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International Journal of Production Research
ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20
Optimal preventive maintenance strategy for
leased equipment under successive usage-based
contracts
Xiaolin Wang, Lishuai Li & Min Xie
To cite this article: Xiaolin Wang, Lishuai Li & Min Xie (2018): Optimal preventive maintenance
strategy for leased equipment under successive usage-based contracts, International Journal of
Production Research
To link to this article: https://doi.org/10.1080/00207543.2018.1542181
Published online: 08 Nov 2018.
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International Journal of Production Research, 2018
https://doi.org/10.1080/00207543.2018.1542181
Optimal preventive maintenance strategy for leased equipment under successive
usage-based contracts
Xiaolin Wanga∗, Lishuai Lia,band Min Xiea,b
aDepartment of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong;
bCity University of Hong Kong Shenzhen Research Institute, Shenzhen, People’s Republic of China
(Received 22 May 2018; accepted 16 October 2018)
In the context of equipment leasing, maintenance service is usually bundled with the leased equipment and offered by the
lessor as an integrated package under a lease contract. The lessor is then responsible to prescribe an effective maintenance
policy to keep the equipment operational in an economical way. This paper investigates upgrade and preventive maintenance
(PM) strategies for industrial equipment during successive usage-based lease contracts with consideration of a warranty
period, from the lessor’s perspective. The accelerated failure time model and age reduction model are adopted to capture the
effect of usage rate and imperfect PM/upgrade on the equipment reliability, respectively. More importantly, since equipment
usage rates may vary across different lease contracts, this study develops an age correspondence framework to characterise
usage rate shifts between successive lease periods. The optimal upgrade degree and the optimal number and level of PM
actions are progressively updated for each upcoming lease period to minimise the total expected lease servicing cost, by
considering the usage rate and maintenance implementation history. Numerical studies show that under given cost structures,
periodical PM activities within each lease period tends to outperform the pre-leasing upgrade actions, though both of them
can reduce the lease servicing cost.
Keywords: Successive leasing; usage-based contract; warranty; upgrade; maintenance management; cost analysis
1. Introduction
1.1. Background and motivation
Businesses require various types of equipment to manufacture their products or to provide services. For instance, mining
companies need excavators and dump trucks to load and transport mining materials. However, rapid technological obso-
lescence and increased complexity of equipment, coupled with high cost of ownership, make leasing instead of owning
equipment an economical option (Nisbet and Ward 2001), especially for small- and medium-sized companies. Meanwhile,
due to the lack of specialised maintenance tools and well-trained maintenance crews, equipment lessees tend to outsource
maintenance activities to lessors (Murthy and Jack 2014). In this context, maintenance service is usually bundled with the
leased equipment and offered by the lessor as an integrated package under a lease contract (Xia et al. 2018). As such,
prescribing an effective maintenance policy, especially preventive maintenance (PM), in the lease contract is under the
responsibility of the lessor, and it is of vital importance to both the lessee and the lessor.
This paper is interested in the maintenance optimisation for leased industrial equipment such as cranes, excavators, and
dump trucks, from the lessor’s perspective. For such equipment, there are three important characteristics in their leasing
problem that should be considered during the determination of an optimal maintenance programme:
(i) Warranty contracts. Nowadays, almost all products are sold with warranty contracts, and industrial equipment is
no exception (He et al. 2017; Xie, Shen, and Zhong 2017; Zhao, He, and Xie 2018). During the warranty period,
any eligible equipment failures are entitled to be rectified by the original equipment manufacturer (OEM). When
planning PM schedule for industrial equipment under lease, it is necessary to take the OEM warranty policy into
account.
(ii) Usage-based lease contracts. Industrial equipment often operates under harsh conditions, and the equipment
reliability is closely related to its usage. In practice, the lease contracts for many industrial equipment are char-
acterised by both lease duration and usage. This type of contract is called usage-based lease contract (Hamidi,
*Corresponding author. Email: xlwang28-c@my.cityu.edu.hk
© 2018 Informa UK Limited, trading as Taylor & Francis Group
2X. Wang et al.
Liao, and Szidarovszky 2016). It is worth mentioning that the usage-based lease contract is a special case of the
two-dimensional lease contract in Iskandar and Husniah (2017), with the usage limit being infinity.
(iii) Successive leasing. The useful life of most industrial equipment is quite long (typically 10–25 years). In this case,
a piece of equipment can be leased multiple times within its useful life, since a single lease period is usually much
shorter than such a long useful life. When a lease contract ceases, the lessor retains ownership of the equipment
and can renew the lease contract if the lessee is interested or lease the same equipment to a new lessee (Murthy and
Pongpech 2008). This is referred to as successive leasing. In fact, such successive leasing of the same equipment
is the core business of many leasing companies that specialise in leasing specific equipment to various lessees
(Ben Mabrouk, Chelbi, and Radhoui 2016b). In the successive leasing context, two essential aspects should be
considered: (a) during a specific lease period, the information about the lease durations and usage rates of future
lease contracts is often unavailable; and (b) when a piece of equipment is leased more than once (i.e. it becomes
a used one), its condition needs to be inspected and upgrade procedures may be required to improve its reliability
(Pongpech, Murthy, and Boondiskulchock 2006).
The three characteristics above raise a challenge to the lessors, namely, how to plan appropriate maintenance schedules
for leased industrial equipment under successive usage-based contracts. The existing approaches and models in the literature
are far from sufficient for assisting the lessors in making such decisions. The present paper intends to bridge this gap.
1.2. Literature review
In the literature, the optimisation of PM strategies for leased equipment has been reported extensively. Jaturonnatee, Murthy,
and Boondiskulchok (2006) made an early attempt to study a general PM policy for leased equipment. Yeh and Chang (2007)
investigated the optimal threshold value of failure rates for leased products with PM actions. Yeh, Chang, and Lo (2011)
jointly studied the optimal lease duration and PM policy for leased equipment to maximise the lessor’s expected profit.
Schutz and Rezg (2013) proposed two maintenance strategies for leased equipment to meet a minimum reliability require-
ment of the lessee. Moreover, a multi-phase PM policy for leased equipment was developed in Zhou et al. (2015), in which
the lease period was divided into multiple phases and different PM frequencies were applied in different phases. Zhou
et al. (2016) investigated a periodical PM strategy for leased equipment subject to continuous internal degradation and
stochastic external shock damage, simultaneously. Ben Mabrouk, Chelbi, and Radhoui (2016a) studied a PM policy for
leased equipment in a context where both PM actions and corrective repairs were imperfect. Hung, Tsai, and Chang (2017)
incorporated random failure penalties into the PM contract of leased equipment, and adopted the Black-Scholes equation
to characterise the expected revenue losses (i.e. the penalty) stemming from overdue repair. Recently, Xia et al. (2017)
developed a lease-oriented opportunistic maintenance methodology for multi-unit leased systems under product-service
paradigm. Some earlier literature on this topic can be referred to Chang and Lo (2011), Pongpech and Murthy (2006),
Pongpech, Murthy, and Boondiskulchock (2006), Yeh, Chang, and Lo (2010), and Yeh, Kao, and Chang (2009), among
others.
The studies above focused predominately on optimising various PM policies within a single time-based lease period.
