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Optimal Condition Based Maintenance Using Attribute
Bayesian Control Chart
Journal:
Part O: Journal of Risk and Reliability
Manuscript ID
JRR-22-0066.R1
Manuscript Type:
Original Research Article
Date Submitted by the
Author:
04-Oct-2022
Complete List of Authors:
Rasay, Hasan; Kermanshah University of Technology, Department of
Industrial Engineering;
Hadian, Seyed Mohammad ; University of Kurdistan
Naderkhani, Farnoosh; Concordia University
Azizi, Fariba; Al Zahra University
Keywords:
Maintenance modelling, System health management, Reliability
engineering, Production System maintenance, Safety engineering
Abstract:
Condition-based maintenance (CBM) has been emerged as a relatively
new trend in maintenance management. Instead of conducting
preventive maintenance actions in specified time intervals, the CBM
program collects information through condition monitoring, then
recommends maintenance actions based on the observed data. On the
other hand, Bayesian control charts use the posterior probability of being
the system in an unhealthy state as the chart statistic. An attribute
Bayesian control chart is employed in this study to monitor a
deteriorating system and plan CBM actions based on a continuous-time
homogeneous Markov chain. The system consists of three states:
healthy, unhealthy, and failure states. A partially observable Markov
decision process (POMDP) is developed, which optimally determines the
sample size, sampling interval, and warning limit to minimize the long-
term expected cost per time unit. Numerical examples and sensitivity
analyses are conducted to clarify the performance of the proposed
attribute control chart. To the best of the authors’ knowledge, this is the
first study of the applications of attribute Bayesian control charts in
condition-based maintenance.
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Optimal Condition Based Maintenance Using Attribute Bayesian Control
Chart
1*Hasan Rasay, Kermanshah University of Technology, Kermanshah, Iran.
2. Seyed Mohammad Hadian, Department of industrial engineering, faculty of engineering, University of
Kurdistan, Sanandaj, Iran.
3 Farnoosh Naderkhani, Concordia Institute for Information System Engineering (CIISE), Concordia
University, Montreal, Canada.
4 Fariba Azizi, Department of Statistics, Faculty of Mathematical Sciences, Alzahra University, Tehran,
Iran.
Abstract
Condition-based maintenance (CBM) has been emerged as a relatively new trend in maintenance
management. Instead of conducting preventive maintenance actions in specified time intervals, the
CBM program collects information through condition monitoring, then recommends maintenance
actions based on the observed data. On the other hand, Bayesian control charts use the posterior
probability of being the system in an unhealthy state as the chart statistic. An attribute Bayesian
control chart is employed in this study to monitor a deteriorating system and plan CBM actions
based on a continuous-time homogeneous Markov chain. The system consists of three states:
healthy, unhealthy, and failure states. A partially observable Markov decision process (POMDP)
is developed, which optimally determines the sample size, sampling interval, and warning limit to
minimize the long-term expected cost per time unit. Numerical examples and sensitivity analyses
are conducted to clarify the performance of the proposed attribute control chart. To the best of the
authors’ knowledge, this is the first study of the applications of attribute Bayesian control charts
in condition-based maintenance.
Keywords: Condition-Based Maintenance, Bayesian Control chart, Markov Decision Process,
Attribute Control Chart.
1 *Corresponding Author, E-mail address: H.Rasay@kut.ac.ir;
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1. Introduction
Generally speaking, maintenance policies can be classified into three main groups: corrective
maintenance (also known as run-to-failure policy), preventive maintenance, and predictive
maintenance (also known as condition-based maintenance). Corrective maintenance is the simplest
policy; implementing maintenance actions only after the equipment fails and stops the operation.
It is well recognized that following this policy is not optimal and significantly increases
maintenance costs. The planning and implementing preventive maintenance (PM) programs play
an important role in maintaining the reliability and availability of modern production systems,
which will deteriorate and fail due to aging. The ultimate goal of any maintenance plan is to predict
and prevent catastrophic system failures before they occur, which in turn improve and ensure
effective system reliability. Under the scheduled PM policy, systems may receive excessive
maintenance and be replaced too early. However, if the condition of the system can be monitored
continuously or even frequently, the PM actions will only be implemented when the failure is
about to occur. This concept is known as condition-based maintenance (CBM). Performance
parameters analysis, vibration monitoring, thermal imaging, or oil analysis are some of the
condition monitoring technologies involved in CBM. In recent works, interest in condition-based
maintenance policies has grown and studies have looked into the concept of utilizing system
information to model its degradation level[1] .
In addition, it is well recognized that the maintenance of industrial equipment and the quality of
the manufactured product are closely related [2], [3]. Given the importance of this subject,
[4]provides a comprehensive literature review. Moreover, it is stated by many authors that
condition monitoring, as an essential step of CBM, and statistical process control (SPC) have close
relationships [5]. There exist cases where the information given by SPC can be employed for
condition monitoring of the system [4], [6]. Despite this fact, maintenance and quality control have
been widely studied as separate issues over the years. Integration of maintenance management and
SPC results in significant cost savings and increases efficiency for a production system.
