurrence of a maintenance action in $i$-th month if and only if $mod\left(⌊x_i⌋,2\right)=1$. This choice of limits for $x_i$ values ensures equal probability for even and odd values.

For cases in which $mod\left(⌊x_i⌋,2\right)=1$, the values of $⌊x_i⌋$ also indicate the intervention level of the maintenance to be performed in the $i$-th month, as $s_j=(s_{⌊x_i⌋}+1)/2$. The time $T_j$ to perform the PM activity is calculated from the fractional part ${\{x}_i\}$, of the form $T_j=30(i+{\{x}_i\})$.

It is noteworthy that, in practice, the intervention level of the maintenance activity can be defined based on the number of tasks performed in the maintenance planning, the time in hours required to perform the planning, and the number of items replaced in maintenance and can be adapted as a function of the system specificity.

3.2.2. Solutions Evaluation

Each genetic algorithm’s candidate solution, which results in a set of times and intervention levels for preventive maintenance, is associated with a total cost along with a fixed user-defined planning horizon, obtained by the sum of preventive and corrective maintenance costs for all cycles. Denoting by $C_{PM}(T_j,s_j)$ and $C_{CM}$ the preventive maintenance and corrective maintenance costs, respectively, the expected total maintenance cost $C_{TOT}$ can be given by Equation (8).

$$C_{TOT}\left(T_j, s_j,c\right)=\sum _{j=1}^c\left[C_{PM}\left(T_j,s_j\right)+E\left[N{\left(t\right)}_j\right]\ast C_{CM}\right]$$

(8)

in which $E\left[N{\left(t\right)}_j\right]$ is the expected number of failures in the cycle $j$ and $s_j$ is the intervention level of the $j$-th PM.

The expected number of failures in each cycle, $E\left[N{\left(t\right)}_j\right]$, is also modified according to the average function $\mu \left(t\right)$, which represents the average number of failures as a function of time. Thus, Equation (8) must be rewritten as Equation (9).

$$C_{TOT}\left(T_j, s_j,c\right)=\sum _{j=1}^c\left[C_{PM}\left(T_j,s_j\right)+\left[\mu \left(T_j\right)-\mu \left(T_{j-1}\right)\right]\ast C_{CM}\right]$$

(9)

The sum of the costs for all maintenance cycles is used as a measure to determine the optimal values of $T_j$ and $s_j$ as well as the optimal number of PM activities.

3.3. Numerical Examples

The experiments performed considered different scenarios regarding the availability of the failure data of systems, including situations where the number of failures is small, by grouping similar subsystems data that enable suitable adjustments of theoretical models.

For the first scenario, the example from Shin et al. [21] was used to provide maintenance data for a central cooler in a nuclear power plant. The example considers five repairable systems observed for 600 days, preventive maintenance carried out at times 150, 250, 500, and 600 (days), with severities equal to 40%, 50%, 90%, and 20%, respectively (simultaneous maintenance in all five systems). A total of 93 failures were observed in this example, randomly distributed among the five systems, as graphically illustrated in Figure 3.

In this first case example, preventive maintenance costs were defined in two scenarios: (a) varying the PM cost as a function of intervention level (

Table 1. Preventive maintenance cost as a function of intervention level for first case example, scenario (a). Scenario (b) considers a constant preventive maintenance cost, equal to USD 4000.

Intervention Level	MP Cost (USD)
20%	2000
40%	4000
50%	9000
90%	12,000

); (b) fixed preventive maintenance cost of USD 4000, regardless of intervention level. The corresponding results of this example allow one to verify that the optimization will return a result close to the expected. As for a fixed cost, the optimal known solution is to perform maintenance as comprehensively as possible, which results in the best improvement factor in the system condition.

In the second case example, a scenario with three identical components subject to independent maintenance actions was considered. In this case, the intervention levels were fixed at 20%, 70%, and 100% with the respective preventive maintenance costs, according to

Table 2. PM costs for the second case example.

Intervention Level	PM Cost (USD)
20%	500
70%	2000
100%	3000

. The corrective maintenance cost was set to be USD 15,000, with a planning horizon of 24 months (730 days). The failure data of the three components are presented in

Table 3. Failure data of the second case example. Past PMAs were performed with an intervention level of 70%. Future PM may have any intervention level as defined in the first column of Table 2.