Despite the importance of warranty contract, only three papers incorporated the warranty contract into the optimisation
of PM policies for leased equipment, i.e. Hajej, Rezg, and Gharbi (2015,2016) and Ben Mabrouk, Chelbi, and Rad-
houi (2016b). Nevertheless, all the three papers considered lessor warranty within the lease period, instead of the OEM
warranty. For the usage-based lease contracts, Hamidi, Liao, and Szidarovszky (2016) was the only work addressing this
issue. They developed a game-theoretic model, under which the lessee determined the optimal lease duration and usage
rate while the lessor was responsible for identifying an optimal PM policy. Furthermore, Ben Mabrouk, Chelbi, and Rad-
houi (2016b) was the sole research that studied the successive leasing problem. In their work, they implicitly assumed
perfect information regarding future lease durations, and then determined optimal upgrade levels between successive leasing
periods to maximise the lessor’s profit over the equipment lifecycle.
Our literature review reveals that the aforementioned three characteristics are either ignored or separately considered
in the literature, and currently no research that integrates all of them in the context of industrial equipment leasing, has
been reported. This paper contributes to the literature by developing an integrated framework to optimise upgrade and PM
strategies under successive usage-based lease contracts considering the OEM warranty.
1.3. Overview
In this paper, both upgrade and PM actions are considered for industrial equipment under successive lease. More specifically,
pre-leasing upgrade actions are carried out between successive lease periods; while imperfect PM actions are periodically
performed within each lease period, and the PM frequency can be altered only when a new lease period starts. The effect of
International Journal of Production Research 3
usage rate and imperfect PM/upgrade on the equipment reliability are captured by the accelerated failure time (AFT) model
and age reduction model, respectively. Since the equipment usually operates under different usage rates during different
lease periods, a statistical-virtual-age-based age correspondence framework is developed in this work. Furthermore, since
the information regarding future lease contracts may be unavailable in the successive leasing context, the ideal case, in
which the optimal PM schedule over the whole lifecycle is identified with pre-known lease durations and usage rates, is not
practical. Thus, this paper proposes to progressively update the PM schedule for each upcoming lease period considering
the usage rate and the upgrade/PM implementation history. Numerical studies show that both upgrade and PM strategies are
beneficial to the lessor in terms of lease cost reduction, while the periodical PM activities within each lease period tend to
be more cost-efficient than the pre-leasing upgrade procedures.
The remainder of this paper is organised as follows. Section 2formulates the model elements, including the equipment
failure model, the upgrade model, and the imperfect PM model. Then, Section 3develops and analyzes the optimisation
model of progressive upgrade and PM updating strategy. Section 4presents numerical examples and sensitivity analyses
to demonstrate the maintenance optimisation model. Finally, Section 5concludes this paper and presents several topics for
future research.
2. Model formulation
2.1. Problem description
Consider that a lessor purchases pieces of new industrial equipment sold with a two-dimensional warranty (W,U), where
Wand Uare time and usage limits, respectively. The OEM commits to rectify any equipment failures during the warranty
period. This warranty contract expires when either the equipment age exceeds Wor the total usage reaches U, whichever
occurs first. This policy is commonly adopted for industrial equipment such as cranes, excavators, and dump trucks (Ye and
Murthy 2016; Wang and Xie 2018). It should be noted that the warranty policy is often determined by the OEM considering
obligation or market competition, and is thus treated as exogenous from the lessor’s perspective.
Within the useful life of a piece of equipment, the lessor earns revenue by sequentially renting it to various lessees. When
a lease contract ceases, the equipment is returned to the lessor, who upgrades it to a better status. Then, the lessor may renew
the lease contract if the lessee is interested (the renewed contract is not necessarily identical to the original one) or lease it
to a new lessee. Denote the parameters of the jth usage-based lease contract by (Lj,rj), where Ljis the length of the lease
period and rjis the usage rate of the equipment (see Figure 1). Note that, since leasing market is typically lessee-oriented,
we suppose that from the lessor’s perspective, Ljand rjare deterministic quantities and their values are pre-specified by the
lessee.
Under a lease contract, the corrective maintenance of the equipment is billed to the lessor, unless the equipment is
protected by the OEM warranty. Besides, each equipment failure over the lease period would incur a penalty cost to the
lessor because a failure will result in downtime and thus production loss for the lessee. In this work, we consider both
upgrade and PM activities implemented to mitigate equipment degradation, and thus to reduce the total lease servicing cost.
The upgrade action is applied at the very beginning of each lease period (except the first one), while imperfect PM actions
are periodically performed within each lease period. In this manner, the decision variables for the jth lease period are the
upgrade level qj(q1=0), the number of PM actions nj, and the corresponding PM level mj(see Figure 1).
It is necessary to point out that due to the successive leasing manner mentioned above, the lessor should progressively
update the maintenance schedule based on the upcoming lease contract and the upgrade/PM implementation history during
previous lease periods. Another thing noteworthy is that in some countries, industrial equipment such as trucks should be
mandatorily discarded after certain years, say, Tlife, or after certain usage, say, Ulife, whichever occurs first. In this case,
before signing the jth lease contract, the lessor has to check whether the equipment will run out of its useful life or not, at
Figure 1. The framework of upgrade and PM during successive lease periods.
4X. Wang et al.
the end of this lease period. In other words, the equipment can be leased for the jth contract if and only if
j
i=1
Li≤Tlife and j
i=1
Liri≤Ulife.(1)
Since the lessor does not have the information regarding future contracts in advance, the maximal number of lease periods
Jis known only when the proceeding condition (1) is violated. That is to say,
J=maxj
j
i=1
Li≤Tlife and j
i=1
Liri≤Ulife .(2)
2.2. Modelling equipment failures
Generally speaking, industrial equipment gradually degrades with age and usage. It is necessary to incorporate the effect of
usage rate into the equipment failure model, especially for the usage-based contracts. In this study, the marginal approach
is employed to model the failure intensity λ(t|r)in terms of time tand usage rate r, in which the usage rate is treated as
a covariate (Jack, Iskandar, and Murthy 2009; Wang and Xie 2018). Specifically, a well-known variation of the marginal
approach, namely, the AFT model, is adopted here; see Iskandar and Husniah (2017), Tong, Song, and Liu (2017), and Ye
et al. (2013), for reference.
An assumption adopted here is that the equipment usage rate is constant within any single lease period, but it may vary
across different lease periods. Further assume that before signing a lease contract, the lessor is able to know the equipment
usage rate in this single period in advance, which is specified by the lessee. The two assumptions above are appropriate for
industrial equipment. For example, excavators in mining plants normally operate in a routine manner, say, eight hours a day.
In this case, the usage rate can be tentatively identified before signing a specific lease contract. Nevertheless, usage rates of
the same equipment during different lease periods may be different, depending on the types and loads of the duties.
Consider a piece of equipment designed for some nominal usage rate r0.IfTr[T0] denotes the time to first failure under
usage rate r[r0], then using the AFT formulation we have (Jack, Iskandar, and Murthy 2009)
Tr
T0=r0
rγ,
where γ>0 is the acceleration factor.
Furthermore, if the distribution function and failure rate function of T0are given by F0(t)and λ0(t), respectively, then
under usage rate r, the distribution function and failure rate function of Trcan be expressed as
F(t|r)=F0tr
r0γ,(3)
and
λ(t|r)=f(t|r)
1−F(t|r)=r
r0γ
λ0tr
r0γ,(4)
where f(t|r)=dF(t|r)/dt. The parameters of F(t|r)and λ(t|r)can be estimated from field failure data and/or warranty
claim data (Dai et al. 2017; Yang, He, and He 2016). Interested readers are referred to Wang and Xie (2018) and Wu (2013)
for a summary of statistical methodologies of two-dimensional warranty data analysis.
