An integrated model for economic design of chi-square control chart and maintenance planning

Abstract

Considered process in this article is a two-stage dependent process. Each item in this process has two quality characteristics as x and y while x and y are related to the stage 1 and 2 respectively. Each stage has two operational states as the in-control state and out-of-control state and transition time from the in-control state to the out-of-control state follows a general continues distribution function. Monitoring of this process is conducted using a chi-square control chart. An integrated model that coordinates the decisions related to the economic design of the used control chart and maintenance planning is presented. For the evaluation of the integrated model performance, a stand-alone maintenance model is also presented and the performance of these two models are compared with each other.

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An integrated model for economic design of chi-
square control chart and maintenance planning
Hasan Rasay, Mohammad Saber Fallahnezhad & Yahia Zare Mehrjerdi
To cite this article: Hasan Rasay, Mohammad Saber Fallahnezhad & Yahia Zare Mehrjerdi (2017):
An integrated model for economic design of chi-square control chart and maintenance planning,
Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2017.1343848
To link to this article: http://dx.doi.org/10.1080/03610926.2017.1343848
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Jun 2017.
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS
, VOL. , NO. , –
https://doi.org/./..
An integrated model for economic design of chi-square control
chart and maintenance planning
Hasan Rasay, Mohammad Saber Fallahnezhad, and Yahia Zare Mehrjerdi
Industrial Engineering Department, Yazd University, Yazd, Iran
ARTICLE HISTORY
Received  December 
Accepted  June 
KEYWORDS
chi-square control chart;
economic design; integrated
model; maintenance
planning.
MATHEMATICS SUBJECT
CLASSIFICATION
D
ABSTRACT
Considered process in this article is a two-stage dependent process.
Each item in this process has two quality characteristics as x and y while
x and y are related to the stage 1 and 2, respectively. Each stage has two
operational states as the in-control state and out-of-control state and
transition time from the in-control state to the out-of-control state fol-
lows a general continues distribution function. The process is monitored
using a chi-square control chart. An integrated model that coordinates
the decisions related to the economic design of the used control chart
and maintenance planning is presented. For the evaluation of the inte-
grated model performance, a stand-alone maintenance model is also
presented, and the performance of these two models is compared with
each other.
1. Introduction
Quality is inversely proportional to variability. This denition implies that if variability in
the important characteristics of a product decreases, the quality of the product increases
(Montgomery 2009). Statistical process control (SPC) is a powerful collection of problem-
solving tools useful in achieving process stability and improving capability through the
reduction of variability. Control chart as a featured tool of SPC has been used extensively by
practitioners to monitor and even reduce process variation by identifying and eliminating
sources of variation (Mehrafrooz and Noorossana 2011). The control chart plays major role
in detecting the assignable causes, so that the necessary corrective action could take place
before non-conforming products are manufactured in a large amount (Charongrattanasakul
and Pongpullponsak 2011).
An important factor in using control chart is the design of the control chart, which means
generally the selection of the sample size, control limits, and frequency of sampling. In the
economic design of control chart, it will be tried to determine the chart parameters such that
the total cost (prot) of using control chart is minimized (maximized). Total cost in the eco-
nomic design of control chart includes sampling cost, production cost when the process is
in control or out of control, and process repairs cost when it goes out of control (Abouei
Ardakan et al. 2016). Duncan (1956) proposed the rst economic model for determining
the three design parameters for the ¯
xcontrol chart that minimizes the average cost when a
CONTACT Mohammad Saber Fallahnezhad fallahnezhad@yazd.ac.ir Industrial Engineering Department, Yazd Univer-
sity, Yazd, Iran.
©  Taylor & FrancisGroup, LLC
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2H. RASAY ET AL.
single out-of-control state exists. The subject of design of control chart has been investigated
in dierent directions so far.
On the other hand, maintenance can be dened as a combination of all technical and
administrative actions including supervision, action intended to retain or restore the system
into a state in which system can perform a required function (Ding and Kamaruddin 2015).