K	Date	Event	s
1	1 January 2020	Start	-
	1 July 2020	PM	0.7
	1 January 2021	PM	0.7
	1 July 2021	PM	0.7
	1 January 2022	PM	0.7
	1 July 2022	End	-
2	1 January 2020	Start	-
	1 May 2020	Failure	-
	1 November 2020	PM	0.7
	1 February 2021	Failure	-
	16 May 2022	Failure	-
	16 November 2022	PM	0.7
	31 December 2022	End	-
3	1 January 2020	Start	-
	1 July 2020	PM	0.7
	1 December 2020	Failure	-
	1 July 2021	PM	0.7
	1 January 2022	PM	0.7
	1 July 2022	End	-

.

In the third case example, a scenario with two components and independent maintenance actions was considered. In this case, the first of the two observed systems did not have failures, and the intervention levels were always 100%. Thus, both the corrective maintenance cost and preventive maintenance cost were considered constant and defined as USD 17,000 and USD 4000, respectively. The planning horizon of the maintenance policy was 48 months (1460 days). The failure data of this example are presented in

Table 4. Failure data of the third case example. Past PMAs were performed with an intervention level of 100%. System k = 1 has no observed failures.

k	Date	Event	s
1	1 January 2020	Start	-
	1 July 2020	PM	1.0
	1 January 2021	PM	1.0
	1 July 2021	PM	1.0
	1 January 2022	PM	1.0
	1 July 2022	End	-
2	1 January 2020	Start	-
	1 May 2020	Failure	-
	1 November 2020	PM	1.0
	1 February 2021	Failure	-
	16 May 2022	Failure	-
	16 November 2022	PM	1.0
	31 December 2022	End	-

.

For the fourth case example, four identical components received preventive maintenance with an intervention level of 50% or 100%, with respective costs of USD 2000 and USD 3000. The corrective maintenance cost was USD 15,000, with a planning horizon of 24 months (730 days). The data of this case are presented in

Table 5. Failure data of the fourth case example.

k	Date	Event	s
1	1 January 2020	Start
	1 July 2020	PM	1.0
	1 January 2021	PM	1.0
	1 July 2021	PM	1.0
	1 January 2022	PM	1.0
	1 July 2022	End
2	1 January 2020	Start
	1 May 2020	Failure
	1 November 2020	PM	0.5
	1 February 2021	Failure
	16 May 2022	Failure
	16 November 2022	PM	0.5
	31 December 2022	End
3	1 January 2020	Start
	1 July 2020	PM	0.5
	1 December 2020	Failure
	1 July 2021	PM	1.0
	1 January 2022	PM	0.5
	1 July 2022	End
4	1 January 2020	Start
	1 September 2020	PM	1.0
	1 December 2020	Failure
	1 August 2021	PM	1.0
	1 April 2022	PM	1.0
	1 June 2022	End

.

For the first case example, additional scenarios were created by varying preventive and corrective maintenance costs to perform a sensitivity analysis of the proposed models.

3.4. Case Study

The focus of this case study was on enhancing the maintenance policy for heat exchangers in a hydroelectric power plant that faces a particular challenge: clogging caused by the rapid growth and encrustation of golden mussels. These invasive mollusks were recently introduced into the river near the plant, leading to frequent failures in the heat exchange system. Since the process relies on river water, the mussels adhere to the exchanger pipes, causing persistent interruptions. The main goal of the proposed approach was to develop a refined maintenance strategy to effectively tackle this problem.

Even though the equipment operates under similar conditions, the position of the equipment on the riverbed can affect the encrustation process of golden mussels. This indicates that the data from heat exchangers will be analyzed individually. The failure data, collected over a 300-day observation period, are provided in

Table 6. Heat exchanger failure data.