In this study, any failures within the lease periods are assumed to be rectified through minimal repairs. The minimal
repair assumption is quite appropriate for complex industrial equipment where an equipment failure occurs due to a com-
ponent failure and the equipment can be made operational by replacing the failed component with a new identical one
(Murthy 1991). As a consequence, the equipment failure rate after repair is nearly the same as that just before failure. Under
this assumption, it is well known that equipment failures over time will occur according to a non-homogeneous Poisson
process (NHPP) with the intensity function having the same form as the failure rate for the time to first failure, i.e. λ(t|r).
2.3. Modelling upgrade and PM actions
In this study, pre-leasing upgrade and post-leasing PM actions are simultaneously considered to improve the equipment
reliability for better operational performance. This section presents the modelling and analysis of the upgrade and imperfect
PM actions in detail.
International Journal of Production Research 5
2.3.1. Modelling imperfect PM actions
In practice, a PM action might include a set of maintenance tasks such as cleaning, lubricating, adjusting/calibrating, system-
atic inspection, and/or component replacements (Ben Mabrouk, Chelbi, and Radhoui 2016b; Zhao et al. 2018). The impact
of such a PM action on the equipment reliability is usually imperfect, which corresponds to imperfect PM. In this work,
the discrete PM modelling framework in Kim, Djamaludin, and Murthy (2004) is employed to characterise the effect of
imperfect PM actions. This discrete PM model has been frequently adopted by many studies; see Ben Mabrouk, Chelbi, and
Radhoui (2016b), Darghouth, Chelbi, and Ait-kadi (2017), Su and Wang (2016b), and Wang and Su (2016), for example.
Assume that the duration required to perform a PM action is very short comparing to a lease period, thus is negligible.
Let njdenote the number of PM actions applied within the jth lease period, then the corresponding PM interval is j=
Lj/(nj+1),j=1, 2, ...,J. Meanwhile, denote the kth, k=1,2, ...,nj, PM instant during the jth lease period by τj,k, then
we have
τj,k=
j−1
i=1
Li+kLj
nj+1,j=1,2,...,J;k=1,2,...,nj,(5)
with τ1,0 =0, τj,nj+1=τj+1,0 =j
i=1Lifor j=1, 2, ...,J(see Figure 1).
The underlying assumption adopted here is that a PM action rejuvenates the equipment so that it can effectively reduce
the equipment’s virtual age. The amount of age reduction is assumed to be proportional to thesupplement of equipment age
since the last PM action (Kijima 1989; Kim, Djamaludin, and Murthy 2004). As a result, within the jth lease period, the
virtual age immediately after the kth PM action is modelled as
vj,k=vj,k−1+δ(mj)(τj,k−τj,k−1),j=1,2, ...,J;k=1, 2, ...,nj,(6)
where mj∈[0,M] is the level of PM effort within the jth lease contract, and δ(mj)is the corresponding age reduction factor.
A PM level mjcorresponds to a specific set of maintenance tasks, and the corresponding δ(mj)can be estimated from
historical maintenance data. Notice that, larger value of mjindicates a greater PM effort, and thus δ(mj)is a decreasing
function with respect to mj. More specifically, if mj=0, then δ(0)=1, which implies that there is no PM; if mj=M
(impossible to achieve in practice), then δ(M)=0, which means that this model does not allow PM effect to be perfect;
while if mj∈(0,M), then the PM action is imperfect. In this study, we assume δ(mj)=(1+mj)e−mj, as in Kim, Djamaludin,
and Murthy (2004).
2.3.2. Modelling upgrade actions
As mentioned earlier, when a piece of equipment is leased more than once, it is no longer a new equipment (but a used
one). In this case, before re-leasing, upgrade effort may be required to improve its initial reliability status to a certain
extent. Although upgrade strategies for used equipment have attracted much attention in the area of warranty management
(Diallo et al. 2017; Khatab, Diallo, and Sidibe 2017; Su and Wang 2016a; Wang, Xie, and Li 2018), this topic still needs
more investigations in the successive leasing context. In this work, we consider imperfect upgrade actions to be performed
between successive lease periods, along with the periodical PM actions within each lease period.
Here, the impact of an imperfect upgrade action is also described by the age reduction approach (Shafiee and
Chukova 2013; Wang, Xie, and Li 2018). Let ˆvj,0,j=2, 3, ...,J, denote the equipment virtual age at the very begin-
ning of the jth lease period (i.e. before upgrade), which will be elaborated later. The effect of an imperfect upgrade action
lies in reducing the equipment virtual age from ˆvj,0 to vj,0 =(1−qj)ˆvj,0,qj∈[0, 1]. In this manner, a larger qjrepresents a
greater upgrade effort, which corresponds to a ‘younger’ equipment virtual age after upgrade.
Following Pongpech, Murthy, and Boondiskulchock (2006), the upgrade cost is modelled as an increasing function of
qj, and is given by
Cu(qj)=Csqjˆvj,0
1−exp−ϕˆvj,0(1−qj),j=2,3,...,J,(7)
where Cs>0 and ϕ>0.
Note that, if qj=0 (i.e. no upgrade), then Cu(0)=0; if qj→1, then Cu(qj)→∞, which implies that it is economically
not possible to upgrade a used equipment to an as-good-as-new state. Notice also that, there is no need to carry out an
upgrade action at the beginning of the first lease period, as the equipment is new. Thus, q1is set to zero, and Cu(q1)=0.
6X. Wang et al.
2.3.3. Age correspondence framework
Since industrial equipment often operates under different usage rates over different lease periods, an age correspondence
framework is developed in this section to characterise the usage rate shifts between successive lease periods.
By combining the upgrade and PM effects together, vj,kin (6) can be further derived as
vj,k=1−qjˆvj,0 +kδ(mj)j,j=1,2, ...,J;k=1, 2, ...,nj,(8)
with v1,0 =0 and q1=0.
In Equation (8), ˆvj,0 is closely related to the equipment’s virtual age at the end of the (j−1)th lease period, which is
vj−1,nj−1+1=vj−1,nj−1+j−1
=1−qj−1ˆvj−1,0 +nj−1δ(mj−1)j−1+j−1,j=2,3, ...,J.(9)
Since the equipment is supposed to operate under different usage rates during the (j−1)th and the jth lease periods,
vj−1,nj−1+1and ˆvj,0 are generally not identical, unless rj−1=rj. To establish an age correspondence for the equipment under
different usage rates, the concept of statistical virtual age in Finkelstein (2007) is adopted in this study. With this concept,
operating the equipment for vj−1,nj−1+1time units under usage rate rj−1is equivalent to operating it for ˆvj,0 time units under
usage rate rj(Wang and Xie 2018). From the reliability perspective, this statement corresponds to
Fvj−1,nj−1+1|rj−1=Fˆvj,0 |rj,j=2, 3, ...,J. (10)
Substituting (3) and (9) into (10), the equipment virtual age (before upgrade) at the very beginning of the jth, j=2, 3, ...,
J, lease period, i.e. ˆvj,0, can be expressed as
ˆvj,0 =vj−1,nj−1+1rj−1
rjγ
=1−qj−1ˆvj−1,0 +nj−1δ(mj−1)j−1+j−1rj−1
rjγ,j=2,3,...,J.
