# Dedicated maintenance and repair shop control for spare parts networks

Chaaben Kouki  Essca School of Management, Department of Operations Management
and Decision Sciences, Angers, France<EMAIL_ADDRESS>Melvin Drent
<EMAIL_ADDRESS>Collin Drent<EMAIL_ADDRESS>M. Zied Babai  mohamed-
<EMAIL_ADDRESS>Eindhoven University of Technology, Department of
Industrial Engineering and Innovation Sciences, Eindhoven, the Netherlands
Kedge Business School, Bordeaux, France

###### Abstract

We study a repairable inventory system dedicated to a single component that is
critical in operating a capital good. The system consists of a stock point
containing spare components, and a dedicated repair shop responsible for
repairing damaged components. Components are replaced using an age-replacement
strategy, which sends components to the repair shop either preventively if it
reaches the age-threshold, and correctively otherwise. Damaged components are
replaced by new ones if there are spare components available, otherwise the
capital good is inoperable. If there is free capacity in the repair shop, then
the repair of the damaged component immediately starts, otherwise it is
queued. The manager decides on the number of repairables in the system, the
age-threshold, and the capacity of the repair shop. There is an inherent
trade-off: A low (high) age-threshold reduces (increases) the probability of a
corrective replacement but increases (decreases) the demand for repair
capacity, and a high (low) number of repairables in the system leads to higher
(lower) holding costs, but decreases (increases) the probability of downtime.
We first show that the single capital good setting can be modelled as a closed
queuing network with finite population, which we show is equivalent to a
single queue with fixed capacity and state-dependent arrivals. For this queue,
we derive closed-form expressions for the steady-state distribution. We
subsequently use these results to approximate performance measures for the
setting with multiple capital goods.

###### keywords:

maintenance , inventory , repair , queues

##  1 Introduction

Spare parts are becoming ubiquitous in modern societies. They are used in
after-sales services and in maintaining operating systems and capital goods
within most companies, which renders their management an important lever to
improve the service to customers and to reduce costs (Al Hanbali & van der
Heijden 2013, Sleptchenko et al. 2018, Topan et al. 2020). The management of
repairable spare parts inventory systems is challenging since it consists not
only in controlling the inventories used to satisfy demand triggered by
planned and unplanned maintenance activities but also in managing the repair
shops that repair the spare parts often with a limited capacity and replenish
the inventory points (Tiemessen & Van Houtum 2013, Arts et al. 2016, Drent &
Arts 2021). The management of such systems require integrated design and
control decisions. The design decisions often relate to the capacity of the
repair shops (e.g., number of repairmen), whereas the control decisions
consist in scheduling the production within the repair shops and managing the
inventory replenishment in the stock points. Examples of companies that
typically make integrated inventory and repair shop decisions for spare parts
are service companies such as transportation companies (e.g., trains, subways,
and airlines) and military organizations (e.g., frigates and cruisers). The
planning of the overhauls or the age replacement policy of the various parts
is generally fixed when capital assets start servicing and taking into account
the age of the parts and their probability of failure since the last
replacement/failure.

The modeling of repairable spare parts systems is very complex and depends
largely on the typology of the repair shop and the allocation of its capacity
repairmen as well as the availability of the required parts for maintenance
activities. The literature dealing with these systems is mainly built on a
research body using the METRIC (Multi-Echelon Technique for Recoverable Item
Control) model that extends since the 1960s (Sherbrooke 1968, Muckstadt &
Sapra 2010). However, it is important to notice that most of this research has
looked at systems triggered by corrective maintenance activities, and research
papers that consider both corrective and preventive maintenance are lacking,
especially when the repair centers have a limited capacity. Moreover, this
literature typically assumes that failures of capital goods follow an
exponential distribution, i.e., the spare parts inventory consumption is given
by Poisson demand processes. The reality is that the exponential distribution
admits a constant failure rate that does not represent the actual failure rate
of a capital good. The use of the exponential distribution is probably due to
the fact that the failure rate of several capital goods together could be
approximated by an exponential distribution, but in no case does the failure
of a single capital good admit an exponential distribution, i.e., a Poisson
process for the part’s demand (Saidane et al. 2013). Note that empirical
goodness-of-fit experiments conducted for spare parts inventories show that
spare parts demand often provides a low fit to the Poisson process (Lengu et
al. 2014, Turrini & Meissner 2019). Our aim in this paper is to fill the
research gaps in the literature presented above.

In this paper, we consider a system with one capital good and one repair shop
having a limited capacity. We assume that unplanned maintenance follows a
general distribution. Moreover, we include planned maintenance by setting an
age for each part installed in the capital good at which the part should be
replaced by a new one from the stock point (if any). Our aim is to determine
the required inventory level and the optimal repair shop capacity as well as
the optimal age-threshold so that the total cost of the system is minimized.
For this system, we perform an exact analysis and we provide the optimal stock
level as well as the optimal capacity of the repair shop and the optimal
threshold age at which planned maintenance should be performed. We then extend
the work by studying a system with multiple capital goods and a repair shop
with limited capacity. We first consider the case that the repair shop does
not differentiate between capital goods and performs the repairs on a first-
come, first-served basis. We then allow for the setting where the repair shop
differentiates between the capital goods according to their criticality and
repairs failed parts from these capital goods according a certain priority
rule. For this system, it is difficult to provide an exact analysis due to the
fact that this network of capital goods and the common repair shop do not
admit a simple steady-state probability of having $n$ spare parts available
for each capital good in the network, unlike Jacksonian network systems. We
therefore analyze this system approximately using a decomposition algorithm
that is based on our exact results obtained in the case a single capital good.
This approximate method appears to be quite accurate. Hence, the contribution
of this paper is twofold.

  1. 1.

This is the first work that analyses a repairable spare parts system composed
of one repair shop with limited capacity and one stock point facing a
generally distributed demand. The cases of one capital good and multiple
capital goods are considered (an exact analysis is developed in the case of
one capital good and an approximate one in the case of multiple capital
goods). We propose an algorithm that enables to optimize the base stock level,
the number of repairables in the system, the age-threshold, and the capacity
of the repair shop.

  2. 2.

We consider both corrective and preventive maintenance strategies in a spare
parts inventory system with a capacitated repair center. To the best of our
knowledge, this is the first paper that makes an integration of both
maintenance strategies within the METRIC modelling framework in addition to
considering a capacitated repair.

The remainder of this paper is organised as follows. In Section 2, we provide
an overview of the literature related to spare parts inventory control with a
focus on the research dealing with repairable inventory systems. Section 3 is
dedicated to the system description ad the analysis in the case of a single
capital good. The analysis is extended in Section 4 to the case of a system
with multiple capital goods. We present in Section 5 the results of the
numerical investigation. The conclusions of our work are provided in Section 6
along with some avenues for further research.

##  2 Literature review

This paper integrates preventive maintenance with spare parts control. As
such, it contributes to two streams of literature. The first stream is on
repairable spare parts inventory systems while the second one focuses on
preventive maintenance. Below we will describe those papers in these fields
and its intersection most related to our research. For a general overview of
spare part inventory control, we refer the reader to Kennedy et al. (2002),
Porras & Dekker (2008), Lengu et al. (2014) and Hu et al. (2018). Likewise,
general overviews on maintenance optimization can be found in Dekker (1996)
and De Jonge & Scarf (2019).

There is an extensive literature on inventory systems of repairable spare
parts. We recall that this literature often considers multi-echelon systems
that include a repair shop that replenishes one or several stock points of
multiple components dedicated to maintaining a capital good or a set of
external customers. The major challenge of this research is to minimize the
inventory holding cost at the stocking points and the operating cost of the
repair facilities as well as providing fast recovery service to the capital
good or customers (Tiemessen & Van Houtum 2013). This stream of literature can
be roughly split into (i) a research area where the stock point is replenished
by the repair shop as a single supply source, and (ii) a research body where
the stock points are replenished by a dual supply, i.e. an expediting source
in addition to the repair shop sourcing.

The earliest contributions on repairable spare parts inventory focus on the
METRIC model proposed by (Sherbrooke 1968) and its variants such as the MOD-
METRIC model (Muckstadt 1973) and the VARI-METRIC model (Slay 1984). The basic
METRIC model consists of a spare parts inventory system consisting of multiple
stock points facing demand for spare parts, which are each replenished by a
central repair center. The failures of the parts at the stock points occur
according to Poisson processes, so that the distribution of parts in the
repair center is again Poisson distributed, independent of the service time
distribution. Moreover, both the stock points and the repair center are
controlled by a base-stock policy. A collection of the research findings
related to the METRIC-related research is given in the books by Muckstadt
(2004) and Muckstadt & Sapra (2010).

It is worth noting that most of the works in the literature on METRIC models
assumes ample (infinite) capacity of the repair facility and an infinite parts
population in the whole system. Some exceptions that deal – like the work in
this paper – with capacitated systems can be found in Gross & Ince (1978),
Gross et al. (1983), Graves (1985), Albright & Gupta (1993) and Diaz & Fu
(1997). With regard to the literature on capacitated systems, most works rely
on closed queueing network models, thereby often being restricted to failure
rates given by Poisson or compound-Poisson processes. Graves (1985) showed the
benefit of considering a finite capacity and assuming a compound-Poisson
demand process, such that demand processes with varying mean-to-variance rates
can be better estimated. This was shown to significantly outperform the basic
METRIC model due to a better estimation of the spare parts requirements in the
system under a high repair shop utilization. Diaz & Fu (1997) developed
extensions where, in addition to the compound Poisson demand process
assumption, general repair distributions are considered and the case of
different classes of parts in the system, each with a different repair
distribution. Zijm & Avşar (2003) considered a two-indenture maintenance
system composed of a capacitated repair shop and a number of identical
installations that are considered as assemblies made up of a number of
repairable components. They modeled the repair shop and the assembly facility
as product form queuing networks, where the component failures of the
assemblies occur according to a Poisson process and the the stock points of
the assemblies are controlled with a base-stock policy. They developed
accurate approximation procedures to evaluate the performance of the system by
means of the fill rate and the work in process level. More recently, Park &
Lee (2011, 2014) analysed the case of an inventory system composed of multiple
stock points that order parts from a central warehouse, which is controlled
with a continuous review $(S,Q)$ inventory policy. The failure of the parts
follows a Poisson process and the repair time in the repair shop that
replenishes the depot follows either an exponential or a two-phase Coxian time
distribution. They provided expressions for the steady-state probabilities of
the system and proposed an approximation method to estimate the parameters
that minimise the costs with a good computational effort. In our paper, we too
consider a capacitated system where the repair shop is controlled according to
a base-stock policy, but contrary too all the works above, we assume a general
distribution for the spare parts demand at the stock points. We also propose
an extension where the inventory system is composed of multiple types of
capital goods.

With regard to the literature that considers that the stock point can be
supplied by a dual sourcing including a repair expediting, we refer to the
research by Arts et al. (2016) and Drent & Arts (2021). This research builds
on the early work on dual source inventory systems developed by Moinzadeh &
Schmidt (1991), Song & Zipkin (2009) among others. A review on the inventory
control of such multiple supply systems is provided by Svoboda et al. (2021).
Arts et al. (2016) analyse a dual sourcing spare parts inventory system with
an expedited sourcing. They make the assumption of a Markov modulated Poisson
process and relax the assumptions that the demand process is a stationary
Poisson process (often considered in the literature).