k = 1			k = 2			k = 3
Time t (Days)	Event	s	Time t (Days)	Event	s	Time t (Days)	Event	s
0.00	Start	-	0.00	Start	-	0.00	Start	-
39.68	Failure	-	29.56	Failure	-	52.13	Failure	-
51.99	Failure	-	48.12	Failure	-	75.27	Failure	-
64.76	Failure	-	60.00	PM	0.8	89.45	Failure	-
80.00	PM	1.0	99.78	Failure	-	105.69	Failure	-
127.74	Failure	-	125.96	Failure	-	120.00	PM	1.0
144.21	Failure	-	140.00	PM	0.8	174.89	Failure	-
154.88	Failure	-	194.21	Failure	-	217.62	Failure	-
160.00	PM	1.0	227.26	Failure	-	233.97	Failure	-
234.88	Failure	-	233.97	Failure	-	240.00	PM	0.8
263.97	Failure	-	240.00	PM	0.8	284.37	Failure	-
285.35	Failure	-	294.88	Failure	-	300.00	End	-
300.00	End	-	300.00	End	-

for three different heat exchangers (k = 1, 2, 3).

Furthermore, the corrective maintenance costs were established at C_CM = 30,000 monetary units, while the cost of preventive maintenance was determined as a function of the intervention level (s) using Equation (10). This equation can be tailored and adjusted to suit the specific requirements of the problem at hand.

$$C_{PM}=1500\times s+500$$

(10)

4. Results

The following presents the results of the numerical examples and the case study from the previous section. All results were obtained by applying the genetic algorithm with the parameters empirically defined according to

Table 7. Genetic algorithm parameters.

	Numerical Examples	Case Study
Parameter	Value	Value
Number of iterations	100	100
Population size	100	100
Mutation rate	5%	1%
Crossover rate	75%	90%
Crossover operator	Two-point	Two-point
Elitism rate	5%	5%

. The results are presented as the average of 10 replications for each example with the respective sampling error $\epsilon$ with a level confidence of 95%, both for the adjusted parameters and the times and intervention levels of preventive maintenance.

4.1. Numerical Examples

Model adjustment results for the first case example are presented in

Table 8. Values of the parameters $\lambda ,\beta$, and $\theta$ for the first case example model.

Replication	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$
#1	117.6538	1.9994	0.9971
#2	128.0954	1.9916	0.6368
#3	109.6726	1.9993	1.2811
#4	100.3383	1.9758	1.6545
#5	103.6600	1.9979	1.5312
#6	111.9697	1.9928	1.1886
#7	103.3317	1.9943	1.5654
#8	113.1235	1.9969	1.1734
#9	124.3926	1.9937	0.8380
#10	101.9077	1.9880	1.5959
Mean	111.4145	1.9930	1.2462
$\epsilon$	5.9536	0.0044	0.2149

. The average values for parameters $\lambda ,\beta$, and $\theta$ were employed to optimize the times and intensities of preventive maintenance in the second optimization stage.

The average values of times and severities in ten replications are presented in

Table 9. Maintenance optimization results for the first case example. The value in bold indicates the best solution obtained in 10 replications. Planning horizon of N = 24 months (730 days).

Replication	$\mathit{c}$	${\mathit{T}}_1$	${\mathit{s}}_1$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_1,{\mathit{s}}_1,\mathit{c})$
#1	1	39	0.2	913.66 × N
#2	1	60	0.2	790.27 × N
#3	1	33	0.2	1036.27 × N
#4	1	30	0.2	1170.97$\times$N
#5	1	34	0.2	1146.51 × N
#6	1	24	0.2	993.19 × N
#7	1	32	0.2	1145.47 × N
#8	1	31	0.2	975.67 × N
#9	1	35	0.2	817.29 × N
#10	1	31	0.2	1162.41 × N
Mean	1	34.90	0.2	1015.17 × N
$\epsilon$	0	5.96	0	88.31

, which shows the optimal number of maintenance cycles ($c$), the optimal PM times $T_j,j=1,\dots ,c$ (in days), the severities $s_j,j=1,...,c$, and the respective total cost, $C_{TOT}(T_j,s_j,c)$.

It is possible to observe that all replications resulted in a single maintenance cycle ($c=1$), with time $T_1\approx 35$ days and an intervention level of 20%, which has the lowest cost among all possible maintenance activities (Table 1).

Since preventive maintenance costs vary as a function of intervention level, according to Table 1, the result that minimizes the total maintenance cost has the lowest intervention level, even though this intervention level promotes the lower improvement in the system intensity of failure. Therefore, optimization is expected to indicate maintenance with maximum levels if preventive maintenance costs are fixed. The results with the preventive maintenance costs set at USD 4000, regardless of the intervention level, are presented in

Table 10. Results of the first case example, with PM cost independent of the intervention level. Planning horizon of N = 24 months (730 days). Bold value represents the best result.