(11)
Accordingly, one can easily obtain vj,kby substituting (11) into (8).
Remark 1 As can be seen from Equation (11), ˆvj,0 decreases as the upgrade level qj−1, the number of PM actions nj−1,
and/or the corresponding PM level mj−1increases. A lower initial virtual age means that the equipment is statistically
Figure 2. Illustration of the effects of upgrade/PMs and usage rate shifts on the intensity function.
International Journal of Production Research 7
‘younger’, and it may require less upgrade and PM services during subsequent lease periods. This is the way that the
upgrade/PM implementation history Hj=(q1=0,n1,m1;...;qj−1,nj−1,mj−1)affects the maintenance optimisation during
subsequent lease periods.
For illustrative purpose, Figure 2visualises the effects of upgrade/PM actions and usage rate shift on the virtual age.
The periodical virtual age reductions within each single lease period are due to the effect of imperfect PM actions; while the
virtual age is updated twice between the first and the second lease periods, which stems from the shift of usage rates and the
upgrade action, respectively. Figure 2shows that operating the equipment for v1,6 time units under usage rate r1corresponds
to operating it for ˆv2,0 time units under usage rate r2. From (11), we have v1,6 <ˆv2,0 since r1>r2. It is worth mentioning
that the area under the curve λ(t|r1)from 0 to v1,6 is identical to that under the curve λ(t|r2)from 0 to ˆv2,0 (Hu, Jiang,
and Liao 2017). Furthermore, the upgrade action reduces the equipment’s virtual age from ˆv2,0 to v2,0, which effectively
improves the equipment reliability.
3. The optimisation model
In this section, the upgrade and PM optimisation model of industrial equipment under successive lease periods is developed.
The lessor’s problem is to progressively determine the optimal upgrade and PM strategy for the jth, j=1, 2, ...,J, lease
period in order to minimise the expected lease servicing cost within this period.
As mentioned before, within the warranty period, only failure penalty cost is incurred to the lessor; while after the
warranty expires, the lessor have to bear both the repair cost and the penalty cost. Thus, the warranty term (W,U)has an
important effect on the lessor’s total lease servicing cost. For the sake of simplicity, hereafter, we only consider the case
where the actual warranty length is shorter than the first lease period, i.e. min{W,U/r1}≤L1. It is not difficult to extend
our optimisation model to a general case, i.e. the OEM warranty terminates within the jth, j=1, 2, ...,J, lease period,
which is briefly introduced in Appendix.
3.1. The first lease period
During the first lease period, n1PM actions are periodically carried out with constant interval 1=L1/(n1+1). Since
equipment failures are minimally repaired with negligible durations, the failure process between any two successive PM
actions follows an NHPP. Thus, the expected number of equipment failures within the first lease period is given by
E[N1(n1,m1)]=
n1
k=0v1,k+1
v1,k
λ(t|r1)dt, (12)
where v1,kis given by (8).
Moreover, given that the actual warranty length Wris shorter than the first lease period, i.e. Wr=min{W,U/r1}≤L1,
the number of PM actions within the warranty period is d=max{k|k1<Wr}. The expected number of equipment
failures within (Wr,L1] can thus be obtained as
EN
1(n1,m1)=v1,d+1
v1,d+Wr−d1
λ(t|r1)dt+
n1
k=d+1v1,k+1
v1,k
λ(t|r1)dt.
Essentially, in the first lease period, the total cost to the lessor is comprised of the following four elements:
(i) Minimal repair cost. As mentioned before, the minimal repair cost for equipment failures over the period (Wr,L1]
is borne by the lessor. Given that the average cost of a minimal repair is Cf, the total expected minimal repair cost
is CfE[N
1(n1,m1)].
(ii) PM cost. As there are n1PM actions within the first lease period, the total PM cost is n1Cp(m1), where Cp(m1)is
the average cost of a PM with level m1.
(iii) Type I penalty cost. The Type I penalty cost is incurred to the lessor if the failed equipment is not restored
to an operational state within a reasonable time (Jaturonnatee, Murthy, and Boondiskulchok 2006). Let Yi
denote the repair duration of the ith failure, 1 ≤i≤N1(n1,m1). Then, the total Type I penalty cost will be
CtN1(n1,m1)
i=1max{0,Yi−¯
t}, where Ctis the Type I penalty cost per unit time and¯
tis a pre-set repair time threshold.
Hence, the total expected cost associated with Type I penalty can be derived as CtE[N1(n1,m1)]∞
¯
t[1 −G(y)]dy,
where G(y)is the distribution function of the repair time.
8X. Wang et al.
(iv) Type II penalty cost. If the number of equipment failures exceeds a pre-set threshold ¯n, a fixed penalty cost Cn
for each additional failure would be incurred to the lessor since this would result in production loss for the lessee
(Jaturonnatee, Murthy, and Boondiskulchok 2006). Without loss of generality, we consider ¯n=0 in this work,
which is reasonable for most industrial equipment. In this case, the total expected cost associated with Type II
penalty is CnE[N1(n1,m1)].
As a result, the lessor’s total expected servicing cost within the first lease period [0, L1] is the sum of these four elements,
and is given by
E[C1(n1,m1)]=CfEN
1(n1,m1)+n1Cp(m1)+ˆ
Ct+CnE[N1(n1,m1)]
=Cfv1,d+1
v1,d+Wr−d1
λ(t|r1)dt+
n1
k=d+1v1,k+1
v1,k
λ(t|r1)dt
+ˆ
Ct+Cnn1
k=0v1,k+1
v1,k
λ(t|r1)dt+n1Cp(m1),
(13)
where ˆ
Ct=Ct∞
¯
t[1 −G(y)]dy.
The lessor’s optimisation problem for the first lease period is thus to determine the optimal number and degree of PM
actions (i.e. n∗
1and m∗
1) to minimise the total expected lease servicing cost, as follows.
minn1,m1E[C1(n1,m1)] s.t. n1∈Z+∪{0},0≤m1≤M. (14)
It is difficult, if not impossible, to obtain closed-form solution to the optimisation problem (14). Fortunately, both n1and m1
are integers, so simple numerical search methods are efficient to identify the optimal solution.
3.2. The jth (j≥2)lease period
In this section, the optimisation model for the jth, j=2, 3, ...,J, lease period is presented, which is different from that of
the first period.
After the expiration of the (j−1)th lease contract, the lessor would rent the same equipment to a new (or the same)
lessee with negotiated lease length Ljand usage rate rj. Before re-leasing, the equipment is subject to an upgrade action to
recover its intrinsic reliability, and the upgrade level is qj. During the jth lease period, the lessor carries out njperiodical PM
actions with level mj. Then, the expected number of equipment failures is given by
ENjqj,nj,mj|Hj=
nj
k=0vj,k+j
vj,k
λ(t|rj)dt,j=2,3,...,J, (15)
where vj,kis given by (8) and Hj=(q∗
1=0,n∗
1,m∗
1;...;q∗
j−1,n∗
j−1,m∗
j−1),j=2, 3, ...,J.
