Manufacturing and Service Operations Management (2022)
Clausius Scientific Press, Canada
DOI: 10.23977/msom.2022.030307
ISSN 2616-3349 Vol. 3 Num. 3
Stochastic Model for Integrated Preventive Maintenance
Planning
Safiye Turgay1,a,*, Samet Koç1,b, Çiğdem Cebeci1,c
1
Department of Industrial Engineering, Sakarya University, Esentepe Kampüsü, Üniversite Cd.,
Kemalpaşa, Serdivan/Sakarya, 54050, Turkey
a
safiyeturgay2000@yahoo.com, bsmtkoc58@gmail.com, ccigdem7cebeci@hotmail.com
*
Corresponding author
Keywords: Stochastic Modelling, Preventive Maintenance, Maintenance Planning,
Mathematical Modelling
Abstract: Preventive maintenance planning management is modelled with stochastic
approach. It is aimed to prevent stoppages and quality disorders due to disturbances,
carriage and maintenance processes in production. Different maintenance policy
alternatives were considered in order to develop and sustain more effective maintenance
policies. The preventive maintenance process includes the cost of inspection and
maintenance status, the cost of repair and other losses in the accidents with operator injuries
and damage to the possible value of the situation. The stochastic model approach is applied
for determine the possible period intervals of the machinery and equipment and the
maintenance process analyzed and discussed in detail. The preventive maintenance
approach in the maintenance planning process is aimed to develop a sustainable
maintenance policy without any disturbance in the quality of production from any
disturbance and disruption of the system in the long term. However, it is aimed to take
preventive maintenance measures as well as to analyze the current system condition and
predict future situations.
1. Introduction
The uncertainty situation in machine failures and stops are very important during the production
activities for this reason we prefer to use stochastic maintenance planning model. The modelling,
uncertainty situation and system performance were analyzed by taking into consideration possible
situations in machine and breakdowns and faults. In the maintenance process, the costs of the
control and maintenance status diagnosis, repair costs and other losses in case of accidents analyzed
by taking into consideration the possible values of the injuries and damage of the operators. It is
aimed to perform the periodical maintenance process with the machinery and equipment where the
maintenance process performed with the stochastic model approach applied.
In this study, production plan activities are evaluated together with the system risk situation and
maintenance plan activities are aimed to be formed considering the current situation and conditions
with integrated process and maintenance functions. Maintenance costs involve a significant part of
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the total operating costs of manufacturing and production facilities. Although, it varies by industry
type, maintenance costs are known to account for between 15% and 60% of the total operating costs.
According to the studies, a large part of the operation and maintenance costs are wasted due to
incorrect, systematic and unplanned maintenance methods. Ineffective maintenance methods can
also have a major impact on the quality of the product produced therefore it directly effects the
increasing of the costs.
The remainder of the paper is organized as follows. Previous studies on preventive maintenance
systems of stochastic structure are reviewed in section 2. The system and integrated preventive
maintenance planning stochastic model is presented in section 3. Then proposed stochastic
maintenance model with illustrative example and analysis are discussed in section 4. Also, the
proposed model and numerical results are given with case study in next section. Section 5 includes
the discussion and some concluding remarks which are provided in conclusion.
2. Literature Survey
The integrated preventive maintenance planning model is aimed to predict the disruption
situations that the system may encounter. The model has been developed in order to prevent
possible unexpected deterioration situations and prevent any such problems and to avoid any
negative problems during the production process with the preventive maintenance operations
carried out considering the machine wear conditions. In conventional maintenance models, the
system and its components are considered to be operating perfectly or fail in two possible situations
[1, 2]. The system and its components are taken into consideration in case of defective production
of parts or defects that may occur in parts during assembly and distortion situations that may occur
in machines during the production of parts. In practice, however, many systems and components
can fail and operate in an intermediate operating state. When some of the components that make up
the component deteriorates, or when the machine forming the component starts to deteriorate, it
may be necessary to either repair the defective machine or replace it with a new one to complete the
performance [3, 4, 5]. Also, the system has been evaluated by considering both the part level and
the machine level. The structure where multi-state components are taken into consideration and the
deterioration conditions of the machinery and equipment used in this structure are taken into
consideration [6-12]. Along with the proposed model, the maintenance model in terms of both
material and machinery produced was examined [13-17].
3. Stochastic Models in Discrete Events
Markov chains are a special type of discrete time stochastic processes. In addition, Markov
chains have the ability to predict the long-term state (equilibrium state) of the system in addition to
its ability to predict the situation at a certain moment. In this study, by considering a multi-state
system consisting of N, multi-state components i (i = 1, 2,., N) in series have K + 1 different states,
each of which is different. The system performance ratio is shown in gk (k = 0, 1,…., K). K
indicates perfect working condition and 0 indicates full failure condition. Transition times between
component states, Markov process approach, and time between transitions are expressed by the
exponential distribution. i, k is the structural distortion or transition rate from the component i [18].
