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Random preventive maintenance policy based on inspection for a multicomponent system using simulation


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In today’s global situation where highly competitive companies demand production efficiently to reduce costs, increase product quality, and customer loyalty, maintenance becomes crucial to achieve this goal by reducing unplanned downtime, reworking of products, and costs. In this sense, the use of models that can represent this type of system, and help managers make decisions more easily, are of vital importance for companies. Thus, a preventive maintenance model for a multicomponent system with different failure mechanisms is proposed in this work. Considering that the objective is to optimize the number and the time of maintenance interventions, that will be done in the system, periodic inspections are carried out in order to minimize the expected costs of maintenance. The optimization was performed with simulation, which proved to be satisfactory, since the decision variables of the model behaved adequately when utilized within the context of an applied case study. In addition, these variables had different performances when analyzed in four different scenarios: the original model of the proposed policy, and three variations attributing costs of penalties. © 2017, Polish Academy of Sciences Branch Lublin. All rights reserved.
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Eksploatacja i NiEzawodNosc – Ma i N t E N a N c E a N d RE l i a b i l i t y Vo l .19, No. 4, 2017
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1. Introduction
Many of today’s technological systems, such as airplanes, nuclear
power plants, military installations, advanced industrial, and medical
equipment, involve high levels of complexity in their maintenance
and operation and require a high level of availability and reliability
[10]. Furthermore, technological companies have been focusing their
activities to achieve operational efficiency in a highly competitive
global economy.
So, maintenance has been highlighted as a tool for this. It can be
said then, that maintenance are technical and administrative actions
that can maintain or restore the state of a system, so that the system
can perform its required functions to keep it in full operation. These
actions classically fall into two categories: preventive maintenance
(PM), and corrective maintenance (CM) [1].
PM activities are typically planned to help prevent the deterio-
ration and failure of the system [24], especially when serious con-
sequences due to failure occurs, which may incur very high costs.
Besides this, maintenance actions can reduce maintenance costs, and
improve the production efficacy of the system, and reliability and
availability. However, excessive PM itself can be expensive and time
consuming. Therefore, it is important to perform adequate and effec-
tive PM practices to balance maintenance activities and cost [27].
An efficient way to perform PM is through inspections. They
can have a variable depth level according to the system, and provide
information about the system’s operating state to perform repairs or
replacements [24, 18]. Zhang et al. [26] developed a PM model based
on inspection for mechanical components in three states with compet-
ing failure modes using an age limit and degradation for the system.
Scarf and Cavalcante [20] established a model based on inspection,
however for a single component that has three states: good, defective
and fail.
Many authors have been using Condition Based Maintenance
(CBM) to create models that take into account inspection, among
them Lam and Banjevic [16], Chen et al. [5], Zhu et al. [30], Dieulle
et al. [9], and Jardine et al. [14]. Other authors have also developed
models for maintenance optimization through inspections in different
Mônica Frank MARSARO
Cristiano Alexandre Virgínio CAVALCANTE
In today's global situation where highly competitive companies demand production eciently to reduce costs, increase product
quality, and customer loyalty, maintenance becomes crucial to achieve this goal by reducing unplanned downtime, reworking of
products, and costs. In this sense, the use of models that can represent this type of system, and help managers make decisions more
easily, are of vital importance for companies. Thus, a preventive maintenance model for a multicomponent system with dierent
failure mechanisms is proposed in this work. Considering that the objective is to optimize the number and the time of mainte-
nance interventions, that will be done in the system, periodic inspections are carried out in order to minimize the expected costs
of maintenance. The optimization was performed with simulation, which proved to be satisfactory, since the decision variables of
the model behaved adequately when utilized within the context of an applied case study. In addition, these variables had dierent
performances when analyzed in four dierent scenarios: the original model of the proposed policy, and three variations attributing
costs of penalties.
Keywords: random preventive maintenance; inspection; simulation; sugarcane plant.
W dzisiejszej sytuacji globalnej, w której przedsiębiorstwa o wysokim stopniu konkurencyjności wymagają efektywnego obniżania
kosztów produkcji, poprawy jakości produktów oraz zwiększania lojalności klientów, konserwacja ma zasadnicze znaczenie dla
osiągnięcia tych celów poprzez redukcję nieplanowanych przestojów, oraz zmniejszenie konieczności usuwania usterek produktów
a także obniżanie kosztów. W tym sensie, wykorzystanie modeli reprezentujących tego typu systemy i ułatwiające menedżerom pode-
jmowanie decyzji , ma kluczowe znaczenie dla rm. W tej pracy zaproponowano model konserwacji zapobiegawczej dla wieloe-
lementowego systemu o różnych mechanizmach uszkodzeń. Biorąc pod uwagę, że celem jest optymalizacja liczby i czasu trwania
zabiegów konserwacyjnych dokonywanych w systemie, przeprowadzane okresowe przeglądy mające na celu zminimalizow-
anie oczekiwanych kosztów utrzymania. Optymalizację przeprowadzono za pomocą symulacji, która okazała się zadowalająca,
ponieważ zmienne decyzyjne modelu zachowywały się odpowiednio przy wykorzystaniu ich w kontekście omawianego studium
przypadku. Dodatkowo, zmienne te przybierały różne wartości dla czterech różnych scenariuszy: pierwotnego modelu proponow-
anej polityki konserwacyjnej i trzech wariantów, w których uwzględniono koszty pracy systemu w stanie awaryjnym.
Słowa kluczowe: losowa konserwacja zapobiegawcza; przeglądy; symulacja; zakład przetwórstwa trzciny
MARSARO MF, CAVALCANTE CAV. Random preventive maintenance policy based on inspection for a multicomponent system using simu-
lation. Eksploatacja i Niezawodnosc – Maintenance and Reliability 2017; 19 (4): 552–559,
Eksploatacja i NiEzawodNosc – Ma i N t E N a N c E a N d RE l i a b i l i t y Vo l .19, No. 4, 2017
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N: Number of inspections in B before τ;
c1: inspection cost of equipment A.
c2: inspection cost of equipment B.
cp: replacement cost of A.
cr: system replacement cost (cp << cr)
cF: failure cost of equipment B – large loss due to the need for
aggressive maintenance, besides the lost profit.
cdt: downtime cost.
cB: cost of defect penalty in B – is counted from the time of the
arrived the defect in B until the moment it is replaced.