It should be noted that all the above described literature assumes, to keep
their analyses based on queueing networks tractable, that the spare parts
demand and the replenishment of the stock are only based on the failure of the
part, which means that it is under a corrective maintenance strategy. The
literature does not consider that spare parts have limited lifetimes and can
potentially be replaced through a preventive maintenance – before they fail –
when the lifetime reaches an age-threshold (Poppe et al. 2017, Drent et al.
2020). This gap in the literature constitutes one of the main contributions of
this paper where we assume that in addition to corrective maintenance due to a
failure that generates the replacement of the parts, there is a preventive
maintenance age-threshold that triggers the replenishment (i.e. the demand) of
parts at the stock point. The age-threshold has an impact on the arrival
process at the repair shop, and should thus be taken into account in
evaluating the performance of the system as a whole.

Finally, it is worth mentioning that the idea of integrating corrective and
preventive maintenance strategies in spare parts inventory systems is not new.
In fact, there is a body of literature that deals with maintenance strategies
and the management of spare parts characterized with a limited lifetime (see,
e.g., Olde Keizer et al. 2017, Poppe et al. 2017, Westerweel et al. 2019,
Castro et al. 2019, Salari & Makis 2020). Poppe et al. (2017), among others,
showed that compared to corrective maintenance, preventive maintenance
policies increase the demand for spare parts, as some remaining useful
lifetime of components may not be used, which increases the inventory and
capacity requirements. However, there is an agreement in the literature that
the cost of the corrective maintenance, especially if the parts are not
available in stock (downtime cost) may be much higher (Eruguz et al. 2018).
This cost is likely to increase even more if the parts are replenished from a
capacitated repair center. This latter effect has not been investigated in the
literature because a common assumption is that spare parts are ordered from an
outside supplier with ample capacity. To the best of our knowledge,
integrating maintenance strategies when minimizing the inventory costs and the
operating costs (the maintenance costs of the capital good and the operating
costs at the repair center) in an inventory system of repairable parts has not
been investigated in the literature. In this paper, we study this intriguing
relationship and fill this gap in the literature.

##  3 System description, modelling and analysis

In this section, we consider the case of one capital good. We describe the
system, we derive the steady state probabilities and we present the cost
function and its properties. The analysis in the case of a system with
multiple capital goods is presented in Section 4.

###  3.1 System description and assumptions

We consider a repairable inventory system consisting of a stock point
responsible for maintaining a single capital good and a repair shop
responsible for repairing components. The capital good consists of a number of
critical components that fail infrequently and independently. These critical
components are crucial for operating the capital good, i.e. the capital good
is down if one of these components fails. The components are at such levels in
the material breakdown structure of the capital good that they can be replaced
as a whole by spare parts. The random lifetime of a critical component,
denoted with $X$, is generally distributed with distribution $\mathbb{P}(X\leq
x)=F(x)$, $f(x)=\frac{\mathrm{d}F(x)}{\mathrm{d}x}$.

All components are controlled by the canonical age-based maintenance policy
introduced by Barlow & Hunter (1960). That is, components are replaced when
their lifetime reaches an age-threshold $\tau>0$, or upon failure, whichever
comes first. Components that fail before reaching $\tau$ have severe
consequences because it is not planned (e.g. think of an assembly line that is
stopped abruptly, trains that stand still, or an airplane crash). On the other
hand, components whose ages reach $\tau$ do not have such severe consequences
as their maintenance practices can be planned for.

Upon replacement of a component (either preventively or correctively), it is
immediately replaced by a new spare part from the stock point. If the required
spare part is not on hand, then the capital good remains down until a spare
part is available again. We assume that replaced parts can always be repaired
(no condemnation) and that these parts are sent to the repair shop
immediately, where, if none of the $K$ repair stations are available, joins a
queue of parts awaiting repair. The repair time is random, and we make the
common assumption (see, e.g., Tian et al. 2021, Li & Tomlin 2022) that this
time is exponentially distributed with rate $\mu_{r}>0$. After the replaced
part is repaired it is immediately shipped back to the stock point, where it
is then available again. The size of the turn-around stock in our system,
$S\in\mathbb{N}_{0}$, of the repairables is determined at time $t=0$ and
cannot be adapted afterwards. Figure 1 illustrates that this repairable
inventory system – for a single capital good – can be viewed as a cyclic
tandem queue network with a fixed number of customers equal to base stock
level $S$.

$\bullet/GI/1$$\bullet/M/K$ Capital good Repair shop Figure 1: Cyclic tandem
queue with a fixed number of customers

To see this, let $n_{o}\in\mathbb{N}_{0}$ and $n_{r}\in\mathbb{N}_{0}$, with
$n_{o}+n_{r}=S$, denote the number of spare parts in stock, including one part
is in operation, and in the repair shop, respectively. Now observe that
repaired spare parts arrive at the stock point of the capital good according
to a Poisson process with state dependent rates $\mu_{r}\cdot\min(S-n_{0},K)$
whenever $n_{r}>0$. If there are no failed components in repair, then all
spare parts are in stock and one is in operation which implies that $n_{o}=S$
and thus the arrivals from the repair shop to the stock point of the capital
good are interrupted. This is indeed a closed queuing network with finite
population $S$. This network admits a stationary probability distribution that
can be written as a product form, only in the unrealistic setting that the
components’ lifetimes follow an exponential distribution.

Fortunately, the queuing network displayed in Figure 1 corresponds precisely
to an $M_{n}/GI/1$ queue with fixed capacity $S$ and state-dependent Poisson
arrival rates $\mu_{r}\cdot\min(S-n_{0},K)$. In fact, Lavenberg (1975) showed
that a (state-dependent) $M/GI/1$ queue with finite capacity can be viewed as
a closed network and the opposite is also true, where the first queue has a
general service time and the the second queue has an exponential service time.
In this paper, we use this analogy to compute performance measures of the
system, thereby allowing for a general lifetime distribution.

Let $N(t)$ denotes the number of orders present in the $M_{n}/GI/1$ queue with
fixed capacity $S$ at time $t$. The steady-state probability of having $n$
orders in this queuing system is denoted by
$P(n|S,\tau,K)=\lim_{t\to\infty}\mathbb{P}(N(t)=n|S,\tau,K)$. Costs are levied
in the following way. We penalize downtime of the capital good at rate
$C_{d}>0$ per time unit. Furthermore, if a component fails before reaching
age-threshold $\tau$, then a cost $C_{u}>0$ is charged. Per unit of turn-
around stock, we pay a cost of $C_{a}$ per time unit, which can be interpreted
as an investment rate. Finally, each part on inventory near the capital good
costs $C_{h}>0$ per time unit, and each unit of capacity $K$ costs $C_{w}>0$
per time unit. We aim to minimize the total sum of these three cost rates.
Hence, the optimization problem that we seek to solve is the following multi-
variable, non-linear, non-convex, mixed integer programming problem (mipp):

$\displaystyle(\textsc{mipp})\quad$ $\displaystyle\min_{\\{S,\tau,K\\}}$
$\displaystyle
C_{u}(1-P(0|S,\tau,K)\mu\mathbb{P}(X<\tau)+C_{d}P(0|S,\tau,K)+C_{a}S+C_{w}K$
(1) subject to $\displaystyle S,K\in\mathbb{N},\quad\tau\geq 0,$

where

$\displaystyle\frac{1}{\mu}=\int_{0}^{\tau}(1-F(x))dx$ (2)

is the mean installation time of a component given that the age-threshold is
equal to $\tau$ (we drop the dependency on $\tau$ for notational clarity).
Observe that the value of the objective function (1) of problem mipp is solely
determined by the steady-state probability distribution $P(n|S,\tau,K)$, which
we will analyze in the next section.

###  3.2 Derivation of the steady-state probabilities

In this section, we focus on how to compute the steady-state probability
$P(n|S,\tau,K)$, which in turn relies on the random installation time of a
component. Due to the nature of the age-based maintenance policy, we must
first express this general distribution as the minimum between the random
lifetime variable $X$ and the age-threshold $\tau$. The following Lemma
provides this distribution and its Laplace transform. Observe that in this
Lemma, we first treat $\tau$ as a random variable so that we can obtain the
installation time distribution by using a degenerate distribution with all its
mass at the value $\tau$.

######  Lemma 1.

Let $f(x)$ and $F(x)$ be the PDF and the CDF, respectively, of the component’s
lifetime. Let $h(y)$ and $H(y)$ be the PDF and the CDF, respectively, of the
random preventive age-threshold, which we denote by the random variable $Y$.
Let $Z\sim min(X,Y)$, then the PDF and the CDF of $Z$, denoted by $g(z)$ and
$G(z)$ respectively, are given by

$\displaystyle G(z)$ $\displaystyle=F(z)+H(z)-F(z)H(z),\mbox{ and,}$
$\displaystyle g(z)$
$\displaystyle=G^{{}^{\prime}}(z)=f(z)+h(z)-f(z)H(z)-F(z)h(z).$

If we let $G^{\ast}$ be the Laplace transform of $g$, then for the particular
case that the preventive maintenance time is a degenerate distribution with
all its mass at the age-threshold $\tau$, then:

$\displaystyle G^{\ast}(s)=\int_{0}^{\tau}f(x)e^{-sx}dx.$

###### Proof.

The proof follows from straightforward probability arguments. Observe that

$\displaystyle 1-G(z)=\mathbb{P}(\min(X,Y)>Z)=\mathbb{P}(X>z)\mathbb{P}(Y>z)$
$\displaystyle=(1-H(z))(1-F(z))$ $\displaystyle=1-F(z)-H(z)+F(z)H(z).$

And for the Laplace transform of $g$, we have
$G^{\ast}(s)=F^{\ast}(s)-\int_{\tau}^{\infty}f(x)e^{-sx}dx=\int_{0}^{\tau}f(x)e^{-sx}dx$.
∎

We now continue with deriving the steady-state probability $P(n|S,\tau,K)$ –
the key ingredients for evaluating the objective function of problem mipp –
and divide our analysis in two cases, depending on the value of the repair
capacity $K$ compared to the turn-around stock $S$:

  1. 1.

$S\leq K$. In this case, there is ample capacity in the repair shop. Hence, as
long as there are $n$ items on stock, the arrival rate at the first queue is
exponential with mean $\lambda_{n}=\mu_{r}(S-n)$.

  2. 2.

$S>K$. In this case, the repair shop is truly capacitated: As long as there
are $n$ items on stock, the arrival rate at the first queue is exponential
with mean $\lambda_{n}=\mu_{r}\min(S-n,K)$.

The analysis in both cases is based on the approach of Kerner (2008) and
Economou & Manou (2015) and makes use of Lemma 1.