Ss (t) = min {s1 (t), s2 (t),…., sN (t)} (1)
In particular, one component falls into a lower state, other neighbouring and / or related
components that affect functionally deterioration rates. Initially, all components are in excellent
working condition and no interaction between components, all degrading at internal degradation
rates [19].
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The state of 1 to k as ik (t) in the state transition rate (t) of the component i (i = 1, 2,., N) can be
expressed as:
(2)
The modified failure rate
consists of two elements: the internal failure rate λi, k, and the
interaction effect on decay is the ratio of f () caused by other failure components. System
performance rate represents the f () at Gs (t) time ts (t) [20].
(Gs (t) = min {g1 (t), g2 (t), gN (t): gi (t) = system performance}) is the number of components
that affect the transition of nI (t) to a lower level. It expresses the state of t in it. Also, the interaction
effect is stochastic in nature, and the system consists of operational conditions, including the state
of environmental / non-critical components. Therefore, we add the δ parameter to capture the
uncertainty. It is caused by random variations in these factors. We assume that the process takes
place with the normal distribution of δ mean zero and standard, deviation of σ [21]. We update the
number of components and affected components f () interaction effect and the transition rates of
all components in the system.
Then, we use this updated transition in accordance with equations. (2) and (3). Accordingly the
term f (・) in the equation (2) can be expressed
(3)
When all components are in perfect condition, i.e. nI (t) = 0 and Gs (t) = GK, f(c) is equal to 1
. In the case of the system (Gs (t) <GK), f (•) value will be less than 1 and
‘s f (•)
and
value will be greater than 1. The uncertainty parameter (δ) and the remaining parameters are
obtained using historical data and subjective input from experts. After modelling the rates of
disruption of multi-state components, the components that can be evaluated considering the
reliability rate is:
(4)
System reliability can then be evaluated as the sum of the following and the possibilities for all
acceptable states of the system:
(5)
Pk (t) is the probability of the state of k at time t and I is an indicator function with a value of 1, a
performance ratio higher than the system demand level (Gs (t) ≥D) and 0 in any other situation.
Zero value is considered a system error in the display function. The possibility of the system state at
time T is given as:
(6)
where
, is the
of the state of each component at time t. At t
time, the state of the system, k, and if only the state of at least one component is k, and the state of
the other is higher than the components k. The state probabilities of each component are calculated
by solving the
Chapman - Kolmogorov differential system.
3.1. Mathematical Model
In this section covers the stochastic preventive maintenance mathematical model.
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Indices:
i
product type
j
machine type used in production
t
production period
Parameters:
D
demand level
gi(t)
performance rate of component i at time t
gi, k
performance rate of component i state k
Gk
system performance rate in its perfect state
Gs(t)
system performance rate at time t
nl(t)
number of influencing components up to time t
Rt
the period of time necessary for the production of the product i during the period t
Ot
the period of overtime in the production of the product i ot t period
Rmax t normal working time available during the period t
Omax t available overtime available during period t
Mjt
preventive maintenance variable for machine j in period t.
if the variable value is 1, maintenance is performed,
if 0, no maintenance is performed.
Wjt
weekend preventive maintenance variable for machine j in period t.
if 1 is weekend maintenance, 0 is not a weekend maintenance.
Zjt
if 1 is the variable that tells the machine j in period t whether the last preventive
maintenance was performed before the period .
If 1, maintenance operation is performed,
if 0, maintenance operation is not performed.
Dit
demand for i product during the period t
Mt
maximum duration of possible preventive maintenance activity performed during the
normal working time during the t period
WMt
maximum duration of preventive maintenance activity possible at the weekend in the t
period
jt
probability of machine j deterioration in period
jt-1
actual deterioration value of machine j in period t-1
CT
total cost
Co
beginning cost
Cm
maintenance cost
Vk
material volume for k th unit
Cok
beginning unit cost
Cm
total maintenance cost
Cq
q th unit cost for improvement
tq
q th unit production period
v
volume
Ckq
q th unit cost of for kth fixed cost
q
Co weight percentage of initial cost
kq
index of damage
T
expected life of the machine
Cq
q th unit’s annual costs;
(7)
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(8)
(9)
(10)
(11)
(12)
This study aims to minimize system costs by minimizing cumulative maintenance costs and
minimizing disruption and quality deterioration in production due to machine downtime and
downtime. We call the opportunistic preventive maintenance model cause of the model considering
reliability threshold status. Also, in case of machine deterioration and stoppages, revision of other
machines in standing system considered and periodic maintenance after certain production levels
are applied to regardless of reliability threshold.