Maintenance planning (MP) and SPC are two key tools for management and control of pro-
ductionprocess.Althoughforyearsthesetwokeytoolsareconsideredandanalyzedsepa-
rately, from the academic and practical point of view, recently some integrated models are
developed for considering MP and SPC jointly. It is mentioned by many authors that there is
a great interaction and interrelation between maintenance planning and SPC that veries the
development of the integrated models. In these models, SPC is usually studied in the aspect
of economic design of control chart.
Therestofthepaperisorganizedasfollows:section2presents a literature review about the
integrated model for the economic design of control chart and MP. In section 3,monitoring
of the process using chi-square control chart is described. Section 4includes the general
structure of the integrated model. The integrated model and stand-alone maintenance model
are developed in sections 5and 6, respectively. Section 7presents an illustrative example
and provides a comparison between the performances of the two models. Finally, section 8
concludes the paper.
2. Literature review
Asthebestofauthor’sknowledge,Tagaras(1988) presented the rst integrated model for
SPC and MP. This subject has been diversied in dierent directions so far. Cassady et al.
(2000) proposed a model for integrating the design of ¯
xcontrol chart and age-replacement
preventive maintenance. The assumed process has two quality states, and it is used as
simulation−optimization approach for determining optimal policy. Linderman, McKone-
Sweet and Anderson (2005) presented a model for process control and maintenance planning
for a process with two operational states as in-control state and out-of-control state. The ¯
x
control chart is used for process monitoring, and three scenarios are assumed for process
evolution in each production cycle. Also, process failure mechanism follows a Weibull distri-
bution function. Zhou and Zho (2008) develop Lindermen et al.’s model by adding another
scenario for the evolution of the process. In this scenario, it is assumed that false alarm
leads to the compensatory maintenance and renews the process. Wang (2010)presented
an integrated model for design of np control chart and maintenance planning in a process
with three states that include two operational states and one failure state. Wu and Makis
(2008) considered economically designing a chi-square control chart for application in the
condition-based maintenance. The process has three states including two operational states
and one failure state, while it is assumed that transition between these states is based on the
exponential distribution. Panagiotidou and Nenes (2009) presented an integrated model for
SPC and maintenance, while a Shewhart chart with variable parameters is used for process
monitoring. Panagiotidou and Tagaras (2010)presentedamodelforintegratingSPCand
maintenance, while x-bar control chart is used for monitoring. The process has three states,
and transition between states are considered as a general continues distribution function.
Panagiotidou and Tagaras (2012) develop the (Panagiotidou and Tagaras 2010)modelby
adding minimal maintenance confronting the out-of-control state, while minimal mainte-
nance restores the process to the good quality state without eecting the equipment age.
Naderkhani and Makis (2015) designed a Bayesian control chart with two sampling intervals
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 3
for condition-based maintenance. The process has three states, and transition between these
statesisbasedontheexponentialdistribution.
Abouei Ardakan et al. (2016)presentedahybridmodelforeconomicdesignofmultivari-
ate exponential weighted moving average (MEWMA) chart and maintenance planning. The
process has two operational states as in-control state and out-of-control state, while transition
betweenthesestatesisbasedontheWeibulldistribution.Yinetal.(2015)developedaninte-
grated model for SPC and maintenance planning based on the delayed monitoring concept.
Delayed monitoring postpones the sampling process till a scheduled time and contributes to
ten-scenarios of the production process. Also, ¯
xcontrol chart is designed for process moni-
toring. Fallhanezhad and Niaki (2011),andFallahnezhad(2014) developed integrated models
for maintenance planning, machine replacement, and production planning using stochastic
dynamic programming.