Control charts are the most powerful process monitoring tools that have been extensively adopted
in manufacturing and other industries to ensure process stability and identify the occurrence of
assignable causes. In designing a control chart, the samples are usually taken from the process
output at equally spaced sampling epochs. Based on the received sample, the value of a control
statistic is calculated and plotted on the control chart. If the value of the statistic is beyond the
control limit, the process is stopped and a search for assignable causes of variation is initiated. If
the assignable cause is found, the repair action is initiated to bring the process back to the “in-
control” state. Otherwise, the process continues without any action. The control charts can be
divided into two groups, namely: (i) attribute control charts, which are used when the quality
characteristic is not measurable and only attribute data are collected, and; (ii) variable control
charts, which are used to monitor the measurable quality characteristic [7].
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Some quality characteristics (QC) may not be appropriately measurable or represented
numerically. For such cases, the items are commonly classified as conforming/nonconforming or
another terminology in quality engineering is defective/non-defective. This type of QCs is called
attribute. For example, number of nonfunctional semiconductor chip in a wafer, the number of
mistakes in completing an application from, and color of an item in carpet or cloth production.
Different types of control charts have been proposed to monitor attribute QCs, which are generally
called attribute control charts. Control charts for monitoring fraction of nonconformity or fraction
of defective items are called p-chart. Sometimes attribute control charts are employed to monitor
the number of defective or nonconforming instead of fraction of nonconforming. These control
charts are commonly called c-chart. Another type of attribute control chart is u-chart developed to
monitor the number of nonconformities in a unit of product. It is used the characteristics of
binomial distribution for development of p-chart, while Poisson distribution is the base of the c-
charts and u-charts. An advantage of attribute control charts over variable ones is that several QCs
can be monitored simultaneity and then the item is classified as nonconforming/defective if it fails
to meet the specifications of any of these QCs. For these cases, if variable control charts are
employed, it will need multivariate control charts or several univariate variable control charts
should be applied [7].
In comparison with variable control charts, attribute control charts have been employed for CBM
only in few studies. In reference [8], an np-control chart is presented for maintenance planning of
different types of deteriorating production processes. Three models are developed. In the first
model, it is assumed exponential distribution characterizes the process deterioration, in the second,
this assumption is relaxed and hence different continuous distributions can be employed. Finally,
another model was developed based on a stochastic geometric process for describing the process
deterioration. Reference [9] employed a cumulative count of conforming (CCC) control chart for
inspection and maintenance planning of a production process while each produced item is
inspected and classified as conforming or nonconforming. Number of inspected items until
observing a nonconformity is used as the chart statistic.
Markov decision processes (MDP), partially observable Markov decision process (POMDP),
dynamic programming and decision trees are among the main operational research methods to
analyze and optimize sequential decisions in stochastic problems, while the results of decision of
the current state affect the subsequent stages. Actions, states, reward functions, transition
probabilities and discount factor are commonly considered as the basic elements of an MDP. To
solve an MDP, different traditional methods based on Bellman optimality equation are proposed,
value iteration and policy iteration algorithms are among others. However, for the large-scale
problems, i.e., problem with many states or actions, the efficacy of the traditional algorithms
decreases. Recently, to cope with the large scale MDPs, new algorithms based on reinforcement
learning have been proposed, e.g., Q-learning, SARSA.
For many production and manufacturing systems, data obtained from condition monitoring (CM)
techniques provide incomplete and partial information about the condition of the system. In other
words, the data of CM are stochastically related to the actual state of machine. For example,
spectrometric oil data or vibration data give only partial information about the operational
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condition of the system. If the data of CM provide complete information regarding system states,
MDPs can be employed. On the other hand, POMDP is usually used for the situations that
partial/incomplete information is obtained from the data of CM.
To optimize the decisions of maintenance, POMDPs and MDPs have been widely employed. In
[11], based on an POMDP a control chart with three critical thresholds is developed for CBM.
Joint optimization of maintenance and sampling is considered using the proposed POMDP. The
optimal policy is characterized to minimize the long-term average cost per time unit. Reference
[12]provides an POMDP for maintenance planning of a deteriorating system with different
operational states and a failure state. It is assumed that data about the quality of produced items
provide incomplete information regarding the system state. A Q-learning algorithm is proposed to
find the optimal policy. To maximize the overall system effectiveness, reference [13] proposed an
POMDP for optimal maintenance and production planning. According to the POMDP, the decision
maker can select optimal maintenance actions and optimal production rate in each time epoch.
Reference [14] employed an POMDP for developing a two-control limit Bayesian control chart
which monitors a three-state system. The proposed Bayesian control chart is characterized by two
sampling intervals and two control thresholds to minimize the long-term average maintenance cost
per time unit. In reference [15], POMDP is used for joint planning of maintenance and quality
control for a degrading production process. It was assumed the quality of the produced items
stochastically related to the machine state. The author derived some characteristics about the
optimal value function, and accordingly provided a value iteration algorithm to maximize the
discounted profit of the system during the planning horizon. Some other applications of POMDP
and MDP in maintenance management can be found in [16]–[18].