Refer to equipment A
k: number of inspections in A (k = 1, 2, ...).
q: number of failures in A.
yA: time to arrival the failure in equipment B.
Refer to equipment B
TB: time to inspection in equipment B (Cumulative).
x: time to arrival of the defect in equipment B.
h: delay time.
yB: time to arrival the failure in equipment B, defined as
Refer to system
SL: System Life
C: Maintenance Cost
Consider a system with two units, or a system composed of two
basic sets, in parallel, in which the system only stops when component
B stops working. Each of the components has a series of particulars
characteristics, as shown below.
Equipment A:
The failures have an abrupt form. That is, there is not a slow 1.
degradation that allows intermediate states before the failure.
Component A is working, and suddenly stops.
The failure of component A is not pronounced, because the 2.
system does not stop working. For this reason, it is only de-
tected by inspection. This type of failure can be considered as
a soft failure [4], because it can reduce the system reliability
and increase the risk of damage due to malfunctioning of the
system, despite not causing immediate failure of the system.
By the failure characteristics presented previously, they can
be classified as hidden failure. According to Taghipour and
Banjevic [22], this kind of failure does not stop the system,
at the moment that it happens; however, it can cause various
types of losses, such as performance (working inefficiently)
or economic (low production level or rework). This is be-
cause there is a time difference between the occurrence of
the failure and its detection [13, 29].
Component A is inspected at times 3. kT, with k = 1, 2, ..., so
there is a schedule of inspections to be performed on it in order
to identify whether it has already failed or not.
The inspection is simple and low cost, and it can be performed 4.
visually without using a complex equipment.
When a failure occurs, the system has a performance loss, in-5.
creasing the costs.
segments of, industry such as considering a high-speed milling tool
wear by Yan et al. [25], in pipes that undergo corrosion by Sahraoui et
al. [21], in reparable multicomponent hospital systems by Golmakani
and Moakedi [13], and for the equipment that are inspected by the
manufacturers in the warranty period with a cost charged to them, and
after this period with the cost charged to the customer [8].
Golmakani and Moakedi [13] were among who developed a main-
tenance cost optimization model based on periodic inspections for a
reparable multicomponent system. Therefore, the main difference be-
tween systems with a single component and systems with more than
one is that there normally exist some kind of dependence, structure or
economic dependence, between them that can be related to the failure
mechanism [4, 15].
The economic dependence is related to the fact that there can
be economic costs when maintenance is performed; the structural
dependence is related to the fact that the components form a part,
and that maintenance in one implies the maintenance of the other
[23]. In addition, the dependency related to the degradation occurs
when the failure rates in one component can affect the incidence in
another [7, 28].
In view of the complexity of the multicomponent systems, the
construction of analytical models becomes a very difficult task. Many
times, it needs simplifications that affect the ability to represent the
main aspects of the real problem. In this sense, what has been seen
is a rise in the use of simulation for maintenance optimization. The
simulation allows modeling a complex behavior and requires fewer
assumptions in relation to analytic modeling.
Although the simulation is well established for representation of
the operation of productive systems in general, it is still developing in
the maintenance area [3]. Simulation has traditionally been used as a
tool to understand and perform experiments with a system. However,
combining the simulation model with the optimization model ensures
better and faster results [2, 11, 12, 18].
The simulation may incur a number of benefits, among them are
experimentation in shorter time, because the model uses a compu-
ter; reduced analytical requirements; and more easy demonstration of
models. But some disadvantages can also be cited: simulation cannot
generate accurate results, when the input date is not accurate as well;
and it may not be easy to generate answers to complex problems [6].
In this way, the present paper intends, through the simulation, to
obtain performance metrics about one policy, in order to evaluate the
effectiveness of a policy that establishes simple procedures for the
maintenance manager. Once variations of this policy are shown to
be effective, we intend to optimize the policy decision variables for
different scenarios, which depict very feasible peculiarities of produc-
tion systems. The maintenance policy is based on inspections and re-
placement taking into account opportunities for a system composed of
two components whose failure characteristics are different, and which
have economic and structural dependency. We used the simulation to
find the lowest expected maintenance cost evaluating some decision
variables of the model described.
This paper is divided as follows: presentation and description of
the proposed maintenance model, with the characteristics of each of
the components, an experimental application of the model through a
case study in the sugarcane industry, showing the main results for the
adopted maintenance policy, and a sensitive analysis about the results
to evaluate the variables behavior.
2. Model Description
For this model, the follow variables and parameters are used:
Decision variables:
T: inspection time in A (Cumulative);
τ: threshold of system age;
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Equipment B:
The failure occurs in a slower way, so it is possible to identify 1.
intermediate states. That is, before the failure, the equipment
presents a defect. In this way, a component type B can undergo
three possible states: (1) Operational, (2) Defective, and (3)
Only it is possible to identify that component B is defective in 2.
state 2 by an inspection.
However, when component B fails, the identification is im-3.
mediate, because this failure is catastrophic, that is, it brings
a damaging consequence to the system. This type of failure is
called hard failure [4].
Component B is opportunistically inspected. This occurs on 4.
inspection of A when it is identified as fail.
The inspection in this component is more complex and incurs 5.
higher costs than the inspection of A.
In view of the characteristics of the equipment that comprises the
system presented previously, and the fact that the inspection of equip-
ment A is cheaper, as well, due to equipment type B presenting de-
fects, these allows the failures in equipment B to be avoided. So, the
following maintenance policy was established:
Equipment A is inspected at times kT, where k = 1, 2, …. If this
equipment is in the failed state, equipment B is inspected. If equip-
ment A does not present failure, then equipment B is not inspected.
