# A Hierarchical Decision-Based Maintenance for a Complex Modular System
Driven by the MoMA Algorithm

María Luz Gámiz
Department of Statistics and O.R., University of Granada, Spain

D. Montoro-Cazorla
Department of Statistics and O.R., University of Jaen, Spain

M.C. Segovia-García
Department of Statistics and O.R., University of Granada, Spain Corresponding
author<EMAIL_ADDRESS>
###### Abstract

This paper presents a maintenance policy for a modular system formed by $K$
independent modules (n-subsystems) subjected to environmental conditions
(shocks). For the modeling of this complex system, the use of the Matrix-
Analytical Method (MAM) is proposed under a layered approach according to its
hierarchical structure. Thus, the operational state of the system (top layer)
depends on the states of the modules (middle layer), which in turn depend on
the states of their components (bottom layer). This allows a detailed
description of the system operation to plan maintenance actions appropriately
and optimally. We propose a hierarchical decision-based maintenance strategy
with periodic inspections as follows: at the time of the inspection, the
condition of the system is first evaluated. If intervention is necessary, the
modules are then checked to make individual decisions based on their states,
and so on. Replacement or repair will be carried out as appropriate. An
optimization problem is formulated as a function of the length of the
inspection period and the intervention cost incurred over the useful life of
the system. Our method shows the advantages, providing compact and
implementable expressions. The model is illustrated on a submarine Electrical
Control Unit (ECU).

Keywords:Modular system; Phase-type distributions; Markovian processes;
Matrix-Analytic Method; Maintenance modeling;

##  1 Introduction

###  1.1 Motivation

Modularity is a concept widely applied in many fields, such as manufacturing
industry, power and energy systems, computer science, smart grid, biology. In
fact, many modern systems consist of modules e.g. aircrafts, rockets,
submarines, automobiles, high-tech medical equipment and others. The
advantages of using modular product design are discussed widely in the
existing literature, which also shows a variety of recent applications to
engineering systems Drossel et al. (2019)-Putri and Buana (2020). When
defining the modules, diverse strategies are found based on different
motivations or design goals Bonvoisin et al. (2016),Baldwin and Clark (2020).
Thus, a modular system can be defined according to either its functioning
requirements Li et al. (2021), manufacturing considerations Shaik et al.
(2015), reliability optimization Dorronsoro (2022) or maintainability targets
Yicong (2016) among others.

In practice, most engineering systems have to fulfill missions that require
high reliability, so they need to be maintained regularly to avoid
catastrophic failures and costly disruptions. In this respect, continuous
system monitoring or inspections over time are key to decide whether a
maintenance action should be performed, which could be preventive or
corrective. Preventive maintenance (PM) actions attempt to maintain the system
in an acceptable condition of operation by avoiding a system failure.
Corrective maintenance (CM) refers to all actions that occur when the system
has already failed. For some systems, continuous monitoring is not feasible
and inspections turn out to be the only alternative tool. Thus, determining
the optimal frequency of inspections to minimize its cost while keeping the
system performance is essential. An extensive amount of literature deals with
multi-component models under different approaches that involve periodic
inspections, age-based maintenance or condition-based maintenance; where
reliability and cost optimization problems are often formulated Sharifi and
Taghipour (2020)-Hashemi et al. (2020).

The present work focuses on multi-component systems designed with a modular
structure. Units (components) within each module are assumed to be similar in
some aspects, e.g. units that perform a certain mission or function in the
system Zhao et al. (2020),Li et al. (2021), units that compose a redundant
structure Li (2016), units making up voting circuits in electrical or computer
systems Liu et al. (2014) and so on. In all cases, a module can be represented
by a subsystem of components. Here, a system designed with $K$-independent
modules (KMS) is studied, where each module is an $n_{i}$-subsystem, and the
system performs its function through its modules. These modules are assumed to
operate under environmental conditions causing shocks that could affect each
module in a different way. The modeling of the shocks is done by Markovian
Arrival Processes (MAPs). The lifetimes of the units within the modules are
assumed to be Phase-type distributed (PH-distributed). This system was already
considered by the authors in Gamiz et al. (2022), where primarily a
reliability analysis was performed, and, maintenance was only treated briefly.
Specifically, the optimal length of the operating time of the system until it
is inspected for the first time was determined based on the probability ($q$)
that the system reaches a critical operating state. In Gamiz et al. (2022),
certain cost-based arguments were used to plan the optimal inspection
schedule, but no detailed analysis based on maintenance actions and cost
specifications as it is done here was carried out.

It is clear that modeling complex systems is a challenging task due to the
large number of units, structures and factors concerned, which lead to an
arduous and large dimensional framework. For this reason, many multi-component
systems are studied in the literature using numerical or simulation techniques
Chen et al. (2022). As an alternative, Markovian processes and the use of the
Matrix-Analytical Method(MAM) has proven to be a useful tool for the study of
complex systems Montoro-Cazorla and Perez-Ocon (2015)-Chakravarthy (2012).
Nevertheless, to the authors’ knowledge, reliability literature on modular
systems is very limited and even more so using the above methodology. Thus,
for modeling this type of system, we propose the use of the MAM under a
layered/leveled approach according to its hierarchical structure where the
whole system is located at the top, the modules are in the middle and the
components are set at the bottom. As a first goal, in Gamiz et al. (2022) the
authors designed the so-called MoMA (Modular Matrix-Analytic Method) Algorithm
to build (bottom-up) the generator of this system and from it the assessment
of the system was carried out. However, the present work focuses on the design
of a maintenance strategy for the system. To this end, our approach is most
appropriate by allowing a state-based hierarchical description of the system
operational conditions, i.e. states at each level capture the operational
health (states) of every element at the level/s below. This provides detailed
information that makes possible to plan maintenance actions in a precise and
optimal way. Regarding the layered description of systems, other works can be
found in the literature under different targets and methodologies. For
instance, the work in Torrado et al. (2021) deals with a multi-level
redundancy assignment problem in a modular system, where the effect of
redundancies (at different levels) on the reliability of the system is
evaluated. For maintenance purposes for a multi-component system, the authors
in Chen et al. (2022) propose a preventive and opportunistic method with a
two-phased inspection: the system is first inspected and, depending on its
condition, in the second phase, the units are inspected or not.

For our system, the states (at each level) can be categorized into groups
according to operational health under progressive deterioration or risk, i.e.
optimal, critical and failure. Thus, we propose the following inspection-
maintenance policy: at the time of the inspection, the operational state of
the system is assessed first (optimal, critical or failure), which determines
whether or not a maintenance action is required. If so, then it becomes
necessary to look at the operational state of each module to make individual
decisions for each. Those modules which are in a critical or failure state are
ascertained. For the rest, no intervention will take place. Thus, maintenance
tasks will be performed accordingly. As can be noted, this maintenance
strategy aligns with system design: system, module and component levels can be
distinguished. It is well justified for those multi-component systems where it
is not feasible or costly to inspect every single component. For example, in a
battery pack system in electric vehicles, voltage checking for the bank of
cells is cheaper than for individual cells, taking into account the assembling
and disassembling costs Chen et al. (2022). In connection with this, existing
literature establishes that in models with positive economic dependencies
maintaining similar units together is cheaper than maintaining them separately
since the set-up costs can be shared as well as a specialized technicians team
when similar maintenance tasks are required Maaroufi et al. (2013),Levi et al.
(2014). In addition, replacing an entire module when it fails is sometimes a
more economical option than replacing each of its components Joo
(2009),Karmarkar and Kubat (1987).

In short, we plan for the system a hierarchical and decision-based maintenance
with periodic inspections, where preventive and corrective actions are
included. These actions are implemented in blocks, i.e. several units could be
repaired at the same time or jointly replaced. The maintenance effect on the
modules is mathematically represented by matrices, which describe the
transitions rules from the occupied state just before the inspection to the
state (to be occupied) just after the inspection. These matrices play an
important role when determining the Markovian process that governs the
evolution of the system over time. As will be shown later, a recursive
procedure allows to describe the Markovian process along each inspection
period starting from the first one, where the system has not yet been
inspected (addressed in Gamiz et al. (2022)). A cost analysis on each
inspection period is performed, based on maintenance costs, e.g. per
inspection, per replacement of each unit, per down-time, and others. Finally,
an optimization problem is formulated as a function of the time between
inspections and the incurred costs along the useful life of the system, where
a maximum number of inspections is allowed. The expressions and calculus are
illustrated on a real system, consisting of the electrical control equipment
of a submarine, formed by four modules. A discussion of the main results and
conclusions are reported.

###  1.2 Contributions

The main contributions of this paper are listed below.

  1. 1.

Modules are key to system reliability and maintainability: with $K$
independent modules, a failure in one module generally does not compromise the
entire system operation, although its performance might degrade. Thus, system
malfunctions or failures can be identified and located more easily than in
non-modular logical structures. As a result, the system maintenance process is
simplified, potentially reducing time and costs. Our approach takes these
aspects of modular design theory into account..

  2. 2.

The method we propose proves to be suitable for modeling this kind of complex
modular systems. Specifically, it allows:

     * •

To give a state-based hierarchical description of the system operational
health, which provides detailed information. Thus, the state of the system
collects information about the state of each of its elements, as each level
captures the state of the elements at the level below. This allows for
accurate assessments of the operational conditions at the time of inspection,
which enables maintenance needs to be precisely established. This is an
important contribution, when compared with those systems that behave like
black boxes at the time of inspection-maintenance.

     * •

To plan a hierarchical condition-based policy accordingly.

     * •

To perform a stepwise analytical study, by levels, where mathematical tools
such as MAPs, PH-distributions and Kronecker operators are essential and lead
to compact and algorithmic expressions that make the model implementable.
Specifically, matrices that are relevant in this maintenance analysis are
novel and easily constructed under this approach.

  3. 3.

For the KMS described, already considered in Gamiz et al. (2022), a
maintenance policy based on regular inspections along its useful life is
provided. A cost minimization problem is formulated in terms of the frequency
of inspection.

  4. 4.