The pioneering works addressing the integration of maintenance and quality control employed an
X-bar control chart to monitor the process [19]–[21]. However, the integrated model in this context
has been developed in different directions, such as the type of control chart used to monitor the
process. To this end, different types of control charts have been employed in the integrated model:
exponential weighted moving average (EWMA) [22], [23], cumulative sum [24], cumulative count
of conforming [9], Hotelling and chi-square [25], [26], attribute control chart [8], and adaptive
control charts [27], among others. One type of control chart is the Bayesian control chart which
computes the posterior probability of being the process in the out-of-control state. This can be used
according to the Bayes rule of the probability theory, the posterior probability is computed and
plotted as the chart statistic. If the posterior probability exceeds a warning limit, i.e., the upper
bound of the control chart, the chart issues an out-of-control signal which necessitates an
investigation to determine its correctness. In comparison with the other types of control charts,
the optimality of the Bayesian control charts has been discussed by many researchers [10], [14].
Also, in the context of SPC, different studies have been conducted on the economic or economic-
statistic design of Bayesian control charts [28]–[30].
In the context of integrated models of maintenance and SPC, some studies have also been
developed while employing the variable Bayesian control charts. In reference [14], an integrated
model of SPC and maintenance was developed while a Bayesian control chart was proposed to
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monitor a degrading system. It was assumed that the chart has two sampling intervals and two
control limits: a warning limit and a maintenance limit. If the posterior probability exceeds the
warning limit, a shorter sampling interval is employed. The observations are assumed to have a
multivariate normal distribution. In reference [31], a multistate Bayesian control chart is developed
to monitor a multistate partially observable system. An POMDP was developed to minimize the
expected cost of the system in a finite time horizon. Reference [32] develops a multivariate
Bayesian control chart for condition-based maintenance of complex systems. The transition
between states of the system was not Markovian. Due to the complexity of the case investigated
in this study, a Monte Carlo simulation approach is employed to obtain the optimal parameters of
the control chart, including inspection interval and control limit.
In reference [33], a multivariate Bayesian control chart was developed for condition-based
maintenance and its performance was compared against a chi-square control chart. Reference [34]
proposed a Bayesian scheme to predict the failure of a partial observable system consisting of two
unobservable operational states and an observable failure state. Deterioration of the system is
based on a hidden three states Markov process. The application of the method was illustrated
using real data from spectrometric analysis. In reference [35], a double-sampling Bayesian control
chart is proposed for CBM and monitoring of a single unit system with two unobservable
operational states and a failure state. A six-state Markov process is derived to analytically model
the proposed system. The critical parameters of the chart, including sample size, sampling interval,
and the preventive maintenance time, are determined to minimize the expected cost per time unit.
As the literature review unveils, there exist some studies which employ variable Bayesian control
charts for CBM or in the context of the integrated models of maintenance and quality control.
However, to the best of the authors’ knowledge, there exists no study into the applications of the
attribute Bayesian control chart for CBM or applications of this chart in the integration of the
decisions of maintenance and quality control. In this study, an attribute Bayesian control chart is
proposed for the CBM of a manufacturing system. At the equidistant sampling interval, sampling
is conducted and the number of the defective items is computed. Accordingly, the posterior
probability that the system is in an unhealthy state is computed. If the posterior probability exceeds
the warning limit, an inspection is conducted on the system which may be followed by maintenance
actions. As data obtained from sampling provide incomplete and partial information about the
actual operational state of the system, a partially observable Markov decision process (POMDP)
model is developed to optimally determine the parameters of the control chart in order to minimize
the expected cost of the system per time unit in a long-term horizon.
The rest of the paper is organized as follows: Section 2 provides the problem statement. In Section
3, a POMDP model is developed for the problem under study. In Section 4, numerical examples
and sensitivity analyses are conducted to show the performance of the proposed model. Finally,
Section 5 concludes the paper.
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2. Problem statement
The following notations are introduced as preliminary to the problem statement and POMDP
model development.
𝐶
𝑠
Sampling cost
,
𝐶
𝐼
𝑇
𝐼
is the cost of the full inspection of the system, which takes time units.
𝐶
𝐼
𝑇
𝐼
𝐶
𝑃𝑀
,
𝑇
𝑃𝑀
is the preventive maintenance cost, which takes time units.
𝐶
𝑃𝑀
𝑇
𝑃𝑀
𝐶
𝐹
,
𝑇
𝐹
is the corrective maintenance cost, which takes time units.
𝐶
𝐹
𝑇
𝐹
𝐶
0
The operational cost of the system at the healthy state per time unit
𝐶
1
The operational cost of the system at the unhealthy state per time unit (
𝐶
1
)
≥
𝐶
0
𝑝
0
,
𝑝
1
The proportion of defective items in the healthy and unhealthy states
respectively ( ).
𝑝
1
≥
𝑝
0
d
A random variable that denotes the number of defective items in a sample
∆
Sampling interval (decision variable)
𝑛
Sample size (decision variable)
w
Warning limit of the Bayesian control chart (decision variable)
Consider a deteriorating production system, while deterioration of it is governed by a continuous-
time homogenous Markov chain. Within a production cycle, the system can be in one of the states:
. States 0 and 1 are operational states which cannot be distinguished from each other
𝑆
=
{0,1,2}
directly, thus they are partially observable. State 0 is the healthy (good) state and state 1 is the
warning (unhealthy) state. State 2 is the failure state of the system which is directly observable
because the system stops in this state. Let denote the state of the system at time point t.