The system is replaced when equipment B failed, or when this is
found in a defective state by an inspection, whichever happens first.
When an inspection reveal that equipment A failed, but the equipment
is in operational state, only equipment A is replaced with a new one.
Taking into account that equipment B failure has repercussions
of serious consequences, some maintenance actions are specified for
the equipment B, in order to not reach the defective state. The mainte-
nance actions that are performed in equipment B are:
Inspection when a failure in equipment A is found;
Preventive replacement when N + 1 successive failures have al-
ready occurred in equipment A;
Preventive replacement at the failure time of A, when on failure
of A the dispositive B already operated for a longer time than τ.
It is observed that despite the existence of three decisions varia-
bles (N, τ, and T), the operation of the model, in practice, is quite sim-
ple: there is no pre-scheduled action for both items. It is only subject
to corrective maintenance, whereas B will be preventively replaced,
when some conditions related to the state of A or the number of fail-
ures presented by the item shows an opportunity to replace B.
Here are the ways that the system is renewed:
1. In failure of B
Figure 1 represents the arrival of the failure moments of equip-
ment A, the time inspection of A (T), the time inspection of B (TB),
the defect arrival of B, and the failure time of B when the system is
The System Life (SL) in this case can be presented as follows in
Equation 1. It occurs only if the defect and failure of component B
happens after the last inspection of B (
). It is also observed that
the condition is linked to the fact that the time of the inspection of A
occurs before or after the limit τ. Therefore, q represents the quantity
of inspections in A, pointing out again that the failure of B is the sum
of the defect time in B and the delay time.
1 , 1, 2,
Ak Bkk
B Bq
k Bkk
SL y if x T q N k
T y T if T
≤ >
= ∩< + =
≤ ≤
We can also study the maintenance costs associated in this case. It
is the sum of the inspection costs of A, the replacement cost of A, the
inspection costs of component B, the failure cost of component B, and
the system replacement cost.
( )
C c q cp c cF=+++
2. When q is equal to N+1
Taking into account that for equipment B to be inspected a fail-
ure in equipment A must have be occurred to generate an opportunity
to inspect B, and the number of times that the equipment B was in-
spected is called N, an excessive number of failure may occur in the
system without find a defect in B. In this way, to avoid performance
losses, in the next failure of A, when q is equal to N+1, the system will
be renewed without perform inspection in B. This case is showed in
Figure 2, where N equals 3.
Fig. 2. Representation the second case of the policy
The SL in this case can be presented in this way:
1 , 1, 2 ,
kk k
SL T if T x T q N kt= ≤>=+=
The system life will be equal to the last time of inspection of A
(Tk). It occurs if the inspections of B did not find a defect in the dis-
positive, and also if the time of the N + 1 inspection of A is less than
the limit τ. We can analyze the maintenance costs for this case, in this
( )
C c N cp c cr=+++
As highlighted in Equation 4, there is no failure cost added, since
it is considered that PM of the system is performed. Consequently, the
Fig. 1. Representation of the first case of the policy
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inspection costs of A and replacement costs of A, together with the
inspection costs of B and the system replacement cost, are all added.
Note that the number of times that the inspection costs of B and the
replacement costs of A are equal to N, because in the N + 1 failure
of A, the policy affirms that the system is replaced, not requiring an
inspection of B.
3. In the rst failure of A after τ
A life threshold τ of the system is specified due to the fact that
there occurs a major performance loss after this time. Besides, it is
observed that the probability is great that a defect in B occurs after
this limit, and, therefore, a preventive replacement is performed in the
first opportunity that A provides. This replacement is performed at the
time that A is inspected, because its failures are only detected in these
conditions. This case of the policy is presented in Figure 3.
Fig. 3. Representation of the third case of the policy
The SL (equation 5) will be equal to the time of the k-th inspection
of A, since it is greater than the time threshold τ, and for this to occur,
the arrival time of the defect and the failure of A should be higher
than the time of the last inspection. We could analyze the maintenance
costs for this case, using the equation 6:
( )( )
C c q cp c cr=+++
In equation 6, we do not consider the failure cost of B, because
we are proposing PM when B reaches a certain age. Accordingly, the
cost is composed of the sum of the inspection costs of A, q − 1 times
the inspection costs of B, and the replacement cost of A (because in
the last inspection of A, the system is replaced, therefore not needing
to inspect B due to A is only changed). Besides that, the replacement
cost of the system is added.
4. When to identify a defect in B
This case is characterized when a defect is found on component
B through inspection, which is caused by a gradual mechanism of
failure that characterizes it. Figure 4 represents this case in which we
can identify the defect arrival and the system replacement being per-
formed at TB3.
The SL for this case can be represented in equation 7 below:
( )
1 , 1, 2,
k Ak kB
SL T if q N y T T x T y kt
= < + < < =
Thus, the SL will be equal to the time of the last inspection of
A, which will give an opportunity to carry out an inspection of B,
and in that inspection, a defect in device B is found. In addition, it is
necessary that the failure of B does not occur. The maintenance costs
associated in this case can be expressed by the following equation:
( )
C c q cp qc c=+++
It is possible to verify that the cost of an inspection in A is the sum
k times the replacement cost of A, added q − 1 times, and the inspec-
tion cost of B is the sum of q times. This is due to the fact that during
the last inspection of B, a defect is found, and therefore the whole sys-
tem is replaced, not needing to replace only A. In addition, the system
replacement cost is added given that there is no failure cost, because
we are working with PM in this case.
According Scarf et al. [21], the search for the optimum value of
the policy considering the cost minimization, can be calculated by
dividing the expected value for the policy cost by the expected value
of the cycle size. Thus, the present paper finds the expected value for
the cost E(C) and the expected value to the size of the cycle E(SL)
by a simulation process (in a free programming software), through
the flowchart shown in Figure 5. In addition, we have used a process
to optimize the three decision variables of the model: N (number of
failures in A), T (time interval of inspections in A), and τ (threshold of
system age), to minimize the total cost of maintenance.