Replication	$\mathit{c}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$
#1	1	48	0.9	847.15 × N
#2	1	40	0.9	1127.42 × N
#3	1	41	0.9	1030.22 × N
#4	1	38	0.9	1228.06 × N
#5	1	38	0.9	1243.24 × N
#6	1	44	0.5	938.48 × N
#7	1	38	0.9	1199.41 × N
#8	1	28	0.5	1163.03 × N
#9	1	45	0.9	884.87 × N
#10	1	39	0.9	1220.14 × N
Mean	1	39.90	0.82	1088.20 × N
$\epsilon$	0	3.35	0.10	93.71

. It can be observed that the most common solution becomes an intervention level of 90%, as expected. It is possible to observe the occurrence of solutions with other levels of intervention, such as 50%. It is noteworthy that variations in the solution are due to the stochastic characteristic of the genetic algorithm itself. Additionally, there is a relatively small difference in the improvement factor and, consequently, in the expected number of failures, for nearby intervention level values (see Figure 4). For all executions under these same conditions, however, solutions with an intervention level of 20% were not obtained.

The 90% intervention level solution obtained in Replication 1 is the one that presented the lowest total cost. In general, the total costs are slightly higher than in the previous case, as the fixed cost of USD 4000 is higher than the cost of a 20% intervention level.

The results of the second case example are presented in

Table 11. Optimal results of the second case example. Planning horizon of N = 24 months (730 days). Bold value represents the best result.

Replication	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{c}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$
#1	1140.0614	1.5543	0.1355	1	140	0.2	19.98 × N
#2	1176.7846	1.5365	0.0087	1	140	0.2	19.71 × N
#3	1173.6136	1.5464	0.0295	1	141	0.2	19.61 × N
#4	1189.1688	1.5465	0.0049	1	140	0.2	19.35 × N
#5	1133.3206	1.4970	0.0013	1	152	0.2	21.18 × N
#6	1204.0306	1.5654	0.0022	1	143	0.2	18.84 × N
#7	1195.5719	1.5634	0.0001	1	141	0.2	19.03 × N
#8	1231.2843	1.5971	0.0140	1	145	0.2	17.95 × N
#9	1180.0496	1.5407	0.0118	1	144	0.2	19.59 × N
#10	1194.1564	1.5550	0.0023	1	118	0.2	19.26 × N
mean	1181.8042	1.5502	0.0210	1	140.4	0.2	19.45 × N
$\epsilon$	17.9246	0.0157	0.0255	0	5.38	0	0.52

. Again, the results of the intervention level and total maintenance cost are quite consistent. The system indicates a single preventive maintenance action around the day T = 140 and an intervention level of 20%.

For the third case example, an N = 48-month planning horizon was considered for two observed systems. It is important to note that, in this case, the joining of the two systems’ data made it possible to consider the first system for which there was no failure data in the period. Additionally, this example only considered preventive maintenance with intervention levels of 100%, which means that the optimization algorithm needed to define only the optimal times to perform the maintenance activities. The respective results are presented in

Table 12. Optimal results of the third case example. Bold value represents the best result.

Replication	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{c}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$
#1	1148.0923	1.9924	0.0118	1	373	1.00	42.72 × N
#2	1157.0846	1.9963	0.0022	1	391	1.00	42.41 × N
#3	1142.0265	1.9916	0.0391	1	372	1.00	42.77 × N
#4	1155.7233	1.9962	0.0111	1	374	1.00	42.37 × N
#5	1140.5708	1.9867	0.0384	1	328	1.00	43.03 × N
#6	1139.0892	1.9956	0.0000	1	369	1.00	43.29 × N
#7	1185.0730	1.9990	0.0325	1	386	1.00	40.81 × N
#8	1146.1480	1.9998	0.0357	1	375	1.00	42.61 × N
#9	1129.0687	1.9951	0.0591	1	387	1.00	43.25 × N
#10	1149.6619	1.9938	0.0216	1	374	1.00	42.56 × N
mean	1149.3829	1.9949	0.0266	1	372.9	1.00	42.57 × N
$\epsilon$	9.8835	0.0024	0.0121	0	11.5	0.00	0.46

.