3.2. Determination of Stochastic Preventive Maintenance Reliability Threshold Level
Determining of the reliability threshold level for component j is considering the reliability
threshold value Rj
(13)
The value hj indicates the hazard ratio function. Tj represents the optimum preventive
gives the stochastic
maintenance time interval for component i. The equation
deterioration probabilities of component j in each preventive maintenance cycle.
The Hj value represents the hazard ratio function. Tj represents the optimum preventive
maintenance interval for the component.
(14)
All these assumptions,
represents the minimum repair cost and preventive maintenance
cost per component time for component j, while representing the cost of dismantling equipment per
unit time. represents the cost of downtime per unit of the system for one system component,
together with downtime due to repair and maintenance.
indicates not only the
deterioration state of component j, but also the default number of disassemblies in the preventive
maintenance period.
while
represents the total minimum stop of the component
in the removal state,
indicates a single minimum repair state
(15).
represents the downtime of the work piece while
represents the cumulative downtime.
represents the total downtime of the system at a standstill. Preventive maintenance interval;
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(16).
The objective of this study is to determine the optimal preventive maintenance interval by
minimizing Cj and determining the Rj reliability threshold.
Preventive maintenance reliability threshold level k reaches the preventive maintenance
threshold; the system is stopped to perform the maintenance process. Preventive maintenance time
intervals are given in a TW time window that are indicated by t = tk and t = tk + TW. When the
number of components is indicated by r, k, the total maintenance cost refers to the combination of
component and G components which is;
(17)
Ck is the maintenance cost for component k and Cη is the maintenance cost for component η in
the G composition. Component k can be repaired or prevented with a minimum maintenance cost
Ck.
(18)
denotes the minimum
Minimum repair presented for Preventive Maintenance in Eq. (18).
cost of the minimum repair condition or preventive maintenance condition for component k, while
the preventive maintenance together with the combination of C with other components.
4. Case Study
In this section, possible necessary maintenance situations were examined, taking into account the
different forms of the machine's operating states. Four different situations of machine operation,
namely, machine operation and non-operation, slow operation of the machine and noise from the
machine were taken into account. At the same time, a possible situation analysis was made by
comparing the machine's malfunction state structure with the applied process states. The feedback
data regarding the disorder status received for a total of 3 months are given in Figure 1. These data
were arranged with state parameters considering Eq. (1) and Eq. (2). Below the alternative cases are
discussed. The operation performed with the parameter D is expressed.
Figure 1: Case dataset
Definitions of states and decisions
S0- The machine works perfectly
S1- The strange sound when the machine is running
S2- The machine sometimes slows down while running
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S3- The machine is at a standstill from time to time
D1-Do nothing
D2- Perform periodic maintenance
D3- Overhaul the machine, change the spare part or change itself.
As a result of the current data analysis, the reduced cost values were included in the model
according to the D1, D2 and D3 state according to the transaction status. Especially by changing,
revising and continuing production of machine parts, took place as the factor reducing cost in the
system the most. Figure 2 shows the correlation status chart. Figure 3 gives the probability
distributions of the situations encountered
Figure 2: Dataset correlation result diagram
Figure 3: Data set state probability distribution
Figure 4 shows the decision results of the situations. As a result of the analysis, S2-D3, in case of
slow operation of the machine, D3 option is selected, the option of replacing the defective part of
the machine or replacing the machine is selected. With the change made in this context, any
disruption or stopping situation that may occur in production will be prevented. The transition state
between states and possible situations and decision chains are given in Figure 5.
Figure 4: Case decision results (C-Case; Y-Decision)
Figure 5: Situations and possibilities of applied maintenance policies
5. Conclusion
In this study, the structure of the integrated preventive maintenance planning problem in
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different situations, different probability and possible situations that may be encountered in
different scenarios are examined. They were examined and aimed to prevent negative situations
such as production stops and faulty product production situations. With the recommended
preventive maintenance, the behaviour of the system was examined; possible downtimes and repair
maintenance operations were analyzed comprehensively in the system. In this way, cost increases
caused by possible deterioration in production were tried to be prevented, and poor quality
production caused by machine defects was tried to be prevented. By the way, the importance of
carrying out preventive maintenance studies and risk management studies together with
maintenance management in later studies has emerged. Thanks to the proposed model, it has been
tried to estimate the state value that the system can take in the next state, taking into account the
possible situations that may be encountered in the system.