####  3.2.1 Steady-state probabilities under ample capacity

This case corresponds to the situation where the repair shop has ample
capacity, which means that the capacity $K$ does not influence the repair time
of the defective parts. Let us first introduce a new random variable $R_{n}$
that represents the distribution of the remaining service time at an arrival
instant of a repaired part who finds $n$ parts in the stock. Let
$F_{n}^{\ast}(s)$ denote the Laplace transform of $R_{n}$. Then, by Theorem
3.1 of Economou & Manou (2015), we can compute $F_{n}^{\ast}(s)$ using the
following recursive scheme:

$\displaystyle
F_{n}^{\ast}(s)=\left\\{\begin{array}[]{ll}G^{\ast}(s)&\mbox{if}\;n=0,\\\
\frac{\lambda_{n}}{s-\lambda_{n}}\left[G^{\ast}(\lambda_{n})\frac{1-F_{n-1}^{\ast}(s)}{1-F_{n-1}^{\ast}(\lambda_{n})}-G^{\ast}(s))\right]&\mbox{if}\;n\geq
1.\\\ \end{array}\right.$ (5)

This recursion initializes with the Laplace transform $G^{\ast}(s)$, which we
introduced in Lemma 1. Using the scheme in (5) and Corollary 3.1 by Economou &
Manou (2015), the expected remaining service time, denoted by
$R(n)=\mathbb{E}(R_{n})$, at an arrival instant of a repaired part who finds
$n$ parts in the system, can also be computed recursively as:

$\displaystyle
R(n)=\left\\{\begin{array}[]{ll}\frac{\frac{1}{\mu}}{1-G^{\ast}(\lambda_{n})}-\frac{1}{\lambda_{n}}&\mbox{if}\;n=1,\\\
\frac{G^{\ast}(\lambda_{n})R(n-1)}{1-F_{n-1}^{\ast}(\lambda_{n})}-\frac{1}{\lambda_{n}}+\frac{1}{\mu}&\mbox{if}\;n\geq
2,\\\ \end{array}\right.$ (8)

where $\lambda_{n}=\mu_{r}(S-n)$ and $1/\mu$ is the mean installation time of
a component given that the age-threshold is equal to $\tau$, see Equation (2).

Using Corollary 4 of Abouee-Mehrizi & Baron (2016), we can now express the
steady-state probabilities as:

$\displaystyle P(n|S,\tau,K)=$ (9)
$\displaystyle\left\\{\begin{array}[]{ll}\left[1+\left(1+\lambda_{S-1}R(S-1)\right)\frac{\lambda_{0}}{\lambda_{S-1}}\prod_{i=0}^{S-2}\frac{1-F_{i}^{\ast}(\lambda_{i+1})}{G^{\ast}(\lambda_{i+1})}\right.\left.+\sum_{j=1}^{S-2}\frac{\lambda_{0}}{\lambda_{j}}\prod_{i=0}^{j-1}\frac{1-F_{i}^{\ast}(\lambda_{i+1})}{G^{\ast}(\lambda_{i+1})}\right]^{-1}&\mbox{if}\;n=0,\\\
\frac{\lambda_{0}P(0|S,\tau,K)}{\lambda_{n}}\prod_{i=0}^{n-1}\frac{1-F_{i}^{\ast}(\lambda_{i+1})}{G^{\ast}(\lambda_{i+1})}&\mbox{if}\leq
n\leq S-1,\\\ 1-\sum_{j=0}^{S-1}P(j|S,\tau,K)&\mbox{if}\;n=S.\\\
\end{array}\right.$ (13)

Using (9), we are now able to efficiently calculate the steady-state
probabilities for the case of ample capacity at the repair shop.

####  3.2.2 Steady-state probabilities for capacitated systems

The derivation of the steady-state probabilities for the case that the repair
shop is capacitated, i.e. when $S>K$, is slightly different from that where
$K\leq S$. In fact, when all repairmen are busy, i.e. when $K$ parts being
repaired at the repair shop, the arrival rate at the stock,
$\lambda_{n}=\mu_{r}min(S-n,K)$, becomes constant and is equal to $K\mu_{r}$.
The probabilities we derived in the previous section cannot be applied
directly because they involve the Laplace transform of $R_{n}$ which turns to
zero if $\lambda_{n}$ becomes constant. Fortunately, Theorem 6 by Oz et al.
(2017) allows us to find $F_{n}^{\ast}(s)$ even for constant arrival rate. We
restate their theorem using our notations. Let $f_{n}(x)=\mathbb{P}(R_{n}=x)\
(x\geq 0)$ denote the probability density function of the random variable
$R_{n}$, Oz et al. (2017) showed that the conditional residual service time
$f_{n}(x)$ satisfies the following recursion:

$\displaystyle f_{n}(x)=\left\\{\begin{array}[]{ll}g(x)&\mbox{if}\;n=0,\\\
\lambda e^{\lambda x}\int_{x}^{\infty}e^{-\lambda
x_{0}}g(x_{0})dx_{0}+\frac{G^{\ast}(\lambda)\left(\lambda e^{\lambda
x}\int_{x}^{\infty}e^{-\lambda
x_{0}}f_{n-1}(x_{0})dx_{0}\right)}{1-F_{n}^{\ast}(\lambda)}&\mbox{if}\;n\geq
1.\\\ \end{array}\right.$ (16)

Using the above recursion at $n=0,...,S-K$ and $\lambda=\lambda_{n}=K\mu_{r}$,
we can rewrite $f_{n}(x)$ and $F_{n}^{\ast}(s)$ for $n=0,1,$ and $2$ as
follows:

$\displaystyle\left\\{\begin{array}[]{l}f_{0}(x)=g(x),\\\
F_{0}^{\ast}(s)=\int_{0}^{\infty}f_{0}(x)e^{-sx}dx=G^{\ast}(s),\\\
f_{1}(x)=\frac{\lambda e^{\lambda x}\int_{x}^{\infty}(e^{-\lambda
x_{0}}g(x_{0})dx_{0}}{1-F_{0}^{\ast}(\lambda)},\\\
F_{1}^{\ast}(s)=\frac{\lambda\int_{0}^{\infty}(\int_{x}^{\infty}e^{-sx_{0}}g(x_{0})dx_{0})dx}{1-F_{0}^{\ast}(\lambda)}=\frac{\lambda\int_{0}^{\infty}xe^{-sx}g(x)dx}{1-F_{0}^{\ast}(\lambda)}=\frac{\lambda}{1-F_{0}^{\ast}(\lambda)}\frac{d\leavevmode\nobreak\
F_{0}^{\ast}(s)}{ds},\\\ f_{2}(x)=\lambda e^{\lambda
x}\int_{x}^{\infty}e^{-\lambda
x_{0}}g(x_{0})dx_{0}+G^{\ast}(\lambda)\frac{\lambda e^{\lambda
x}\int_{x}^{\infty}e^{-\lambda
x_{1}}f_{1}(x_{1})dx1}{1-F_{1}^{\ast}(\lambda)},\\\
F_{2}^{\ast}(s)=-\lambda\frac{dG^{\ast}(s)}{ds}-\frac{\lambda^{2}G^{\ast}(\lambda)}{(1-F_{1}^{\ast}(\lambda))F_{0}^{\ast}(\lambda)}\frac{d^{2}F_{0}^{\ast}(s)}{2ds^{2}},\\\
\end{array}\right.$ (23)

and for the general case, we can write for $n\geq 1$:

$\displaystyle
F_{n}^{\ast}(s)=\frac{\lambda}{(1-F_{n-1}^{\ast}(\lambda))}\left[-G^{\ast}(\lambda)\frac{dF_{n-1}^{\ast}(s)}{ds}+\frac{dG^{\ast}(s)}{ds}(F_{n-1}^{\ast}(\lambda)-1)\right],$

where,

$\displaystyle\left\\{\begin{array}[]{l}\frac{dF_{n-1}^{\ast}(s)}{ds}=\frac{\lambda}{(1-F_{n-2}^{\ast}(\lambda))}\frac{1}{2}\left[-G^{\ast}(\lambda)\frac{d^{2}F_{n-2}^{\ast}(s)}{ds^{2}}+\frac{d^{2}G^{\ast}(s)}{ds^{2}}(F_{n-2}^{\ast}(\lambda)-1)\right],\\\
\frac{d^{2}F_{n-2}^{\ast}(s)}{ds^{2}}=\frac{\lambda}{(1-F_{n-3}^{\ast}(\lambda))}\frac{1}{3}\left[-G^{\ast}(\lambda)\frac{d^{3}F_{n-3}^{\ast}(s)}{ds^{3}}+\frac{d^{3}G^{\ast}(s)}{ds^{3}}(F_{n-3}^{\ast}(\lambda)-1)\right],\\\
\vdots\\\
\frac{d^{n-1}F_{1}^{\ast}(s)}{ds^{n-1}}=\frac{\lambda}{(1-F_{0}^{\ast}(\lambda))}\frac{1}{n}\left[-G^{\ast}(\lambda)\frac{d^{n}F_{0}^{\ast}(\lambda)}{ds^{n}}+\frac{d^{n}G^{\ast}(s)}{ds^{n}}(F_{0}^{\ast}(\lambda)-1)\right].\\\
\end{array}\right.$ (28)

In summary, for a repair shop with limited capacity $K<S$, the Laplace
transform of the random variable that represents the distribution of the
remaining service time at an arrival instant of a customer who finds $n$
customers in the system, $R_{n}$ can be written as:

$\displaystyle
F_{n}^{\ast}(s)=\left\\{\begin{array}[]{ll}G^{\ast}(s)&\mbox{if}\;n=0,\\\
\frac{\lambda}{(1-F_{n-1}^{\ast}(\lambda))}(-G^{\ast}(\lambda)\frac{dF_{n-1}^{\ast}(s)}{ds}+\frac{dG^{\ast}(s)}{ds}(F_{n-1}^{\ast}(\lambda)-1)&\mbox{if}\;s=\lambda_{n},\\\
\frac{\lambda_{n}}{s-\lambda_{n}}(G(\lambda_{n})\frac{1-F_{n-1}(s)}{1-F_{n-1}(\lambda_{n})}-G(s))&\mbox{if}\;s\neq\lambda_{n},\\\
\end{array}\right.$ (32)

and the steady-state probability is again given by Equation (9), where
$F_{n}^{\ast}(s)$ is now given by (32).

Now that we have derived expressions for computing the steady-state
probabilities, we turn our attention back again to the objective function of
problem mipp. This is the aim of next section.

###  3.3 Cost function properties and optimizing algorithms

In this section we investigate the objective function of problem mipp in more
detail. Recall that this objective function is given by

$\quad\min_{\\{S,\tau,K\\}}C_{u}(1-P(0|S,\tau,K)\mu\mathbb{P}(X<\tau)+C_{d}P(0|S,\tau,K)+C_{a}S+C_{w}K,$

with $\\{S,K\\}\in\mathbb{N}$ and $\tau\in\mathbb{R}^{+}$. Unfortunately, we
are unable to show that this function is convex on one of these parameters,
namely $S$, $\tau$ or $K$, which would have eased the procedure of finding the
optimal values significantly. Fortunately, however, we are able to demonstrate
other desireable properties concerning the monotonicity of the probability of
being in the state zero, i.e. $n=0$. Specifically, as the following Lemma
illustrates, we have that $P(0|S,\tau,K)$ is decreasing (in the weak sense) in
$S$ and $K$.

######  Lemma 2.

The following two statements hold:
(i) $P(0|S,\tau,K)$ is non-increasing in $S$.
(ii) $P(0|S,\tau,K)$ is non-increasing in $K$.