All the aforementioned models are about a system with one component. Besides developing
the integrated models of SPC and MP for a system with only one component, there are few
models in the literature of this subject for the system with more than one component. As the
best of author’s knowledge, only two models are presented for the system with more than one
component. Liu et al. (2013) presented a model for a series system consisting of two identical
series units. Each unit has three states including two operational state plus a failure state, while
transition between these states is based on the exponential distribution. Also, ¯
xcontrol chart
is used for process monitoring. The second model for system with more than one component
is a model proposed by Zhong and Ma (2015). This model is about a two-stage-dependent
process. The process has two states, and Shewhart individual-residual joint control chart is
used for process monitoring.
Based on the above literature review, the main contribution of this paper is to develop
an integrated model for designing chi-square control chart and MP for a two-stage process,
while no restriction is assumed about the failure mechanism in each stage. Also this model
has a general structure such that it can be applied for any types of inspection policy. Moreover,
a model that only considers maintenance planning is also presented, and the performances of
these two models are compared with each other.
3. Process monitoring using chi-square control chart
Consider a two-stage dependent process while one type of product is produced in this system.
Figure 1 shows this process. Each item has two quality characteristics as x and y, and the
quality characteristic x and y are related to the rst and second stage, respectively. Also, the
relationshipbetweenxandycanbeexpressedbasedonthelinearregressionmodel,andfor
item j the paired observation Xj=(xj,yj) has a bivariate normal distribution.
Two types of assignable causes may aect the process. Assignable cause 1 and 2 are related
to the rst and second stage, respectively. Also, time of occurrence of the assignable causes
follows a general continuous distribution function with non-decreasing failure rate. Hence,
in each time point, each stage can be in one of the two operational states: in-control state
denoted by 0 and out-of-control state denoted by 1. Hence, the process state, in each time
point, is denoted by (u,v), if stage 1 operates in state u (u =0,1) and stage 2 operates in state v
(v =0,1). Thus, the process operates in state (0,0) if no assignable cause eects on the process.
Figure . The considered process.
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4H. RASAY ET AL.
The state (1,0) indicates that only assignable cause 1 has eects on the process, while the state
(0,1) indicates that only assignable cause 2 has eects on the process. Also, the state (1,1)
shows the situation that both stages are out of control, or in the other word, both assignable
causes have eect on the process.
It is assumed that when the process is in state (0,0), Xjfollows a bivariate normal distribu-
tion with parameter (μ0,
0).Alsoweassumethattheoccurrenceoftheassignablecauseswill
not aect the covariance matrix 0but leads to the change in the process mean. At each sam-
pling time i(i =1,2, …,m-1), a sample with size n is taken from the process, and the value of
the statisticχ2
i=n(Xi−μ0)−1
0(Xi−μ0)is plotted on the chi-square control chart. Where
Xi=n
j=1Xij
n,i=1,2, ..., m−1.Whentheprocessisinthein-controlstate,thenthestatis-
tic χ2
ifollows central chi-square distribution with 2 degrees of freedom. If the process oper-
ates in the state (u,v) (u+v=0), Xjfollows a bivariate normal distribution with parameter,
(μu,v,
0)and the statistic χ2
ifollows non-central chi-square distribution with non-centrality
parameter δu,v=n(μu,v−μ0)−1
0(μu,v−μ0)(denoted by χ2
i(δu,v)) with 2 degrees of
freedom.
The parameters of the Chi-square control chart are upper control limit (UCL), sampling
inspection times, and sample size. The probability of type I error or αis determined as
follows:
α=∞
UCL
f(x)dx (1)
Where f(x) is a chi-square distribution with 2 degree of freedom.
If the process operates in the out-of-control state (u,v) (u+v=0), the probability of type II
error or βis computed as follows:
βu,v=UCL
0
fδu,v(x)dx (2)
While fδ(x)is a non-central chi-square distribution with δas a non-centrality parameter
and 2 degree of freedom. It is worth mentioning that for real application, the values of μ0,μ
u,v
and 0must be estimated from the past data of process. In the case that number of observa-
tion used in estimation is not large enough, Hotelling’s T2control chart should be considered
instead of chi-square control chart (Wu and Makis 2008).