References
[1] Zhu, H., Liu, F., Shao, X., Liu, Q., Deng, Y. (2011) A cost-based selective maintenance decision-making method for
machining line, Qual Reliab Eng Int, 27 (2) pp. 191–201
[2] Jia, Q.-S. (2010) A structural property of optimal policies for multi-component maintenance problems IEEE Trans
Autom Sci Eng, 7 (3), pp. 677–680
[3] Alimian, M., Saidi-Mehrabad, M., Jabbarzadeh, A. (2019)A robust integrated production and preventive
maintenance planning model for multi-state systems with uncertain demand and common cause failures, Journal of
Manufacturing Systems 50, 263–277
[4] Shahraki, A.F., Yadav, O.P., Vagiatzzis, C.(2020), Selective maintenance optimization for multi-state systems
considering stochastically dependent components and stochastic imperfect maintenance actions, Reliability
Engineering&System Safety, 196, 106738.
[5] Zhou, X., Huang,K., Xi, L., Lee, J.(2015) Preventive maintenance modeling for multi-component systems with
considering stochastic failures and disassembly sequence, Reliability Engineering & System Safety, 142, 231-237
[6] Dao, C.D., Zuo, M.J. (2017) Selective maintenance of multi-state systems with structural dependence, Reliability
Engineering and System Safety, 159, 184-196.
[7] Di, M., Dio, R. Iannone, S. Miranda and S. Riemma, (2013), A framework for the choice of the opportunistic
maintenance policy in industrial contexts, IEEE International Conference on Industrial Engineering and Engineering
Management, pp. 1716-1720, doi: 10.1109/IEEM.2013.6962703.
[8] Pham, H., Wang, H. (2000) Optimal (τ, T) opportunistic maintenance of a k-out-of-n: G system with imperfect PM
and partial failure, Naval Research Logistics (NRL), 47 (3), 223–239.
[9] Hou, W., & Jiang, Z. (2013). An opportunistic maintenance policy of multi-unit series production system with
consideration of imperfect maintenance. Applied Mathematics & Information Sciences, 7(1L), 283–290.
[10] Alrabghi, A., Tiwari, A. (2015) State of the art in simulation-based optimisation for maintenance systems,
Computers & Industrial Engineering, 82, 167–182
[11] Triska, Y., Agostino, İ.C.R.S., Penna, P.M., Braghirolli, L.F., Frazzon, E.M. (2021) Integrated production and
maintenance planning method with simulation-based optimization, IFAC-PapersOnLine, Volume 54, Issue 1, 349-354,
ISSN 2405-8963,
[12] Chang, Q., Ni, J., Bandyopadhyay, P., Biller, S., Xiao, G., Maintenance opportunity planning system, Journal of
Manufacturing Science and Engineering, 129 (3) (2007), pp. 661– B.
[13] Dekker, R., Reliability Engineering and System Safety, 51, 229(1996)
[14] Portioli-Staudacher, A., Tantardini, M. (2012),"Integrated maintenance and production planning: a model to
include rescheduling costs", Journal of Quality in Maintenance Engineering, Vol. 18, 1, 42 – 59
[15] Creţu, A., Peptan, E (2003), “Uncertainty and optimum portfolios”, ASE Press, Bucharest
[16] Jonge, B., Scarf, P.A., A review on maintenance optimization, European Journal of Operational Research, Volume
285, Issue 3, 805-824,
[17] Gholizadeh, H., Chaleshigar, M., Fazlollahtabar, H.(2022) Robust optimization of uncertainty-based preventive
maintenance model for scheduling series–parallel production systems (real case: disposable appliances
production) ,ISA Transactions, Volume 128, Part B, pp. 54-67.
[18] Hillier, S.Frederick, Gerald J. Lieberman (2001), Introduction to Operations Research, 7th. Edition. McGraw-Hill,
New York.
[19] Liu, Y., Chen, Y., Jiang, T. (2018) On Sequence Planning for Selective Maintenance of Multi-State Systems under
Stochastic Maintenance Durations, European Journal of Operational Research, 268, 113-127.
66
[20] Pistikopolos, E.N., Vassiliadis, C.G., Arvela, J. and Papageorgiou, L.G. (2001), “Interactions of maintenance and
production planning for multipurpose processplants – a system effectiveness approach”, Industrial and Engineering
Chemistry Research, Vol. 40, pp. 3195-207.
[21] Wang, N., Hu, J., Ma, L., Xiao, B., Liao, H. (2020) Availability Analysis and Preventive Maintenance Planning for
Systems with General Time Distributions, Reliability Engineering and System Safety, 201, 106993.
67