###### Proof.

We first rewrite $P(0|S,\tau,K)$. From Equation 3.23 of Economou & Manou
(2015) and Equation 1 in Ross (2006), we know that for all $n\geq 2$

$\displaystyle\sum_{j=n}^{S}P(j|S,\tau,K)=P(n-1|S,\tau,K)\lambda_{n-1}R(n-1)+\mu\sum_{j=n}^{S}\lambda_{j}P(j|S,\tau,K).$

For $n=2$, we know that
$\sum_{j=1}^{S}P(j|S,\tau,K)=1-P(0|S,\tau,K)-P(1|S,\tau,K)$, and also that
$\lambda_{S}=0$ for $n=S$. Hence,

$\displaystyle\sum_{j=1}^{S}P(j|S,\tau,K)=1-P(0|S,\tau,K)-P(1|S,\tau,K)=P(1|S,\tau,K)\lambda_{1}R(1)+\frac{1}{\mu}\sum_{j=1}^{S-1}\lambda_{j}P(j|S,\tau,K).$

Using the expression of $R(1)$ and $P(1|S,\tau,K)$ from Equation (8) and (9)
yields

$\displaystyle P(0|S,\tau,K)=\left[1+\frac{\lambda_{0}}{\mu
G^{\ast}(\lambda_{1})}+\frac{\lambda_{0}}{\mu}\sum_{n=2}^{S-1}\prod_{i=0}^{n-1}\frac{1-F_{i}^{\ast}(\lambda_{i+1})}{G^{\ast}(\lambda_{i+1})}\right]^{-1}.$

Consider now two node cyclic networks as represented in Figure 1: One with $S$
and one with $S+1$ customers. All other parameters are the same for both
networks. By Theorem 1 (iii) of Shanthikumar & Yao (1989) we know that the
average output rate at node $i,i=1,2$ of network one with a population of $S$
customers is less than the average output rate at of the same node of network
two with a population of $S+1$. We can rewrite this result as

$\displaystyle\sum_{j=0}^{S-1}\lambda_{j}P(j|S,\tau,K)=\mu(1-P(0|S,\tau,K)\leq\sum_{j=0}^{S}\lambda_{j}P(j|S+1,\tau,K)=\mu(1-P(0|S+1,\tau,K)),$

which proves Assertion (i). Similarly, to prove Assertion (ii), we consider
two node cyclic networks: One with server capacity $K$ and one with server
capacity $K+1$, respectively, at node 2. Then, by Theorem 3 of Shanthikumar &
Yao (1989), we have:

$\displaystyle\mu(1-P(0|S,\tau,K)\leq\mu(1-P(0|S,\tau,K+1)),$

which completes the proof of Assertion (ii). ∎

Lemma 2 is not only intuitive – indeed, increasing repair capacity and/or the
turn-around stock will not lower the probability of downtime – it also helps
us in designing algorithms to compute optimal values of $K$ and $S$ for a
fixed choice of $\tau$. Specifically, from Equation (1) and Lemma 2, it is
clear that $C_{u}(1-P(0|S,\tau,K)+C_{a}S$ is increasing in $S$. Moreover, this
expression tends to infinity as $S$ tends to infinity. Based on these
observations, we propose our first algorithm, Algorithm 1, which enables to
find the optimal $S^{*}$ (for a fixed value of $K$ and $\tau$).

Based on Algorithm 1, we propose Algorithm 2 that makes use of Algorithm 1,
that is guaranteed to find the optimal value of $K^{*}$ and $S^{*}$ for a
fixed value of $\tau$. Indeed, according to Lemma 1, the steady-state
probability $P(0|S,\tau,K)$ is decreasing as $K$ increases. Therefore, we can
use the Algorithm 1 recursively to find the optimal value of $K^{*}$ for a
given $S$ (and hence also for $S^{*}$).

Observe that in both algorithms, we repeatedly need to compute the steady-
state probabilities for fixed choices of $S$, $K$, and $\tau$. For these
computations, we make use of the closed-form formulas developed in Section
3.2.

Result: Optimal $S^{*}$

Step 1: Fix $S:=0,S^{*}:=0$ and $TC_{min}=\infty$

Step 2: Compute
$TC=C_{u}(1-P(0|S,\tau,K)\mu\mathbb{P}(X<\tau)+C_{d}P(0|S,\tau,K)+C_{a}S+C_{w}K.$

if  _$TC <TC_{min}$_ then

$TC_{min}=TC$;

$S:=S+1$;

Go to step 3;

end if

Step 3: Compute $C_{u}(1-P(0|S,\tau,K)\mu\mathbb{P}(X<\tau)+C_{a}S+C_{w}K$

if  _$C_{u}(1-P(0|S,\tau,K)\mu\mathbb{P}(X <\tau)+C_{a}S+C_{w}K<TC_{min}$_
then

Go to Step 2;

else

Stop.

end if

Algorithm 1 Finding the optimal stock level $S^{*}$ for a given number of
repairmen $K$.

Result: Optimal $S^{*}$ and $K^{*}$

Step 1: Fix $K:=1$ use Algorithm 1 to find $S^{*}$ and its corresponding
optimal total cost $TC(S^{*},1,\tau)$. Set $TC_{min}=TC$

Step 2: Set $K:=K+1$, use Algorithm 1 and find the new $S^{*}$ and it
corresponding optimal total cost $TC(S^{*},K,\tau)$.

if  _$TC <TC_{min}$_ then

$TC_{min}=TC$;

Go to step 3;

end if

Step 3: Compute
$C_{u}(1-P(0|S^{*},\tau,K)\mu\mathbb{P}(X<\tau)+C_{a}S^{*}+C_{w}K$

if  _$C_{u}(1-P(0|S^{*},\tau,K)\mu\mathbb{P}(X
<\tau)+C_{a}S^{*}+C_{w}K<TC_{min}$_ then

Go to Step 2;

else

Stop.

end if

Algorithm 2 Finding the optimal stock level $S^{*}$ and the optimal capacity
$K^{*}$ .

##  4 Extension to multiple capital goods

In this section, we use our exact analysis for the single capital good, as
described in the previous sections, to design an approximation for the case
where we have several capital goods.

We now consider a network in which each capital good is characterized by
different failure rates and age-thresholds, but where they share a common
repair shop with limited repairman resources. This network is shown in Figure
2. Because the repair shop resources are shared among the capital goods, we
need to decide upon the repair prioritization policy in the repair shop. We
consider both a first-come, first-served (FCFS) discipline, where parts are
repaired in the order in which they arrive at the repair shop, and a priority
scheduling policy with priority classes. When considering a priority
scheduling policy, the case where the number of repairman is greater than one
is unfortunately not tractable to analyse. In that setting, the steady-states
probabilities for each class of capital goods cannot be computed due to the
curse of dimensionality caused by multiple classes and multiple servers.
Therefore, we only treat the case with priorities under the assumption of a
single repair resource at the repair shop, while for the FCFS discipline, we
also treat the case with multiple repair resources.

In summary, we study two systems:

  1. 1.

A system for which the repair shop is an $\bullet/M/K$ queue with FCFS
discipline.

  2. 2.

A system for which the repair shop is an $\bullet/M/1$ with priorities between
classes.

$\bullet/GI/1/S_{1}$$\bullet/GI/1/S_{2}$$\vdots$$\bullet/GI/1/S_{n}$$\bullet/M/K$

Figure 2: Cyclic maintenance network with multiple capital goods

The networks we consider do not have a steady-state probability that can be
written as product form of state probabilities of each node, so we use the
approximation method of Marie (1979) to obtain these probabilities. The method
developed by Marie (1979) and extended by Baynat & Dallery (1996) consists of
replacing general service time queues with state-dependent exponential service
time queues. The exponential service time is obtained by analysing each
station with a general service time in isolation but with a state-dependent
arrival rate. The method is iterative and appears to be a good technique for
providing different performance measures (indeed, we later compare this method
with a simulation which shows that the method performs very well). Baynat &
Dallery (1996) extended Marie’s work to multiple classes by simply decomposing
the original network into multiple networks, each of which handles only one
class. The dependency between classes is taken into account when analysing
isolated multi-class stations.

###  4.1 Repairing under a first-come, first-served discipline

We first replace for each capital good $i$ the $\bullet/G_{i}/1/S_{i}$ by an
$\bullet/M(n)/1/S_{i}$ queue with state dependent service rate
$\mu_{i}(n),n=1,...,S_{i}$, and the repair shop $\bullet/M/K$ queue by
$\bullet/M_{n}/K$ node node with state dependent service rate
$\mu_{i,0}(n),n=1,...,S_{i}$. Our network associated with capital good $i$ is
now a two nodes Jackson network as shown in Figure 3. This network does admit
a product stationary probability, which can be written as follows:

$\displaystyle\tilde{P}_{i}(n)=T_{i}\left(\prod_{k=1}^{n}\frac{1}{\mu_{i}(k)}\right)\prod_{k=1}^{S_{i}-n}\frac{1}{\mu_{i,0}(k)},i=1,...,J;n=0,...,S_{i}$
(33)

where, $T_{i}$ is the normalising constant of the node of capital good $i$,
$n$ are the number of items in the queue $\bullet/G_{i}/1/S_{i}$ and $S_{i}-n$
are the number of failed parts at the repair shop node. Observe that this sub-
network in isolation is similar to the model we analysed in Section 3.2.

Since capital good $i,i=1,...,J$, can only have capital good $i$ items, the
marginal steady-state probability to have $n$ items at capital good $i$ is
$\tilde{P}_{i}(n)$. For the repair shop node, the marginal steady-state
probability to have $n$ items of class $i$ is then simply
$\tilde{P}_{i}(S_{i}-n)$.

$\bullet/M_{n}/1/S_{i}$$\bullet/M_{n}/K$ Capital good Repair shop Figure 3:
Sub-network for capital good $i$.

For this Jackson network, we can readily obtain the state dependent arrival
rate $\lambda_{i}(n)$ at each node $i,i=1,...,J$ and the the state dependent
arrival rate at the repair shop (node 0) $\lambda_{0,i}(n)$ for each class
$i,i=1,...,J,$ using the following equations:

$\displaystyle\lambda_{i}(n)$
$\displaystyle=\mu_{i}(n+1)\frac{\tilde{P}_{i}(n+1)}{\tilde{P}_{i}(n)},\text{
and, }\lambda_{0,i}(n)$
$\displaystyle=\mu_{0,i}(n+1)\frac{\tilde{P}_{i}(S_{i}-n-1)}{\tilde{P}_{i}(S_{i}-n)},i=1,...,J.$
(34)

However, the scheme above in (34) can only be used if the service rates
$\mu_{i}(n)$ and $\mu_{0,i}(n)$, $n=1,...,S_{i}$ are known. It turns out that
these service rate can be obtained by analysing each node in isolation for a
given values of $\lambda_{i}(n)$ and $\lambda_{0,i}(n)$, for which we have
derived exact expressions in Section 3.2.