𝑋
𝑡
Instantaneous transition rates are defined as follows:
,
𝑞
𝑖𝑗
=
𝑙𝑖𝑚
𝑃
(
𝑋
ℎ
=
𝑗
│
𝑥
0
=
𝑖
)
ℎ
;𝑖
≠
𝑗
ℎ→0
,
𝑞
𝑖𝑖
=
―
∑
𝑖
≠
𝑗
𝑞
𝑖𝑗
(1)
where . The matrix of instantaneous transition rates can be represented as follows:
𝑖,𝑗
∈
{0,1,2}
.
𝑄
=
[
―
(
𝑞
01
+
𝑞
02
)
𝑞
01
𝑞
02
0
―
𝑞
12
𝑞
12
0
0
0
]
(2)
As the sojourn times in the healthy and unhealthy states are exponentially distributed, the transition
probability matrix during time unit can be computed based on the Kolmogorov backward
𝑡
differential equations. The result is as follows:
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,
𝑃
=
[
𝑃
𝑖𝑗
(
𝑡
)
]
=
[
𝑒
―
𝑣
0
𝑡
𝑞
01
(
𝑒
―
𝑣
1
𝑡
―
𝑒
―
𝑣
0
𝑡
)
𝑣
0
―
𝑣
1
1
―
𝑒
―
𝑣
0
𝑡
―
𝑞
01
(
𝑒
―
𝑣
1
𝑡
―
𝑒
―
𝑣
0
𝑡
)
𝑣
0
―
𝑣
1
0
𝑒
―
𝑣
1
𝑡
1
―
𝑒
―
𝑣
1
𝑡
0
0
1
]
(3)
where . Let } be the observable failure time of the
𝑣
0
=
𝑞
01
+
𝑞
02
;
𝑣
1
=
𝑞
12
𝜉
=
inf
{𝑡
∈
𝑅
+
:
𝑋
𝑡
=
2
system.
Let and denote the proportion of defective items in states 0 and 1 respectively. It is assumed
𝑝
0
𝑝
1
that the proportion of defective items in state 1 is larger than the corresponding value in state 0,
i.e., . Each production cycle starts while the system is in state 0. An attribute Bayesian
𝑝
1
>
𝑝
0
control chart is employed to monitor the operational states of the system. The Bayesian control
chart monitors the posterior probability that the system shifts to the unhealthy state. More
precisely, at equidistant intervals with length , i.e., at time points , a random sample
Δ
Δ
, 2
Δ
, 3
Δ
, ….
with size n is taken. The number of defective items denoted by d is counted and accordingly, the
posterior probability that the system is in state 1 is computed. If the posterior probability exceeds
a warning limit, the system is stopped and a full inspection is conducted. If the inspection confirms
that the system is in the unhealthy state, then a preventive maintenance action is implemented
which renews the system.
Assume a sampling epoch that starts from the healthy state. The expected time the system spends
in the healthy state, which is denoted by , and the expected time that the system spends in the
𝜏
0,0
unhealthy state, which is denoted by , are computed using the following equations:
𝜏
0,1
,
𝜏
0,1
(∆)
=
𝑞
01
𝑣
0
―
𝑣
1
×
[
𝑣
0
(1
―
𝑒
―
𝑣
1
Δ
)
―
𝑣
1
(1
―
𝑒
―
𝑣
0
Δ
)
𝑣
0
𝑣
1
]
(4)
.
𝜏
0,0
(∆)
=
1
―
𝑒
𝑣
0
∆
𝑣
0
(5)
Now, consider a sampling epoch that starts from the unhealthy state. If denotes the expected
𝜏
1,1
duration of time that the system spends in the unhealthy state, then the following equation holds:
.
𝜏
1,1
(
∆
)
=
1
―
𝑒
𝑣
1
∆
𝑣
1
(6)
Let denote the probability that the system is in state 1 given the observations of
𝜋
𝑡
=
𝑃(
𝑋
𝑡
=
1|
𝑆
𝑡
)
time t denoted by . Using the Bayes formula, the posterior probability is updated as in the
𝑆
𝑡
following:
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. (7)
𝜋
𝑚
Δ
=
(
𝑛
𝑑
)
𝑝
𝑑
1
(1
―
𝑝
1
)
𝑛
―
𝑑
[
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)(1
―
𝜋
(𝑚
―
1)
Δ
)
]
(
𝑛
𝑑
)
𝑝
𝑑
1
(1
―
𝑝
1
)
𝑛
―
𝑑
[
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)(1
―
𝜋
(𝑚
―
1)
Δ
)
]
+
(
𝑛
𝑑
)
𝑝
𝑑
0
(
1
―
𝑝
0
)
𝑛
―
𝑑
(1
―
𝜋
(
𝑚
―
1
)
Δ
)
𝑝
00
(
Δ
)
After dividing the numerator and denominator by , Equation (7) is simplified
(
𝑛
𝑑
)
𝑝
𝑑
1
(1
―
𝑝
1
)
𝑛
―
𝑑
as in the following:
. (8)
𝜋
𝑚∆
=
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)(1
―
𝜋
(𝑚
―
1)
Δ
)
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)(1
―
𝜋
(𝑚
―
1)
Δ
)
+
(
𝑝
0
𝑝
1
)
𝑑
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
―
𝑑
(1
―
𝜋
(
𝑚
―
1
)
Δ
)
𝑝
00
(
Δ
)
During a production cycle, once the system shifts to the failure state, a corrective maintenance
action is performed. According to the descriptions presented, each production cycle terminates due
to one of the following reasons: (1) implementing the corrective maintenance; (2) if the posterior
probability exceeds the warning limit and the inspection indicates that the system is in the
unhealthy state which is followed by preventive maintenance; (3) if the posterior probability
erroneously exceeds the warning limit. According to the renewal theory, for any stationary policy
characterized by warning limit (w), sample size (n), and sampling interval ( , the long-run
𝛿
Δ
)
expected cost per time unit (ECT) can be obtained by dividing the expected total cost of a
production cycle (TC) over the expected length of a production cycle (CL):
.