Fig. 5. Flowchart of the simulation process
3. Case Study
For this paper we consider a sugarcane plant, whose production
of sugar is 40 tons by an hour. For the extraction of the broth of sug-
arcane, it is first necessary to perform a washing process, then it is
minced and defibrated; generally, this set of operations is considered
as the process of preparation of the raw material. A machine, namely
a sugarcane shredder, is used to open a sugarcane cell, so that the
extraction of the broth in the next step of the production process is
carried out with greater efficiency.
Fig. 4. Representation of the fourth case of the policy
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The sugarcane shredder is formed by a rotor, to which a set of
rotating hammers is attached, so as to force the passage of the sugar-
cane through a small aperture along a shredder plate. For this study,
the set of hammers is called equipment A, whereas the rotor is called
equipment B.
The system treated here involves the manufacturing process that
uses a perishable raw material, the sugarcane, and as the time interval
between the harvest and the beginning of processing increases, the
quality of the processed product decreases. After 10 hours, the de-
terioration begins to have a much steeper curve. Thus, a system shut
down by a failure in the set of the system that we are studying can
cause a loss of production, both due to the loss of raw materials and
the loss of production of the finished product. We can verify the need
for periodic inspections of these systems.
A failure in component A may be caused by foreign materials en-
tering with the sugarcane, which was not properly removed during
the washing at the beginning of the process, in addition to the natural
wear by the effort made by the hammers to defibrate the cane. This
failure can generate excessive vibrations and imbalance due to the
hammers, causing problems in other equipment such as oil pumps
or the rupture of welds in pipes that may lead to loss of steam and
oil, and can also cause the accumulation of mass on the sides of the
equipment. So, these failures can be characterized by hidden failures,
because it can only detected by an inspection on hammers.
Due to the structural dependency between the two components,
a failure of the hammer opens an opportunity for the rotor to be in-
spected to check the existence of a defect. Whereas the failure of the
rotor causes the total shutdown of the system, it is the equipment that
causes the assembly to rotate for the passage of the sugarcane that will
be defibrated. So, due to the characteristics cited above, like the sig-
nificant failure mechanism, the maintenance cost significant and the
dependency, this set is analyzed separately of the whole system.
So, some assumptions of the system are presented:
The failure of equipment A occurs by an exponential distribu-
tion, with parameter λ1, in which the unit of measurement of
failure arrival is months;
The defect of equipment B is a random variable X that occur by
a Weibull distribution, with parameters η (scale) and β (shape),
evaluated in months;
The delay time of equipment B is a random variable H that oc-
curs by an exponential distribution, with parameter λ2 evaluated
The failure of equipment B occurs
The inspections, for both equip-
ment are perfect;
The inspections times are not sig-
A failure in equipment B is imme-
diately identified and corrected by
a replacement of new equipment,
renewing the system;
On the replacement of equipment B, the system is restored to a
new condition.
For the more proper analysis of the maintenance policy shows
here, some scenarios were created that are considered possible events
in the system presented above. These scenarios can be characterized
in this way:
Scenario 1: The failure of equipment A does not cause a loss of
performance in the system. This is the general model of the policy,
incurring costs according to what showed in the description previ-
ously presented.
Scenario 2: The failure in equipment A causes a loss of perfor-
mance in the system. For this scenario, we consider that equipment A
is working in a failure state, that is, there may be some hammer in the
set that is not performing the job correctly, causing an accumulation
of sugarcane in the corners, as well as not defibrating it correctly. This
can result in a loss of quality and quantity of the final product, result-
ing in more maintenance-related costs.
Scenario 3: The failure in equipment A does not cause a loss of
performance (or can be considered null given the low expressiveness),
but the defect in equipment B causes a loss of performance. Because
of this, the equipment will work much more in this state, with major
costs being the penalty for this situation, for example, rotation set with
minor velocity.
Scenario 4: The failure in equipment A causes a loss of perfor-
mance and a defect in B, too.
The data corresponding to the parameters of the model were col-
lected, and are summarized in Table 1. The units of measurement for
the parameters related to time were months and for the monetary costs
units per month ($/month).
The optimal results for the maintenance policy for the four sce-
narios is in Table 2.
For Scenario 1, the optimum policy will be dealing with a cost
$4.759/month, doing periodic inspections in equipment A in a time
interval of 0.9 months, where N would be equal to 4 and τ equal to
4.5 months. Therefore, for Scenario 2, in that the failure of A causes
losses during the process, it is considered that the best policy would
have a maintenance cost of $4.931/month, with inspection intervals of
0.4 months, where N would be equal to 9 and τ equal to 4.9 months.
For Scenario 3, the optimum would be to have a maintenance cost
of $5.856/month, and inspections should be done at intervals of 0.8
months, where N is equal to 3 and the limit of τ equal to 3.2 months.
In addition, for Scenario 4, the best value of cost is $6.017/month,
an inspection interval of A equal to 0.4 months, N equal to 7, and a
threshold τ equal to 3.2 months.
With this, it is possible to identify when the failure of the hammer
assembly results in an additional cost, such as a penalty for working
Table 1: Values of the model parameters
Parameter Value Unit
c1 0.1cp $/month
c2 0.5cp $/month
cp 1 $/month
cr 10cp $/month
cF 25cp $/month
cdt 1 $/failure time
cB 10 $/defect time
λ11 Failure/month
λ20.66 Failure/month
Table 2. Results
Scenario 1 Scenario 2 Scenario 3 Scenario 4
(C*) Parameters Cost
(C*) Parameters Cost
(C*) Parameters Cost
(C*) Parameters
Values 4.759
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in a failure state, the expected optimum maintenance cost increases,
and by contrast, the interval between inspections is shortened. This is
because the cost of working with failed equipment is greater than the
cost of doing the inspection, which is natural in these types of cases.