PH-distributions and MAPs extend the Exponential distribution and Poisson
arrival process, respectively, among others, often assumed in engineering
studies. For example, the submarine ECU studied in Liu et al. (2014) assumes
all lifetimes and repair times exponentially distributed; such study also
reveals a complex framework due to the large number of states given for the
system, which could be improved using the MAM, as we propose in this paper.
Additionally, each module in the system is assumed to fail independently of
the others based on two sources of failure: units wear-out and external
shocks. This situation fits many modular systems in practice, where each
module can be exposed to certain environmental or working conditions depending
on the assigned function to perform in the system or depending on its physical
location, e.g. modules underwater as considered in the paper referenced
before.

  5. 5.

A case example concerning the Electrical Control Unit (ECU) of a submarine
illustrates numerically, for different cost scenarios, the optimal length of
the inspection period and, from this, the total number of inspections to be
performed over the useful life of the system.

Further contributions can be made by applying our method to the study of other
complex models, such as the following ones: 1) systems that are decomposed
into independent subsystems (modules) for simplification aims (see Liang et
al. (2022)); 2) modular systems where module is derived in another way,
depending on the purpose or context of application. For example, a module can
be interpreted as a group of components to be maintained in the same way (see
Putri and Buana (2020)). In summary, our methodology is a powerful and
flexible tool for the analysis of complex systems. Therefore, it is suitable
for addressing a wide variety of reliability and maintainability problems
taking advantage of the modular design. For example, by adding or updating
functions, or adding redundancy in the system by means of modules, the
modeling of the system (after these changes) could be easily adapted.

The remainder of this paper is organized as follows. Section 2 provides the
relevant definitions for application of the MAM. Section 3 establishes the
assumptions of the system. Sections 4 and 5 are devoted to the proposed
maintenance policy for the system and the associated cost model. In these
sections, the key mathematical elements for modeling the system maintenance
and to formulate a cost optimization problem are defined. Section 6
illustrates the model on a real system, a submarine ECU comprising four
modules. A discussion of the main results and some conclusions are presented
in Section 7.

##  2 Preliminaries

In this section we present the nomenclature used and define some basic concept
that will be needed for the rest of the paper.

###  2.1 Acronyms $\&$ Notation

KMS |  |  $K$-Modular System
---|---|---
PH-distribution |  | Phase-Type distribution
MAP |  | Markovian Arrival Process
MAM |  | Matrix-Analytical Method
MoMA |  | Modular Matrix-Analytic Alagorithm
ECU |  | Electrical Control Unit
SEM |  | Subsea Electrical Module
BOP |  | Blowout Preventer
Table 1: Acronyms

In Table 2 we detail the notation used in the paper.

$K$ |  | Number of modules in the system
---|---|---
$n_{i}$ |  |  Number of units in module $i=1,\ldots,K$
$\mathbb{Q}_{sys}$ |  | Infinitesimal generator of the system
$U$ |  | Set of operational states of the system
$U_{1}$ |  | Set of optimal states of the system
$U_{2}$ |  | Set of critical states of the system
$D$ |  | Set of down states of the system
$s_{ji}$ |  |  State of unit $j$ of module $i$
$\widetilde{s}_{i}$ |  |  state of module $i$
$\widetilde{\bf s}$ |  | state of the system
$M^{j}_{i}$ |  |  Matrix of maintenance-effect on unit $j$ in module $i$
$\beta_{j,i}$ |  |  Vector of state-probability for unit $j$ in module $i$ after maintenance action
${\bf M}_{i}$ |  |  Matrix of maintenance-effect on module $i$
$\beta_{i}$ |  |  Initial probability vector for module $i$ after replacement of the module
$\tau_{{\rm a}}$ |  |  ${\rm a}th$ inspection time
$A$ |  | Maximum number of inspections
$X(t)$ |  |  State of the system at time $t$, $0<t<\tau$ (without maintenance)
$({\rm a}\tau,({\rm a}+1)\tau]$ |  |  ${\rm a}$-th operational cycle of the system, ${\rm a}=0,\ldots,A-1$
$\widetilde{\alpha}({\rm a})$ |  |  System state probability vector after ${\rm a}$-th inspection
$\widetilde{X}({\rm a},t)$ |  |  State of the system at time ${\rm a}\tau+t$, $0<t<\tau$, ${\rm a}=1,\ldots,A$
$\widetilde{\alpha_{i}}({\rm a})$ |  |  Vector of state probabilities for module $i$ after ${\rm a}$-th inspection.
$\widetilde{X}_{i}({\rm a},\tau)$ |  |  State of module $i$ at the end of the ${\rm a}$-th operational cycle
$\beta$ |  | Initial probability vector of the system after full replacement
${\bf C}_{i}$ |  |  Matrix of maintenance cost of module $i$
$f_{s}({\rm a},t)$ |  |  Lifetime density of the system after ${\rm a}$-th inspection
$Cost({\rm a},\tau,i)$ |  |  Total cost of maintenance on module $i$ at the ${\rm a}$-th inspection
$EC_{sys}({\rm a},\tau)$ |  |  Total expected cost of maintenance of the system in the interval $({\rm a}\tau,({\rm a}+1)\tau]$
Table 2: Notation.

###  2.2 Basic concepts

PH-distributions, MAPs, and Kronecker operations play an important role in
this paper. They are the basic elements in the application of (MAMs), and are
formally defined below to provide a better understanding of this work. For
further details, readers are referred to (Bellman (1970); Graham (1981); Neuts
(1981)).

###### Definition 1

PH-distribution

Consider a finite Markov chain with $m$ transient states and one absorbing
state with the infinitesimal generator $Q$ partitioned as

$Q=\left(\begin{array}[]{ c c}T&T^{0}\\\ 0&0\end{array}\right).$

where $T$ is a matrix of order $m$ and $T^{0}$ is a column vector such that
$Te+T^{0}=0$. The vector $e$ is a column vector of ones. For eventual
absorption into the absorbing state, starting from the initial state it is
necessary and sufficient that $T$ be nonsingular. Suppose that the initial
state of the Markov chain is chosen according to the probability vector
$(\alpha,a_{m+1})$. Let $X$ denote the time until absorption; then, $X$ is a
random variable taking non-negative values, with probability distribution
function given by $F(x)=1-\alpha e^{Tx}e,$ for $x\geq 0.$

We then denote $X$ as following a $PH(\alpha,T)$-distribution of order $m$.

###### Definition 2

Markovian Arrival Process
Suppose that $D=\left(d_{ij}\right)$ is the generator of an irreducible Markov
chain with $m$ states. At the end of a sojourn time in state $i$ that is
exponentially distributed with parameter $\lambda_{i}$, one of the following
two events could occur: with probability $p_{ij}^{(1)}$, the transition
corresponds to an arrival and the underlying Markov chain is in state $j$ with
$1\leq i,j\leq m$, while with probability $p_{ij}^{(0)}$ the transition
corresponds to no arrival and the state of the Markov chain is $j,j\neq i$. We
can define matrices $D_{0}=\left(d_{ij}^{(0)}\right)$ and
$D_{1}=\left(d_{ij}^{(1)}\right)$ such that $d_{ii}^{(0)}=-\lambda_{i},$
$d_{ij}^{(0)}=\lambda_{i}p_{ij}^{(0)},$ for $j\neq i$ and
$d_{ij}^{(1)}=\lambda_{i}p_{ij}^{(1)},$ $1\leq i,j\leq m$. By assuming
$D_{0\text{ }}$ to be a nonsingular matrix, the inter-arrival times are finite
with a probability of one, and the arrival process does not terminate. Hence,
it can be seen that $D_{0}$ is a stable matrix. The generator $D$ is then
provided by $D=D_{0}+D_{1}.$ Let $\alpha$ be the initial probability vector of
the underlying Markov chain.

Then, $D_{0}$ governs the transitions corresponding to no arrival and $D_{1}$
governs those corresponding to an arrival. It can be shown that MAP is
equivalent to Neuts’ versatile Markovian point process. The point process
described by the MAP is a special class of semi-Markov processes with their
transition probability matrix provided by

$\int_{0}^{x}e^{D_{0}t}dtD_{1}=\left[I-e^{D_{0}x}\right](-D_{0})^{-1}D_{1},\text{
}x\geq 0$

This MAP is represented by the MAP $(D_{0},D_{1})$ of order $m$.

###### Definition 3

Kronecker product of matrices
If $A$ and $B$ are rectangular matrices with dimensions $m_{1}\times m_{2}$
and $n_{1}\times n_{2}$, respectively, their Kronecker product $A\otimes B$ is
a matrix with the dimensions $m_{1}n_{1}\times m_{2}n_{2}$, which can be
written in compact form as $(a_{ij}B).$

###### Definition 4

Kronecker sum of matrices
If $A$ and $B$ are square matrices with dimensions $m_{1}$ and $n_{1}$,
respectively, their Kronecker sum, denoted by $A\oplus B$, is a matrix defined
by $A\otimes I_{n_{1}}+I_{m_{1}}\otimes B$, where $I_{m_{1}}$, $I_{n_{1}}$ are
identity matrices with dimensions $m_{1}$ and $n_{1}$, respectively.

##  3 System description

A system formed by $N$ units grouped in $K$ modules is considered. The failure
of each module can be due to the wear-out of its units or to a shock that
affects the module. The lifetime of each unit follows a PH-distribution.
Shocks arrive at the module (system) following a MAP. The generator of such
system was built in Gamiz et al. (2022). All the expressions appearing in this
section were derived in Gamiz et al. (2022). Here we summarize the assumptions
and the results needed for the paper.

###  3.1 Model assumptions

  1. 1.

The $i$-th module is formed by $n_{i}$ units. The lifetime of unit $j$ in the
module follows a PH-distribution $PH(\alpha_{j,i},T_{j,i})$, with $m_{j,i}$
phases where $i=1,2,\cdots,K$, $j=1,2,\cdots,n_{i}$ and
$N=\sum_{i=1}^{K}n_{i}$. The units within a module are considered independent.

  2. 2.

Shocks arrive to module $i$ following a $MAP(D_{0,i},D_{1,i})$, of order
$b_{i}$ where $D_{i}=D_{0,i}+D_{1,i}$ is the infinitesimal generator. Matrix
$D_{0,i}$ governs the inter-arrival times between shocks that affect the
module and matrix $D_{1,i}$ contains the transition rates between the phases
of the MAP when a shock arrives.