Since the lessor has to bear the upgrade cost, PM cost, minimal repair cost, and failure penalty costs during the jth, j=
2, 3, ...,J, lease period, the total expected lease servicing cost can be derived as
ECjqj,nj,mj|Hj=Cf+ˆ
Ct+CnENjqj,nj,mj|Hj+Cu(qj)+njCp(mj)
=Cf+ˆ
Ct+Cnnj
k=0vj,k+j
vj,k
λ(t|rj)dt+Csqjˆvj,0
1−exp−ϕˆvj,0(1−qj)+njCp(mj).(16)
Therefore, the lessor’s optimisation problem for the jth lease contract is to determine the optimal upgrade level, the optimal
number and level of PM actions (i.e. q∗
j,n∗
j, and m∗
j) to minimise the total expected lease servicing cost, as follows.
minqj,nj,mjECjqj,nj,mj|Hj s.t. qj∈[0,1],nj∈Z+∪{0},0≤mj≤M,j=2,3, ...,J. (17)
As can be seen from (17), as the lessor sequentially leases the same equipment to a series of lessees, the optimal upgrade
and PM policy should be progressively updated for an upcoming lease period jbased on the contract information (Lj,
International Journal of Production Research 9
Figure 3. The progressive updating framework of maintenance optimisation.
rj) and the maintenance history Hj. This progressive nature is due to the unavailability of future contract informa-
tion in the successive leasing context. In summary, the proposed maintenance optimisation framework is illustrated in
Figure 3.
After the last lease period, the total expected servicing cost within the whole lifecycle of the equipment can be
determined as follows.
ETC q∗,n∗,m∗=
J
j=1
ECjq∗
j,n∗
j,m∗
j|Hj,
where q∗=(q∗
1=0,q∗
2,...,q∗
J),n∗=(n∗
1,n∗
2,...,n∗
J), and m∗=(m∗
1,m∗
2,...,m∗
J).
3.3. Model analysis
3.3.1. Special cases
Two special cases of the proposed upgrade and PM strategy are discussed, namely, only upgrade strategy and only PM
strategy, respectively.
(a) nj=0 and mj=0 (i.e. only upgrade strategy). In this case, only upgrade action is performed at the beginning of
each lease period. Then, the lessor’s total expected lease servicing cost at the jth lease period is given by
ECjqj,0,0 |Hj=⎧
⎪
⎪
⎨
⎪
⎪
⎩
CfL1
Wr
λ(t|r1)dt+ˆ
Ct+CnL1
0λ(t|r1)dt,j=1,
Cf+ˆ
Ct+Cnvj,0+Lj
vj,0
λ(t|rj)dt+Csqjˆvj,0
1−exp−ϕˆvj,0(1−qj),j=2,...,J,(18)
where vj,0 =(1−qj)ˆvj,0 =(1−qj)[(1−qj−1)ˆvj−1,0 +Lj−1](rj−1/rj)γ,j=2, 3, ...,J.
(b) qj=0 (i.e. only PM strategy). In this case, periodical PM actions are carried out within each lease period, while no
upgrade actions are performed between successive lease periods. Then, the lessor’s total expected lease servicing cost at the
10 X. Wang et al.
jth lease period is given by
ECj(0,nj,mj|Hj)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Cfv1,d+1
v1,d+Wr−d1
λ(t|r1)dt+
n1
k=d+1v1,k+1
v1,k
λ(t|r1)dt
+ˆ
Ct+Cnn1
k=0v1,k+1
v1,k
λ(t|r1)dt+n1Cp(m1),j=1,
Cf+ˆ
Ct+Cnnj
k=0vj,k+j
vj,k
λ(t|rj)dt+njCp(mj),j=2,...,J,
(19)
where vj,0 =ˆvj,0 =[ˆvj−1,0 +nj−1δ(mj−1)j−1+j−1](rj−1/rj)γ,j=2, 3, ...,J.
3.3.2. When is an upgrade action beneficial?
It is beneficial for the lessor to apply an upgrade procedure if the total expected leasing cost with both upgrade and PM
actions is less than that with only PM actions (qj=0). Let no
jand mo
jdenote the optimal number and level of imperfect PM
actions such that E[Cj(0,nj,mj|Hj)] in (19) is minimised. Then, the upgrade procedure is beneficial if
ECjq∗
j,n∗
j,m∗
j|Hj<ECj0,no
j,mo
j|Hj,forj=2,3, ...,J.
3.3.3. When are PM actions beneficial?
It is beneficial for the lessor to implement scheduled PM actions if the total expected leasing cost with both upgrade and
PM actions is less than that with only upgrade actions (nj=0 and mj=0). Let qo
j=0 denote the optimal upgrade degree
such that E[Cj(qj,0,0 |Hj)] in (18) is minimised. Then, the PM actions are beneficial if
ECjq∗
j,n∗
j,m∗
j|Hj<ECjqo
j,0,0 |Hj,forj=1,2, ...,J.
4. Numerical examples and sensitivity analyses
In this section, numerical examples are presented to demonstrate the applicability of the proposed maintenance optimisation
model and strategy. Also, detailed sensitivity analyses are performed on key model parameters to study the effect of input
uncertainty on the results.
4.1. Numerical study
Here the industrial equipment of interest are excavators made in China, which are sold with a warranty period of W=12
months and U=2000 hours, whichever occurs first. The excavator failures under nominal usage rate r0follow a two-
parameter Weibull distribution with scale parameter αand shape parameter β. Thus, using the AFT formulation (4), the
failure intensity under actual usage rate rcan be derived as
λ(t|r)=β
αt
αβ−1r
r0γβ. (20)
It is known that the excavators are designed to operate under nominal usage rate r0=0.167 ×103hours/month. Based
on warranty data analysis, the model parameters in (20) are obtained as α=1.24, β=1.20, and γ=3 (Yang, He, and
He 2016).
Within the useful life of an excavator, the lessor earns revenue by successively renting it to various lessees. Under a lease
contract, an excavator failure incurs an average minimal repair cost of Cf=$100 to the lessor. In addition, each excavator
failure would result in a Type II penalty cost of Cn=$100. The repair time of a failure follows Weibull distribution G(y),
with scale and shape parameters being 0.5 and 0.5, respectively, so that the mean repair time is one hour. If the repair time
exceeds t=2 hours, then there is a penalty Ct=$300 per additional hour. As a result, periodical imperfect PM actions
during each lease period are carried out by the lessor to improve the excavator reliability and thus reduce the repair and
penalty expenses. We consider δ(mj)=(1+mj)e−mjfor mj=0,1, ..., 5, and the corresponding PM costs are Cp(0)=$0,
International Journal of Production Research 11
Cp(1)=$10, Cp(2)=$30, Cp(3)=$60, Cp(4)=$100, and Cp(5)=$160, as in Kim, Djamaludin, and Murthy (2004). At
the same time, upgrade procedure is implemented at the beginning of each lease period to further improve the excavator’s
health status. The parameters of the upgrade cost are Cs=10 and ϕ=0.01. In this numerical study, the units for time and
usage rate are months and 1000 hours/month, respectively, while the unit for cost is US dollar ($), unless noted specifically.
In this example, we consider three successive lease contracts (L1,r1)=(36,0.151),(L2,r2)=(48,0.130), and
(L3,r3)=(30,0.173). For instance, the third lease contract lasts for 30 months and the negotiated usage rate is 173
hours/month (approximately 8 hours per working day). The grid search method is then adopted to obtain the optimal upgrade
level and the optimal number and level of PM actions in each lease period. It should be mentioned that q∗
j,j=2,3,...,J,
is searched with a step of 0.01. The computation time is quite short in this setting (several seconds). In this manner, the
solution quality and computation speed are good enough for practical use.