In addition, when analyzing an inclusion of cost penalties for the
set of rotors to work in a defective state, there is an increase in the
expected maintenance cost compared with the original model and the
second scenario. This is because sometimes the system will operate
with this set in the defective state, for the reason that this defect is
only identified in an inspection. Therefore, the time between inspec-
tions that was 1.9 months in Scenario 1 decreases to 0.8 in Scenario 3,
that is, equipment A (Hammers) is further inspected to provide more
opportunities of inspections for equipment B (Rotor) to decrease this
added penalty cost. By decreasing the inspection time interval, the
amount of times the rotors are inspected may be higher.
If both sets of equipment are working poorly, in a failure state (for
A) and in a defective state (for B), this generates additional costs, and
the optimal cost expected for the policy increases considerably. The
time between inspections decreases in only half a month, because of
the fact that the inspection cost is well below the penalty cost.
4. Sensitive analysis
To analyze the variation of the optimal values of the policy, one of
the fixed parameters was maintained and the other two were varied.
The results of these variations are presented below.
Figure 7 presents a variation for the optimal values of the policy
for Scenario 1. It is necessary to understand that the interval time be-
tween inspections is dependent on the time of τ and N, once there is
sufficient time before τ so that N + 1 inspections can be carried out, as
previously shown in policy description. In this way, when we change
the N value for 3, the maximum time between inspections will be
equal 1.1 months, and for N equal to 4, the time should be lower, until
0.7 months. It can be verified in Figure 6 that there is an apparent vari-
ation in the optimum expected cost of maintenance, with the variation
of only one N for more and one for less. This is not so evident given
the variation of the limit of τ, where the curves overlap.
Fig. 6. Sensitive analysis of Scenario 1
As can be seen in Figure 7, the variation in Scenario 2 behaves
differently than Scenario 1. By varying the value of N, the cost curves
end up overlapping, with a very small variation in cost. When the
value of τ is changed, a significant difference is observed when its
value decreases, while when it increases, the expected value of the
cost is very close to the optimum value. What we can observe is that
despite the modifications, the lowest cost remains for the optimally
presented value of T, N, and τ.
Behavior similar to this is observed in Scenario 4 (Figure 9).
When you change the value to N, there is an overlap of the expected
cost curves. While the variation of τ gives an apparent difference in
the curve, especially for a smaller value.
Figure 8 shows the variation for the expected cost curves for Sce-
nario 3, in which its behavior is similar to that presented in the Sce-
nario 1 analysis. There is a distance between the curves for both the N
variation and the variation of τ, but the latter when it is increased, is
closer to the optimum curve.
It should be emphasized that when there is a penalty cost, a greater
variation in costs is obtained when the decision variables are modi-
fied. In addition, it should be noted that the curves have different sizes
when related to T values, since the decision variables are related to
each other; increasing or decreasing one of them implies restricting
values to the others.
Table 3 shows the behavior of the model according to variations
in its input parameters.
It can be emphasized that, as expected, when changing cost val-
ues (cp, cB, or cdt), only the maintenance cost value for the scenario
in which this cost appears is modified. For example, by varying the
value of the penalty cost of B, the behavior of the model remains the
same for Scenarios 1 and 2, whereas for Scenario 3 when it is in-
creased, a decrease in the time between inspections is observed as the
limit τ. While for Scenario 4, the time between inspections remains
the same, but the limit τ and N are decreased. A similar fact occurs in
the variation for the cost penalty related to equipment A.
Also worth mentioning is the behavior related to parameter λ1 that
represents the number of failures that occur per month. When this
quantity was increased, that is, if there were more failures of A in one
month, it was observed that the expected cost of maintenance and the
time between inspections suffered decay. It is believed that this occurs
because since A fails more, there is greater opportunity to perform
inspections in B, thus reducing the time it works in the defective state,
in addition to preventing the occurrence of failure at the same time,
which has a high cost impact for the model, whereas, the limit τ and
N increased.
In relation to the variations made in the parameters of B, it can be
verified that when β is increased there is a tendency to decrease the
intervals between inspections, concentrating the maintenance actions
in a shorter lifetime. This is due to the characteristics of this param-
eter, in that when increasing its value, they are more concentrated on
the origin in the time axis and more elongated in the axis of the val-
ues. In relation to the scale parameter of the Weibull distribution that
Fig. 7. Sensitivity Analysis of Scenario 2
Fig. 8. Sensitivity analysis of Scenario 3
Fig. 9. Sensitivity analysis of Scenario 4
Eksploatacja i NiEzawodNosc – Ma i N t E N a N c E a N d RE l i a b i l i t y Vo l .19, No. 4, 2017
c i E N c E
represents the arrival of the defect for this equipment, a standard in
the modification of the results regarding the decision variables is not
observed. This can be a result of the randomness characteristic of the
distribution, although decreasing the value of η shows an increase in
maintenance costs, since the arrival of the defect of B occurs before,
and a decrease in that cost when this parameter is increased.
Still evaluating parameters related to equipment B we have λ2,
which represents the mean time between the arrival of the defect and
the failure of this equipment through an exponential distribution. It
can then be verified that as the arrival time of the fault increases, the
maintenance cost decreases and, generally, the interval between in-
spections increases, as well as the limit τ. In relation to N, sometimes
it holds and other times it decreases, because it is dependent on the
values of the other decision variables as justified above.
5. Conclusion
The present paper showed maintenance based on random mainte-
nance policy and simulation. It was possible verify that when a penal-
ty cost is involved, the time required to perform a periodic inspection
in the components is lowered, and consequently, the expected main-
tenance costs for the policy are higher. It was verified that the results
obtained through a simulated procedure of parameters are satisfactory
and occurs as expected, thus emphasizing the possibility of producing
models for maintenance policies using simulated processes.
This was verified by applying the model in a real case, in which
the behavior of this model was evaluated and tested. This real evalu-
ation is important because it can be said that when the costs deal with
large amounts of money, any savings realized could be a gain for the
company, even gains that are not measurable such as consumer satis-
faction through non-fulfillment of rework.