𝐸𝐶𝑇
=
𝐸
𝛿
(𝑇𝐶)
𝐸
𝛿
(𝐶𝐿)
(9)
It is reasonably assumed that , because means that the best policy is to
𝐶
𝐹
>
𝐶
𝑃
+
𝐶
𝐼
𝐶
𝐹
<
𝐶
𝑃
+
𝐶
𝐼
wait to observe the failure state and conduct the corrective maintenance. In the development of the
POMDP, computing the conditional reliability function is needed. Thus, the following equation is
obtained in this regard: for any , the conditional reliability function is given by
𝑡
∈
𝑅
+
.
𝑅
(
𝑡
│
𝜋
𝑚
Δ
)
=
(
1
―
𝜋
𝑚
Δ
)
(
1
―
𝑝
02
(
𝑡
)
)
+
𝜋
𝑚
Δ
(1
―
𝑝
12
(
𝑡
)
)
(10)
3. Development of POMDP
At time points , sampling is conducted. According to the information of the
𝑚
Δ
, 𝑚
=
1,2, …
sample, the posterior probability that the system is in state 1 is computed. Suppose that at the
sampling time point , the system has not failed, i.e., . The posterior probability interval
𝑚
Δ
𝜉
>
𝑚
Δ
[0,1] is partitioned into K subintervals. In the POMDP, the system is in state k if the current value
of belongs to the interval . If the posterior probability is below the warning limit, the
𝜋
[
𝑘
―
1
𝐾
,
𝑘
𝐾
)
state space of the POMDP belongs to set , where . For the case that posterior
𝐿
1
𝐿
1
=
{𝑖:𝑖
<
𝑤}
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probability exceeds the warning limit, the state space of POMDP belongs to set , where
𝐿
2
𝐿
2
. After that, a full inspection is conducted. If the inspection concludes that the system
=
{𝑖:𝑤
<
𝑖}
is in state 1, the preventive maintenance (PM) action is implemented. This case is denoted by
𝐿
3
. The case that the system enters the observable failure state is denoted by . Hence
=
{𝑃𝑀}
𝐿
4
=
{𝐹}
the state space of the POMDP is .
𝐿
=
{0,
𝐿
1
,
𝐿
2
,
𝐿
3
,
𝐿
4
}
The parameters of the POMDP are defined as in the following:
The probability of transition from state r to state k during a decision epoch;
𝑃
𝑟,𝑘
:
𝑟,𝑘
∈
𝐿
The expected sojourn time in state r
𝜏
𝑟
:
The expected cost incurred until the next decision epoch given the current state is r
𝐶
𝑟
:
To solve the POMDP and determine the optimal policy, a policy iteration algorithm is used which
is a well-known approach to solving the POMDP models. The proposed policy iteration algorithm
leads to the following linear equations:
,
𝑢
𝑟
=
𝐶
𝑟
―
𝑔
(
𝑤,𝑛,
Δ
)
𝜏
𝑟
+
∑
𝑘
∈
𝐿
𝑢
𝑘
𝑃
𝑟,𝑘
; 𝑓𝑜𝑟 𝑟
∈
𝐿
.
𝑢
0
=
0
(11)
In the following, different terms of this linear equations are computed.
3.1. Computing the expected cost and the mean sojourn time
The expected cost until the next decision epoch, given that is computed as in the following:
𝑖
<
𝑤
(12)
𝐶
𝑖
=
𝐸
(
𝑐𝑜𝑠𝑡
│
𝑖
)
=
𝐶
𝑠
×
𝑅
(
Δ
│
𝜋
𝑚
Δ
)
+
(
1
―
𝜋
𝑚
Δ
)
×
(
𝐶
0
𝜏
0,0
(
∆
)
+
𝐶
1
𝜏
0,1
(
∆
)
)
+
𝜋
𝑚
Δ
𝐶
1
𝜏
1,1
(
∆
)
;
∀𝑖
∈
𝐿
1
If the posterior probability exceeds the warning limit, the full inspection with cost is carried out.