Based on the parameter settings above, the optimal upgrade and/or PM decisions under the three maintenance strate-
gies are listed in Table 1. The proposed strategy produces optimal solutions (n∗
1,m∗
1)=(6,5),(q∗
2,n∗
2,m∗
2)=(0.12,4,4),
and (q∗
3,n∗
3,m∗
3)=(0.47,6,4)for the three lease periods, respectively, with the total expected lifecycle lease cost being
$36771.7. For illustrative purpose, Figure 4shows the existence of the optimal maintenance decisions for the second lease
period. As can be observed, the optimal solution can be identified by the grid search method, though there is no closed form.
On the other hand, the only upgrade strategy leads to q∗
2=0.33 and q∗
3=0.54, with the total expected leasing cost being
$46924.7. While the only PM strategy leads to (n∗
1,m∗
1)=(6,5),(n∗
2,m∗
2)=(4,4), and (n∗
3,m∗
3)=(5,4), with the total
expected leasing cost being $37389.7. Notice that, the total expected leasing cost would be $48586.7, if neither upgrade nor
PM actions are carried out. Therefore, in this case, it is worthwhile to perform both upgrade and PM actions to reduce the
lease servicing cost, from the lessor’s perspective. This is not surprising since the only PM and only upgrade strategies are
special cases of the proposed upgrade and PM strategy, as shown in Section 3.3.1. In other words, their feasible domains
are subsets of that of the upgrade and PM strategy, thus their optimal solutions cannot be better than that of the combined
strategy.
To obtain more insights on the effect of input uncertainty on the results, comprehensive sensitivity analyses are presented
in the following sections, by varying one or two parameters at a time while holding the others unchanged.
Table 1. Summary of the optimal maintenance decisions.
Upgrade & PM Only upgrade Only PM
j=1j=2j=3j=1j=2j=3j=1j=2j=3
q∗
j– 0.12 0.47 – 0.33 0.54 – – –
n∗
j64 6 – – –64 5
m∗
j54 4 – – –54 4
E[Cj] 9256.8 10548.1 16966.7 11691.7 13795.1 21437.9 9256.8 10562.2 17570.7
TC 36771.7 46924.7 37389.7
Figure 4. The existence of the optimal upgrade and PM decisions for the second lease period.
12 X. Wang et al.
4.1.1. Sensitivity analysis of Cfand Cn
It is interesting to investigate the impacts of minimal repair cost Cfand penalty cost Cnon the results. The optimal upgrade
and/or PM decisions for various combinations of Cfand CnaresummarisedinTable2. It is necessary to point out that
the Type I cost Cthas similar effects on the optimal solutions to the Type II cost Cn, which can be easily known from
Equations (13) and (16). Therefore, we focus only on Cnin this analysis.
From Table 2, we have the following observations:
(i) The optimal upgrade degree q∗
j, optimal number of PM actions n∗
j, optimal PM degree m∗
jand the corresponding
lease servicing cost during the jth lease period tend to increase as the minimal repair cost Cfand/or the Type II
penalty cost Cnincreases. This is to be expected since the lease servicing cost tends to increase when the repair
cost and/or penalty cost increases. Hence, the lessor has to enhance the upgrade and PM efforts to improve the
excavator reliability.
(ii) The proposed upgrade and PM strategy always leads to the lowest total lifecycle cost among the three maintenance
strategies. This demonstrates the necessity of applying both upgrade and PM programmes for lease cost reduction
purpose. Also, under the given cost structures, the only PM strategy results in lower servicing costs than the only
upgrade strategy.
(iii) The optimal upgrade levels under the only upgrade strategy are higher than or equal to those under the combined
strategy, i.e. qo
j≥q∗
j,j=2, 3. This can be explained by the fact that without periodical PM activities, the upgrade
efforts under the only upgrade strategy should be enhanced to mitigate the equipment degradation. This is in line
with our intuition.
4.1.2. Sensitivity analysis of Csand ϕ
We then perform sensitivity analysis with respect to the upgrade cost parameters Csand ϕ. The results are summarised in
Table 3. Note that, the optimal PM decision under the only PM strategy is independent of Csand ϕ. Thus, the optimal PM
decision of this strategy, which has been listed in Table 1, is omitted in Table 3.
The following findings can be drawn:
(i) The optimal upgrade degree q∗
j, optimal number of PM actions n∗
j, and optimal PM degree m∗
jduring the jth lease
period tend to decrease as Csincreases, while increase as ϕincreases. This is reasonable since when Csincreases
and/or ϕdecreases, the lessor would lower the upgrade degree so as to avoid heavy upgrade expenses.
(ii) The total lease servicing costs under the only upgrade strategy are higher than that under the only PM strategy
($37389.7; see Table 1), which implies that periodical PM activities within each lease period is more cost-efficient
than the one-time upgrade procedure at the beginning of the lease period. This may be due to the maintenance
insufficiency in the first lease period (no upgrade and PM), which cannot prevent the equipment degradation during
subsequent lease periods effectively. This finding is reflected by the current maintenance practice in the sense that
industrial equipment should be subject to regular PM programmes, which include a set of maintenance tasks such
as cleaning, lubricating, adjusting, and/or replacing degraded components.
4.1.3. Sensitivity analysis of the OEM warranty term
As discussed earlier, the warranty term might have an important influence on the lessor’s total servicing cost. Figure 5
shows the total lifecycle lease cost under various warranty terms. It can be seen that the total lifecycle lease cost gradually
decreases with the length of the OEM warranty period, regardless of the maintenance strategies applied. The implication
of this finding is that the lessor should take the OEM warranty into account when making optimal maintenance decisions,
since the corrective maintenance cost under warranty is borne by the OEM.
4.1.4. Sensitivity analysis of the number of lease periods
The numerical analyses above are performed based on three lease periods, i.e. J=3. It is meaningful to investigate the
effectiveness of the three maintenance strategies under different numbers of lease periods (i.e. J). Figure 6shows the total
lifecycle lease cost of the three maintenance strategies under different J, with (L1,r1)=(36,0.151),(L2,r2)=(48,0.130),
(L3,r3)=(30,0.173), and (L4,r4)=(48,0.130). It is intuitive to observe that the total lifecycle cost increases as the number
of lease period Jincreases. Moreover, the total lifecycle cost of the combined upgrade and PM strategy is always lower than
those of the other two strategies, and the cost difference between them shows upward trend as Jincreases. This observation
International Journal of Production Research 13
Table 2. Comparison of the three maintenance strategies with respect to Cfand Cn.