  3. 3.

The MAP process affecting module $i$ is independent of the other MAP processes
affecting the rest of the modules.

  4. 4.

Following the bottom-up strategy explained in Gamiz et al. (2022), the state
of the system is ultimately determined by the states of the units that are
part of each module. We consider the following notation. Let $E$ be the set of
states of the system. We denote $\widetilde{\bf s}$ one particular element of
$E$, which we describe as follows

$\widetilde{\bf
s}=(\widetilde{s}_{1};\widetilde{s}_{2};\ldots;\widetilde{s}_{K}),$

where $\widetilde{s}_{i}=(s_{i1},s_{i2},\ldots,s_{in_{i}})$ denotes the state
of module $i$ and is given by a particular combination of the states of the
units within the module. That is $s_{ji}$ is the state of the $j$-th unit of
the module $i$, with $j=1,\ldots,n_{i}$; and $i=1,\ldots,K$. Notice that we
use “,” to discriminate states of the units within a module and “;” to
distinguish the states of two different modules. Finally, let $n$ denote the
total number of states of the system.

  5. 5.

A shock may or may not cause the failure of a module. Let
$p_{1,i},i=1,\cdots,K$ be the probability that a shock causes the failure of
module $i$, and $p_{0,i}$ the probability that the module does not fail due to
the shock. Additionally, $p_{0,i}+p_{1,i}=1$.

  6. 6.

The system might fail even if some modules are still operational.

Below we provide the expression of the system generator ($\mathbb{Q}_{sys}$).
This generator is given in terms of the number of modules that have not
failed, i.e. the macro-states in the generator refer to the number of modules
that remain operational.

$\mathbb{Q}_{sys}=\left(\begin{array}[]{c c c c c c}{\bf
Q}_{K}^{\prime}&\widetilde{\bf Q}_{K}^{\prime}&0&0&\cdots&0\\\ 0&{\bf
Q}_{K-1}^{\prime}&\widetilde{\bf Q}_{K-1}^{\prime}&0&\cdots&0\\\
0&0&\ddots&\ddots&\cdots&0\\\ \vdots&\vdots&&\ddots&\ddots&\vdots\\\
0&0&0&0&{\bf Q}_{K-(K-1)}^{\prime}&\widetilde{\bf Q}_{K-(K-1)}^{\prime}\\\
0&0&0&0&0&0\end{array}\right).$ (1)

In this generator, the transitions between the macro-states are given by:

  * •

${\bf Q}_{K-l}^{\prime}$ is a matrix describing the transitions when $l$
modules out of $K$ modules have failed and the rest remain operational.
Transitions in this matrix have to consider all possible combinations of $l$
modules failing out of $K$, therefore this matrix is composed of
$\binom{K}{l}\times\binom{K}{l}$ blocks of matrices. The blocks in the
diagonal contain the transitions within the same state i.e. from the state
where $l$ specific modules have failed to the state where exactly the same
modules have failed. The off-diagonal blocks in this matrix are given by
matrices of zeros of appropriate dimensions, as these transitions mean that a
different set of modules have failed and that is not possible. The diagonal
blocks are akin to the following one:

${\bf Q}_{1}^{s}\oplus\cdots\oplus{\bf Q}_{h_{1}-1}^{s}\oplus{\bf
Q}_{h_{1}+1}^{s}\oplus\cdots\oplus{\bf Q}_{h_{l}-1}^{s}\oplus{\bf
Q}_{h_{l}+1}^{s}\oplus\cdots\oplus{\bf Q}_{K}^{s}.$ (2)

with $h_{r}$ in $\\{1,2,\cdots,K\\}$, $r=1,\cdots,l$ and $\oplus$ is the
Kronecker sum.

In (2) matrix ${\bf Q}_{i}^{s}$ provides the transitions between the
operational states in module $i$. These transitions could be due to the change
of phase of its units (wear-out of the units) or to the arrival of a shock
that does not affect the module. This matrix is given by,

${\bf Q}_{i}^{s}={\bf Q}_{i}\oplus(D_{0,i}+p_{0,i}D_{1,i}),$

where matrix ${\bf Q}_{i}$ is the matrix that represents the internal changes
in module $i$, i.e. changes between the operational phases of the units, that
do not lead to the module failure.

  * •

Matrix $\widetilde{\bf Q}_{K-l}^{\prime}$ is formed by
$\binom{K}{l}\times\binom{K}{l+1}$ blocks of matrices that describe the
transitions when a module fails, e.g. module $p$, out of the remaining
operational ones, i.e. from $K-l$ specific operational modules to $K-(l+1)$,
where $l$ of the modules have previously failed. These transitions can be
described by matrices like the following one:

$I_{\sum_{j=1}^{n_{1}}m_{j,1}}\otimes\cdots\otimes
I_{\sum_{j=1}^{n_{p-1}}m_{j,p-1}}\otimes(\widetilde{\bf Q}_{p}^{s}{e})\otimes
I_{\sum_{j=1}^{n_{p+1}}m_{j,p+1}}\otimes\cdots\otimes
I_{\sum_{j=1}^{n_{K}}m_{j,K}},$ (3)

with ${e}$ a column vector of 1s of the appropriate dimension. The rest of the
transitions in matrix $\widetilde{\bf Q}_{K-l}^{\prime}$ are given by matrices
of zeros of appropriate dimension.

Note that in this case module $p$ fails out of the remaining operational
modules, being module $p$ not necessarily consecutive to any of the previously
failed ones.

In (3) matrix $\widetilde{\bf Q}_{i}^{s}$ represents the transitions when the
module fails due to the failure of one of its units or due to a shock. Matrix
$\widetilde{\bf Q}_{i}^{s}$ is given by:

$\widetilde{\bf Q}_{i}^{s}=\left(\begin{array}[]{c|c}\widetilde{\bf
Q}_{i}\otimes I&I\otimes p_{1,i}D_{1,i}\end{array}\right).$

where matrix $\widetilde{\bf Q}_{i}$ is the matrix that represents the
transitions to a failure state due to an internal failure, i.e., the failure
of its units as a consequence of the wear-out process.

  * •

Finally, for any $i$ in $\\{1,...,K\\}$, matrix ${\bf Q}_{i}^{*s}$ is the
generator of module $i$ considering that it might fail due to the wear-out
process or due to the arrival of a shock. This generator is given by the
following expression:

${\bf Q}_{i}^{*s}=\left(\begin{array}[]{c|c }{\bf Q}_{i}^{s}&\widetilde{\bf
Q}_{i}^{s}\\\ \hline\cr 0&0\\\ \end{array}\right).$

### Example. A $k$-out-of-$K$ system

The macro-states of the system can be grouped into operational and failure
states depending on its arrangement, for the particular case of a $k$-out-
of-$K$ system, $K-k+1$ modules must fail for the system to fail. Let
$\mathbb{Q}_{sys}^{\prime}$ be the matrix that describes the transitions
between the operational states, and $\widetilde{\mathbb{Q}}_{sys}^{\prime}$
the matrix that describes the transitions between the operational and failure
states. These matrices are given by,

$\mathbb{Q}_{sys}^{\prime}=\left(\begin{array}[]{c c c c c c}{\bf
Q}_{K}^{\prime}&\widetilde{\bf Q}_{K}^{\prime}&0&\cdots&0\\\ 0&{\bf
Q}_{K-1}^{\prime}&\widetilde{\bf Q}_{K-1}^{\prime}&\cdots&0\\\
0&0&\ddots&\ddots&\cdots\\\ \vdots&\vdots&&\ddots&\widetilde{\bf
Q}_{k+1}^{\prime}\\\ 0&0&0&0&{\bf Q}_{k}^{\prime}\\\ \end{array}\right),$

and,

$\widetilde{\mathbb{Q}}_{sys}^{\prime}=\left(\begin{array}[]{c}0\\\ 0\\\
\vdots\\\ 0\\\ \widetilde{\bf Q}_{k}^{\prime}\end{array}\right).$

Therefore, the generator of the system can be described in a compact form as
follows:

$\mathbb{Q}_{sys}=\left(\begin{array}[]{c|c}\mathbb{Q}_{sys}^{\prime}&\widetilde{\mathbb{Q}}_{sys}^{\prime}\\\
\hline\cr 0&0\\\ \end{array}\right),$

As it can be seen, this generator is given in terms of 2 macro-states, the
operational state (up state) that will be denoted by U and the failure state
(down state) that will be denoted by D.

As for the system, we can also grouped the states of each module into
operational and failure states. In Gamiz et al. (2022) $Q_{i}^{*}$ is the
generator that describes how the module might fail solely due to the wear-out
of its units, and is given in terms of the operational units, in a similar way
to the description of the failure of the system in terms of the operational
modules see (1). More specifically,

${\bf Q}_{i}^{*}=\left(\begin{array}[]{c c c c c
c}Q_{n_{i}}&\widetilde{Q}_{n_{i}}&0&0&\cdots&0\\\
0&Q_{n_{i-1}}&\widetilde{Q}_{n_{i-1}}&0&\cdots&0\\\
0&0&\ddots&\ddots&\cdots&0\\\ \vdots&\vdots&&\ddots&\ddots&\vdots\\\
0&0&0&0&Q_{1}&\widetilde{Q}_{1}\\\ 0&0&0&0&0&0\end{array}\right),$

The macro-states of this generator can be in turn grouped into operational and
failure macro-states depending on the configuration of the units within the
module, e.g. if the units in the module are set in series then

${\bf Q}_{i}=Q_{n_{i}}$ and $\widetilde{\bf Q}_{i}=\widetilde{Q}_{n_{i}}$.

where the rest of the blocks in matrix ${\bf Q}_{i}^{*}$ are given by matrices
of $0s$ of appropriate dimensions.

