Proceedings of the 41st International Conference on Computers & Industrial Engineering
MAINTENANCE STRATEGY FOR LEASE EQUIPMENT
Jérémie Schutz, Nidhal Rezg
LGIPM – COSTEAM Project, UFR MIM, Ile du Saulcy, France
{schutz|rezg}@univ-metz.fr
Abstract - In case of leasing, the user rents equipment for a predetermined time. During this period, all
maintenance actions are performed by the lessor. In this research paper, the aim consists to determine an
optimal maintenance policy for ensuring a minimum reliability, required by the customer. Two strategies
are proposed: the first consists to perform preventive actions whenever the system reliability reaches a
predefined reliability threshold. These actions are characterized by a reduction of the system age. The
objective is therefore to determine the efficiency factor of the optimal maintenance minimizing
maintenance costs. For the second strategy, "improvement” actions replace the corrective actions during
an interval to be determined to minimize maintenance costs. The first strategy will be solved using a
numerical procedure and the second strategy uses an algorithm of discrete event simulation.
Keywords - Leasing, Preventive maintenances, Optimization, Simulation
1 INTRODUCTION
Recently, the most economically powerful
countries have been affected by financial crises.
These latest financial crises require a new of
strategies for managing industrial. It is undeniable
that the socio-economic development is rather
unfavorable for the growth of new businesses.
Also, we note an increase in lease strategies applied
to the industrial field. Rather than purchase a costly
production system, young companies prefer to rent
these facilities to assess the markets and their
prospects.
Thus, we remark in the literature several recent and
emerging works that emphasize lease strategies.
The early work on this area can be attributed to
Smith et al. in 1985, where the authors are only
interested by a financial vision. The first scientific
work, that include maintenance aspects, date of five
years. Jaturonnatee et al. (2006) deal the aspect of
leasing through the optimization of a maintenance
policy of sequential type. Indeed, they search the
optimal number of preventive actions, the date and
the efficiency factor for each preventive action. The
preventive maintenance effects are modeled by a
reduction of failure intensity.
During the past five years, many works have been
proposed by R.H. Yeh. In Yeh et al. (2007), the
main aim consists to minimize the maintenance
costs. In this research paper, the maintenance
policy consists to perform minimal corrective
actions when failures occur. The preventive
maintenance actions are realized when the failure
rate reaches a preset threshold, denoted . When
these preventive actions are performed, the
instantaneous failure rate is reduced of a preset
amount
. The minimization of maintenance
costs is done by determining the optimal values of
the parameters and . Furthermore, the authors
showed, for this maintenance policy, that the
optimal threshold of failure rate
is equal to the
efficiency degree of preventive maintenance .
In Yeh et al. (2009), the authors draw on works
conducted in 2007. The main difference lies in
preventive maintenance intervals. In this paper,
planning is based on a sequential maintenance
policy.
Therefore,
the
authors
examine
simultaneously the number of preventive actions to
perform and the time intervals between two
consecutive preventive actions. Moreover, they
proved that this maintenance policy is equivalent or
more effective than the policies developed by
Jaturonnatee et al. (2006) and Pongpech et al.
(2006).
Very recently, Yeh et al. (2011B) used another
approach to model the effect of preventive
maintenance activities. Indeed, in this work, the
effect of preventive maintenance is modeled by an
age reduction of the system. So after the th
maintenance, the age of the system is reduced by an
amount
(time unit). As in their work in 2009, a
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sequential maintenance policy is used. Depending 2 PROBLEM STATEMENT
of the Weibull distribution parameters, ie when the Before to describe precisely the problem, it seems
failure rate is increasing, the authors showed that appropriate to specify the different notations
the optimal efficiency degree of maintenance, employed in the remainder of this paper.
denoted , is the same for each preventive activity.
Moreover, they proved, for this particular case, that 2.1 Notations
this factor
is also the optimal time interval The various concepts and notations used in this
article are described below:
between two successive preventive actions.
: Lease period;
All these works have been resolved through the
: Number of preventive maintenance
development of efficient algorithms and properties actions;
of these maintenance policies.