Upgrade & PM Only upgrade Only PM
(Cf,Cn)j=1j=2j=3j=1j=2j=3j=1j=2j=3
(50, 50) q∗
j– 0.00 0.22 – 0.00 0.31 – – –
n∗
j533–––533
m∗
j434–––433
E[Cj] 5266.0 5986.7 9986.5 6277.8 7443.3 12132.0 5266.0 5986.7 10035.7
TC 21239.2 25853.1 21288.4
(100, 50) q∗
j– 0.00 0.34 – 0.06 0.43 – – –
n∗
j534–––533
m∗
j444–––444
E[Cj] 6317.0 7671.4 12468.7 7729.0 9607.4 15397.5 6317.0 7671.4 12638.1
TC 26457.1 32733.9 26626.5
(150, 50) q∗
j– 0.02 0.43 – 0.23 0.49 – – –
n∗
j535–––534
m∗
j544–––544
E[Cj] 7336.0 9113.4 14858.1 9180.2 11724.8 18423.0 7336.0 9113.6 15208.8
TC 31307.5 39328.0 31658.4
(50, 100) q∗
j– 0.02 0.43 – 0.23 0.49 – – –
n∗
j535–––534
m∗
j544–––544
E[Cj] 8271.7 9113.4 14858.1 10240.5 11724.8 18423.0 8271.7 9113.6 15208.8
TC 32243.2 40388.3 32594.2
(100, 100) q∗
j– 0.12 0.47 – 0.33 0.54 – – –
n∗
j646–––645
m∗
j544–––544
E[Cj] 9256.8 10548.1 16966.7 11691.7 13795.1 21437.9 9256.8 10562.2 17570.7
TC 36771.7 46924.7 37389.7
(150, 100) q∗
j– 0.17 0.50 – 0.40 0.58 – – –
n∗
j857–––855
m∗
j544–––544
E[Cj] 10218.3 11838.6 19000.1 13142.9 15831.5 24430.0 10218.3 11871.2 19806.3
TC 41057.0 53404.4 41895.7
(50, 200) q∗
j– 0.17 0.50 – 0.40 0.58 – – –
n∗
j857–––855
m∗
j544–––544
E[Cj] 11091.1 11838.6 19000.1 14203.3 15831.5 24430.0 11091.1 11871.2 19806.3
TC 41929.8 54464.8 42768.5
(100, 200) q∗
j– 0.26 0.53 – 0.46 0.61 – – –
n∗
j868–––856
m∗
j544–––544
E[Cj] 12032.8 13324.1 21087.3 15654.5 17841.6 27386.9 12032.8 13400.4 22389.2
TC 46444.2 60882.9 47822.4
(150, 200) q∗
j– 0.31 0.58 – 0.50 0.63 – – –
n∗
j867–––867
m∗
j545–––544
E[Cj] 12974.5 14784.0 23253.5 17105.6 19830.4 30341.3 12974.5 14927.9 24784.4
TC 51012.0 67277.3 52686.8
14 X. Wang et al.
Table 3. Comparison of the three maintenance strategies with respect to Csand ϕ.
Upgrade & PM Only upgrade
(Cs,ϕ) j=1j=2j=3j=1j=2j=3
(1, 0.01) q∗
j– 0.80 0.86 – 0.86 0.87
n∗
j667–––
m∗
j545–––
E[Cj] 9256.8 9577.1 14372.5 11691.7 12045.7 18391.2
TC 33206.5 42128.7
(5, 0.01) q∗
j– 0.46 0.64 – 0.60 0.69
n∗
j657–––
m∗
j544–––
E[Cj] 9256.8 10323.0 15918.0 11691.7 13271.8 20179.0
TC 35497.8 45142.5
(10, 0.01) q∗
j– 0.12 0.47 – 0.33 0.54
n∗
j646–––
m∗
j544–––
E[Cj] 9256.8 10548.1 16966.7 11691.7 13795.1 21437.9
TC 36771.7 46924.7
(1, 0.05) q∗
j– 0.92 0.94 – 0.94 0.95
n∗
j678–––
m∗
j545–––
E[Cj] 9256.8 9063.1 13575.7 11691.7 11304.6 17528.1
TC 31895.6 40524.5
(5, 0.05) q∗
j– 0.80 0.86 – 0.86 0.87
n∗
j667–––
m∗
j545–––
E[Cj] 9256.8 9593.5 14381.8 11691.7 12150.4 18433.7
TC 33232.1 42275.9
(10, 0.05) q∗
j– 0.69 0.80 – 0.78 0.81
n∗
j657–––
m∗
j545–––
E[Cj] 9256.8 9918.7 15005.1 11691.7 12722.6 19100.7
TC 34180.6 43515.1
(1, 0.10) q∗
j– 0.95 0.96 – 0.96 0.96
n∗
j678–––
m∗
j545–––
E[Cj] 9256.8 8919.8 13386.7 11691.7 11100.3 17308.3
TC 31563.3 40100.3
(5, 0.10) q∗
j– 0.87 0.90 – 0.90 0.91
n∗
j668–––
m∗
j545–––
E[Cj] 9256.8 9338.2 13982.3 11691.7 11787.5 17989.9
TC 32577.3 41469.1
(10, 0.10) q∗
j– 0.80 0.86 – 0.86 0.87
n∗
j667–––
m∗
j545–––
E[Cj] 9256.8 9614.5 14393.6 11691.7 12295.4 18489.3
TC 33264.9 42476.4
International Journal of Production Research 15
Figure 5. Impact of the warranty term on the total lifecycle lease servicing cost.
Figure 6. Impact of the number of lease periods, J, on the total lifecycle lease servicing cost.
is essential for lessors because the cost benefit of implementing upgrade and PM activities becomes larger when a piece of
equipment goes through more lease contracts (and its total lease duration becomes longer).
4.2. Additional numerical experiments
In this section, additional numerical experiments are conducted, based on different and randomly-generated model
parameters. This will help us further validate the proposed maintenance model and strategy, and gain more managerial
insights.
In this experiment, we focus on three sets of model parameters, i.e. Weibull parameters (αand β), minimal repair and
penalty costs (Cfand Cn+ˆ
Ct), and upgrade cost parameters (Csand ϕ), which are key parameters associated with the
maintenance optimisation problem. Here, we select three levels for each parameter set. In total, there are 27 combinations
of these key parameters; see Table 4for details. Moreover, the parameter settings of PM level δ(mj)and PM cost Cp(mj)
are the same as those in Section 4.1. Other parameters are arbitrarily set as γ=1, r0=1, W=1, U=2, (L1,r1)=(3,0.8),
(L2,r2)=(4,1.2), and (L3,r3)=(2,0.8). In this experiment, the units for time and usage are changed to year and 104
kilometres, respectively.
The optimal lifecycle leasing costs of the proposed upgrade and PM strategy under different parameter combinations are
shown in Figure 7, along with those of the only PM and only upgrade strategies. The following findings can be obtained:
(i) The total expected lifecycle leasing costs of the upgrade and PM strategy are always lower than those of the only
PM and only upgrade strategies. This further demonstrates the cost effectiveness of this combined strategy, as
discussed earlier.
16 X. Wang et al.
Table 4. Combinations of the model parameters.
Index αβCfCn+ˆ
CtCsϕ
1 1.5 1.5 85 120 6.5 0.04
2 3.5 0.04
3 3.5 0.08
4 85 150 6.5 0.04
5 3.5 0.04
6 3.5 0.08
7 150 150 6.5 0.04
8 3.5 0.04
9 3.5 0.08
10 1.5 2.5 85 120 6.5 0.04
11 3.5 0.04
12 3.5 0.08
13 85 150 6.5 0.04
14 3.5 0.04
15 3.5 0.08
16 150 150 6.5 0.04
17 3.5 0.04
18 3.5 0.08
19 1 2.5 85 120 6.5 0.04
20 3.5 0.04
21 3.5 0.08
22 85 150 6.5 0.04
23 3.5 0.04
24 3.5 0.08
25 150 150 6.5 0.04
26 3.5 0.04
27 3.5 0.08
Figure 7. Total lifecycle lease servicing costs for different combinations of model parameters.