###### Remark 1

Matrix $\mathbb{Q}_{sys}$ is the generator that governs the evolution of the
system until it fails. In the described system we have considered that
independent MAPs can affect individual modules. Alternatively, in Gamiz et al.
(2022) the case where a shock that arrives following a MAP and affects the
system as a whole is also considered. We do not describe this particular case
in the present paper but in the next sections, matrix $\mathbb{Q}_{sys}$ might
represent the evolution of the system standing shocks that affect the system
as a whole as well.

###  3.2 The MoMA algorithm

The MoMA algorithm was developed in Gamiz et al. (2022) to built the system
generator using a bottom up approach, i.e. starting from the units generators.
In this algorithm first the generator of module i is built considering the
units generators, then the system generator is built combining the module
generators as well as the generators corresponding to the shocks arrival.
Three scenarios are considered: 1. There are no shocks affecting the system,
2. A shock could affect the system as a whole causing its failure and 3. $K$
independent shock processes might affect the modules causing their failure.

##  4 Maintenance policy

For the above described system we propose the following maintenance policy:
the system is inspected at times $\tau_{\rm a}={\rm a}\tau$, where $\tau$ is
fixed and determined considering maintenance costs as it will be shown later
on. Additionally, maintenance actions depend on the state of the system at
inspection time $\tau_{\rm a}$. For this system we distinguish between optimal
operational states, critical states (the performance of the system has
degraded, it could mean that although it has not failed it might be
approaching the end of its life) and down states (the system has failed). The
set of states that are optimal are labeled by $U_{1}$, the set of critical
states are labeled by $U_{2}$ and the set of down states are labeled by $D$,
although we are only considering one down state and do not distinguish failure
modes.

As mentioned above, the maintenance policy looks at the state of the system at
the inspection time and proceeds as follows:

  * •

If the system is in an optimal state $\left(U_{1}\right)$, no intervention
takes place, leaving the system in the same state.

  * •

If the system is in a critical state $\left(U_{2}\right)$, we look at the
modules. The failed modules are replaced and for the rest we distinguish
between modules in a critical state (approaching their failure) and those in
an optimal state, for the former ones the failed units are maintained, for the
latter ones no intervention takes place, i.e. the wear-out state of the module
remains the same.

  * •

Finally, if the system has failed we replace it.

As it can be seen, we only inspect the modules if the system is in a critical
state at inspection times. Otherwise, if the system is in an optimal state we
do nothing, and if the system has failed we replace it following initial
probability vector $\beta$. Additionally, when the system is in a critical
state, maintenance actions restore the modules to an optimal state.

Figure 1 describes the transitions between the set of states $U_{1}$, $U_{2}$
and $D$ considering wear-out, shocks and the maintenance policy. The described
transitions replicate at module level.

Figure 1: Modular system transitions. (1) Units wear-out. Shock processes.
Modules might fail. (2) Failure of $s$th module. (3) Failure of $r$th module.
System failure. (4) System failure due to a shock. (5) Maintenance. Restore
failed units and modules. (6) Maintenance. Replace system.

###  4.1 Construction of maintenance matrices ${\bf M}_{i}$

Let us define matrix ${\bf M}_{i}$ to describe how the maintenance policy
affects module $i$ when the system is in a critical state. We describe this
matrix by blocks. In general, let us denote $M_{A_{1},A_{2}}$ the sub-matrix
describing the transitions from states in subset $A_{1}$ to states in subset
$A_{2}$, for any $A_{1},A_{2}\subset E_{i}$ where $E_{i}=U_{i1}\cup U_{i2}\cup
D_{i}$ is the state space for module $i$, for $i=1,...,K$. At inspection time
$\tau_{a}$:

  * •

If the module is in an optimal state, then no intervention takes place.
Therefore $M_{U_{i1},U_{i1}}=I$.

  * •

If the module is in a critical state, $U_{i2}$, all the failed units are
maintained, the rest remain unchanged. Within $U_{i2}$ we have different
combinations of units in operational and failed state. For example, let us
consider a 2-out-of-4 module, this module will be in a critical state when 2
of its units have failed, we have $\binom{4}{2}$ combinations of failed and
operational units that should be taken into account to determine the states
within $U_{i2}$. In 4.2 some detailed examples are given.

In general, for a specific combination of failed units in module $i$ and after
a maintenance action, module $i$ re-initiates its operation considering the
following expression:

$M^{1}_{i}\otimes M^{2}_{i}\otimes\cdots\otimes M^{n_{i}}_{i}$.

where $M^{j}_{i}=I$ if unit $j$ is in an operational state. If the unit is in
a failed state it re-initiates its operation following a maintenance action
vector $\beta_{j,i}$, where $\beta_{j,i}$ contains the probabilities of the
maintenance action restoring the unit to any of its operational phases.

As an example, if unit $j,j=1,\cdots,n_{i}$ has failed then the corresponding
block of probabilities in sub-matrix $M_{U_{i1},U_{i2}}$ is given by:

$I_{m_{1,i}}\otimes\cdots\otimes I_{m_{j-1,i}}\otimes\beta_{j,i}\otimes
I_{m_{j+1,i}}\otimes\cdots\otimes I_{m_{n_{i},i}}$.

  * •

If the module has failed, then the module is replaced and it re-initiates its
operation following maintenance action vector $\beta_{i}$. Therefore,
$M_{{D},U_{i1}}=\beta_{i}$. Eventually one can take $\beta_{i}=\alpha_{i}$
(the initial probability distribution), however, to be general, we consider
here that the probability of restoring module $i$ to any operational state
might be different from the provided by initial probability distribution.

  * •

The rest of the blocks within this matrix are given by matrices of zeros.

In summary, each row in matrix ${\bf M}_{i}$ is a vector describing the
probabilities to restore the module from an specific state to any operational
state after a maintenance action takes place.

###  4.2 Examples

Let us illustrate matrix ${\bf M}_{i}$ with a couple of examples.

  1. 1.

First, we consider that module $i$ is formed by 2 units set in parallel, both
units are Phase-type distributed with two phases each, phases 0 and 1 are the
operational phases and phase 2 is the absorption one, i.e. the unit has
failed. We denote the absorption phase by F from now on. We define the
operational, critical and failed states of the module in terms of the units
phases as follows: $U_{i1}=\\{(0,0),(0,1),(1,0),(1,1)\\}$,
$U_{i2}=\\{(0,F),(1,F),(F,0),(F,1)\\}$, and $D_{i}=\\{(F,F)\\}$. In this
particular example the module is in a critical state when one of its units has
failed. Then, matrix $M^{i}$ is given by,

${\bf M}^{i}=\bordermatrix{&U_{i1}&&U_{i2}&&D_{i}\cr
U_{i1}&I&\vrule&0&\vrule&0\cr\hline\cr
U_{i2}&\begin{matrix}I\otimes\beta_{2,i}\cr\beta_{1,i}\otimes
I\end{matrix}&\vrule&0&\vrule&0\cr\hline\cr
D_{i}&\begin{matrix}\beta_{i}=\beta_{1,i}\otimes\beta_{2,i}\end{matrix}&\vrule&0&\vrule&0}.$

.

  2. 2.

Another example is given by a module composed by 3 units, each of them Phase-
type distributed with 1 phase, being 0 the operational phase and F the
absorption phase, i.e. they are exponentially distributed.

Module $i$ is a 2-out-of-3 module. Therefore the optimal states of the module
are given by $U_{i1}=\\{(0,0,0)\\}$ and the critical states are given by
$U_{i2}=\\{(0,0,F),(0,F,0),(F,0,0)\\}$. The down state encompasses states
$\\{(0,F,F),(F,0,F),(F,F,0),(F,F,F)\\}$, although we do not distinguish
failure modes and consider it as a unique state, as mentioned above.

The expression of matrix ${\bf M}^{i}$ is given by,

${\bf M}^{i}=\bordermatrix{&U_{i1}&&U_{i2}&&D_{i}\cr
U_{i1}&1&\vrule&0&\vrule&0\cr\hline\cr
U_{i2}&\begin{matrix}e\end{matrix}&\vrule&0&\vrule&0\cr\hline\cr
D_{i}&\begin{matrix}\beta_{i}=1\end{matrix}&\vrule&0&\vrule&0}.$

where $e$ is a all-ones $3\times 1$ vector.

###  4.3 Consequences of the maintenance actions

Let $X(t)$ be the state of the system before the first inspection takes place.
Let us recall that the system generator is given by $\mathbb{Q}_{sys}$ and let
us say that $\alpha_{sys}$ is the system initial probability distribution.

Let us consider $\\{\widetilde{X}({\rm a},t),0\leq t\leq\tau,{\rm
a}=0,1,2,\cdots,A\\}$, where $\widetilde{X}({\rm a},t)$ is the state of the
system at time $t\in[0,\tau]$ after the ${\rm a}$-th inspection. In other
words, $\widetilde{X}({\rm a},t)$ represents the state of the system at time
${\rm a}\tau+t$. Additionally, $A$ is the maximum number of inspections, i.e
at the $Ath$ inspection the system is replaced, and $\widetilde{X}(0,t)=X(t)$,
$0<t\leq\tau$.

Let $E=U_{1}\cup U_{2}\cup D$. In general, the order of the states in matrix
$\mathbb{Q}_{sys}$ allows us to directly assign the states to $U_{1}$, $U_{2}$
and $D$ without having to reorganize these states. However, this might not be
always the case, when this happens we can in practice re-label the states to
work with them. In what follows we assume that we do not need to reorganize
nor re-label these states.

After the ${\rm a}$-th inspection the system is in one of the states in set
$U_{1}$, i.e. $\widetilde{X}({\rm a},0)\in U_{1}$. The new state depends on
the state of the system prior to the inspection and the performed maintenance
action, as shown in Figure 2, then $\widetilde{X}({\rm a},0)$ is a function of
$\widetilde{X}({\rm a}-1,\tau)$ and the performed maintenance action, which is
determined by matrices ${\bf M}_{1},{\bf M}_{2},...,{\bf M}_{K}$. Therefore,
at the beginning of the ${\rm a}$-th period we have an initial probability
distribution over the system states, let us denote this distribution
$\widetilde{\alpha}({\rm a})$; if ${\rm a}=0$ then $\widetilde{\alpha}({\rm
a})=\alpha_{sys}$. To determine $\widetilde{\alpha}({\rm a})$ we can
distinguish three situations (see Figure 2).