: Efficiency factor of preventive
maintenance actions;
Usually, the maintenance policies are mainly based
: Optimal efficiency factor of preventive
on time intervals. In literature, we find often two maintenance based on the variable ;
types of maintenance policies: periodic or
: Functional age of the equipment after the
sequential. Sequential Maintenance policies are th
preventive maintenance action;
characterized by search of the optimal number of
: Duration of preventive interval, i.e. time
maintenance actions to be performed during a given
th
th
and
preventive
between
the
period and the optimal intervals between two
maintenances;
preventive maintenance activities. Periodic policy
th
: Duration for each
preventive
is characterized exclusively by the search of
number of preventive actions. In this case, the times maintenance interval, during which failures are
intervals are obtained by dividing the time horizon fixed by minimal actions;
: Hazard rate function;
by the number of preventive maintenance actions
(Barlow et al., 1965). In their paper, Zhou et al.
seuil : Reliability threshold required to perform
(2007) propose a new policy of preventive preventive activities;
maintenance. The latter is based on system
seuil : Reliability threshold at which corrective
reliability. The authors seek the optimal reliability actions are replaced by “improving” actions;
threshold to reach in order to perform preventive
: Total cost of maintenance;
actions.
: Cost of correction action;
: Cost of preventive activity based on
Our research work is based on the maintenance
efficiency
factor;
policy developed by Zhou et al. Unlike the author,
: Fixed cost of preventive action;
the reliability threshold is considered as a data
: Variable cost of preventive action;
given by the customer and we seek to determine the
: Cost of “improving” activity based on
efficiency degree of preventive maintenance.
Indeed, this degree is a variable to determine the efficiency factor;
: Fixed cost of “improving” action;
cost of preventive maintenance actions. Moreover,
we propose a new threshold, named “improvement”
: Variable cost of “improving” action;
reliability threshold. After this threshold, minimal
: Floor of (partie entière);
corrective actions are replaced by imperfect
: Ceil of .
maintenances.
The rest of this paper will be organized as follows.
Section 2 gives a description of the studied system.
Section 3 presents the mathematical formulation
and the used methodology to solve this problem.
Section 4 presents a numerical approach to
illustrate our methodology. Finally, section 5 gives
the conclusion and the future works.
2.2 Problem description
In this paper, we consider new equipment available
to a customer. During the lease period, the lessor
must perform all corrective and preventive actions.
Compared to the duration of the lease period, the
times related to maintenance actions, corrective or
preventive, are considered negligible.
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
equipment works. It is therefore coherent that the
cost of preventive action is divided into a fixed cost
(displacement, ...) and a variable cost depending on
the desired efficiency (qualified maintenance
personnel, duration ...).The cost model can be
expressed as follows:
seuil
(2)
Préventive maintenances
Corrective maintenances
Figure 1. Representation of the first
maintenance policy
Corrective actions are intended to return the
equipment in working condition and the failure rate
is unchanged. Therefore, it corresponds to the
minimal repair model "As Bad As Old" (Barlow et
al., 1965).
Preventive maintenance are characterized by an age
reduction of the system according to the model of
type II, developed by Kijima (1989) also known as
ARA∞ model (Arithmetic Reduction of Age
Memory ∞) (Doyen et al., 2004). Unlike the model
of type I, where the effect of maintenance is to
reduce the functional age by an amount
proportional to the time elapsed since the last
maintenance, the model of type II is characterized
by a reduction of the functional age itself. After the
th
preventive maintenance, the functional age of
the system can be expressed as:
This proposed maintenance strategy aims to ensure
to the customer a minimum reliability of its
equipment. The aim of this paper is to find the total
cost of maintenance actions for a minimum desired
reliability and consequently, to determine the
optimum dates of preventive maintenance. It is also
possible to determine the minimum reliability
required to minimize the total maintenance costs.
In a second step, improvements will be added to the
maintenance strategy described above. We seek an
during which the
interval
seuil
seuil
corrective maintenance actions will be replaced by
maintenance actions called "improving". The
effects of these “improving” actions are equivalent
to the effect of preventive actions, i.e. based on a
reduction of the age following the efficiency factor
. As these "improving" actions are not planned,
and they improve the state of the system, the cost
will exceed any corrective and preventive actions.