4. General structure of the integrated model and notations introduction
Inthissection,thegeneralstructureoftheintegratedmodelisdescribed,andthenotations
used in development of the models are introduced.
4.1. Structure of the integrated model
At specied point of time, t1,t2,…,t
m-1 that are decision variables of the integrated model, a
sample with size n is selected, and the observations Xj=(xj,yj)are obtained for each item.
Based on these observations, the value of the statistic χ2
iis calculated and plotted on the
chi-square control chart. If the value of this statistic exceeds the UCL, the chart releases an
alarm. To verify the correctness of this alarm, an error-free inspection is conducted on the
process. This inspection is called maintenance inspection to distinguish it from the sampling
inspection. The time and cost of the maintenance inspection are tIand CI,respectively.Ifthis
inspection concludes that the released alarm form the control chart is incorrect, the process
continues to its operation without any further interruption. If the maintenance inspection
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 5
Figure . Different scenarios for the evolution of the process in each production cycle.
indicates that the alarm is correct, then the reactive maintenance (RM) is implemented on the
process,andtheprocessisrenewed.Itisworthnotingthatatt
mthere is no sampling inspec-
tion but maintenance inspection is conducted on the process, regardless of the process state.
Eight scenarios are possible for the evolution of the process in each production cycle.
These scenarios are illustrated in Figure 2 and described as follows. If the process operates
in state (0,0) until the end of the production cycle, tm,thenatt
m,preventivemaintenance
(PM) is implemented on the process (scenrios1 and 5). If the process operates in the out-
of-control state (u,v) (u+v=0) and the control chart releases an alarm before time tm-1 ,then
reactive maintenance form type (u,v) (RM(u,v)) is conducted on the process (scenarios 2,3,4).
Besides these scenarios, it is possible to occur the situations that the process operates in the
out-of-control state (u,v) but the control chart cannot release an alarm for this state in the
following sampling inspection period. In these situations, at time point tm,themaintenance
inspection is conducted on the process, true state of the process is determined, and RM(u,v)
is implemented on the process (scenarios 6,7,8).
4.2. The notations
Notation Description
Ru,v Expected revenue of the process per time unit when the stage  is in state u and stage  is in state v (u =,;
v=,).
CQC Sampling inspection cost
CPM cost of performing preventive maintenance
CRM(u,v) cost of conducting RM when stage  is in state u and stage  is in state v.
CICost of performing maintenance inspection
tPM Time to perform preventive maintenance
tRM(u,v) TimetoperformRMwhenstageisinstateuandstageisinstatev
tITime to perform maintenance inspection
fk(t) Density function of time of quality shift for stage k (k =,)
Fk(t) Cumulative distribution function (c.d.f) of time of quality shift for stage k Fk(x)=1−Fk(x)
ti(i =,..,m) Time points of inspections (decision variables)
αProbability of type I error
βu,vProbability of type II error for the control chart if stage  operates in state u and stage  operates in state v
UCL Upper control limit of control chart (decision variables)
tmTime of conducting PM on the process (decision variables)
m Number of inspection in each production cycle (decision variables)
n Sample size (decision variables)
E[Tu,v ] Expected time that the process operates in state (u,v) in each production cycle
E[QC] Expected number of sampling inspection in each production cycle
E[α] Expected number of false alarm in each production cycle
PPM Probability of terminating the production cycle due to performing PM
PRM(u,v) Probability of terminating the production cycle due to performing RM(u,v)
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6H. RASAY ET AL.
5. Integrated model development
The described integrated model in the section 4.1 can be considered as a renewal reward
process consisting of the stochastic and independent identical cycles. Hence, EPT is obtained
as follows:
EPT =E[P]
E[T](3)
Before computing E[P] and E[T] in equation 3,itisnecessarytocalculatesomeequations
and mention some preliminaries. For the evolution of the process in the arbitrary inspection
interval [ti-1,ti], 10 dierent scenarios are possible to occur. These scenarios, their probabili-
ties, and amount of time that process operates in each state are illustrated in Tabl e 1 .Notice
that the calculated probabilities in this table are conditional probability given the state of the
process in the start of the inspection time (ti,t
i-1). Also in the gures of this table, the process
state at ti-1 is the state of the process just after inspection at ti-1, while the process state at tiis
the state of the process just before inspection at ti.