Let ${\rm a}>0$, and, for a particular state of the system $\widetilde{\bf
s}\in E$, let us denote

$\widetilde{\alpha}(\rm a,\widetilde{\bf s})=\mathbb{P}\/(\widetilde{X}(\rm
a,0)=\widetilde{\bf s}).$

It is clear that $\widetilde{\alpha}(\rm a,\widetilde{\bf s})>0$ only for
$\widetilde{\bf s}\in U_{1}$. Then we have:

  1. 1.

Denote $\widetilde{\alpha}_{U_{1}}(\rm a)$, the initial law at the $\rm a$-th
interval given that the state of the system at the end of the previous
inspection time is optimal, i.e. $\\{\widetilde{X}(\rm a-1,\tau)\in U_{1}\\}$
. In this case, no maintenance action is performed leaving the system as it
was before the inspection, consequently, for $\widetilde{\bf s}\in U_{1}$,

$\widetilde{\alpha}_{U_{1}}(\rm a,\widetilde{\bf
s})=\mathbb{P}\/(\widetilde{X}(\rm a-1,\tau)=\widetilde{\bf
s}\mid\widetilde{X}(\rm a-1,\tau)\in U_{1}),$

and $\widetilde{\alpha}_{U_{1}}(\rm a,\widetilde{\bf s})=0$, for
$\widetilde{\bf s}\notin U_{1}$.

  2. 2.

Denote $\widetilde{\alpha}_{U_{2}}(\rm a)$, the initial law at the $\rm a$-th
interval given that the state of the system at the end of the previous
inspection time is critical, i.e $\\{\widetilde{X}(\rm a-1,\tau)\in U_{2}\\}$.

If at the time of inspection the system is in a critical state, we look at the
modules and act upon the modules to restore the system to $U_{1}$. In this
sense, we can describe the probability distribution after the maintenance
action in terms of the modules, more specifically, let $\widetilde{X}_{i}({\rm
a},\tau)$ be the state of module $i$ at the ${\rm a}$-th inspection, then,

$\widetilde{\alpha_{i}}({\rm a})=\widetilde{\alpha_{i}}({\rm a}-1)exp({\bf
Q}_{i}^{*s}\tau){\bf M}_{i},i=1,\cdots,K.$

being ${\bf Q}_{i}^{*s}$ the generator of module $i$, with $i=1,\cdots K$.

Considering the initial probability distribution for each module at the ${\rm
a}$-th inspection we obtain the initial probability distribution of the system
as follows

$\widetilde{\alpha}_{U_{2}}({\rm a})=\bigl{(}\widetilde{\alpha}_{U_{11}}({\rm
a})\otimes\widetilde{\alpha}_{U_{21}}({\rm
a})\otimes\ldots\otimes\widetilde{\alpha}_{U_{K1}}({\rm
a}),0,\cdots,0\bigr{)}.$

given that vector $\widetilde{\alpha_{i}}({\rm a})$ can be partitioned as
$(\widetilde{\alpha}_{U_{i1}}({\rm a})\ \vrule\
\widetilde{\alpha}_{U_{i2}}({\rm a})\ \vrule\ \widetilde{\alpha}_{D_{i}}({\rm
a}))$. Notice that also in this case we have that $\widetilde{\alpha}_{2}(\rm
a,\widetilde{\bf s})=0$, for $\widetilde{\bf s}\notin U_{1}$.

  3. 3.

Finally, denote $\widetilde{\alpha}_{D}(\rm a)$, the initial law at the $\rm
a$-th interval given that the system has failed at the end of the previous
inspection time, i.e. $\\{\widetilde{X}(\rm a-1,\tau)\in D\\}$. In this case,
the system it is replaced following maintenance action vector $\beta$, then
$\widetilde{\alpha}_{3}(\rm a)=\beta$.

So, the initial probability distribution for the system at inspection time
$\tau_{\rm a}={\rm a}\tau$ taking into account these situations is given by

$\displaystyle\widetilde{\alpha}({\rm a})=\widetilde{\alpha}_{U_{1}}({\rm
a})\mathbb{P}\/(\widetilde{X}({\rm a-1},\tau)\in U_{1})$ $\displaystyle\hskip
35.56593pt+\widetilde{\alpha}_{U_{2}}({\rm a})\mathbb{P}\/(\widetilde{X}({\rm
a-1},\tau)\in U_{2})$ $\displaystyle\hskip
35.56593pt+\widetilde{\alpha}_{D}(\rm a)\mathbb{P}\/(\widetilde{X}({\rm
a-1},\tau)\in D).$ (4)

Figure 2: Maintenance scheme for the system at inspection time

##  5 Maintenance cost

Every maintenance action has an associated cost that comprises the inspection
cost, the cost associated to the performed maintenance action, the set-up
cost, etc. Additionally, the maintenance action cost depends on the performed
maintenance action and this in turn depends on the state of the system at the
inspection time $\tau_{{\rm a}}$.

### Case 1: The system is in an optimal state

If the system is in an optimal state, no maintenance action takes place,
therefore, we just have an inspection cost given by $C_{I}$.

### Case 2: The system is in a critical state

In this situation we define matrix ${\bf C}_{i}$ the cost matrix associated to
the maintenance actions performed to each module at the ${\rm a}$-th
inspection. This matrix is described by blocks in a similar way to matrix
${\bf M}_{i}$:

  * •

If the module is in an optimal state, then no intervention takes place and we
just have the inspection cost. Therefore $C_{U_{i1},U_{i1}}=C_{I}I$, where $I$
is an identity matrix with the same dimensions as $U_{i1}$.

  * •

If the module is in a critical state, all the failed units are maintained, the
rest remain unchanged. We have an associated cost for each of the failed
units, that depends on the unit state and also on the performed maintenance
action.

Within block $C_{U_{i2},U_{i1}}$ we have different combinations of units in
operational and failed state. For a specific combination we have a cost
associated to the maintenance action performed on the failed units, i.e. the
total cost to restore the module to an optimal state would be given by the sum
of the costs associated to each of the failed units.

In practice, to build this block in matrix ${\bf C}_{i}$, we consider a cost
vector that includes the cost associated to restore each unit to one of the
optimal phases plus the inspection cost. Then, for each specific combination
of failed and operational units we multiply the cost vector for a vector that
contains 1 if the unit has failed and the transition to the optimal phase is
possible, and 0 otherwise. For example, let us consider a module with 3 units
set in parallel, the first unit is of Phase-type with 2 phases (0, 1 and F
being the absorption phase), and the second and third units are of Phase-type
with one phase (0 and F being the absorption phase). Let us consider the
situation where the first and third unit have failed, we want to calculate the
cost of maintaining this module in a critical state. We have two
possibilities:

    1. 1.

We could go from state (F,0,F) to state (0,0,0), in this case the cost would
be given by
$c_{F0}^{1}+c_{F0}^{3}=(1,0,0,1)(c_{F0}^{1},c_{F1}^{1},c_{F0}^{2},c_{F0}^{3})^{t}$
where $c_{F0}^{1}$ is the cost associated to restoring unit 1 from failure to
phase 0 plus the inspection cost.

    2. 2.

We could go from state (F,0,F) to state (1,0,0), in this situation the cost
would be given by
$c_{F1}^{1}+c_{F0}^{3}=(0,1,0,1)(c_{F0}^{1},c_{F1}^{1},c_{F0}^{2},c_{F0}^{3})^{t}$
where $c_{F1}^{1}$ is the cost associated to restoring unit 1 from failure to
phase 1 in addition to the inspection cost.

  * •

If the module has failed then it is replaced with a cost replacement vector
$C_{RM,i}$. This vector contains the cost of re-initializing the module to any
of its optimal phases after the maintenance intervention plus the inspection
cost. See the application in Section 6.

### Example. 2-units parallel module

To illustrate matrix ${\bf C}_{i}$ we consider a previous example of a module
formed by 2 units set in parallel, being both units Phase-type distributed
with two phases. For this particular example, the cost associated to
maintaining the units is given by the cost vector
$c_{i}=(c_{F0}^{1},c_{F1}^{1},c_{F0}^{2},c_{F1}^{2})^{t}$. We have that matrix
${\bf C}_{i}$ is given by

${\bf C}_{i}=$

$\bordermatrix{&U_{i1}&&U_{i2}&&D_{i}\cr
U_{i1}&C_{I}I&\vrule&0&\vrule&0\cr\hline\cr
U_{i2}&\begin{matrix}(0,0,1,0)c_{i}&(0,0,0,1)c_{i}&0&0\cr
0&0&(0,0,1,0)c_{i}&(0,0,0,1)c_{i}\cr(1,0,0,0)c_{i}&0&(0,1,0,0)c_{i}&0\cr
0&(1,0,0,0)c_{i}&0&(0,1,0,0)c_{i}\cr\end{matrix}&\vrule&0&\vrule&0\cr\hline\cr
D_{i}&\begin{matrix}C_{RM,i}\end{matrix}&\vrule&0&\vrule&0}$

equivalently,

$\bordermatrix{&U_{i1}&&U_{i2}&&D_{i}\cr
U_{i1}&C_{I}I&\vrule&0&\vrule&0\cr\hline\cr
U_{i2}&\begin{matrix}c_{F0}^{2}&c_{F1}^{2}&0&0\cr 0&0&c_{F0}^{2}&c_{F1}^{2}\cr
c_{F0}^{1}&0&c_{F1}^{1}&0\cr
0&c_{F0}^{1}&0&c_{F1}^{1}\cr\end{matrix}&\vrule&0&\vrule&0\cr\hline\cr
D_{i}&\begin{matrix}C_{RM,i}\end{matrix}&\vrule&0&\vrule&0}$

### Case 3: The system has failed

Finally, if the system is in a failed state and needs to be replaced, the cost
associated is given by the inspection cost $C_{I}$, the replacement cost
$C_{SR}$ and the loss of production cost due to the the system being down
during certain interval of time. For the latter we calculate the expected loss
of production cost, or down cost. Let us define $c_{down}({\rm a},t)$ a
function that provides the cost due to the system downtime at the ${\rm a}$-th
interval $({\rm a}\tau,({\rm a}+1)\tau]$. If the system has failed at time
${\rm a}\tau+t$, where $0<t<\tau$, then the expected down cost in the interval
$(0,\tau]$ is given by:

$EC_{down}({\rm a},\tau)=\int_{0}^{\tau}f_{s}({\rm a},t)c_{down}({\rm
a},t){\rm d}t.$ (5)

where $f_{s}({\rm a},t)=\widetilde{\alpha}({\rm
a})exp(Q_{sys}t)(-Q_{sys}\textbf{e})$ is the density function of the lifetime
of the system.