Specifically, for nodes $i,i=1,...,J$, the steady-state probabilities, which
we now index by $i$, can be used as a way to approximate the state dependent
service rate at node $i$. The principle of this approximation is to set the
arrival rate equal to $\lambda_{i}(n),n=0,...,S_{i}$ for node $i,i=1,...,J$.
Then by Equation (9), we can approximate the conditional throughput (average
output rate at node $i$), denoted by $\upsilon_{i}$ as

$\displaystyle\upsilon_{i}(n)=\lambda_{i}(n-1)\frac{P_{i}(n-1)}{P_{i}(n)},n=1,...,S_{i},$

Now, the approximate state dependent service rate at node $i$ is set to

$\displaystyle\mu_{i}(n)=\upsilon_{i}(n),n=1,...,S_{i}$ (35)

We now need to find the state dependent service rate $\mu_{0,i}(n)$ at the
repair shop (node 0). To do so, we again assume that we know the state
dependent arrival rates $\lambda_{0,i}(n)$. We can then obtain $\mu_{0,i}(n)$
in a similar way as for nodes $1,...,J$.

Since there are $J$ classes of items to be repaired at node 0, the repair shop
can be viewed as a multi-class queueing system. If it has $K$ repairman, then
the steady-state probability of this queueing system is given by:

$\displaystyle
P_{0}(n_{1},...,n_{J})=\begin{cases}\frac{1}{T_{0}}(1/\mu_{r})^{\sum_{i=1}^{J}n_{i}}\prod_{i=1}^{J}\frac{\prod_{k=0}^{n_{i}-1}\lambda_{0,i}(k)}{n_{i}!},&\sum_{i=1}^{J}n_{i}\leq
K,\\\
\frac{1}{T_{0}}\frac{\left(\sum_{i=1}^{J}n_{i}\right)!(1/\mu_{r})^{\sum_{i=1}^{J}n_{i}}}{K!K^{\left(\sum_{i=1}^{J}n_{i}\right)-K}}\prod_{i=1}^{J}\frac{\prod_{k=0}^{n_{i}-1}\lambda_{0,i}(k)}{n_{i}!},&\sum_{i=1}^{J}n_{i}>K,\end{cases}$
(36)

where $T_{0}$ is the normalising constant and $n_{i},i=1,...,J,$ are the
number of failed parts of capital good $i$ at the repair shop.

The marginal steady-state probability of having $n_{i},i=1,...,J,$ parts at
the repair shop can be expressed as

$\displaystyle P_{0}(n_{i})=\sum_{n_{j}\leq S_{j},j\neq
i}P_{0}(n_{1},...,n_{J}),n_{i}=0,...,S_{i}.$ (37)

From the above equation, the conditional throughput of class $i$ at the repair
shop node is

$\displaystyle\upsilon_{0,i}(n)=\lambda_{0,i}(n_{i}-1)\frac{P_{0}(n_{i}-1)}{P_{0}(n_{i})},n_{i}=1,...,S_{i},$

and the approximate state dependent service rate of class $i$ at node $0$ is
set as

$\displaystyle\mu_{0,i}(n)=\upsilon_{0,i}(n),n=1,...,S_{i}.$ (38)

We are now ready to provide an algorithm, Algorithm 3, to compute the steady-
state probabilities of the original network of Figure 2 and provide an
approximate solution for finding the best $S_{i}^{*},i=1,...J.$ that minimizes
the total cost. This algorithm makes use of Algorithm 2, we developed in
Section 3.2.

Result: Optimal $S_{i}^{*}$

Step 1: for  _i=1:J_ do

Set $\mu_{0,i}(n_{i})=n_{i}\mu_{r}$ and $\mu_{i}(n)=\mu_{i}$;

This is an initialisation for service rate at each node, $\mu_{i}$ is given by
Equation (2);

Set $v=1$ and $\mu_{0,i}^{v}(n_{i})=\mu_{0,i}(n_{i})$ ,
$\mu_{i}^{v}(n)=\mu_{i}(n)$ and fix some tolerance $\epsilon$;

end for

Step 2: for  _i=1:J_ do

Compute $\tilde{P}_{i}(n),n=0,...,S_{i}$ by Equation (33) and using
$\mu_{0,i}^{r}(n_{i})$ and $\mu_{i}^{r}(n)$;

Compute $\lambda_{0,i}(n_{i}),n_{i}=0,...,S_{i}-1$ and
$\lambda_{i}(n),n=0,...,S_{i}-1$ using Equation (34);

end for

Step 3: for  _i=1:J_ do

Compute $P_{i}(n),n=0,...,S_{i}$ using Equation (9);

Compute $P_{0}(n_{i}),n_{i}=0,...,S_{i}$ using Equations (36) and (37) ;

Compute $\mu_{0,i}(n_{i}),n_{i}=0,...,S_{i}-1$ and
$\mu_{i}(n),n=0,...,S_{i}-1$ using Equations (35) and (38);

Set $v=v+1$ $\mu_{0,i}^{v}(n_{i})=\mu_{0,i}(n_{i})$ and
$\mu_{i}^{v}(n)=\mu_{i}(n)$

end for

if  _$\mu_{0,i}^{v}(n_{i})-\mu_{0,i}^{v-1}(n_{i}) >\epsilon$_ then

Go to step 2;

else

Use Algorithm 2 to find the optimal $S_{i}^{*},i=1,...,J$.

end if

Algorithm 3 Finding the optimal $S_{i}^{*},i=1,...,J,$ for a given number of
repairman $K$.

###  4.2 Repairing under a priority discipline

We now treat the case where the repair station is an $\bullet/M/1$ queue with
multiple priorities. We consider the case where the station station a single
server (repairman) and that the failed parts are ranked in their order of
priority. We assume that class 1 has the highest priority and class $J$ has
the lowest priority. When a failed component of type $i<j$ arrives at the
repair station and the part $j$ is being repaired, the server immediately
stops repairing the component $j$ and starts with part $i$. The part $j$
resumes the repair at the point where it stopped if no part of a higher class
is being repaired or in the queue. That is, we assume a preemptive priority
rule.

A queue with multiple priorities and a preemptive service discipline has been
analysed very well in the literature. The majority of works approximate the
stationary probability or provides some performance measures without computing
the steady-state probabilities (e.g. mean response time). We refer the reader
to the work of Wu (2014), where the most important works have been largely
summarized under the assumption that the arrival rate is independent of the
state of the system.

For state-dependent arrival rates, little work has been done. The most recent
is that of Brandwajn & Begin (2017), who developed an algorithmic procedure to
approximate the stationary probability of each priority class. Only the work
by Bitran & Caldentey (2002) gives the exact calculation of the stationary
probabilities with two priority classes. We summarize their procedure in the
following Lemma.

######  Lemma 3.

(Bitran & Caldentey 2002), For a two class $M_{n}/M/1$ queue with state
dependent arrival rates, the steady-state probabilities are:

$\displaystyle
P_{1}(n)=\begin{cases}\left[\sum_{n=1}^{S_{1}}\left((1/\mu_{r})^{k}\prod_{k=0}^{n-1}\lambda_{1}(k)\right)\right]^{-1}&n=0\\\
P_{1}(0)(1/\mu_{r})^{k}\prod_{k=0}^{n-1}\lambda_{1}(k)&n=1,...,S_{1}\end{cases}$

and

$\displaystyle
P_{2}(n)=\frac{e^{\prime}C_{k}}{\sum_{n=0}^{S_{2}}e^{\prime}C_{k}}$

where:
$C_{0}$ is the right eigenvector of $B_{0}$ associated to the eigenvalue 0,
$B_{k}=A_{k}-\lambda_{2}(k)e_{1}e^{\prime},k=0,...,S_{2}$
$e^{\prime}$ is the transpose of a column matrix with ones elements and size
$S_{1}+1$,
$e_{1}$ is a column matrix of size $S_{1}+1$ with one in its first element and
zeros otherwise,
$B_{k}=A_{k}-\lambda_{2}(k)e_{1}e^{\prime},k=0,...,S_{2}$

$\displaystyle A_{k}=\begin{pmatrix}a_{0,k}&-\mu_{r}&0&\cdots&0\\\
-\lambda_{1}(0)&a_{1,k}&-\mu_{r}&0\cdots&0\\\ &\ddots&\ddots&\ddots\\\
&&-\lambda_{1}(S_{1}-2)&a_{S_{1}-1,k}&-\mu_{r}\\\
0&\cdots&\cdots&-\lambda_{1}(S_{1}-1)&a_{S_{1},k}\\\ \end{pmatrix}$

$\displaystyle a_{i,j}=\begin{cases}\lambda_{1}(0)+\lambda_{2}(0)&i=j=0\\\
\lambda_{1}(0)+\lambda_{2}(j)+\mu_{r}&i,j>0\\\
\lambda_{2}(j)+\mu_{r}&i=S_{1},j\geq 0\\\
\lambda_{1}(i)+\lambda_{2}(j)+\mu_{r}&\\\ \end{cases}$

and $C_{k}=\lambda_{2}(k-1)B_{k}^{-1}C_{k-1},k=1,...,S_{2}$

We now provide an extension of Lemma 3 to more than two classes. Observe that
class $j$ $(2<j\leq J)$, sees all priorities classes $i<j$ as a single
priority class with a Poisson arrival rate

$\displaystyle\lambda=\sum_{i=1}^{j-1}\sum_{i=0}^{S_{i}}\lambda_{i}(n)P_{i}(n).$

Therefore, to find the steady-state probability
$P_{j}(n),n=0,...,S_{j},2<j\leq J$, we can apply Lemma 3 by setting
$\lambda_{1}(n)=\lambda$ and $S_{1}=\infty$. We will use this observation in
computing the steady-state probabilities for the system for which the repair
shop is an $\bullet/M/1$ with priorities between classes.

##  5 Numerical investigation

In this section we report on an extensive numerical investigation. We first
discuss our experiments with one capital good, and then discuss those
involving multiple capital goods. The objectives of the numerical experiments
differ per setting.

###  5.1 One capital good

We start with evaluating the performance of preventive maintenance in the
context of a single capital good through comparative statics. We first report
on Figures 5-7 the total cost as a function of $\tau$ and the capacity of the
repair shop $K$. We vary $\tau$ between 0 and 2 and we consider values of $K$
between 1 and 5. In order to analyze the sensitivity to the unit costs, we
consider values of $C_{u}\in\\{10,20\\}$, $C_{d}\in\\{20,40\\}$ and
$C_{w}=0.75$.

Figure 4: Total cost variation as a function of $\tau$ and the capacity $K$,
$C_{u}=10$, $C_{d}=20$, $C_{w}=0.75$.

Figure 5: Total cost variation as a function of $\tau$ and the capacity $K$,
$C_{u}=10$, $C_{d}=40$, $C_{w}=0.75$.

Figure 6: Total cost variation as a function of $\tau$ and the capacity $K$,
$C_{u}=20$, $C_{d}=20$, $C_{w}=0.75$.

Figure 7: Total cost variation as a function of $\tau$ and the capacity $K$,
$C_{u}=20$, $C_{d}=40$, $C_{w}=0.75$.