If the process operates in the state (0,0) just before inspection at ti, then it certainly oper-
ates in the state (0,0) after inspection at ti. If the process operates in state (u,v) (u+v=0) just
before inspection at ti, then with the probability βu,v, it continues to operate in this state, and
with the probability 1 −βu,vthe control chart releases an out-of-control signal, RM(u,v) is
implemented on the process, and the process is renewed.
Let dene Puv
tias the probability of operation of the process in state (u,v) just after inspection
at ti. These probabilities are obtained using equations 4,5,6,and7.
First P0,0
tiis computed based on the following equation:
P0,0
ti=F1(ti)F2(ti);1≤i≤m(4)
This equation is obtained based on the fact that the process operates in state (0,0) if and only
if the failure time for both stages becomes more than ti.P1,0
tiis obtained using the following
equation:
P1,0
ti=β1,0P0,0
ti−1×P(bti−1)+P1,0
ti−1×P(hti−1);1≤i≤m−1(5)
Considering Table 1 ,thesumofthetwotermsinsidethesquarebracketsistheprobability
that the process operates in state (1,0) just before inspection at ti. Also, if process operates in
state (1,0), before inspection at ti, with the probability β1,0, the control chart cannot release
this state, and the process continues to operate in this state after inspection at ti.
By considering Tab l e 1 and similar to the derivation of the equation 5,thefollowingtwo
equations are obtained for P0,1
tiand P1,1
ti.
P0,1
ti=β0,1P0,0
ti−1×P(cti−1)+P0,1
ti−1×P(fti−1);1≤i≤m−1(6)
And
P1,1
ti=β1,1P0,0
ti−1×P(dti−1)+P0,0
ti−1×P(eti−1)+P0,1
ti−1×P(gti−1)+P1,0
ti−1×P(iti−1)+P1,1
ti−1×1
1≤i≤m−1(7)
Also,thisequationisheldatthestartofeachproductioncycle:
P0,0
0=1;P0,1
0=P1,0
0=P1,1
0=0(8)
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 7
Tab le . Different scenarios for the evolution of the process in each inspection interval in the integrated
model.
Scenario figure Probability of occurring Ti
00 Ti
10 Ti
01 Ti
11
aP(ati−1
)=F1(ti)
F1(ti−1)
F2(ti)
F2(ti−1)ti-ti- 
bP(bti−1
)=
F2(ti)
F2(ti−1)ti
ti−1
f1(t)dt
F1(ti−1)
t-ti- ti-t  
cP(cti−1
)=
F1(ti)
F1(ti−1)ti
ti−1
f2(t)dt
F2(ti−1)
t-ti- t
i-t 
dP(dti−1
)=
ti
ti−1
f1(t)
F1(ti−1)ti
t
f2(t)
F2(t)dtdt
t-ti- t−tti−t
Continued on next page
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8H. RASAY ET AL.
Tab le . (continued).
Scenario figure Probability of occurring Ti
00 Ti
10 Ti
01 Ti
11
e
P(eti−1
)=ti
ti−1
f2(t)
F2(ti−1)
ti
t
f1(t)
F1(t)dtdt =
1−p(ati−1
)−p(bti−1
)−
p(cti−1
)−p(dti−1
)
t-ti- t−tt
i−t
fP(fti−1
)=F1(ti)
F1(ti−1)t
i-ti- 
gP(gti−1
)=1−p(fti−1
)t-t
i- ti-t
hP(hti−1
)=F2(ti)
F2(ti−1)t
i-ti- 
Continued on next page
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 9
Tab le . (continued).