In the future, it is possible to suggest the aggregation of this model
with other types of analysis, such as spare parts, besides considering
imperfect maintenance through inspection failures.
The authors of the paper thank for CAPES – Coordenação de Aper-
feiçoamento de Pessoal de Nível Superior – for the financial support
to the research, and FAPEMA – Fundação de Amparo à Pesquisa e
ao Desenvolvimento Científico e Tecnológico do Maranhão.
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1 0.66 2 5 0.1Cp 0.5Cp 1 10 25 10 1 4.759 0.9 4.5 4 4.931 0.4 4.9 9 5.856 0.8 3.2 3 6.017 0.4 3.2 7
0.5 4.418 1.4 5.7 3 4.606 0.4 5.7 10 5.649 0.8 3.2 2 5.816 0.4 3.6 7
1 5.267 0.9 3.6 3 5.427 0.4 4 7 6.172 0.7 2.8 3 6.334 0.4 3.2 6
3 4.475 0.5 5.6 10 4.634 0.4 5.7 10 5.199 0.3 4.8 10 5.304 0.3 5.2 10
2 4.562 0.5 5.6 10 4.685 0.3 5.8 10 5.413 0.5 4 7 5.513 0.2 4.5 10
0.66 4.849 2.3 4.7 1 5.042 0.5 4 7 6.07 0.9 2.7 2 6.252 0.5 2.5 4
0.5 4.925 2.3 4.7 1 5.123 0.6 3.6 5 6.252 0.8 2.4 2 6.413 0.6 2.4 3
4 5.62 2 4 1 5.888 0.5 4 7 6.999 0.8 2.4 2 7.18 0.5 2.5 4
4.5 5.168 2.2 4.5 1 5.36 0.4 4.4 10 6.361 0.7 2.8 3 6.539 0.3 3 9
5.5 4.417 1 5.1 4 4.568 0.5 5.6 10 5.428 0.6 3.6 5 5.571 0.4 3.6 6
6 4.09 0.8 5.7 6 4.232 0.3 5.8 10 5.032 0.7 3.5 4 5.171 0.4 4 9
1.5 5.574 1.3 5.3 3 6.926 0.7 5.7 7 8.342 1 4 3 8.553 0.5 4.5 8
2.5 4.292 0.8 4 4 4.443 0.4 4.4 8 5.206 0.7 2.8 3 5.361 0.4 3.2 7
3 4.024 0.8 4 4 4.159 0.3 4.2 10 4.817 0.6 3 4 4.96 0.5 3 5
0.5 2.373 1.2 4.9 3 2.502 0.3 5.2 10 3.376 0.6 2.4 3 3.508 0.3 2.7 7
2 9.493 0.9 4.5 4 9.706 0.6 4.9 7 10.644 0.9 3.6 3 10.854 0.6 3.6 5
3 14.265 0.9 4.5 4 14.491 0.8 4.9 5 15.445 0.8 4 4 15.665 0.8 4 4
5 4.755 0.9 4.5 4 4.922 0.4 4.9 9 5.322 0.9 3.6 3 5.496 0.4 4 9
15 4.755 0.9 4.5 4 4.922 0.4 4.9 10 6.315 0.7 2.8 3 6.482 0.4 2.8 6
0.5 4.755 0.9 4.5 4 4.853 0.5 5.1 9 5.84 0.8 3.2 3 5.943 0.3 3 5
5 4.755 0.9 4.5 4 5.179 0.2 5.3 10 5.84 0.8 3.2 3 6.292 0.2 3.6 9
Eksploatacja i NiEzawodNosc – Ma i N t E N a N c E a N d RE l i a b i l i t y Vo l .19, No. 4, 2017
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Mônica Frank MARSARO
Department of Mechanical and Production Engineering
State University of Maranhão – UEMA
University City Paulo VI – post office box 09, São Luís – MA
– Brazil – Zip Code: 65055-970
Cristiano Alexandre Virgínio CAVALCANTE
Department of Production Engineering
Federal University of Pernambuco – UFPE
Professor Moraes Rego Avenue, 1235 – University
City, Recife – PE – Brazil – Zip Code: 50670-901
... In their last work [124], the authors presented the maintenance policy based on inspections and replacement, taking into account opportunities for a system composed of two components, whose failure characteristics are different. They used a simulation to obtain the minimal expected maintenance costs. ...
... To that end, we develop a Discrete Event Simulation (DES) based method to reproduce the GRP-queue system and obtain the perfor-mance indicators of interest. Simulation models allow for analysis of systems with complex behavior, require fewer assumptions when compared with analytical models and are used as tools to perform experimentations with systems (Marsaro & Cavalcante [21]). Due to present model's complexity, an analytical solution is unfeasible. ...
Due to its advanced technology, maintenance services of healthcare equipment have been commonly executed by the original equipment manufacturer (OEM), which can be characterized as a monopolist. In this context, hospitals require high availability of their equipment at a reasonable servicing cost, whereas OEM aims to maximize its profit by selling extended warranty (EW) services for multiple consumers. The issue of drawing a maintenance contract between OEM and hospitals has already been treated by adopting a Stackelberg’s game. However, the “as good as new” and “as bad as old” assumptions are usually considered, which are rather difficult to observe in practice, especially for healthcare institutions and their technology-intensive equipment. Thus, we here adopt generalized renewal processes (GRP) for modelling imperfect repairs, and we develop a discrete event simulation method for finding the best strategies of each player: OEM sets the prices for EW and on-demand maintenance that optimize its profit, while hospitals choose which option they should hire. We also present an application example with real data gathered from an angiography device, which is used for mapping blood vessels and diagnosing heart diseases. © 2018, Polish Academy of Sciences Branch Lublin. All rights reserved.