As a particular case we can consider $c_{down}({\rm a},t)$ a linear function
of the non-operational time since the failure of the system until $\tau$, then
$c_{down}({\rm a},t)=(\tau-t)C_{down}$, where $C_{down}$ is a constant.

Considering all the possibilities, we obtain the expected maintenance cost at
the $\rm a$-th inspection time. Firstly, the cost incurred due to the
maintenance of module $i$, if the system is in a critical state at the ${\rm
a}$-th time, is determined as follows

$Cost({\rm a},\tau,i)=\widetilde{\alpha}_{i}({\rm a}-1)exp({\bf
Q}_{i}^{*s}\tau){\bf D_{i,U_{2}}}({\bf M}_{i}\odot{\bf C}_{i})e,i=1,\cdots,K.$
(6)

where $\odot$ represents the Hadamard product or element wise product.
Besides, for the system to be in a critical state, only certain states of the
module $i$ are allowed for the expression (6). More specifically, for
$i=1,\ldots,K$, let $(\widetilde{\bf s};\widetilde{s}_{i})$ denote a state of
the system in which the $i$-th module is in state $\widetilde{s}_{i}$,
regardless the rest of the modules of the system. We denote ${\bf
D_{i,U_{2}}}$ a diagonal matrix of appropriate dimension, which has 1 in the
diagonal only in those positions corresponding to states $\widetilde{s}_{i}$
for which there exists a combination of states of the rest of modules such
that $(\widetilde{\bf s};\widetilde{s}_{i})\in U_{2}$. That is, to calculate
the cost incurred by the maintenance action of module $i$ in this case, we
only consider when module $i$ is in one of those states that lead to a
critical state of the system. Therefore, we need to consider the combination
of the modules states to determine the system state.

Then, the expected maintenance cost for the system is given, at inspection
${\rm a}$-th, by

$\displaystyle EC_{sys}({\rm a},\tau)$ $\displaystyle=$ $\displaystyle
C_{I}\mathbb{P}\/(\widetilde{X}({\rm a}-1,\tau)\in U_{1})$
$\displaystyle+\left(\sum_{i=1}^{K}Cost({\rm
a},\tau,i)\right)\mathbb{P}\/(\widetilde{X}({\rm a}-1,\tau)\in U_{2})$
$\displaystyle+\left(C_{I}+EC_{down}({\rm
a}-1,\tau)+C_{SR}\right)\mathbb{P}\/(\widetilde{X}({\rm a}-1,\tau)\in D).$

Finally, assuming a maximum number of $A$ inspections to perform along the
useful life of the system, the total maintenance cost for that period is given
by

$EC_{sys}(\tau)=\sum_{a=1}^{A}EC_{sys}({\rm a},\tau).$

Thus, an optimization problem can be formulated as a function of the incurred
cost and the frequency of inspection $\tau$. We will address this problem
numerically in the next section when the model is illustrated by means of a
real case study.

##  6 Numerical application

Our numerical example is inspired by the case study presented in Liu, Liu and
Cai (2014) (Liu et al. (2014)). The authors carry out a reliability analysis
of the electrical control system of a subsea Blowout Preventer (BOP) stack
based on Markov methodology. Specifically, two voting schemes on system
performance are evaluated based on Markov modeling. The electrical control
system of subsea BOP system is a redundant system. As it is described in Liu
et al. (2014), two control pods are above the BOP stack, affording redundant
control of subsea functions. Each pod contains a Subsea Electrical Module
(SEM), which is contained in domed containers made of thick steel under
external water pressure.

The SEMs contain processor modules, input modules, and output modules. Other
secondary components of the system are considered completely reliable. Three
processors are used to form a processor subsystem (module). Each processor
runs the same application programs, processing data and sending command
signals. The processor subsystem performs 2-out-of-3 voting, which means the
system can hence detect and correct a single failure. For the sake of
simplicity, in this paper we assume that the input and output modules also
performs 2-out-of-3 voting. In the surface, 2 control panels are installed
working in parallel. If control panels fail, they will be repaired immediately
without removing the BOP system from the water. The subsea components namely
input modules, output modules and processors will be pulled to the surface and
repaired when the control system cannot work due to their failure.

We consider the same values of transition rates of Liu et al. (2014) for
processors, input and output modules. That is, the lifetime of each of the 3
processors in the corresponding module has Exponential distribution with rate
$\lambda_{p}$ as specified in Table 3. We treat the input and output modules
similarly. However, for the control panels, we consider a more general case.
Specifically we assume that each follows a Phase-type distribution with the
parametrization given in Table 3. In this case we have chosen the parameters
of the Phase-type distribution with the same mean time as the Exponential
distribution considered in Liu et al. (2014) for the control panels.

Module | Distribution | Parameters
---|---|---
Processor | Exponential | $\lambda_{p}=1.820$
Input | Exponential | $\lambda_{in}=.9798$
Output | Exponential | $\lambda_{out}=.9780$
Control Panel | Phase type | ${\bf Q_{c}}=\left(\begin{array}[]{cc}-6.304&6.304\\\ 0&-6.304\end{array}\right);\ \alpha_{c}=(1,0)$
Table 3: Lifetime distribution of the units per module. Time is scaled by 1e-5
time units.

###  6.1 Description of the modules

Following Liu et al. (2014) the SEM system consists of the 4 modules described
above (control panel, processors, input and output) arranged in series. Figure
3 shows the corresponding reliability block diagram.

Figure 3: Reliability Block Diagram of the SEM system

Let $X(t)$ denote the state of the system (SEM) at time $t$. We then have that
$\\{X(t),t>0\\}$ is a continuous-time Markov chain and we can use the
algorithm MoMA described in Gamiz et al. (2022) to derive the system
generating matrix. We first analyze each module separately.

In this system we have two types of modules, for each module we obtain its
generator. In generator ${\bf Q}_{i}$ we only provide the transitions between
optimal and critical states, given that the transitions to the down state can
be deduced from them.

  * •

$M_{1}$: Control panel. This module is a two-units parallel structure. Each
unit has Phase-type distribution lifetime with parameters $(\alpha_{c},{\bf
Q}_{c})$ given in Table 3. The state of $M_{1}$ along time is described by a
Markov process $\\{X_{1}(t),t>0\\}$. At any time $t>0$ each unit of the module
can be in one of three states: 0, if the unit is in optimal state; 1, if the
unit is operative but critical; and, $F$, if the unit is in failure. Then
after applying MoMA we get the set of all possible states of the module, that
is, $E_{1}=\\{(0,0),(0,1),(1,0),(1,1),(0,F),(1,F),(F,0),(F,1),(F,F)\\}$. Given
the parallel structure of this module we can split the state space into a
subset of optimal states $U_{11}=\\{(0,0),(0,1),(1,0),(1,1)\\}$; a subset of
critical states $U_{12}=\\{(0,F),(1,F),(F,0),(F,1)\\}$, and a unitary subset
with the failure state $D_{1}=\\{(F,F)\\}$. The lifetime of the module has
Phase-type distribution with parameters given as

$\displaystyle{\bf Q}_{1}=$
$\displaystyle\left(\begin{array}[]{rrrrrrrr}-12.608&6.304&6.304&0.000&0.000&0.000&0.000&0.000\\\
0.000&-12.608&0.000&6.304&6.304&0.000&0.000&0.000\\\
0.000&0.000&-12.608&6.304&0.000&0.000&6.304&0.000\\\
0.000&0.000&0.000&-12.608&0.000&6.304&0.000&6.304\\\
0.000&0.000&0.000&0.000&-6.304&6.304&0.000&0.000\\\
0.000&0.000&0.000&0.000&0.000&-6.304&0.000&0.000\\\
0.000&0.000&0.000&0.000&0.000&0.000&-6.304&6.304\\\
0.000&0.000&0.000&0.000&0.000&0.000&0.000&-6.304\end{array}\right),$

and initial vector $\alpha_{1}=(1,0,0,0,0,0,0,0)$.

  * •

$M_{2}$: Processor. This module is a 2-out-of-3 structure. It is composed of 3
units that are independent and identically distributed (i.i.d.) with
Exponential lifetime with rate $\lambda_{p}=1.820$. In this case the state
space is simpler than $M_{1}$, that is, only two states are considered for
each unit: 0, if the unit is operative, and $F$ if the unit is in failure. Let
$\\{X_{2}(t),t>0\\}$ be the Markov chain describing at any time the module
state. The set of states can be partitioned into $E_{2}=U_{21}\cup U_{22}\cup
D_{2}$, where $U_{21}=\\{(0,0,0)\\}$ is the optimal state;
$U_{22}=\\{(0,0,F),(0,F,0),(F,0,0)\\}$ are the critical states; and,
$D_{2}=\\{(0,F,F),(F,0,F),(F,F,0),(F,F,F)\\}$ are the failure states. We
remind that we do not consider transitions between the failure states so we
actually treat the subset $D_{2}$ as a unitary set. Thus, the model considered
has only 5 states in total. Using the MoMA algorithm, we obtain the module
lifetime is Phase-type with parameters given by

${\bf Q}_{2}=\left(\begin{array}[]{rrrr}-5.46&1.82&1.82&1.82\\\
0.00&-3.64&0.00&0.00\\\ 0.00&0.00&-3.64&0.00\\\ 0.00&0.00&0.00&-3.64\\\
\end{array}\right),$

and the initial vector is given by $\alpha_{2}=(1,0,0,0)$.