Scenario figure Probability of occurring Ti
00 Ti
10 Ti
01 Ti
11
iP(iti−1
)=1−P(hti−1
)t-t
i- t
i-t
jP(jti−1
)=1t
i-ti-
Based on the description presented about the integrated model so far, the following two
equations are obtained for E[T] and E[P]:
E[P]=
v=0,1
u=0,1
R(u,v)E[Tu,v]−
v=0,1
u=0,1
CRM(u,v)PRM(u,v)−CPMPPM −CQCE[QC]
−CIE[α]−CI(9)
E[T]=
v=0,1
u=0,1
E[Tu,v]+
v=0,1
u=0,1
tRM(u,v)PRM(u,v)+tPMPPM +tIE[α]+tI(10)
Inthefollowing,weproceedtocomputeeachtermintheequations9and 10.Theexpected
time that the process operates in state u,v, Tu,v is obtained based on the equations 11 and 13.
First, E[T00] is computed using the following equation:
E[T00]=tmF1(tm)F2(tm)+tm
0
tf
1(t)F2(tm)dt +tm
0
tf
2(t)F1(tm)dt (11)
Using the fact t hat f1(t)dt =−dF1(t)and integrating tm
0tf
1(t)F2(tm)dt by parts, this
leads to the following simpler equation for E[T00]:
E[T00]=tm
0
F1(t)F2(t)dt (12)
If Ti
u,vis dened as the expected time that the process operates in state (u,v) in the interval
(ti-1,ti), then Tu,v can be obtained by the following equation.
E[Tu,v]=
m

i=1
Ti
u,v,u=0,1;v=0,1;v+u= 0 (13)
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10 H. RASAY ET AL.
Ti
1,0is obtained as follows:
Ti
1,0=P0,0
i−1×F2(ti)
F2(ti−1)ti
ti−1
f1(t)
F1(ti−1)(ti−t)dt +ti
ti−1
(t−t)f1(t)
F1(ti−1)ti
t
f2(t)
F2(t)dtdt 
+P1,0
i−1F2(ti)
F2(ti−1)(ti−ti−1)+ti
ti−1
(t−ti−1)f2(t)
F2(ti−1)dt(14)
If the state of the process is (0,0) at ti-1 andscenariobordoccurs,thentheprocessoperates
in state (1,0) in one part of the interval (ti-1,ti). Also, if the state of the process is (1,0) at ti-1,
and scenario h or i occur, the process operates in state (1,0) in one part of the interval (ti-1,ti)
as illustrated in Tabl e 3
Also, the following two equations are obtained similarly, for T0,1
iand T1,1
i:
Ti
0,1=P0,0
i−1×F1(ti)
F1(ti−1)ti
ti−1
f2(t)
F2(ti−1)(ti−t)dt +ti
ti−1
(t−t)f2(t)
F2(ti−1)ti
t
f1(t)
F1(t)dtdt 
+P0,1
i−1F1(ti)
F1(ti−1)(ti−ti−1)+ti
ti−1
(t−ti−1)f1(t)
F1(ti−1)dt(15)
And
Ti
1,1=P1,1
i−1(ti−ti−1)+P0,1
i−1ti
ti−1
f1(t)
F1(ti−1)(ti−t)dt +P1,0
i−1ti
ti−1
f2(t)
F2(ti−1)(ti−t)dt
+P0,0
i−1ti
ti−1
(ti−t)f1(t)
F1(ti−1)ti
t
f2(t)
F2(t)dtdt +ti
ti−1
(ti−t)f2(t)
F2(ti−1)ti
t
f1(t)
F1(t)dtdt 
(16)
If Pi
RM(u,v)is dened as the probability of conducting RM(u,v) after the inspection at ti,
thentheprobabilityofterminatingtheproductioncycleduetotheRM(u,v)isobtainedas
following:
PRM(u,v)=
m

i=1
Pi
RM(u,v);u=0,1;v=0,1;u+v= 0 (17)
Pi
RM(u,v)can be obtained using equations 18,19,20,and21.Derivationoftheseequations
is as follows:
Pi
RM(1,0)=1−β1,0P0,0
ti−1P(bti−1)+P1,0
ti−1P(hti−1),1≤i≤m−1 (18)
In equation 18,sumofthetwotermsinsidethesquarebracketsistheprobabilitythat
the process operates is state (1,0) just before inspection at ti. On the other hand, if the process
operates in state (1,0) before inspection at ti, with probability (1-β1,0) the control chart releases
this state, RM(1,0) is implemented on the process and the process is renewed.