𝐶
𝐼
Hence, for set L2, the following equation holds:
𝐶
𝑖
=
𝐶
𝐼
; 𝑖
∈
𝐿
2
(13)
Finally, for sets L3 and L4 as the preventive maintenance and corrective maintenance are carried
out respectively, the following equations hold:
𝐶
𝑖
=
𝐶
𝑃𝑀
; 𝑖
∈
𝐿
3
(14)
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𝐶
𝑖
=
𝐶
𝐹
; 𝑖
∈
𝐿
4
(15)
The mean sojourn time when the posterior probability is below the warning limit is computed as
follows:
𝜏
𝑖
=
∫
Δ
0
𝑅
(
𝑡
│
𝜋
𝑚
Δ
)
𝑑𝑡
=
(
1
―
𝜋
𝑚
Δ
)
×
(
𝜏
0,0
(∆)
+
𝜏
0,1
(∆)
)
+
𝜋
𝑚
Δ
𝜏
1,1
(∆); ∀𝑖
∈
𝐿
1
(16)
When the posterior probability is above the warning limit, the full inspection is carried out, which
takes time units. Hence, the following equation is obtained:
𝑇
𝐼
𝜏
𝑖
=
𝑇
𝐼
; ∀𝑖
∈
𝐿
2
(17)
Finally, for sets and which the preventive and corrective maintenance actions are carried out,
𝐿
3
𝐿
4
the following equations are obtained:
𝜏
𝑖
=
𝑇
𝑃𝑀
; ∀𝑖
∈
𝐿
3
(18)
𝜏
𝑖
=
𝑇
𝐹
; ∀𝑖
∈
𝐿
4
(19)
3.2. Computing the transition probability
The transition probabilities are computed as in the following.
1. The following equation holds for transition probability where i and k are below the warning
limit:
.
𝑃
𝑖,𝑘
=
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
| 𝜉
>
𝑚
Δ
,𝑖
)
𝑅(
Δ
|𝑖)
(20)
It should be noted that the second term of Equation 20 is the conditional reliability computed
using Equation 10. Computing the first term needs more mathematical manipulations which are
provided in the following, after presenting the transition probabilities between the other states of
the POMDP.
2. For the k below the warning limit and j above the warning limit, the following equation
holds.
.
𝑃
𝑘,𝑗
=
𝑃
(
𝑗
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑗
𝐾
|𝜉
>
𝑚
Δ
,𝑘
)
𝑅(
Δ
|𝑘)
(21)
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3. From the inspection state, the system either transits to state 0 if a false alarm occurs or
transits to state PM if a true alarm occurs. The transition probabilities for the two cases are
as follows:
𝑃
ℎ,0
=
1
―
ℎ
―
0.5
𝐾
𝑃
ℎ,𝑃𝑀
=
ℎ
―
0.5
𝐾
(22)
4. Once the system transits to the observable failure state, a mandatory corrective
maintenance action is implemented. Thus, the following equations hold.
𝑃
𝑖,𝐹
=
(
1
―
𝜋
𝑚
Δ
)
𝑃
02
(
𝑡
)
+
𝜋
𝑚
Δ
𝑃
12
(
𝑡
)
𝑃
𝑃𝑀,0
=
1
𝑃
𝐹,0
=
1
(23)
In the following, we proceed to develop mathematical manipulations to compute the first term in
Equations 20 and 21. It is computed as follows:
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
|𝜉
>
𝑚
Δ
,𝑖
)
=
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
0
)
×
𝑃
(
𝑋
𝑚
Δ
=
0
│
𝑖,𝜉
>
𝑚
Δ
)
+
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
1
)
×
𝑃
(
𝑋
𝑚
Δ
=
1
│
𝑖,𝜉
>
𝑚
Δ
)
(24)
The right-hand side of Equation 24 has four main terms. Computing the second and the fourth
terms are provided in the following:
,
𝑃
(
𝑋
𝑚
Δ
=
0
│
𝑖,𝜉
>
𝑚
Δ
)
=
𝑃
00
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
𝑃
00
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
+
𝑃
01
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
+
𝑃
11
(
Δ
)
𝜋
(𝑚
―
1)∆
(25)
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.
𝑃
(
𝑋
𝑚
Δ
=
1
│
𝑖,𝜉
>
𝑚
Δ
)
=
𝑃
01
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
+
𝑃
11
(
Δ
)
𝜋
(𝑚
―
1)∆
𝑃
00
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
+
𝑃
01
(
Δ
)
(
1
―
𝜋
(𝑚
―
1)∆
)
+
𝑃
11
(
Δ
)
𝜋
(𝑚
―
1)∆
(26)
For the first term of Equation 24, we have:
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
0
)
=
𝑃
(
𝑘
―
1
𝐾
≤
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)
(
1
―
𝜋
(
𝑚
―
1
)
Δ
)
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)
(
1
―
𝜋
(
𝑚
―
1
)
Δ
)
+
(
𝑝
0
𝑝
1
)
𝑑
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
―
𝑑
(
1
―
𝜋
(
𝑚
―
1
)
Δ
)
𝑝
00
(
Δ
)
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
0
)
(27)
To simplify the notations, let’s define and
𝛼
=
𝑝
11
(
∆
)
𝜋
(
𝑚
―
1
)
Δ
+
𝑝
01
(
Δ
)(1
―
𝜋
(𝑚
―
1)
Δ
)
𝛽
=
(1
―
. Thus, Equation 27 is simplified as in the following:
𝜋
(
𝑚
―
1
)
Δ
)
𝑝
00
(
Δ
)
.