From 5-7 we observe that the optimal costs can vary quite substantially in the
capacity of the repair shop $K$, especially when the preventive maintenance
threshold is set low. For example, Figures 5 and 5 indicate that the optimal
costs increase significantly when $K$ decreases from $5$ to $1$ for low values
of $\tau$. Figures 5-7 furthermore show that for low capacity $K$, the optimal
policy prescribes a high threshold limit to balance the cost of an overloaded
repair shop. Conversely, when the repair shop has more capacity, the optimal
policy prescribes a lower age-threshold, allowing the asset to take advantage
of preventive maintenance and reduce downtime due to breakdowns. This is
possible because the repair shop is less congested, allowing parts to be
repaired in less time than with a similar system of lower capacity. From these
figures we can also deduce that the cost savings due to preventive maintenance
can be quite significant. Indeed, everything else fixed, we see that
decreasing $\tau$ can lead to significant decreases in costs, which implies
that coordinated preventive maintenance (i.e. low $\tau$) has benefit over
coordinated corrective (i.e. high $\tau$).

In Figures 9-11 we plot the optimal value of inventory level $S$ as a function
of $\tau$ and the capacity $K$. We vary $\tau$ between 0 and 2 and $K$ between
1 and 5. We consider the values $C_{u}\in\\{10,20\\}$, $C_{d}\in\\{20,40\\}$
and $C_{w}=0.75$.

Figure 8: Optimal stock level $S$ variation as a function of $\tau$ and the
capacity $K$, $C_{u}=10$, $C_{d}=20$, $C_{w}=0.75$.

Figure 9: Optimal stock level $S$ variation as a function of $\tau$ and the
capacity $K$, $C_{u}=10$, $C_{d}=40$, $C_{w}=0.75$.

Figure 10: Optimal stock level $S$ variation as a function of $\tau$ and the
capacity $K$, $C_{u}=20$, $C_{d}=20$, $C_{w}=0.75$.

Figure 11: Optimal stock level $S$ variation as a function of $\tau$ and the
capacity $K$, $C_{u}=20$, $C_{d}=40$, $C_{w}=0.75$.

We observe that the optimal inventory level $S$ is rather small when the
repair shop has low capacity and a low age-threshold. If the capacity $K$
increases, we see that the inventory level $S$ also increases. However, as the
age-threshold $\tau$ increases, we see that the inventory level $S$ reaches a
fixed value that becomes less sensitive to the capacity $K$ of the repair
shop. The optimal policy is therefore to choose the value of $S$ that
minimizes the total cost for a given value of $K$ with less dependence on the
value of the age-threshold. Furthermore, as $K$ increases, the optimal policy
prescribes a lower inventory level $S$ since there is less need to compensate
for the congestion time when items are in repair.

Our numerical investigation has so far focused on the effect of varying
parameters on the optimal policy and the optimal cost. We did so under the
assumption that the underlying lifetime distribution is correct. However, in
practice, there may be misspecfication in this lifetime distribution as it is
often estimated from historical data. Fortunately, as our model permits a
general lifetime distribution, we can quantify how robust our model is with
respect to such misspecification. To do this, we proceed as follows. We will
compare two capital goods with two estimates of the lifetime distribution. In
the first case, we assume that the true litetime distribution is a gamma
distribution, but that the decision-maker wrongly assumes that it is a Weibull
distribution. In the second case, we do the opposite, i.e., the true lifetime
distribution is a Weibull distribution while the decision maker assumes it is
gamma distributed. Both lifetime distributions admit the same mean and the
same coefficient of variation. We are now interested in the impact of this
error on the total cost. That is, we compute the cost of the optimal policy
obtained under the wrong lifetime distribution when implemented in the system
with the true lifetime distribution, and then compare this cost with the
optimal cost for that latter system.

Our test bed consists of 3200 instances by permuting various values for
$C_{u}$, $C_{d}$, $1/\mu_{r}$, $K$, as well as the coefficient of variation
$C_{v}$ of the true lifetime distribution and the estimated one. For both
distributions the average lifetime is fixed to 2 time units. Table 1 contains
all parameter values. In Table 2 we provide the average of the observed
percentage cost difference as well as the maximum percentage difference over
all instances for a specific parameter value. The true lifetime distribution
can be found in the first column of the Table 2 (i.e. Weibull or Gamma); the
lifetime distribution as assumed by the decision maker will then be the other.

Figure 12: Example: Maximum percentage difference between total costs when the
lifetime is Weibull versus Gamma distribution Table 1: Test bed

Parameters | Values
---|---
$C_{d}$ | $\\{5,10,20,40\\}$
$C_{u}$ | $\\{1,5,10,20\\}$
$C_{v}$ | $\\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\\}$
$1/\mu_{r}$ | $\\{1,2,3,4\\}$
$K$ | $\\{1,2,3,4,5\\}$

On average over all instances, we observe that the percentage difference when
using a wrong estimation of the true lifetime distribution does not have much
impact on the total cost. While the maximum difference can go up to 8%, on
average it is only 0.23% (0.21%) when the true lifetime distribution is a
gamma (Weibull) distribution while the decision-maker assumes that it is
Weibull (gamma) distributed. The difference increases with the value of
$C_{u}$ and $K$ and decreases with the value of $C_{d}$, $C_{v}$ and
$1/\mu_{r}$. Particular attention should be paid to these cases, especially
when $C_{d}$, $C_{v}$ and $1/\mu_{r}$ are low because the system then requires
different threshold parameters $\tau$ and a stock level $S$ than those found
using the real lifetime distribution. The good performance over all instances
is due to the role of the threshold $\tau$ which offsets in most cases the
impact in total cost when using a wrong lifetime distribution instead of the
real one.

###  5.2 Multiple capital goods

While we developed an exact approach for the single capital good setting, we
relied on an approximative procedure for the multiple capital goods setting.
Our first objective is therefore to quantify how accurate our approximative
procedure is. To that end, we consider a setting with two capital goods 1 and
2 whose failure follows an Erlang distribution with a mean $\mu_{1}=1$ and
$\mu_{2}=2$ and a number of phases respectively equal to 3 and 6. The cost
parameters are fixed as follows: $C_{u}\in\\{10,20\\}$, $C_{d}\in\\{20,40\\}$,
$C_{a}=0.25$ and $C_{w}=0.75$. We also set
$\tau\in\\{0.25,0.5,1,1.25,1.5,1.75,2\\}$, $K\in\\{1,2,3,4,5\\}$ and
$\mu_{r}=1$. For this setting, we assess the value of our approximative
procedure by comparing our approximations with simulated values, both for the
first come first served and the priority repair shop discipline. The
simulation model is built in Arena Rockwell. The simulation length is set to
100000 units, which guarantees to obtain a response within an accuracy of
$10^{-3}$. Tables 3-5 provides the results of the comparison between our
approximative procedure and simulation. Based on these tables we conclude that
our approximations are quite accurate: the difference between the costs of our
model and the costs obtained through simulation is less than 2% for all
instances considered. This shows that our approximative procedure that
decomposes the multiple capital goods setting into single capital goods for
which we use our exact results performs well.

Now that we found that our approximations are accurate, the second objective
of our numerical investigation is to study the effect of using a priority
discipline in the repair shop on the overall performance. The presence of a
scheduling policy based on criticality and costs allows to reduce the downtime
and the dedicated stock levels for the most critical capital assets (i.e.
those with highest costs). We study two capital goods, and give priority to
capital good 1 over capital good 2. We do not add any additional cost due to
the priority to the repair shop. Since capital good 1 has higher priority, we
set the capital cost of capital good 1 higher than capital good 2. The results
of the comparison are presented in Table 6 and 7.

As the cost of downtime for the priority captial good is high and the cost of
failure before threshold is low, and for the non-priority capital good the
cost of downtime is low and the cost of failure is high, the potential savings
from prioritizing capital good 1 over capital good 2 can reach up to 19% over
first comes first served. This is especially true when the age threshold of
capital good 2 decreases. Indeed, this leads to more repair shop utilization
for capital good 2, which would delay repair for capital good 1 if served
under a first comes first served discipline. We also find that the age
threshold for capital good 1 has less impact on the performance of the
prioritized system than the non-prioritized system. Indeed, priority plays
somewhat the role of age-threshold to reduce the waiting time and thus the
total cost of the system.

Figure 13: Example: Priority versus non priority: $C_{u}=20$ and $C_{d}=40$
for the capital good with the highest priority

##  6 Conclusions

We have considered a repairable spare parts inventory system consisting of a
stock point for a single and multiple capital goods and a capacitated repair
shop. Parts are replaced either when there is a failure or when their lifetime
reaches an age threshold. We have modelled the system as as a cyclic tandem
queue network where the number of customers is fixed and equal to base stock
level. In the case of a single capital good, we have derived a closed form
expression of the steady-state probability of the number of orders in the
queuing system. After deriving some convexity properties of the steady state
probabilities and the cost function, we have proposed an algorithm to optimize
the number of repairables in the system, the age-threshold, the capacity of
the repair shop and the base stock level. In the case of multiple capital
goods, since a closed form expression of the steady state probability cannot
be derived, we provide an approximate expression by replacing in the queuing
network the general service time queues with state-dependent exponential
service time queues where the latter is obtained by analyzing each station
with a general service time in isolation but with a state-dependent arrival
rate. We have also considered multiple priorities in the repair shop and a
preemptive service discipline, which means that when a failed part with higher
priority arrives at the repair shop and another part with a lower priority is
being repaired, the server immediately stops repairing the latter and starts
with the former.

The numerical investigation reveals that the cost savings due to preventive
maintenance can be substantial. We have shown that in the system with single
capital good, when the capacity of the workshop increases, a lower age-
threshold should be used, which allows the asset to take advantage of
preventive maintenance and hence reduce costs. The results also show that the
system uses a low base stock level when the repair shop has a low capacity and
a low age-threshold. This base stock level increases with the capacity of the
repair shop until it reaches a fixed value that becomes less sensitive to the
capacity. Another finding of the numerical analysis is the robustness of our
model if there is a misspecification of the lifetime distribution instead of
the general distribution. Moreover, in the case of multiple capital goods, we
show the high quality of the derived approximation after comparing it to that
obtained through a simulation model and we also show the performance
improvement that results from using the priority discipline in the repair
shop.

It is important to notice that the analysis developed in this paper assumes
exponential service times in the repair shop. Hence, it would be interesting
to extend this work by considering generally distributed service times in the
repair shop in addition to the generally distributed lifetime of the parts.
Another restrictive assumption made in the case of multiple capital goods is
the consideration of a single repairmen in the repair shop, which facilitates
the priority rule in the system. Therefore, an interesting avenue for further
research would be to consider the case of multiple repairmen, which is complex
to analyze even in the case of exponentially distributed service times due to
the complexity related to the consideration of a scheduling policy for the
parts in the repair shop.

##  Appendix A Gamma versus Weibull distribution

Table 2: Percentage difference between total costs when the lifetime is Gamma
versus Weibull distribution.