... The article [7] uses fuzzy variables (fuzzy sets), while paper [22] examines multistate systems with components requiring minimal repair. Using simulation methods for testing preventive renewals was presented in paper [20]. This paper examines operation systems in which the technical object may at a given moment appear in one of the n states. ...
Preventive replacement is applied to improve the device availability or increase the profit per unit time of the maintenance system. In this paper, we study age-replacement model of technical object for n-state system model. The criteria function applied in this paper describe profit per unit time or coefficient of availability. The probability distribution of a unit‘s failure time is assumed to be known, and preventive replacement strategy will be used over very long period of time. We investigate the problem of maximization of profit per unit time and coefficient availability for increasing the failure rate function of the lifetime and for a wider class of lifetime. The purpose of this paper is to obtain conditions under which the profit per unit time approaches a maximum. In this paper we shows that the criteria function (profit per unit time or coefficient availability) can be expressed using the matrix calculation method. Finally, a numerical example to evaluate an optimal replacement age is presented. © 2018, Polish Academy of Sciences Branch Lublin. All rights reserved.
The objective of this Chapter is to present delay-time based maintenance models for multi-unit systems performing in various reliability structures (series, parallel, nk-out-of-n types) for the two cases of perfect and imperfect inspection performance. First, the necessity of the DT modelling for multi-unit systems performing in various reliability structures is discussed. The research gap is underlined and structure of the Chapter is presented. Moreover, preliminary simulation DT models are characterized. They focus on availability and maintenance costs of technical systems performing in series and non-series structures with perfect and imperfect inspection. Their sensitivity analysis is carried out and main results are discussed. Moreover, first decision rules for maintenance managers are specified. Later, the short overview of recent developments on delay-time based maintenance modelling for systems in non-series reliability structures is presented and structure for the next Sections is given. In Sect. 5.2 new analytical delay-time maintenance models are developed. The implemented maintenance policy bases on Block Inspection policy implementation and the analytical model is presented for the perfect inspection case. The solution is based on renewal reward theory use and is given for the first inspection cycle. The main decision variable is the time period between successive system inspection actions performance Tin, while the purpose of the developed model is to estimate the expected maintenance cost per unit of system time. As a result, the main assumptions for the DT models are defined. This gives the possibility to obtain the main reliability functions for the modelled system performing in various reliability structures. Later, the expected maintenance cost model is introduced. Section 5.3 provides the reader with extended delay-time maintenance models for systems performing in two reliability structures—series and parallel ones. The analysed system is a two-component system (including component A and component B). The main assumptions are defined and compatible with the ones determined in Sect. 5.2. The main difference is connected with the possibility of planned and unplanned inspection actions performance during the time interval (0, Tin). First, the number of event scenarios is presented and analysed. Later, the probability functions of system failure and preventive replacement are presented. This gives the possibility to obtain the expected maintenance cost model. In the next Sections the author discusses and analyses the possibilities of use of the obtained simple delay-time based maintenance models for determining the best inspection time interval. First, the convergence of the chosen analytical model with the simulation model is given. Later, the optimization modelling issues are discussed and the new cost coefficient is proposed. The Chapter is completed by concluding remarks that contain the most important conclusions obtained from the performed research developments.
This paper studies the selective maintenance problem for multi-state systems with structural dependence. Each component can be in one of multiple working levels and several maintenance actions are possible to a component in a maintenance break. The components structurally form multiple hierarchical levels and dependence groups. A directed graph is used to represent the precedence relations of components in the system. A selective maintenance optimization model is developed to maximize the system reliability in the next mission under time and cost constraints. A backward search algorithm is used to determine the assembly sequence for a selective maintenance scenario. The maintenance model helps maintenance managers in determining the best combination of maintenance activities to maximize the probability of successfully completing the next mission. Examples showing the use of the proposed method are presented.
Condition-based maintenance (CBM) is a maintenance strategy that collects and assesses real-time information, and recommends maintenance decisions based on the current condition of the system. In recent decades, research on CBM has been rapidly growing due to the rapid development of computer-based monitoring technologies. Research studies have proven that CBM, if planned properly, can be effective in improving equipment reliability at reduced costs. This paper presents a review of CBM literature with emphasis on mathematical modeling and optimization approaches. We focus this review on important aspects of the CBM, such as optimization criteria, inspection frequency, maintenance degree, solution methodology, etc. Since the modeling choice for the stochastic deterioration process greatly influences CBM strategy decisions, this review classifies the literature on CBM models based on the underlying deterioration processes, namely discrete- and continuous-state deterioration, and proportional hazard model. CBM models for multi-unit systems are also reviewed in this paper. This paper provides useful references for CBM management professionals and researchers working on CBM modeling and optimization.
A general (N, T, τ) maintenance model is developed for multicomponent systems with two types of components, namely main and auxiliary components. The main component suffers from evident failures, which are assumed to be found or detected as soon as they occur. Auxiliary components with protective or standby functions are modeled by a k-out-of- n: F subsystem, in which failures are hidden and assumed to be detected and fixed only at inspections. Although the shutdown of a subsystem may not halt the system, it could cause a potential risk to the system or financial losses. In this model, the whole system can be renewed at the Nth failure of the main component or at time T, whichever occurs first. Further, incomplete periodic inspections and the optimal number of repairs before replacement are also considered in opportunistic maintenance. Incomplete periodic inspections can efficiently overcome the drawbacks of existing maintenance based on periodic inspections and opportunistic maintenance. For the case of exponential lifetime distribution, an explicit analytical expression of the maintenance cost rate in a renewal cycle is derived by applying the Laplace transform to recursive equations. By setting parameters N,T,τ to tend to infinity, respectively, special properties are derived and a comparison with several maintenance models is performed. For a given life expectancy of the system (namely T), the existence of optimal parameters N and τ is proven. Numerical examples are presented to show the effectiveness of the proposed model.