For the fourth case example, the experiments considered four equivalent systems subject to PMA intervention levels of 50% or 100%. The results are presented in

Table 13. Optimal results of the fourth case example. Bold value represents the best result.

Replication	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{c}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$
#1	843.3548	1.8067	2.9728	1	348	0.5	29.85 × N
#2	836.6138	1.8040	2.9818	1	347	0.5	30.16 × N
#3	835.1684	1.7999	2.9952	1	348	0.5	30.27 × N
#4	835.9370	1.8063	2.9922	1	347	0.5	30.14 × N
#5	838.6593	1.8007	2.9894	1	347	0.5	30.12 × N
#6	833.4663	1.8056	2.9937	1	347	0.5	30.26 × N
#7	838.8566	1.8049	2.9937	1	348	0.5	30.04 × N
#8	843.2426	1.7997	2.9824	1	348	0.5	29.95 × N
#9	837.7752	1.8042	2.9814	1	348	0.5	30.11 × N
#10	840.8179	1.8033	2.9876	1	348	0.5	29.99 × N
mean	838.3892	1.8035	2.9870	1	347.6	0.5	30.09 × N
$\epsilon$	2.0510	0.0016	0.0045	0	0.3	0.0	0.08

.

It is noteworthy that in all simulated scenarios, the proposed approach indicated a PM planning with a single maintenance activity over the time horizon. This result persisted even with changes in the corrective and preventive maintenance costs, as shown in Figure 5, for the first case example. It was observed that an increase in corrective maintenance costs caused an anticipation of the maintenance activity to reduce the total cost, but the impact was relatively small. This result is because the failure intensity function tended to have a constant value over time in the examples, according to Figure 1. Under this condition, the improvement promoted by MP was relatively small, independent of the intervention level, and did not significantly affect the total maintenance cost. When the preventive maintenance has minimal impact on extending the component or system’s lifespan, conducting regular maintenance becomes less efficient, especially if the maintenance cost outweighs the benefits gained.

On the other hand, the cost of preventive maintenance has a higher impact on the maintenance time, and higher costs extend the maintenance action. This characteristic can be better observed in Figure 6.

In general, maintenance activities with an intervention level of 20% were indicated in most cases. However, more comprehensive maintenance activities were indicated in cases where the times of such activities were small. This result indicates that, from the point of view of costs, it can be advantageous to perform more severe maintenance actions in advance.

These results were obtained considering all $C_{PM},C_{CM}$ combinations, varying from USD 200 to USD 10,000 (increments of USD 200) and from USD 500 to USD 50,000 (increments of USD 500), respectively. The average values of the parameters defined in Table 8 were used.

Another set of experiments considered different values of the parameter $\beta$, which is related to the expected number of failures over time. Higher values of this parameter increase the cost with corrective maintenance, as the expected number of failures grows over time. In these cases, more frequent preventive maintenance may be required. The experiments consider $\beta$ values ranging from 2.5 to 5.0, with steps of 0.5, as presented in

Table 14. Preventive maintenance optimization regarding the different expected numbers of failures over time. Illustration of first case example.

$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{c}$	$\widehat{\mathit{p}}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$
111.3703	2.5	1.2348	1	1.00	33	0.2	2461.28 × N
107.6006	3.0	1.3936	4	0.50	34, 74, 139, 353	0.9, 0.9, 0.2, 0.2	5604.27 × N
109.7674	3.5	1.2675	5	0.50	32, 70, 145, 270, 443	0.9, 0.9, 0.9, 0.2, 0.2	9790.07 × N
110.6717	4.0	1.2472	6	0.55	47, 91, 138, 224, 338, 493	0.9, 0.9, 0.9, 0.9, 0.2, 0.2	16,371.01 × N
103.8392	4.5	1.5262	8	0.35	29, 60, 125, 236, 339, 454, 519, 608	0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.2, 0.2	21,828.61 × N
113.7472	5.0	1.1006	7	0.30	24, 53, 109, 199, 292, 401, 538	0.9, 0.9, 0.9, 0.9, 0.9, 0.4, 0.4	54,680.71 × N

for the first case example, allowing us to evaluate the assertiveness of the PM optimization. In the table, the average results are presented regarding the most frequent number of cycles observed in 10 replications (mode), including a column with the observed frequency, named $\widehat{p}$.