  * •

$M_{3}$: Input. This module is a 2-out-of-3 structure. The units have i.i.d.
exponential lifetimes with rate $\lambda_{in}=.9798$. Let $\\{X_{3}(t),t>0\\}$
be the Markov chain describing the state of the module. The set of states of
the module is equal to $M_{2}$ and we can consider a similar partition. That
is, $E_{3}=U_{31}\cup U_{32}\cup D_{3}$, with $U_{31}$ the optimal state;
$U_{32}$, the critical states, and $D_{3}$ the failure state of the module.
The lifetime of $M_{3}$ has Phase-type distribution with parameters

${\bf Q}_{3}=\left(\begin{array}[]{rrrr}-2.939&0.980&0.980&0.980\\\
0.000&-1.960&0.000&0.000\\\ 0.000&0.000&-1.960&0.000\\\
0.000&0.000&0.000&-1.960\\\ \end{array}\right),$

and $\alpha_{3}=(1,0,0,0,0)$.

  * •

$M_{4}$: Output. This module is a 2-out-of-3 structure. The units have i.i.d.
Exponential lifetimes with rate $\lambda_{out}=.9780$. Let
$\\{X_{4}(t),t>0\\}$ be the Markov chain describing the state of the module.
The set of states of the module is equal to $M_{2}$ and we can consider a
similar partition. That is, $E_{4}=U_{41}\cup U_{42}\cup D_{4}$, with $U_{41}$
the optimal state; $U_{42}$, the critical states, and $D_{4}$ the failure
state of the module. The lifetime of $M_{4}$ has Phase-type distribution with
parameters

${\bf Q}_{4}=\left(\begin{array}[]{rrrr}-2.934&0.978&0.978&0.978\\\
0.000&-1.956&0.000&0.000\\\ 0.000&0.000&-1.956&0.000\\\
0.000&0.000&0.000&-1.956\\\ \end{array}\right),$

and the initial vector is given by $\alpha_{4}=(1,0,0,0)$.

###  6.2 System model without maintenance

Let $\\{X(t);t>0\\}$ be the Markov chain describing the system behaviour
through time. A graphical description of the system is given in Figure 3. The
MoMA algorithm has been run again to assemble $M_{1}$, $M_{2}$, $M_{3}$ and
$M_{4}$ in a series structure to compose the system. We obtain the state of
the system as a function of the states of the modules, i.e.
$X(t)=\Phi(X_{1}(t),X_{2}(t),X_{3}(t),X_{4}(t))$, with $\Phi$ defined in terms
of the series structure of the system.

We denote the state space $E=U\cup D$, partitioned in two subsets, which are
up states and down states. After running MoMA, we obtain 512 operative states
as the size of set $U$, and 133 down states. Similar to the module cases, we
do not take into account transitions between the down states, thus we consider
the set of down states $D$ as a unitary set. The system has Phase-type
distribution with parameters $({\alpha}_{sys},{\mathbb{Q}}_{sys})$, which we
do not specifically write here due to space limitations. Moreover, we assume
that at time $t=0$ all the units in the system are in optimal conditions so,
all components of vector ${\alpha}_{sys}$ are equal to 0 except the first one,
which is equal to 1.

Similar to the modules we distinguish optimal states and critical states in
the system. That is $U=U_{1}\cup U_{2}$, where in particular
$U_{1}=\\{(0.0:0.0.0:0.0.0:0.0.0),(0.1:0.0.0:0.0.0:0.0.0),(1.0:0.0.0:0.0.0:0.0.0),(1.1:0.0.0:0.0.0:0.0.0)\\}$
and the rest of operative states are critical. We denote a generic state
$\widetilde{i}$ of the system as

$\widetilde{s}=(s_{11},s_{12};s_{21},s_{22},s_{23};s_{31},s_{32},s_{33};s_{41},s_{42},s_{43})$

where $s_{ji}$ denotes the state of the $j$-th unit in module $i$ being
$i=1,2,3,4$ and $j=1,2,3$.

The mean lifetime of the system is $\mu=24000\ hours$. The reliability
function is represented in Figure 4.

Figure 4: Reliability function of SEM system

###  6.3 Maintenance

Maintenance policy is explained in Section 4. The system is assumed to be
inspected regularly at times $\tau_{\rm a}={\rm a}\tau$, with ${\rm
a}=0,1,\ldots,A$ and $\tau>0$. For ${\rm a}=0$ that is $\tau_{0}=0$ we assume
that the system is operating in optimal conditions.

To describe the state of the system (with maintenance) we introduced the
stochastic process $\\{\widetilde{X}({\rm a},t)\\}$ in Section 4 , where
$\widetilde{X}(\rm a,t)$ gives the state of the system at time $t+{\rm
a}\tau$, with $0\leq t\leq\tau$. Let us call $({\rm a}\tau,({\rm a}+1)\tau]$
the ${\rm a}$-th operating cycle of the system ($\rm a$-th interval).

Focusing on the first inspection period, the concrete intervention carried out
depends on the state of the system at time $\tau$, and in any case, consists
of restoring the system to a desired performance level. The new state for the
system is chosen according to a pre-specified maintenance rule (see Section
4). After maintenance a new operating cycle starts, and the state of the
system is described by $\widetilde{X}(1,t)$, with $0<t\leq\tau$. This policy
is used at the end of every $\tau$-cycle. Each maintenance action involves a
particular cost that we specify in Table 4 for our particular SEM system.

| Value | Description
---|---|---
$\boldsymbol{\beta}_{j,i}$ | $(0.5,0.5)^{t}$ | Vector of probabilities of restoring
|  | failed unit $j$ of module $i$ to operational phase
$C_{I}$ | 1 | Cost of inspection (${\tt mu}$)
$c_{F0}$ | 1 | Cost of restoring a unit to its optimal
|  | operational phase (${\tt mu}$)
$C_{F1}$ | 0.5 | Cost of restoring a unit to a
|  | non-optimal operational phase (${\tt mu}$)
$C_{RM}$ | $3C_{F0}$ | Cost of replacement of a module (${\tt mu}$)
$C_{SR}$ | $3C_{RM}$ | Cost of replacement of the system (${\tt mu}$)
| Sc. 1: 1e-3 |
$C_{down}$ | Sc. 2: 1e-2 | Cost of non productivity
| Sc. 3: 1e-1 | per unit time $({\tt mu}/{\tt h})$
| Sc. 4: 1 |
Table 4: Specification of values for parameters of the model for the simulated
SEM system, considering several scenarii, given by 4 different values of
parameter $C_{down}$. Note: ${\tt mu}$=monetary unit.

In this particular case we assume that the replacement costs of the different
modules are the same, as well as the costs associated to restoring each unit
to an operational phase. We consider 5 different cases for the non-operational
cost, $C_{down}$. See Section 4.

The initial state in a new operating cycle, i.e. $\widetilde{X}(1,0)$ is
selected according to a probability vector $\widetilde{\alpha}(1)$ which is
defined according to equation (4.3). We provide here a reminder.

### Case 1. The system is in optimal conditions

If $\widetilde{X}(0,\tau)\in U_{1}$, the system is not maintained, and then

$\mathbb{P}\/(\widetilde{X}(1,0)=\widetilde{s})=\mathbb{P}\/(\widetilde{X}(0,\tau))=\widetilde{s})=\mathbb{P}\/(X(\tau)=\widetilde{s}).$

where $\widetilde{s}$ has been defined above. That is, the initial vector for
the second operating cycle $\widetilde{\alpha}(1)$ is given by the occupation
probability vector at time $\tau$. In this case the only cost involved is the
inspection cost $(C_{I})$.

### Case 2. The system is in failure

If $\widetilde{X}(0,\tau)\in D$, the full system is replaced. Then
$\widetilde{\alpha}(1)=\beta$. In this case the involved cost is, on the one
hand $C_{SR}$ plus the expected cost due to non-productivity cost given in
equation (5).

### Case 3. The system is critical

If $\widetilde{X}(0,\tau)\in U_{2}$, the system is maintained. As explained in
Section 4, every module of the system is inspected separately; and depending
on the state of the module an action is undertaken, as explained below. For
$i=1,2,3,4$, let us denote $\\{\widetilde{X}_{i}({\rm a},t);{\rm
a}=0,1,\cdots,A;0<t\leq\tau\\}$ the process that represents the state of
module $M_{i}$ when maintenance is considered. In this case to determine the
initial probability law we consider the state selected for each module after
being maintained, $\widetilde{\alpha}_{i}(1)$, for $i=1,2,3,4$ and that will
be the starting point of the module for the second $\tau$-period. The initial
vector of probabilities at the system level is then

$\widetilde{\alpha}(1)=\widetilde{\alpha}_{1}(1)\otimes\widetilde{\alpha}_{2}(1)\otimes\widetilde{\alpha}_{3}(1)\otimes\widetilde{\alpha}_{4}(1).$

Note that we only focus here on the transitions to the optimal set of states
$U_{1}$ given that the rest of the transitions are not possible.

  * •

Module $M_{1}$. This is a 2-unit parallel module. The units are Phase-type
with 2 operative phases.

    * –

The module state is critical if one unit has failed $(U_{12})$. This unit is
then restored to one of the two operative phases according to a vector of
probabilities $\beta_{j,1}$, $j=1,2$. The operative unit remains unchanged.

    * –

The module has failed if both units have failed. In this case, the module is
restored to the optimal state class $U_{11}$ according to the vector of
probabilities $\beta_{1,1}\otimes\beta_{2,1}$.

In the particular case of the example we take $\beta_{1}=\beta_{2}=(0.5,0.5)$,
then the maintenance action on this module is ruled by matrix ${\bf M}_{1}$
given in (7).

Each maintenance action involves a particular cost determined by the
corresponding transition of the matrix ${\bf M}_{1}$. If the module state is
such that $\widetilde{X}_{1}(0,\tau)\in U_{11}$, the only cost is due to
inspection. If $\widetilde{X}_{1}(0,\tau)\in U_{12}$, then the failed units
are maintained and we have to consider the cost of restoring these units to an
operational state in addition the inspection cost. If
$\widetilde{X}_{1}(0,\tau)\in D_{1}$, then the module is restored to $U_{11}$,
the incurred cost here is the inspection cost plus the renewal cost which
depends on the state of $U_{11}$ that is eventually chosen.