(ii) It can be observed that under parameter combination #3, the lifecycle leasing cost of the only upgrade strategy is
even lower than that of the only PM strategy. This is because that under this combination, the equipment failure
probability is relatively low, the repair and penalty costs are small, and the upgrade cost is small (with small Cs
and large ϕ). In this case, pre-leasing upgrade action at the beginning of each lease period is effective enough for
reducing the total leasing expenses. Nevertheless, this scenario may be rare in real applications.
(iii) Moreover, when the Weibull scale parameter αbecomes smaller and/or the shape parameter βbecomes larger
(corresponding to higher equipment failure rate), the lease cost growth under the upgrade and PM strategy is much
International Journal of Production Research 17
less significant than those under the only upgrade and only PM strategies, as shown in Figure 7. This also indicates
the superiority of this combined strategy.
(iv) Furthermore, Figure 7shows that the lifecycle lease servicing cost increases as Cfand/or Cn+ˆ
Ctincreases; while
it decreases as Csdecreases and/or ϕincreases. These observations are consistent with those before.
5. Concluding remarks
In this work, for the first time, the OEM warranty, the usage-based lease contract and the successive leasing manner are inte-
grated into the maintenance optimisation problem of leased industrial equipment. A modelling and optimisation framework
of a progressive upgrade and PM updating strategy is proposed to assist equipment lessors in making optimal decisions
on upgrade and PM activities during successive lease periods. Since industrial equipment generally operates under dif-
ferent usage rates during different lease periods, a new PM modelling framework is developed by combining the AFT
model, the age reduction model with the concept of statistical virtual age. Essentially, the progressive nature of the proposed
maintenance optimisation problem due to the successive leasing manner is highlighted.
Overall, this paper, on the one hand, provides a quantitative maintenance modelling and optimisation framework and,
on the other hand, draws several managerial insights/findings from the numerical studies, to facilitate lessors’ maintenance
decision-making process. The main finding is that it is worthwhile to implement both pre-leasing upgrade and post-leasing
PM strategies to minimise the total lease servicing cost. Nevertheless, under the given cost structures, the periodical PM
actions within each lease period appear to be more effective than the one-time upgrade procedure at the beginning of the
lease period, especially when the equipment failure rate is high. Consequently, the lessors should focus more on regular
maintenance tasks such as cleaning, lubricating, adjusting, and/or replacing degraded components, within each lease period.
Furthermore, the cost benefit of applying upgrade and PM activities becomes larger, when the equipment failure rate is
higher, the equipment goes through more lease contracts, and/or the OEM warranty is incorporated. These findings and
insights would be of importance to equipment lessors who seek the minimum cost to be generated from their leased industrial
equipment.
This study can be extended in several directions.
•In this paper, we considered that periodical PM actions are performed within each lease period; while aperiodic
PM policy with decreasing PM interval is of potential interest to further reduce the leasing cost.
•Besides, it is more realistic to consider that PM actions have non-negligible durations.
•Relaxing the minimal repair assumption and considering imperfect repairs also have practical value.
•Finally, this paper made the assumption that no information regarding the future usage rates will be known in
advance. However, it may be more interesting to consider the future usage in certain ways, e.g. the expected usage,
if the lessor manages a large fleet of industrial equipment for leasing purpose. This will help the lessor take into
account the whole equipment lifecycle, instead of focusing solely on each single lease period, when making optimal
maintenance decisions.
Acknowledgements
The authors are grateful to the associate editor and the anonymous referees for their helpful comments and suggestions to the original
version of this paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by the Research Grants Council of Hong Kong under a Theme-based Research Fund [grant number T32-101/15-
R] and a General Research Fund [grant number CityU 11203815], and also by the National Natural Science Foundation of China [grant
number 71532008, 71601166].
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Appendix
In Section 3, the optimisation model is derived in the case where the actual warranty length is shorter than the first lease period, i.e.
min{W,U/r1}≤L1. In this appendix, we present the optimisation model in a general case where the warranty period terminates within
the hth lease period, i.e. h−1
i=1Li<W<h
i=1Liand h−1
i=1Liri<U<h
i=1Liri,h=1, 2, ...,J; see Figure A1. In this case, hcan
be obtained by
h=min⎧
⎨
⎩
j
j
i=1
Li>Wor j
i=1
Liri>U⎫
⎬
⎭
.
As a result, the actual warranty length Wrin the time dimension should be
Wr=⎧
⎪
⎨
⎪
⎩
W,ifrh≤¯rh,
h−1
i=1
Li+U−
h−1
i=1
Liri rh,ifrh>¯rh,
where ¯rh=(U−h−1
i=1Liri)/(W−h−1
i=1Li).
It is worth mentioning that hand Wrcan be known only after the (h−1)th lease period. That is to say, we cannot know hand Wrin
advance, e.g. at the very beginning of the equipment lifecycle. This is also due to the successive leasing manner, which is the same as the
scenario of determining J.
When the warranty period terminates within the hth lease period, the number of PM actions implemented within the warranty period
[0,Wr]is h−1
i=1
ni+d=
h−1
i=1
ni+maxk
h−1
i=1
Li+kh<Wr.(A1)
Then, the expected number of equipment failures within the period (Wr,h
i=1Li]isgivenby
EN
h(qh,nh,mh|Hh)=vh,d+h
vh,d+Wr−h−1
i=1Li−dh
λ(t|rh)dt+
nh
k=d+1vh,k+h
vh,k
λ(t|rh)dt,(A2)
where dcan be obtained from Equation (A1).
20 X. Wang et al.
Figure A1. Illustration of the expiry of the warranty period in a general case.
Therefore, when the OEM warranty terminates within the hth lease period, the lessor’s total expected servicing cost within the jth, j
=1, 2, ...,J, lease period is given by
ECjqj,nj,mj|Hj=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ˆ
Ct+CnE[N1(n1,m1)]+n1Cp(m1),j=1,
ˆ
Ct+CnENjqj,nj,mj|Hj+Cu(qj)+njCp(mj),2≤j≤h−1,
ˆ
Ct+CnENjqj,nj,mj|Hj+Cu(qj)
+njCp(mj)+CfEN
jqj,nj,mj|Hj,j=h,
ˆ
Ct+Cn+CfENjqj,nj,mj|Hj+Cu(qj)+njCp(mj),h+1≤j≤J,
(A3)
where Cu(qj)is given by (7), E[N1(n1,m1)] by (12), E[Nj(qj,nj,mj|Hj)], j=2, ...,J, by (15), and E[N
h(qh,nh,mh|Hh)]by(A2).
The interpretation of (A3) goes like:
(i) when j≤h−1, the jth lease period is totally covered by the warranty period, then the lessor’s servicing cost includes only the
upgrade cost (except for the first period), PM cost and failure penalty costs;
(ii) when j=h, this lease period is partially overlapped with the warranty period, then the lessor has to pay for the minimal repair
cost within (Wr,h
i=1Li], in addition to the upgrade cost, PM cost and failure penalty costs; and
(iii) when h+1≤j≤J, all of the upgrade cost, PM cost, minimal repair cost, and failure penalty costs within the jth lease period
are borne by the lessor.
Finally, the optimisation problem for the jth lease period is also given by (17), by replacing E[Cj(qj,nj,mj|Hj)]with(A3).