The various costs are defined as follows:
(3)
(4)
(1)
The costs of « improving » actions is based on
When the efficiency factor is equal to 0 and 1, preventive costs, i.e. with fixed and variables costs.
preventive
maintenances
are
respectively They are defined by:
considered as minimal and perfect. For all values in
(5)
the interval ]0, 1[, preventive activities are regarded
as imperfect actions. Consequently, the equipment
is restored to an intermediate state, between AGAN For this new maintenance policy, the decision
(As Good As New) and ABAO (As Bad As Old). As variables will be seuil et .
the efficiency factor is included in the interval ]0,
1[, it is logical that the cost of preventive actions 3 MATHEMATICAL FORMULATION
depends on the effectiveness of these preventive For the first maintenance policy, the aim consists to
actions. However, it would be illogical to consider determine, for the lease period, the relationship
a proportional cost to the efficiency factor. For between the effectiveness of maintenance actions
example, in a lease strategy, maintenance crews and total cost of maintenance. Subsequently, we
associated with the lessor, are not located where the
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
seek the optimal value of efficiency factor to
minimize maintenance costs.
For the lease period , the total cost of maintenance
is expressed, with the factor , by the following
equation:
Drawing on the work of Yeh (2007), one possibility
would be to consider
and
as two decision
variables and to cancel the two following partial
derivatives:
(6)
(11)
In eq. 6,
represents the average number of
failure for the lease period . The number of
preventive maintenance activities is computed as
follows:
(7)
In this first policy, the efficiency factor is constant.
Therefore, the intervals
are constant and
equal to
.
For the maintenance strategy, based on an age
reduction of the system, the average number of
failures is expressed by:
(8)
(12)
However, we note that the two decision variables
are incorporated into the function of failure rate.
Solving these two equations require the use of a
given distribution. To be generic, we opted for a
numerical solution. From equation (7), it appears
that the value of
is based on the value of .
Therefore, for a given , there is a particular
interval for the value of efficiency factor. From
Figure 1, we note that the value of is linked to the
duration of the last preventive maintenance
interval. As this duration belongs to the interval
, we obtain:
(13)
(9)
(14)
As preventive actions are performed when the
reliability reaches the threshold seuil , we obtain
the following equation:
Consequently, for a fixed , the values of
the interval below:
(10)
The optimal value of efficiency factor
is
obtained by canceling the partial derivative of cost
with respect to the efficiency factor. However, the
value of
appears in the determination of the
number (Eq. 7) and consequently in the floor.
are in
(15)
We just determine the interval of the efficiency
factor
based on the variable . Currently, we
seek to minimize the tests to perform about the
number of preventive actions, namely . The
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
number of preventive actions is minimal when the
effectiveness of maintenance actions tends to 1.
Also, we obtain:
Given ,
Compute
seuil
et
inc
from eq. 10
(16)
inc
By applying the same methodology to determine
the maximum value for , we obtain
when
. To avoid this problem, we will ensure
the existence of a local minimum, which validate
the existence of an optimal number of preventive
maintenance.
Compute and store
yes
no
if
yes
(17)
n
Figure 2. numerical procedure
The proof of this lemma is given by Appendix A.
Knowing that exists a
which satisfies the
relationships expressed by Eq. 17, it is possible to
use a numerical procedure (Figure 2).
seuil
seuil
(19)
(18)
The second maintenance policy differs from the
previous by adding “improving” actions. These
actions replace corrective actions when the
reliability
is
included
in
the
interval
.
If
the
reliability
of
the
equipment
seuil
seuil
reaches directly the value
seuil , imperfect
preventive maintenance is performed similarly to
the first policy.
The average number of failures, represented by
seuil , is given by:
For this policy, the total cost of maintenance is
expressed as follows:
seuil
else
(20)
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
Similarly to equation (10),
the following relation:
is linked to
seuil
Given parameters :
( , seuil , …)
,
,…
by
Generate
Compute
(21)
Unlike the first policy where the th preventative
maintenance intervals (for
) were constant,
they can be variable in this second policy.
ndeed, “improving” actions can restore the system
to a previous state and a new interval can begin…
as shown in Figure 3.
no
yes
no
seuil
seuil
yes
Preventive Maintenances
Corrective Maintenances
« Improving » maintenances
Interval
during
which
« improving » actions replace
corrective maintenances.
yes
Figure 3. Representation of the second
maintenance policy
no
The
relationship
previously established, i.e.
is no longer verifiable.