Equations for computing Pi
RM(0,1)and Pi
RM(1,1)are derived similar to the Pi
RM(1,0).Hence,the
following two equations are obtained for them:
Pi
RM(0,1)
=(1−β01 )P0,0
ti−1P(cti−1)+P0,1
ti−1P(fti−1),1≤i≤m−1 (19)
And
Pi
RM(1,1)=(1−β11 )P0,0
ti−1P(dti−1)+P0,0
ti−1P(eti−1)+P1,0
ti−1P(iti−1)+P0,1
ti−1P(gti−1)
+P1,1
ti−1P(jti−1),1≤i≤m−1 (20)
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 11
At tmthere is no sampling inspection, and hence Pi
RM(u,v)is computed based on this equa-
tion:
Pm
RM(1,0)=P0,0
tm−1P(btm−1)+P1,0
tm−1P(htm−1)
Pm
RM(0,1)=P0,0
tm−1P(ctm−1)+P0,1
tm−1P(ftm−1)
Pm
RM(1,1)
=P0,0
tm−1P(dtm−1)+P0,0
tm−1P(etm−1)+P1,0
tm−1P(itm−1)+P0,1
tm−1P(gtm−1)+P1,1
tm−1P(jtm−1)
(21)
Each production cycle is terminated due to perform RM or PM. The probability for ter-
minating the production cycle due to implementing RM is computed. Hence, the following
equation is obtained for terminating the production cycle due to implementing PM:
PPM =1−
v=0,1
u=0,1
PRM (u,v)(22)
If Pi
QC is dened as the probability of performing sampling inspection at ti,thenE[QC]is
computed as follows:
E[QC]=
m−1

i=1
Pi
QC (23)
And Pi
QC canbeobtainedusingthisfollowingequation:
Pi
QC =P0,0
ti−1+P1,1
ti−1+P0,1
ti−1+P1,0
ti−1;1≤i≤m−1 (24)
It is worth mentioning that sum of these probabilities in the right side of equation 24 is not
equal 1 because there is always the probability that the process is renewed by conducting each
type of RM before reaching to time ti.
If we dene Pi
αastheprobabilityofreleasingafalsealarmfromthecontrolchartinthe
inspection at ti,thenE[α]isobtainedasfollowing:
E[α]=
m−1

i=1
Pi
α(25)
Where Pi
αis obtained as follows:
Pi
α=F1(ti)F2(ti)α;1≤i≤m−1 (26)
6. Stand-alone maintenance model
In this model, it is assumed that, in each production cycle, the process starts its operation from
the in-control state, and there is no inspection and sampling from the process. The process
continues to its operation until time point tm.Inthismodel,t
mis the only decision variable.
At tm, maintenance inspection is conducted on the process, and true state of the process is
identied. At tm, if the process is in state (0,0), then PM is conducted on the process. On the
other hand, at tm, if the process is in state (u,v; u+v=0), RM(u,v) is conducted on the process.
Similar to the integrated model, also in this model, the production cycles of this model
can be considered as a renewal reward process consisting of the stochastic and independent
identical cycles. Hence, EPT can be computed using equation 3.
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12 H. RASAY ET AL.