Distribution | Parameters | Average | Maximum
---|---|---|---
setting | $K=1$ | $K=2$ | $K=3$ | $K=4$ | $K=5$ | $K=1$ | $K=2$ | $K=3$ | $K=4$ | $K=5$
Gamma | $C_{d}$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 8 | 6
5 | 0 | 0 | 0 | 0 | 0 | 2 | 7 | 7 | 8 | 5
10 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 4 | 6 | 6
20 | 0 | 0 | 0 | 0 | 0 | 3 | 5 | 5 | 7 | 5
40 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 5 | 7 | 5
Weibull | $C_{d}$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 6 | 5
5 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 5 | 5 | 4
10 | 0 | 0 | 0 | 0 | 0 | 3 | 7 | 7 | 6 | 5
20 | 0 | 0 | 0 | 0 | 0 | 4 | 5 | 5 | 4 | 3
40 | 0 | 0 | 0 | 0 | 0 | 4 | 5 | 4 | 4 | 4
Gamma | $C_{u}$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 8 | 6
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1
5 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 3 | 2 | 2
10 | 0 | 0 | 0 | 0 | 0 | 4 | 3 | 3 | 3 | 3
20 | 0 | 0 | 1 | 1 | 1 | 3 | 7 | 7 | 8 | 6
Weibull | $C_{u}$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 6 | 5
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1
5 | 0 | 0 | 0 | 0 | 0 | 3 | 2 | 2 | 2 | 2
10 | 0 | 0 | 0 | 0 | 0 | 4 | 5 | 4 | 3 | 3
20 | 0 | 0 | 1 | 0 | 1 | 4 | 7 | 7 | 6 | 5
Gamma | $Cv$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 8 | 6
0.1 | 0 | 0 | 0 | 0 | 1 | 3 | 5 | 5 | 4 | 4
0.2 | 0 | 1 | 0 | 0 | 0 | 2 | 7 | 5 | 3 | 4
0.4 | 0 | 0 | 0 | 0 | 0 | 1 | 5 | 2 | 4 | 2
0.6 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 3 | 3 | 4
0.8 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 2 | 2 | 4
Weibull | $Cv$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 6 | 5
0.1 | 0 | 0 | 1 | 1 | 1 | 0 | 7 | 7 | 6 | 5
0.2 | 0 | 1 | 0 | 0 | 0 | 4 | 5 | 5 | 3 | 3
0.4 | 0 | 0 | 0 | 0 | 0 | 4 | 3 | 4 | 3 | 2
0.6 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 4 | 2 | 2
0.8 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 2 | 4
Gamma | $L$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 8 | 6
1 | 0 | 0 | 0 | 0 | 0 | 4 | 5 | 7 | 8 | 6
2 | 0 | 0 | 0 | 0 | 0 | 2 | 7 | 5 | 4 | 4
3 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 2 | 4 | 3
4 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 3 | 3 | 4
Weibull | $L$ | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 6 | 5
1 | 0 | 0 | 0 | 0 | 0 | 4 | 7 | 7 | 6 | 5
2 | 0 | 0 | 0 | 0 | 0 | 4 | 3 | 5 | 5 | 4
3 | 0 | 0 | 0 | 0 | 0 | 1 | 5 | 4 | 2 | 3
4 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 3 | 3 | 3

##  Appendix B Comparison with Simulation

Table 3: Comparison with simulation under a fixed capacity $K=1$ and First
come first served repair shop discipline.

Input parameters | Our model | Simulation | Difference
---|---|---|---
$C^{2}_{u}$ | $C^{2}_{d}$ | $\tau_{1}$ | $\tau_{2}$ | $S_{1}$ | $S_{2}$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | in (%)
10 | 20 | 0.5 | 0.5 | 4 | 1 | 0.63 | 0.48 | 0.9 | 0.1 | 47.79 | 0.63 | 0.47 | 0.9 | 0.1 | 47.78 | 0
10 | 20 | 1 | 0.5 | 6 | 1 | 0.35 | 1.2 | 0.92 | 0.08 | 44.31 | 0.36 | 1.16 | 0.92 | 0.08 | 44.39 | 0.2
10 | 20 | 2 | 0.5 | 7 | 1 | 0.21 | 2.12 | 0.91 | 0.09 | 44.29 | 0.2 | 2.08 | 0.91 | 0.09 | 44.22 | -0.2
10 | 20 | 0.5 | 1 | 1 | 4 | 0.88 | 0.12 | 0.28 | 1.27 | 44.11 | 0.88 | 0.13 | 0.28 | 1.23 | 44.04 | -0.2
10 | 20 | 0.5 | 2 | 2 | 6 | 0.8 | 0.22 | 0.05 | 3.13 | 40.24 | 0.8 | 0.21 | 0.04 | 3.09 | 40.1 | -0.4
10 | 20 | 1 | 1 | 4 | 1 | 0.39 | 0.98 | 0.79 | 0.21 | 42.56 | 0.4 | 0.96 | 0.79 | 0.21 | 42.53 | -0.1
10 | 20 | 1 | 2 | 2 | 5 | 0.67 | 0.39 | 0.07 | 2.57 | 38.37 | 0.66 | 0.38 | 0.07 | 2.5 | 38.24 | -0.3
10 | 20 | 2 | 1 | 4 | 2 | 0.38 | 1.05 | 0.65 | 0.4 | 42.37 | 0.38 | 1 | 0.65 | 0.39 | 42.38 | 0
10 | 20 | 2 | 2 | 2 | 5 | 0.6 | 0.49 | 0.07 | 2.67 | 38.27 | 0.59 | 0.48 | 0.06 | 2.59 | 38.15 | -0.3
20 | 20 | 0.5 | 0.5 | 4 | 1 | 0.63 | 0.48 | 0.9 | 0.1 | 47.8 | 0.63 | 0.47 | 0.9 | 0.1 | 47.79 | 0
20 | 20 | 1 | 0.5 | 6 | 1 | 0.35 | 1.2 | 0.92 | 0.08 | 44.32 | 0.36 | 1.16 | 0.92 | 0.08 | 44.4 | 0.2
20 | 20 | 2 | 0.5 | 7 | 1 | 0.21 | 2.12 | 0.91 | 0.09 | 44.3 | 0.2 | 2.08 | 0.91 | 0.09 | 44.23 | -0.2
20 | 20 | 0.5 | 1 | 1 | 4 | 0.88 | 0.12 | 0.28 | 1.27 | 44.72 | 0.88 | 0.13 | 0.28 | 1.23 | 44.65 | -0.1
20 | 20 | 0.5 | 2 | 2 | 4 | 0.77 | 0.25 | 0.14 | 1.82 | 43.27 | 0.78 | 0.24 | 0.13 | 1.74 | 43.19 | -0.2
20 | 20 | 1 | 1 | 4 | 1 | 0.39 | 0.98 | 0.79 | 0.21 | 42.74 | 0.4 | 0.96 | 0.79 | 0.21 | 42.71 | -0.1
20 | 20 | 1 | 2 | 3 | 3 | 0.56 | 0.58 | 0.29 | 1.1 | 41.29 | 0.56 | 0.56 | 0.29 | 1.03 | 41.33 | 0.1
20 | 20 | 2 | 1 | 4 | 2 | 0.38 | 1.05 | 0.65 | 0.4 | 42.67 | 0.38 | 1 | 0.65 | 0.39 | 42.68 | 0
20 | 20 | 2 | 2 | 3 | 4 | 0.51 | 0.68 | 0.18 | 1.62 | 41.15 | 0.51 | 0.65 | 0.18 | 1.51 | 41.14 | 0
Fixed parameters: $C^{1}_{u}=20$, $C^{1}_{d}=40$,$C^{1}_{a}=C^{1}_{a}=0.25$,$C_{w}=0.5$. |

Table 4: Comparison with simulation under a fixed capacity $K=5$ and First
come first served repair shop discipline.

Input parameters | Our model | Simulation | Difference
---|---|---|---
$C^{2}_{u}$ | $C^{2}_{d}$ | $\tau_{1}$ | $\tau_{2}$ | $S_{1}$ | $S_{2}$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | in (%)
10 | 20 | 0.5 | 0.5 | 7 | 6 | 0 | 4.51 | 0.01 | 3.7 | 14.33 | 0.01 | 4.44 | 0.01 | 3.61 | 14.29 | 0
10 | 20 | 1 | 0.5 | 5 | 5 | 0 | 3.55 | 0.01 | 2.96 | 20.3 | 0 | 3.66 | 0.01 | 2.94 | 20.14 | 0
10 | 20 | 2 | 0.5 | 4 | 5 | 0.01 | 2.73 | 0.01 | 2.97 | 24.59 | 0 | 2.96 | 0.01 | 2.96 | 24.35 | 0
10 | 20 | 0.5 | 1 | 6 | 4 | 0 | 3.8 | 0 | 2.95 | 14.16 | 0 | 3.82 | 0 | 2.95 | 14.08 | 0
10 | 20 | 0.5 | 2 | 6 | 3 | 0 | 3.83 | 0 | 2.37 | 16.33 | 0 | 3.85 | 0 | 2.41 | 16.25 | 0
10 | 20 | 1 | 1 | 4 | 4 | 0.01 | 2.66 | 0 | 2.98 | 20.49 | 0.01 | 2.72 | 0 | 2.98 | 20.35 | 0
10 | 20 | 1 | 2 | 4 | 3 | 0.01 | 2.66 | 0 | 2.37 | 22.69 | 0.01 | 2.71 | 0 | 2.41 | 22.57 | 0
10 | 20 | 2 | 1 | 4 | 4 | 0.01 | 2.76 | 0 | 2.98 | 24.82 | 0 | 2.98 | 0 | 2.97 | 24.7 | 0
10 | 20 | 2 | 2 | 4 | 3 | 0.01 | 2.76 | 0 | 2.37 | 27.02 | 0 | 2.98 | 0 | 2.4 | 26.9 | 0
20 | 20 | 0.5 | 0.5 | 7 | 6 | 0 | 4.51 | 0.01 | 3.7 | 14.41 | 0.01 | 4.44 | 0.01 | 3.61 | 14.38 | 0
20 | 20 | 1 | 0.5 | 5 | 5 | 0 | 3.55 | 0.01 | 2.96 | 20.38 | 0 | 3.66 | 0.01 | 2.94 | 20.23 | 0
20 | 20 | 2 | 0.5 | 4 | 5 | 0.01 | 2.73 | 0.01 | 2.97 | 24.67 | 0 | 2.96 | 0.01 | 2.96 | 24.44 | 0
20 | 20 | 0.5 | 1 | 6 | 4 | 0 | 3.8 | 0 | 2.95 | 15.01 | 0 | 3.82 | 0 | 2.95 | 14.94 | 0
20 | 20 | 0.5 | 2 | 6 | 3 | 0 | 3.83 | 0 | 2.37 | 19.62 | 0 | 3.85 | 0 | 2.41 | 19.55 | 0
20 | 20 | 1 | 1 | 4 | 4 | 0.01 | 2.66 | 0 | 2.98 | 21.35 | 0.01 | 2.72 | 0 | 2.98 | 21.2 | 0
20 | 20 | 1 | 2 | 4 | 3 | 0.01 | 2.66 | 0 | 2.37 | 25.99 | 0.01 | 2.71 | 0 | 2.41 | 25.87 | 0
20 | 20 | 2 | 1 | 4 | 4 | 0.01 | 2.76 | 0 | 2.98 | 25.67 | 0 | 2.98 | 0 | 2.97 | 25.56 | 0
20 | 20 | 2 | 2 | 4 | 3 | 0.01 | 2.76 | 0 | 2.37 | 30.32 | 0 | 2.98 | 0 | 2.4 | 30.2 | 0
Fixed parameters: $C^{1}_{u}=20$, $C^{1}_{d}=40$,$C^{1}_{a}=C^{1}_{a}=0.25$,$C_{w}=0.5$. |

Table 5: Comparison with simulation under a fixed capacity $K=1$ and priority
rule at the repair shop.