Obviously, with the increase in the parameter $\beta$, which is directly proportional to the expected number of failures, the total costs increased proportionally to the number of maintenance actions. The difficulty of the problem also seemed to grow with the parameter $\beta$, which explains the increased variability of the results ($\widehat{p}$ from 1.0 to 0.3). This limitation can be circumvented by adjusting the AG parameters, such as the size of the population and the number of generations.

Finally, the results of the intervention level should be highlighted. As observed before, in the results shown in Figure 5 and Figure 6, the solutions in Table 14 consistently indicate a greater intervention level in the first maintenance activity, reducing this intervention level in the final maintenance activity. Such a result is largely due to the form of parameterization of the improvement factor, which generated a lower increase in the improvement values as the intervention level increased, as shown in Figure 7.

4.2. Results of the Case Study

First, we present the results of model adjustment of the three heat exchangers discussed in this case study.

Table 15. Model adjustment results.

k	Measure	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{\theta }$
1	Mean	100.0816	1.9865	0.9690
	$\epsilon$	0.1176	0.0038	0.0189
2	Mean	100.4445	1.9834	1.1862
	$\epsilon$	0.2237	0.0108	0.0333
3	Mean	100.7491	1.9796	1.6559
	$\epsilon$	0.3356	0.0109	0.0408

displays the average values of parameters λ, β, and θ from ten runs. These values were later utilized for optimizing the predictive maintenance times and intervention levels in the second stage of the optimization process. The model adjustment outcomes revealed similarities in the values of λ and β across the three equipment types, but there was considerable variation in parameter θ. It is essential to note that this parameter directly influenced the preventive maintenance improvement factor, indicating a higher effectiveness of preventive maintenance in this scenario. A more in-depth analysis of this result can be based on the optimization outcomes for the time and intervention levels of preventive maintenance, as presented in

Table 16. Optimization results of preventive maintenance times and intervention levels, for a planning horizon of N = 180 days. Best results achieved in ten model replications, along with the mean and standard deviation (std) of the total cost.

k	$\mathit{c}$	${\mathit{T}}_{\mathit{j}}$	${\mathit{s}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{T}\mathit{O}\mathit{T}}({\mathit{T}}_{\mathit{j}},{\mathit{s}}_{\mathit{j}},\mathit{c})$	Mean	Std
1	4	40, 76, 115, 139	1.0, 1.0, 1.0, 1.0	$309.77\times N$	$319.18\times N$	5.44
2	5	30, 61, 91, 118, 142	1.0, 1.0, 1.0, 1.0, 1.0	$274.21\times N$	$284.44\times N$	9.42
3	6	22, 43, 71, 94, 121, 154	0.7, 1.0, 1.0, 1.0, 1.0, 1.0	$224.36\times N$	$233.73\times N$	6.53

. The table displays the best results achieved in ten model replications, along with the mean and standard deviation of the total cost.

As the parameter θ increased, the recommended number of maintenances also rose, leading to values of 4, 5, and 6 for k = 1, 2, and 3, respectively (with θ values of 0.9690, 1.1862, and 1.6559). This effect was also noticeable in the intervention levels allowed for each equipment. Heat exchangers 1 and 2 typically required a 100% intervention, while equipment k = 3 allowed interventions of 70%, for example. Interestingly, for k = 3, lower intervention levels were suggested at the beginning of the planning horizon, possibly due to the higher reliability during the initial analysis period and the increased effectiveness of preventive maintenance in this scenario (as indicated by the higher value of θ).

Furthermore, it is crucial to discuss the results concerning the total maintenance costs in each case. It became apparent that a higher number of preventive maintenance actions led to a reduction in the overall cost, which was directly influenced by the effectiveness of these actions. This behavior can be attributed to the decrease in the expected number of failures following preventive maintenance, which also holds true for maintenance actions with lower intervention levels, even though the effect may be less pronounced in these instances. Clearly, there exists a threshold beyond which reducing the total cost per unit of time by increasing the number of maintenance actions is no longer beneficial.