𝑃
(
𝑘
―
1
𝐾
≤
𝜋
𝑚
Δ
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
0
)
=
𝑃
(
𝑘
―
1
𝐾
≤
𝛼
𝛼
+
(
𝑝
0
𝑝
1
)
𝑑
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
―
𝑑
𝛽
<
𝑘
𝐾
|𝑖,
𝑋
𝑚
Δ
=
0
)
(28)
After some mathematical manipulations, Equation 28 is simplified as in the following:
.
𝑃
{
log
(
(
𝛼
𝛽
)(
𝐾
―
(𝑘
―
1)
𝑘
―
1
)
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
)
log
(
𝑝
0
(1
―
𝑝
1
)
𝑝
1
(1
―
𝑝
0
)
)
≤
𝑑
<
log
(
(
𝛼
𝛽
)(
𝐾
―
𝑘
𝑘
)
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
)
log
(
𝑝
0
(1
―
𝑝
1
)
𝑝
1
(1
―
𝑝
0
)
)
|𝑖,
𝑋
𝑚
Δ
=
0
}
(29)
The number of defective items in a sample with size n follows a binomial distribution as in the
following:
; while the system is in healthy state,
𝑑~𝐵𝑖𝑛
(
𝑛,
𝑝
0
)
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; while the system is in unhealthy state.
𝑑~𝐵𝑖𝑛
(
𝑛,
𝑝
1
)
Let define and as in the following:
Λ
1
Λ
2
,
Λ
1
=
⌊
log
(
(
𝛼
𝛽
)(
𝐾
―
(𝑘
―
1)
𝑘
―
1
)
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
)
log
(
𝑝
0
(1
―
𝑝
1
)
𝑝
1
(1
―
𝑝
0
)
)
⌋
(30)
.
Λ
2
=
⌊
log
(
(
𝛼
𝛽
)(
𝐾
―
𝑘
𝑘
)
(
1
―
𝑝
0
1
―
𝑝
1
)
𝑛
)
log
(
𝑝
0
(1
―
𝑝
1
)
𝑝
1
(1
―
𝑝
0
)
)
⌋
(31)
Thus, for the first and the third terms of Equation 24 can be computed as follows:
𝜁
=
0,1,
𝑃
(
Λ
1
≤
𝑑
<
Λ
2
│
𝑖,
𝑋
𝑚
Δ
=
𝜁
)
=
𝑃
(
𝑑
<
Λ
2
│
𝑖,
𝑋
𝑚
Δ
=
𝜁
)
―
𝑃
(
𝑑
<
Λ
1
│
𝑖,
𝑋
𝑚
Δ
=
𝜁
)
=
Λ
2
―
1
∑
𝑗
=
0
(
𝑛
𝑗
)
𝑝
𝑗
𝜁
(
1
―
𝑝
𝜁
)
𝑗
―
Λ
1
―
1
∑
𝑗
=
0
(
𝑛
𝑗
)
𝑝
𝑗
𝜁
(
1
―
𝑝
𝜁
)
𝑗
(32)
4. Numerical examples and sensitivity analyses
To show the performance of the proposed attribute control chart, in this section, numerical
examples are presented and sensitivity analyses are conducted. The data of the example are shown
in Table 1. MATLAB2019 is used to run the code of POMDP. Decision variables, i.e., sample size
(n), inspection interval ( ), and warning limit (w) are discretized as in the following:
Δ
. For example, it means that for sample size, the values
𝑛
=
10:10:50; ∆
=
1:1:5;𝑤
=
0.1:0.1:0.9
of 10, 20, 30, 40, and 50 are searched. It should be noted that, in Table 1, sampling cost is declared
as a function of sample size: 1+0.05 , while 0.05 is the variable sampling cost and 1 is the fixed
×
𝑛
sampling cost. Also, K is set at 60, which means that the posterior probability interval [0,1] is
partitioned into 60 subintervals. According to the analyses, for the larger values of K, no significant
change in ECT is observed while the run time of the MATLAB code increases drastically.
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Table 1. Data of the example
𝑇
𝐼
𝐶
𝐼
𝐶
𝑃𝑀
𝑇
𝑃𝑀
𝐶
𝐹
𝑇
𝐹
𝐶
𝑠
𝑝
0
𝑝
1
3
200
1000
4
1500
10
1+0.05
×
𝑛
0.05
0.2
𝑞
01
𝑞
02
𝑞
12
𝐶
0
𝐶
1
0.04
0.02
0.3
10
50
The results are as follows . Accordingly, every 2 time units
𝑛
=
20, ∆
=
2, 𝑤
=
0.3, 𝐸𝐶𝑇
=
61.41
(for example every 2 hours), a sample with size 20 is taken and the posterior probability is
computed using Equation 8. If the posterior probability exceeds 0.3, the system is stopped and the
investigation is carried out to determine the possible unhealthy state of the system. Using this
policy optimizes the average cost of the system per time unit which is 61.41.
In the following, some sensitivity analyses are carried out in order to get some insight into the
performance of the POMDP. As the first step, the impact of change in the operational costs in the
healthy and unhealthy states is investigated. To this end, the value of is decreased from 50 of
𝐶
1
the base example to 40, 30, and 20. The results are shown in Table 2. The most noticeable impact
of this decrease is a significant increase in the warning limit. The warning limit increases from 0.3
of the base example to 0.4, 0.6, and finally to 0.9 while . Decreasing the also increases
𝐶
1
=
20
𝐶
1
sampling interval, while decreasing the sample size. These changes can be attributed to the fact
that the decrease of C1 while C0 does not change means that the difference between the operational
costs in the health and unhealthy states decreases. Thus, the parameters of the control chart can be
chosen looser. Alternatively, the results show that as the differences of operational costs of the
system in the healthy and unhealthy states increase, a tighter inspection policy should be employed,
i.e., warning limit and sampling interval decrease, while the sample size increases.