Input parameters | Our model | Simulation | Difference
---|---|---|---
$C^{2}_{u}$ | $C^{2}_{d}$ | $\tau_{1}$ | $\tau_{2}$ | $S_{1}$ | $S_{2}$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | $P_{1}(0)$ | $E[I_{1}]$ | $P_{2}(0)$ | $E[I_{2}]$ | $TC(S,\tau,K)$ | in (%)
10 | 20 | 0.5 | 0.5 | 3 | 1 | 0.54 | 0.62 | 0.99 | 0.01 | 46.69 | 0.54 | 0.62 | 0.99 | 0.01 | 46.64 | 0
10 | 20 | 1 | 0.5 | 5 | 1 | 0.25 | 1.62 | 0.99 | 0.01 | 43.02 | 0.25 | 1.64 | 0.99 | 0.01 | 42.95 | 0
10 | 20 | 2 | 0.5 | 7 | 1 | 0.11 | 3.2 | 0.98 | 0.02 | 43.72 | 0.1 | 3.35 | 0.97 | 0.03 | 43.36 | 0
10 | 20 | 0.5 | 1 | 1 | 3 | 0.68 | 0.32 | 0.69 | 0.4 | 45.45 | 0.68 | 0.32 | 0.69 | 0.4 | 45.41 | 0
10 | 20 | 0.5 | 2 | 1 | 4 | 0.68 | 0.32 | 0.48 | 0.85 | 42.96 | 0.68 | 0.32 | 0.48 | 0.86 | 42.79 | 0
10 | 20 | 1 | 1 | 2 | 2 | 0.38 | 0.82 | 0.82 | 0.21 | 42.45 | 0.38 | 0.82 | 0.82 | 0.22 | 42.52 | 0
10 | 20 | 1 | 2 | 1 | 5 | 0.56 | 0.44 | 0.31 | 1.52 | 39.48 | 0.56 | 0.44 | 0.3 | 1.54 | 39.35 | 0
10 | 20 | 2 | 1 | 2 | 3 | 0.31 | 0.98 | 0.73 | 0.37 | 42.18 | 0.3 | 0.98 | 0.73 | 0.38 | 42.2 | 0
10 | 20 | 2 | 2 | 1 | 6 | 0.51 | 0.49 | 0.22 | 2.11 | 39.08 | 0.51 | 0.49 | 0.22 | 2.14 | 38.95 | 0
20 | 20 | 0.5 | 0.5 | 3 | 1 | 0.54 | 0.62 | 0.99 | 0.01 | 46.69 | 0.54 | 0.62 | 0.99 | 0.01 | 46.64 | 0
20 | 20 | 1 | 0.5 | 5 | 1 | 0.25 | 1.62 | 0.99 | 0.01 | 43.02 | 0.25 | 1.64 | 0.99 | 0.01 | 42.95 | 0
20 | 20 | 2 | 0.5 | 7 | 1 | 0.11 | 3.2 | 0.98 | 0.02 | 43.72 | 0.1 | 3.35 | 0.97 | 0.03 | 43.37 | 0
20 | 20 | 0.5 | 1 | 1 | 3 | 0.68 | 0.32 | 0.69 | 0.4 | 45.71 | 0.68 | 0.32 | 0.69 | 0.4 | 45.67 | 0
20 | 20 | 0.5 | 2 | 1 | 4 | 0.68 | 0.32 | 0.48 | 0.85 | 44.66 | 0.68 | 0.32 | 0.48 | 0.86 | 44.51 | 0
20 | 20 | 1 | 1 | 2 | 2 | 0.38 | 0.82 | 0.82 | 0.21 | 42.61 | 0.38 | 0.82 | 0.82 | 0.22 | 42.67 | 0
20 | 20 | 1 | 2 | 1 | 5 | 0.56 | 0.44 | 0.31 | 1.52 | 41.76 | 0.56 | 0.44 | 0.3 | 1.54 | 41.65 | 0
20 | 20 | 2 | 1 | 2 | 3 | 0.31 | 0.98 | 0.73 | 0.37 | 42.41 | 0.3 | 0.98 | 0.73 | 0.38 | 42.43 | 0
20 | 20 | 2 | 2 | 2 | 4 | 0.31 | 0.98 | 0.54 | 0.77 | 41.52 | 0.3 | 0.98 | 0.55 | 0.8 | 41.5 | 0
Fixed parameters: $C^{1}_{u}=20$, $C^{1}_{d}=40$,$C^{1}_{a}=C^{1}_{a}=0.25$,$C_{w}=0.5$. |

##  Appendix C Value of Priority versus age-threshold

Table 6: Value of Priority vs age-threshold

Cost parameters | Optimal policy | Optimal policy | Difference
---|---|---|---
Item 1 | Item 2 | with priority | without priority | in
$C_{u}$ | $C_{d}$ | $C_{u}$ | $C_{d}$ | $\tau_{1}$ | $\tau_{1}$ | $S_{1}$ | $S_{2}$ | $Z_{1}$ | $\tau_{1}$ | $\tau_{2}$ | $S_{1}$ | $S_{2}$ | $Z_{2}$ | %
1 | 20 | 1 | 5 | 5 | 5 | 7 | 1 | 9.55 | 5 | 5 | 7 | 1 | 9.92 | -4
1 | 20 | 5 | 5 | 5 | 2 | 7 | 1 | 9.65 | 5 | 2 | 8 | 1 | 10.43 | -8
1 | 40 | 1 | 5 | 5 | 5 | 10 | 1 | 11.18 | 5 | 5 | 12 | 1 | 12.65 | -13
1 | 40 | 5 | 5 | 5 | 2 | 10 | 1 | 11.25 | 5 | 3 | 12 | 1 | 13.04 | -16
5 | 20 | 5 | 5 | 5 | 2 | 6 | 1 | 13.23 | 5 | 2 | 6 | 1 | 13.74 | -4
5 | 40 | 1 | 5 | 5 | 5 | 9 | 1 | 14.9 | 5 | 5 | 11 | 1 | 16.18 | -9
5 | 40 | 5 | 5 | 5 | 2 | 10 | 1 | 14.98 | 5 | 3 | 12 | 1 | 16.6 | -11
1 | 10 | 20 | 5 | 5 | 1 | 5 | 1 | 8.56 | 5 | 1 | 5 | 1 | 8.95 | -5
1 | 20 | 10 | 5 | 5 | 1 | 7 | 1 | 9.69 | 5 | 1 | 8 | 1 | 10.74 | -11
1 | 20 | 20 | 5 | 5 | 1 | 7 | 1 | 9.72 | 5 | 1 | 8 | 1 | 10.87 | -12
1 | 40 | 10 | 5 | 5 | 1 | 10 | 1 | 11.28 | 5 | 2 | 13 | 1 | 13.33 | -18
1 | 40 | 1 | 10 | 5 | 5 | 9 | 1 | 15.95 | 5 | 5 | 10 | 1 | 16.55 | -4
1 | 40 | 20 | 5 | 5 | 1 | 10 | 1 | 11.3 | 5 | 1 | 13 | 1 | 13.43 | -19
1 | 40 | 5 | 10 | 5 | 4 | 10 | 1 | 16.04 | 5 | 5 | 11 | 1 | 17.01 | -6
5 | 20 | 10 | 5 | 5 | 1 | 6 | 1 | 13.28 | 5 | 1 | 7 | 1 | 14.05 | -6
5 | 20 | 20 | 5 | 5 | 1 | 6 | 1 | 13.31 | 5 | 1 | 7 | 1 | 14.19 | -7
5 | 40 | 10 | 5 | 5 | 1 | 10 | 1 | 15.01 | 5 | 2 | 12 | 1 | 16.89 | -13
5 | 40 | 20 | 5 | 5 | 1 | 10 | 1 | 15.03 | 5 | 1 | 12 | 1 | 16.99 | -13
5 | 40 | 5 | 10 | 5 | 4 | 9 | 1 | 19.75 | 5 | 5 | 10 | 1 | 20.5 | -4
1 | 20 | 20 | 10 | 5 | 1 | 7 | 1 | 14.55 | 5 | 1 | 7 | 1 | 15.1 | -4
1 | 40 | 10 | 10 | 5 | 2 | 10 | 1 | 16.12 | 5 | 2 | 11 | 1 | 17.46 | -8
1 | 40 | 20 | 10 | 5 | 1 | 10 | 1 | 16.18 | 5 | 1 | 12 | 1 | 17.91 | -11
5 | 40 | 10 | 10 | 5 | 2 | 9 | 1 | 19.84 | 5 | 2 | 11 | 1 | 20.97 | -6
5 | 40 | 20 | 10 | 5 | 1 | 9 | 1 | 19.9 | 5 | 1 | 11 | 1 | 21.44 | -8

Table 7: Priority versus non priority

|  |  | Captial good 1 with high priority
---|---|---|---
|  | $C_{u}$ | 1 | 5 | 10 | 20
| $C_{u}$ | $C_{d}$ | 5 | 10 | 20 | 40 | 5 | 10 | 20 | 40 | 5 | 10 | 20 | 40 | 5 | 10 | 20 | 40
Captial good 2 with low priority | 1 | 5 | 4 | -1 | -4 | -13 | 13 | 3 | 0 | -9 | 19 | 7 | -1 | -5 | 23 | 13 | 2 | -4
10 |  | 6 | 0 | -4 |  | 7 | 2 | -1 |  | 16 | 3 | 0 |  | 24 | 8 | 0
20 |  |  | 7 | 1 |  |  | 7 | 2 |  |  | 8 | 4 |  |  | 17 | 3
40 |  |  |  | 8 |  |  |  | 8 |  |  |  | 7 |  |  |  | 8
5 | 5 |  |  |  |  | 10 | 1 | -4 | -10 | 15 | 4 | -2 | -7 | 17 | 9 | 0 | -5
10 |  |  |  |  |  | 7 | 0 | -4 |  | 14 | 2 | -2 |  | 21 | 7 | -1
20 |  |  |  |  |  |  | 6 | 1 |  |  | 7 | 2 |  |  | 15 | 3
40 |  |  |  |  |  |  |  | 7 |  |  |  | 7 |  |  |  | 7
10 | 5 |  |  |  |  |  |  |  |  | 12 | 2 | -4 | -8 | 14 | 6 | -1 | -6
10 |  |  |  |  |  |  |  |  |  | 13 | 1 | -3 |  | 19 | 6 | -2
20 |  |  |  |  |  |  |  |  |  |  | 7 | 1 |  |  | 16 | 2
40 |  |  |  |  |  |  |  |  |  |  |  | 6 |  |  |  | 7
20 | 5 |  |  |  |  |  |  |  |  |  |  |  |  | 10 | 3 | -3 | -6
10 |  |  |  |  |  |  |  |  |  |  |  |  |  | 16 | 3 | -4
20 |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 16 | 1
40 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 7

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