4.3. Sensitivity Analysis of the Case Study

A sensitivity analysis of the solution was conducted concerning the preventive and corrective maintenance costs, with the latter usually being higher due to unplanned downtime expenses. In this case, experiments were carried out considering all combinations of values for $C_{PM}$ and $C_{CM}$, ranging from USD 1000 to USD 5000 (in increments of USD 500) and from USD 10,000 to USD 30,000 (in increments of USD 500), respectively.

The optimal number of preventive maintenance actions was observed for each combination of cost values, as well as the most frequent intervention level for these actions, as shown in Figure 8. It can be observed that the optimal number of preventive maintenance actions, denoted as c, was strongly influenced by variations in the cost of these actions, with these variables being inversely proportional. Therefore, the optimal number of preventive maintenance actions is higher when the cost of these actions is lower, a relationship that becomes more pronounced when corrective costs increase.

For this example, performing maintenance with a lower level of intervention was considered in a few cases, especially when both preventive and corrective maintenance costs were lower and θ was higher (k = 3). However, by reducing the level of intervention, the preventive maintenance actions needed to be more frequent to achieve the same system improvement factor.

On the other hand, the impact of corrective maintenance cost on c was less significant when the difference between $C_{PM}$ and $C_{CM}$ was smaller. This is the scenario, for instance, where $C_{PM}=\mathrm{U}\mathrm{S}\mathrm{D} 5000$ and $C_{CM}=\mathrm{U}\mathrm{S}\mathrm{D} \mathrm{10,000}$, in which only one preventive maintenance action is indicated.

Simplified one-dimensional illustrations of these results are shown in Figure 9, with the number of preventive maintenance actions for a few values of $C_{PM}$ and $C_{CM}$. Overall, the outcomes for different systems are similar, yet they consistently demonstrate an increasing trend in the number of preventive maintenance actions concerning the parameter θ, alongside the consideration of preventive maintenance with lower intervention levels, especially for k = 3, with lower values of parameters $C_{PM}$ and $C_{CM}$.

5. Conclusions

In this study, we proposed an approach for the maintenance optimization of repaired systems subjected to imperfect preventive maintenance activities. Reliability models consider repairable systems in which the failure rate is described by a non-homogeneous Poisson process, in which the fault intensity function varies with time. An age-based reduction factor, which restores the system’s effective age to an intermediate state between “as good as new” and “as bad as old” after PM activities, is defined as a function of independent severities. This means that the intervention level is treated as a decision variable in the optimization problem and independent, therefore, of maintenance times.

The preventive maintenance policy is based on a non-periodic approach, with the optimal times determined by the proposed method. So, the proposed maintenance strategy determines not only the optimal number of maintenance activities but also defines the optimal times for PMAs and the respective optimal intervention level that minimizes the total maintenance cost along a fixed and user-defined planning horizon.

The proposed preventive maintenance strategy was tested using data from various examples and a case study from a hydroelectric power plant in southern Brazil, considering similar systems and maintenance with different intervention levels.

In situations in which the expected number of failures was approximately constant, preventive maintenance had a small impact on system condition, regardless of the corrective and preventive maintenance costs. In such cases, only one MP was indicated in all tested examples. On the other hand, when the expected number of failures increased, it was more advantageous to increase the number of preventive maintenance activities with lower intervention levels than to increase the intervention level of the maintenance activities. It is worth noting, however, that any additional costs caused by the system’s unavailability to perform maintenance activities were not considered, even if maintenance with a higher intervention level tended to require a longer stopping time.

The case study centered on heat exchangers showcased the feasibility and efficacy of the proposed approach. It offers valuable insights into the correlation between preventive maintenance effectiveness and the optimal frequency of maintenance actions. Additionally, this study revealed that even maintenance actions with lower intervention levels could prove advantageous in minimizing the total maintenance costs, contingent upon the equipment’s characteristics and maintenance effectiveness. However, by reducing the level of intervention, the preventive maintenance actions need to be more frequent to achieve the same system improvement factor.

In addition to addressing multi-component systems, which is reported as a limitation in a recent study [20], the proposed approach flexes the maintenance policy by treating the intervention level as a decision variable and allowing balance in the trade-off between system reliability and maintenance cost. In the next step of the research, the introduction of stochastic costs for corrective maintenance can be evaluated, since this can better reflect realistic situations.