Table 2. Sensitivity analyses for operational cost of the unhealthy state
n
∆
w
ECT
Base Example
(;)
𝐶
0
=
10
𝐶
1
=
50
20
2
0.3
61.41
𝐶
0
=
10
;𝐶
1
=
40
10
2
0.4
61.05
𝐶
0
=
10
;𝐶
1
=
30
10
5
0.6
60.4
𝐶
0
=
10
;𝐶
1
=
20
10
5
0.9
59.7
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The impact of change in the instantaneous transition rates is shown in Table 3. To this end, the
values of in the base example decrease in two steps. The results illustrate that decrease of
𝑞
𝑖𝑗
𝑞
𝑖𝑗
dramatically decreases ECT. Also, with decreasing , sampling interval and warning limit
𝑞
𝑖𝑗
increase while the sample size decreases. These findings are intuitive to some extent because the
decrease of means that the system deteriorates slower which leads to a decrease in the ECT
𝑞
𝑖𝑗
meaning a looser inspection policy can be employed to monitor the operational states of the system.
Table 3. Impact of change in the instantaneous transition rates
n
∆
w
ECT
Base Example
( )
𝑞
02
=
0.02,
𝑞
01
=
0.04,
𝑞
12
=
0.3
20
2
0.3
61.41
𝑞
02
=
0.01,
𝑞
01
=
0.02,
𝑞
12
=
0.2
10
2
0.4
41.91
𝑞
02
=
0.005,
𝑞
01
=
0.01,
𝑞
12
=
0.1
10
4
0.4
28.66
Table 4 shows the impact of change in the inspection cost . From of the base example,
𝐶
𝐼
𝐶
𝐼
=
200
the inspection cost first decreases to 150 and then increases to 250. The results given from Table
4 indicate that inspection cost has a dramatic impact on the warning limit. Also, there is a direct
relationship between the inspection cost and the warning limit. As the inspection cost increases,
the warning limit increases and vice versa.
Table 4. Sensitivity analyses for inspection costs
n
∆
w
ECT
Base Example
( )
𝐶
𝐼
=
200
20
2
0.3
61.41
𝐶
𝐼
=
150
10
2
0.1
59.74
𝐶
𝐼
=
250
10
5
0.8
61.96
The effect of corrective and preventive maintenance costs is studied in Table 5. Corrective
maintenance cost has a major impact on the warning limit, as the results of Table 5 show. Also, an
increase of considerably increases ECT. Another finding from the cases of Table 5 is that as
𝐶
𝐹
the difference between the costs of corrective maintenance and preventive maintenance increases,
the warning limit decreases.
Table 5. Sensitivity analyses for corrective maintenance cost
n
∆
w
ECT
Base Example ( )
𝐶
𝐹
=
1500,
𝐶
𝑃𝑀
=
1000
20
2
0.3
61.41
𝐶
𝐹
=
1400,
𝐶
𝑃𝑀
=
1000
10
5
0.9
58.54
𝐶
𝐹
=
1600,
𝐶
𝑃𝑀
=
1000
20
2
0.1
63.64
𝐶
𝐹
=
2000,
𝐶
𝑃𝑀
=
1000
20
2
0.1
71.42
𝐶
𝐹
=
1500,
𝐶
𝑃𝑀
=
500
20
1
0.2
53.18
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Finally, the effect of change in the proportions of defective items in the healthy and unhealthy
states is shown in Table 6. As the table shows, remains unchanged, while in the first case
𝑝
0
𝑝
1
decreases from 0.2 to 0.1, in the second case increases to 0.3 and then to 0.4. The results show that
an increase of leads to a decrease in the sample size, and an increase in the sampling interval
𝑝
1
and warning limit.
Table 6. Impact of change in the defective items proportions
n
∆
w
ECT
Base Example
( )
𝑝
0
=
0.05,
𝑝
1
=
0.2
20
2
0.3
61.41
𝑝
0
=
0.05,
𝑝
1
=
0.1
10
3
0.2
61.62
𝑝
0
=
0.05,
𝑝
1
=
0.3
10
3
0.3
61.37
,
𝑝
0
=
0.05
𝑝
1
=
0.4
10
5
0.3
61.35
5. Conclusion
The main novelty of the current study is illustrating the applications of attribute Bayesian control
charts for CBM. To this end, a POMDP model is developed. To monitor the system and collect
information as a condition monitoring technique in CBM, at equidistant intervals, a sample with
size n is taken and the number of defective items is counted. Accordingly, the posterior probability
of the Bayesian control chart is computed. If the posterior probability exceeds the warning limit,
the full inspection is carried out which may be followed by preventive maintenance action. The
model optimally determines the inspection intervals, warning limit of the chart, and sample size to
minimize the long-term costs of the system per time unit. Numerical examples and sensitivity
analyses are conducted to show the performance of the proposed Bayesian control chart.
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