Display cost
seuil
Thus, it is difficult to express mathematically the
Figure 4. Simulation procedure
value of . For this reason, the determination of
decision variables
and
will be
seuil
computed by discrete event simulation algorithm 4 NUMERICAL EXAMPLE
To illustrate the research work presented in this
whose structure is given in Figure 4
paper, numerical examples will be based on the
following data:

(time unit)




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(Weibull distribution)
seuil
(money unit)
Proceedings of the 41st International Conference on Computers & Industrial Engineering

Total cost of maintenance actions
From the given data, it is possible to determine the
value which characterizes the duration
. This
duration represents the time between two
consecutive maintenances when the equipment is
considered as new. In our case, when the equipment
reaches the reliability of 0.7, .this duration
is
equal to 59.72 tu.
10000
9000
14
15
> 15
0,53
0,5
0,47
5551,28
5785,164
6020,23
In Table 2, we remark that the total maintenance
costs decrease as the threshold reliability decreases.
Indeed, the number of preventive actions is also
reduced. However, when the number of preventive
actions decreases, the value of the optimal
efficiency factor increases. This observation may
be explained by the fact that the cost of preventive
activities is relatively high compared to the costs of
corrective actions.
8000
Tableau 2. Cost variation based
on the parameter e
7000
6000
5000
e
0,95
0,9
0,85
0,8
0,75
0,7
0,65
0,6
0,55
0,5
0,45
0,4
0,35
0,3
0,25
0,2
0,15
4000
0,2
0,4
0,6
0,8
1
Efficiency factor of preventive maintenances
Figure 5. Cost variation based on the efficiency
factor of preventive maintenance
Figure 5 shows the variations of the total cost of
maintenance based on the efficiency factor. The
aim consists to ensure a minimum reliability of 0.7.
The minimum total cost (4860.69 mu) is obtained
when the efficiency factor is equal to 0.82. For this
value, there are 9 preventive maintenance activities.
Moreover, we note that the curve of total cost is
discontinuous. This discontinuity can be explained
by the number of preventive actions, which is
constant (i.e. an integer) on certain intervals. The
table below gives the number of preventive actions
for the various intervals of
Table 1. Interval of efficiency factor based
on the variable
8
9
10
11
12
13
[0,53 ; 0,56]
[0,50 ; 0,52]
]0,00 : 0,49]
[0,93 ; 1]
[0,92 ; 0,82]
[0,74 ; 0,81]
[0,69 ; 0,73]
[0,62 ; 0,67]
[0,57 ; 0,61]
0,93
0,82
0,74
0,69
0,62
0,57
5095,67
4860,69
4877,37
5035,92
5151,04
5337,63
0,73
0,76
0,77
0,8
0,76
0,82
0,83
0,75
0,79
0,84
0,77
0,85
0,78
0,89
0,82
0,99
0,9
1,421261
1,984437
2,427006
2,786974
3,248515
3,45712
3,7587
4,337675
4,524785
4,698711
5,29729
5,346565
6,003125
5,953941
6,660774
6,346025
7,275737
29
19
15
12
11
9
8
8
7
6
6
5
5
4
4
3
3
11241,96
7972,31
6634,77
5776,27
5245,43
4860,69
4582,13
4364,96
4195,79
4044,03
3951,85
3837,03
3798,51
3705,84
3697,59
3673,78
3649,99
For the second strategy, Figure 6 shows changes in
the total cost depending on the threshold of
reliability seuil . Although “improving” actions is
more expensive than corrective and preventive
actions, we find that for a value of seuil less than
0.76, the total cost of maintenance actions is
minimal. Indeed, when the system is near to
preventive action and it fails, it is interesting that
maintenance crews will move once to restore the
system to a previous state that twice (corrective
action and action preventive) within a relatively
short time. Instead, we remark that it is not useful
to make too early these “improving” actions
because the age reduction does not prove sufficient
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
Total cost of maintenance
actions
to cover costs incurred by these actions in relation
to corrective actions.