In the present paper, a system with components subject to soft and hard failures is considered. It is assumed that hard failures are revealed and fixed immediately and present an additional opportunity for inspection (opportunistic inspection), but soft failures are hidden and only corrected at periodic inspections. The objective is to find the optimal maintenance policy for all components and the optimal periodic inspection for the entire system. Two models are considered in this context. In the first model, hard-type and soft-type components are subject to minimal repair or corrective replacement, and soft-type components undergo opportunistic inspections. In the second model, in addition to the assumptions of the first model, hard-type components may be preventive replaced at periodic inspections. In our models, we base the maintenance decision for the soft-type components on the optimal number of minimal repairs until replacement, and for the hard-type components – on the optimal age before replacement. A recursive equation is provided for deriving the required expected values. Hidden failures preclude us from expressing the terms of the objective function in closed form. For this reason, the optimal periodic inspection interval for the system minimising its total expected life cycle cost is found for both models using simulation.
Existing approaches for modelling maintenance rely on oversimplified assumptions which prevent them from reflecting the complexity found in industrial systems. In this paper, we propose a novel approach that enables the modelling of non-identical multi-unit systems without restrictive assumptions on the number of units or their maintenance characteristics. Modelling complex interactions between maintenance strategies and their effects on assets in the system is achieved by accessing event queues in Discrete Event Simulation (DES). The approach utilises the wide success DES has achieved in manufacturing by allowing integration with models that are closely related to maintenance such as production and spare parts systems. Additional advantages of using DES include rapid modelling and visual interactive simulation. The proposed approach is demonstrated in a simulation based optimisation study of a published case. The current research is one of the first to optimise maintenance strategies simultaneously with their parameters while considering production dynamics and spare parts management. The findings of this research provide insights for non-conflicting objectives in maintenance systems. In addition, the proposed approach can be used to facilitate the simulation and optimisation of industrial maintenance systems.
Optimization of inspection and maintenance strategies based on machinery degradation could improve the operation safety and manufacturing efficiency. In this paper, a novel non-fixed periodic inspection strategy will be proposed and applied in precognitive maintenance to monitor and repair a multistage degradation process. The overall cost function of each degradation stage can be individually constructed based on its existing cost factors, and optimal solutions of the above-applied inspection intervals will be obtained by minimizing the corresponding cost function. Due to the high breakdown risk in the final stage, a warning threshold is tailored to divide it into two parts, where two additional inspection intervals are applied, accordingly. The effectiveness of our cost-optimal non-fixed periodic inspection strategy is testified for the inspection and maintenance of an industrial tool wear experiment in a high-speed milling machine, which can save 37% of the total maintenance cost without accuracy loss of the wear estimation and prediction, compared with the conventional periodic inspection strategy.
The degradation of mechanical components is non-stationary and their degradation level can only be detected through offline discrete inspection in most cases. Aiming at the effect of the non-stationary feature and the delay of state detection caused by discrete inspection on long-run operation cost, this paper focuses on proposing an optimal inspection-based maintenance policy for three-state mechanical components subject to competing failure modes. A double-Wiener-process degradation model is established to describe the two operation states, which includes two Wiener process models under the same law but with different parameters. In addition, a preventive maintenance policy including a degradation threshold, an age threshold and a degradation control limit are adopted. A five-scenario probability-based model is proposed to describe the state evolution during one inspection interval and then the analytical model of the impact of delay on detecting state-transition and degradation-level on preventive maintenance policy is proposed. An optimization model of an inspection- based maintenance is established. Finally, sensitivity analysis in terms of the decision variables for state-transition time distribution and parameters in wear-out state is carried out based on numerical examples and the efficiency and superiority of the proposed policy is demonstrated by comparing with the current two-state maintenance policy.
We propose an inspection and replacement policy for a single component system that successively executes missions with random durations. The failure process of the system can be divided into two states, namely, normal and defective, following the delay time concept. Inspections are carried out periodically and immediately after the completion of each mission (random inspections). The failed state is always identified immediately, whereas the defective state can only be revealed by an inspection. If the system fails or is defective at a periodic inspection, then replacement is immediate. If, however, the system is defective at a random inspection, then replacement will be postponed if the time to the subsequent periodic inspection is shorter than a pre-determined threshold, and immediate otherwise. We derive the long run expected cost per unit time and then investigate the optimal periodic inspection interval and postponement threshold. A numerical example is presented to demonstrate the applicability of the proposed maintenance policy.
This paper integrates production, maintenance, and quality for an imperfect process in a multi-period multi-product capacitated lot-sizing context. The production system is modeled as an imperfect machine, whose the status is considered to be either in-control or out-of-control. When the machine is out of control, it produces a fraction of nonconforming items. During each period, this machine is inspected and imperfect preventive maintenance activities are simultaneously performed to reduce its age, proportional to the preventive maintenance level. The objective is to minimize the total cost, while satisfying the demand for all products. Our optimization model allows for a joint selection of the optimal values of production plan, and the maintenance policy, while taking into account quality related costs. A solution algorithm is developed and illustrative numerical examples are presented. It is found that the increase in PM level leads to reductions in quality control costs. Furthermore, if the cost of performing PM is high to the point where it is not compensated for by reductions in the quality related costs, then performing PM is not justifiable. Finally, using non-periodic preventive maintenance with the possibility of different preventive maintenance levels may result in an improvement of the total cost.
In this paper, we propose a decision policy for condition-based maintenance that schedules inspections according to the current health of the system, optimized myopically over the next inspection interval. In traditional condition-based maintenance practices, regular inspections are considered a given requirement. Most measurements, however, confirm that the machine in question is in good working order and that no maintenance is necessary. We consider the possibility that there are savings to be had in the number of inspections that are undertaken. As such, we use a proportional hazards model for risk of failure and a Markovian process to model the system covariates. The cost and time of inspections are incorporated into our model, and an optimal decision for one interval is made. This process can be repeated for each decision point, resulting in a decision policy that produces an inspect-or-replace result and an optimal time for the next inspection. Applied over the life of an asset, this policy can be used as a complete condition monitoring program supporting evidence based asset management.