The matrices of maintenance action ${\bf M}_{1}$ and incurred cost ${\bf
C}_{1}$ are obtained as

${\bf M}_{1}=\left(\begin{array}[]{cccc}1.00&0.00&0.00&0.00\\\
0.00&1.00&0.00&0.00\\\ 0.00&0.00&1.00&0.00\\\ 0.00&0.00&0.00&1.00\\\
0.50&0.50&0.00&0.00\\\ 0.00&0.00&0.50&0.50\\\ 0.50&0.00&0.50&0.00\\\
0.00&0.50&0.00&0.50\\\ 0.25&0.25&0.25&0.25\\\ \end{array}\right),\hskip
8.5359pt{\bf C}_{1}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 3&4.5&0&0\\\ 0&0&3&2.5\\\ 3&0&2.5&0\\\ 0&3&0&2.5\\\
3&2.5&2.5&2\\\ \end{array}\right)$ (7)

In addition to the costs due to unit repair, unit replacement or module
replacement (whichever intervention has been carried out) the inspection cost
has to be taken into account.

  * •

Modules $M_{2}$, $M_{3}$, $M_{4}$. The rest of modules in the SEM are similar,
each consisting of 2-out-of-3 of exponentially distributed units, in all cases
the optimal set of states is a unitary set where all three units in the module
are functional. In all cases, the maintenance matrix reduces to a column
vector of 1; and the corresponding cost matrix also reduces to a column vector
such that ${\bf
c}_{i}=(C_{I},C_{I}+c_{F0}^{1},C_{I}+c_{F0}^{2},C_{I}+c_{F0}^{3},C_{I}+C_{RM,i})^{t}$.
Considering the values in Table 4, we have ${\bf c}_{i}=(1,2,2,2,4)^{t}$ for
all $i=2,3,4$.

At the end of the first cycle, the cost incurred due to maintenance of ${\bf
M}_{i}$ when the system is critical is given by equation (6), i.e.

$Cost(1,\tau,i)=\widetilde{\alpha}_{i}(0)exp({\bf Q}_{i}^{*s}\tau){\bf
D}_{1,U_{2}}({\bf M}_{i}\odot{\bf C}_{i})e,\quad i=1,2,3,4;$

and this expression is used to calculate the total cost incurred in the system
maintenance action at time $\tau$ as it is expressed in Section 4. Finally,
the initial distribution of the $i$-th module for the next period is given by
equation 2

$\widetilde{\alpha_{i}}(1)=\widetilde{\alpha_{i}}(0)exp({\bf
Q}_{i}^{*s}\tau){\bf M}_{i},\quad i=1,2,3,4.$

Let us assume that the useful lifetime of the system (with maintenance) is
denoted by $\mathbb{T}$, that is $\mathbb{T}=A\tau$, where $A$ is the maximum
number of maintenance inspections carried out. The goal is to determine the
optimal length of the operative cycles, i.e. $\tau$, in the sense of
minimizing the total cost incurred by all maintenance actions implemented at
the end of each $\tau$-cycle during the whole period $(0,\mathbb{T}]$.

To select the optimal value of $\tau$, and consequently of $A$, given that
duration of useful lifetime of the system is prespecified, we consider the
following Montecarlo simulation algorithm.

Algorithm

  1. 1.

Define an equispaced grid of $M$ values $\\{\tau_{1},\ldots,\tau_{M}\\}$ with
$\tau_{M}=\mu)$; and set $m=0$;

  2. 2.

Put $m=m+1$ and:

    1. 2.1.

Define $Cost^{(m)}$, an empty vectors of size $R$. Set $r=0$;

    2. 2.2.

Put $r=r+1$ and randomly simulate one path of the process in the interval
$(0,\tau_{m})$, that is
$\\{(\widetilde{i}_{1},\widetilde{t}_{1}),\ldots,(\widetilde{i}_{n(\tau_{m})},\widetilde{t}_{n(\tau_{m})})\\}$,
where $\widetilde{i}_{k}$ are the successive visited states and
$\widetilde{t}_{k}$ are the corresponding simulated jump times, so that
$\widetilde{t}_{n(\tau_{m})}\leq\tau_{m}$;

    3. 2.3.

Given $\widetilde{i}_{n(\tau_{m})}$, consider the following cases:

    4. (a)

If $\widetilde{i}_{n(\tau_{m})}\in U_{1}$ then $Cost^{(m)}_{r}=C_{I}$;

    5. (b)

If $\widetilde{i}_{n(\tau_{m})}\in U_{2}$ then
$Cost^{(m)}_{r}=C_{I}+C_{M_{1}}+C_{M_{2}}+C_{M_{3}}+C_{M_{4}}$;

    6. (c)

If $\widetilde{i}_{n(\tau_{m})}\in D$ then
$Cost^{(m)}_{r}=C_{I}+\left(\tau_{m}-\widetilde{t}_{n(\tau_{m})}\right)C_{Down}+C_{SR}$.

    7. 2.4.

Go to step 2.2 until $r=R$

  3. 3.

Define $Av\\_cost^{(m)}=(1/R)\sum_{r=1}^{R}Cost^{(m)}_{r}$;

  4. 4.

Define $A^{(m)}=\mu/\tau_{m}$;

  5. 5.

Go to step 2 until $m=M$;

  6. 6.

Select $m_{0}=\underset{m}{\arg\min}\\{Av\\_cost^{(m)}A^{(m)}\\}$, and fix
$\tau=\tau_{m_{0}}$.

Figure 5 displays the results with the specifications of Table 4. The optimal
values are given in Table 5. Left panel of Figure 6 displays the trend of
total cost increase as the cost per unit time for non productivity of the
system increases. Right panel of Figure 6 displays the curve of optimal
inspections times that is obtained for each scenario.

The computational time of this experiment is not an issue of significant
concern even when a large number of iterations are considered; that is for a
large value for $R$. For instance, a simulation with the specifications of
Table 4 for the first scenario (i.e. $c_{down}=0.001$, taking $R=50000$ and
$M=100$) using a 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80GHz 2.80 GHz takes
approximately 11.66 min.

Figure 5: Maintenance strategy with 4 different specifications for $C_{down}$. Scenario | 1 | 2 | 3 | 4
---|---|---|---|---
$C_{down}$ | 0.0010 | 0.0100 | 0.1000 | 1.0000
$\tau$ (hours) | 8300 | 4390 | 2200 | 980
$Cost\ ({\tt mu})$ | 17.4791 | 24.4786 | 42.0189 | 87.0051
$A$ | 6 | 11 | 23 | 51
Table 5: Optimal values considering 4 different scenarii. Note: ${\tt
mu}$=monetary unit.

Figure 6: Left panel: Total incurred cost of maintenance ($Cost$) for each
value of non-operation cost. Right panel: Optimal time of inspection ($\tau$)
for each value of non-operation cost.

##  7 Conclusions

In the present paper we propose a maintenance policy for complex modular
systems, where the modules in the system consist of a number of units. We
assume that the system might fail due to the wear-out of its units or due to
external shocks that represent changes in its environment. Additionally, the
units lifetimes follow PH-distributions, where the phases represent different
levels of degradation in the unit. The modeling of this system is based on MAM
and it was addressed in a previous paper.

Here we focus on the modeling of a maintenance policy for such a system, as
well as the optimization of the cost associated to this policy, through the
lifetime of the system. The proposed maintenance policy is a hierarchical
decision-based policy where the system is inspected at regular intervals of
time. At the time of inspection the system could be in an optimal, critical or
failed state. If it is in an optimal state no maintenance action takes place.
If it is in a failed state, the system is replaced. Finally, if it is in a
critical state its modules are inspected. We check the individual modules for
an optimal, critical or failed state. In this case, the same policy as for the
system is applied, i.e. the module will be replaced if it is in a failed
state. Alternatively, if it is in an optimal state no intervention will take
place. If it is in a critical state, the state of units will be checked and
the failed units will be replaced.

For the proposed maintenance policy, we have defined maintenance matrices
associated to the modules that describe the transition probabilities to an
optimal state at inspection times, considering possible maintenance actions.
Related to these matrices we have defined other matrices that provide the cost
associated to the maintenance actions (described in the maintenance matrices).
Additionally, we depict the stochastic process that describes the evolution of
the system in time, considering performed maintenance actions. We have also
provided the expressions to determine the initial probability vector, after
every inspection, as well as the expected maintenance cost. From this, we can
optimize the cost in terms of the frequency of inspections.

The main contribution of this paper is the use of MAM to design a maintenance
policy for complex modular systems. We have obtained closed-form expressions
related to the proposed maintenance policy. The methodology here is applicable
to other complex systems, based on layered composition/decomposition method.
Finally, we have demonstrated the use of the methodology on a real use-case.
Specifically we have considered the ECU system presented in Liu et al. (2014),
where no maintenance analysis is carried out. This aspect is new in our study
of this system. As we do not have true knowledge about the parameters involved
in the cost analysis (i.e. $C_{I}$, $C_{RM}$, etc), we have chosen all the
values of Table 4 which represent the different types of cost involved in our
maintenance policy, arbitrarily and actually with no fully real meaning. Had
we had expert guidance in this respect we could have built a meaningful
maintenance-cost model, more interpretable from a practical decision-making
sense in real life. Nevertheless, after running some simulations which are not
shown in the paper, we have noticed that the most important index in our
maintenance plan is the cost of non productivity per unit time, that is
$C_{down}$. In other words, the parameter with respect to which the system is
most sensitive is $C_{down}$, and then we have considered different scenarii
according to the particular value of this parameter.

As future work, we want to incorporate dependency between the modules that are
part of the system, presently our model considers that the modules fail
independently. Additionally, we currently do not distinguish failure modes for
the modules nor for the system. Therefore we would also consider different
failure modes and related maintenance actions. Finally, we aim to use
empirical data to demonstrate the proposed methodology.

## Acknowledgements

This work is jointly supported by the Spanish Ministry of Science and
Innovation—State Research Agency through grants PID2020-120217RB-I00 and
PID2021-123737NB-I00.

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