5500
5400
5300
5200
5100
5000
4900
4800
4700
0,71 0,74 0,77 0,8 0,83 0,86 0,89 0,92 0,95 0,98
Reliability threshold at which corrective actions are
rep aced by “ mprov ng” act on
Figure 6. Cost variation based
on the parameter e
5 CONCLUSION & PROSPECTS
In this paper we are interested in lease strategy.
Unlike existing work, performing of preventive
actions was determined from the minimum
acceptable level of reliability by the customer. Two
maintenance strategies have been studied. The first
strategy is characterized by use of preventive
actions when reliability of the system reaches a
threshold defined by the customer. We therefore
determine the optimal efficiency factor which
minimizes the total cost of maintenance. The
second strategy, based on the first, is characterized
by performing of "improving" actions. These
"improving" actions replace corrective actions
when system reliability is included in the interval
. The effectiveness of these actions
seuil
seuil
is similar to preventive maintenance. Indeed, after
an "improving" or preventive activity, the system is
restored to a previous age, following the value of
the efficiency factor , based on the Kijima model
of Type II. Both strategies have been solved using
numerical procedures and discrete event simulation
algorithms. These methods of resolution have been
chosen to allow a generic resolution, applicable to
any distribution (unlike analytical models
developed by Yeh et al. (2007, 2009, 2011A,
2011B).
In future, this work may be extended by
determining, for each strategy, optimal efficiency
factors for each preventive or "improving" action.
In these cases, the number of decision variables
becomes high. Also, the resolution of these issues
will be based on the development of heuristics or
the use of meta-heuristics. These meta-heuristics
have already solved works whose theme is fairly
near (Schutz et al., 2010).
6 REFERENCES
Barlow, R.E., Proschan, F. (1965). Mathematical
theory of reliability. John Wiley & Sons.
Doyen, L., Gaudoin, O. (2004). Classes of
imperfect repair models based on reduction of
failure intensity or virtual age. Reliability
Engineering and System Safety, 84(1), pp. 45–
56.
Jaturonnatee,
J.,
Murthy,
D.N.P.,
R.
Boondiskulchok, R. (2006) Optimal preventive
maintenance of leased equipment with
corrective minimal repairs. European Journal of
Operational Research, 174(1), pp. 201-215.
Kijima, M. (1989) Some results for repairable
systems with general repair. Journal of Applied
Probability, 26 pp. 89-102.
Pongpech, J., Murthy , D.N.P. (2006) Optimal
periodic preventive maintenance policy for
leased equipment, Reliability Engineering &
System Safety, 91(7), pp. 772-777.
Schutz, J., Rezg, N., Léger, J.-B. (2010) Periodic
and sequential preventive maintenance policies
over a finite planning horizon with a dynamic
failure
law,
Journal
of
Intelligent
Manufacturing, 10.1007/s10845-009-0313-7.
Smith, C.W., Macdonald Wakeman, J.-R.,
Macdonald Wakeman, L. (1985) Determinants
of Corporate Leasing Policy, The Journal of
Finance, 40(3), pp. 895-908.
Yeh, R.H., Chang, W.L., (2007) Optimal threshold
value of failure-rate for leased products with
preventive maintenance actions. Mathematical
and Computer Modelling, 46(5-6), pp. 730-737.
Yeh, R.H., Kao, K.-C., Chang, W.L., (2009)
Optimal preventive maintenance policy for
leased equipment using failure rate reduction.
Computers & Industrial Engineering, 57(1), pp.
304-309.
Yeh, R.H., Lo, H.-C., Yu, R.-Y., (2011a) A study
of maintenance policies for second-hand
products. Computers & Industrial Engineering,
60(3), pp. 438–444.
Yeh, R.H., Kao, K.-C., Chang, W.L., (2011b)
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products under various maintenance costs.
Expert Systems with Applications, 38(4), pp.
3558-3562.
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Proceedings of the 41st International Conference on Computers & Industrial Engineering
Zhou, X., Xi, L., Lee, J. (2007) Reliability-centered
predictive maintenance scheduling for a
continuously monitored system subject to
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7 APPENDIX A
The proof of the existence of
is given by:
Moreover, the limits to
gives:
constante
constant
So, there is a number
following relations:
constante
which satisfies the
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