Tab le . Different scenarios for the evolution of the process in the maintenance model in each production
cycle.
scenario Stage Stage probability reaction
Scenario In control until tmIn control until tmF1(tm)F2(tm)Implementing PM
at tm
Scenraio Shifts to the
out-of-control state
prior to tm
In control until tmF1(tm)F2(tm)Implementing
RM(,) at tm
Scenraio In control until tmShifts to the
out-of-control state
prior to tm
F1(tm)F2(tm)Implementing
RM(,) at tm
Scenraio Shift to out-of-control at t Shift to out-of-control at
t(tm>t>t>0)tm
0tm
tf1(t)f2
(t)dtdt
Implementing
RM(,) at tm
Scenraio Shift to out-of-control at
t(tm>t>t>0)
Shift to out-of-control
at t tm
0tm
tf2(t)f1
(t)dtdt
Implementing
RM(,) at tm
E[T] is calculated as following:
E[T]=tm+F1(tm)F2(tm)tpm +F1(tm)F2(tm)trm(01)
+F2(tm)F1(tm)trm(10)+F1(tm)F2(tm)trm(11)+tI(27)
Computing E[P] is more complicated. For deriving E[P], note that ve scenarios are pos-
sible for the evolution of the process in each production cycle. These scenarios are illustrated
in Table 2 .
If we dene P(Si) as the probability of occurring the scenario Si(S1,S2,..S5), and dene
E[P|Si] as the expected prot in each production cycle conditioned on the occurrence of the
scenario Si, then E[P] is derived as following:
E[P]=
5

i=1
P(Si)E[P|Si]−CI(28)
In the equation 28,C
Iis subtracted from the prot because there is always the maintenance
inspection at tm, regardless of the process state.
The equations for computing E[P|Si] are obtained as following:
E[P|S1]=R0,0tm−CPM (29)
E[P|S2]=R0,0tm
0
tf
1(t|t<tm)dt +R1,0tm−tm
0
tf
1(t|t<tm)dt−CRM(1,0)
(30)
E[P|S3]=R0,0tm
0
tf
2(t|t<tm)dt +R0,1tm−tm
0
tf
2(t|t<tm)dt−CRM(0,1)
(31)
E[P|S4]=R0,0tm
0
tf
1(t|t<tm)dt tm
t
f2(t|t<t<tm)dt
+R1,0tm
0
(t−t)f1(t|t<tm)dt tm
t
f2(t|t<t<tm)dt
+R1,1tm
0
(tm−t)f1(t|t<tm)dt tm
t
f2(t|t<t<tm)dt−CRM(1,1)(32)
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 13
Tab le . The parameters of the basic example.
Factor δ δ δ µµvR
 R R R
value  .        
Factor C fCvCICPM CRM(1,0)CRM(0,1)CRM(1,1)tItPM tRM(1,0)tRM(0,1)
value        .   
Factor tRM (1,1)
value 
E[P|S5]=R0,0tm
0
tf
2(t|t<tm)dt tm
t
f1(t|t<t<tm)dt
+R0,1tm
0
(t−t)f2(t|t<tm)dt tm
t
f1(t|t<t<tm)dt
+R1,1tm
0
(tm−t)f2(t|t<tm)dt tm
t
f1(t|t<t<tm)dt−CRM(1,1)(33)
7. The illustrative examples and sensitivity analysis
In this section, the comparison study between the performances of the two developed models,
the integrated model and stand-alone maintenance model, is conducted. Moreover, the sen-
sitivity analysis for the two models is performed. For this purpose, rst we consider a basic
example that its parameters are indicated in Tab l e 3. Then, by changing the parameters of
this example, eight dierent cases are produced. Tab le 4 shows the changed parameters in
each case. Tab l e 5 illustrates the results of the two models: optimization for the basic example
alongsideeightproducedcases.Intheanalysisofthissection,itisassumedthatfailuremech-
anism for each stage follows a Weibull distribution. Though the inspection times in the inte-
grated model (t1,t2,…,t
m-1) can be any arbitrary values and developed integrated model could
be applied for any types of inspection policy, it is supposed that constant sampling interval is
applied for determining the inspection times. Also sampling cost at each inspection time is
equal to n×Cv+Cf,whileC
fand Cvare the xed and variable sampling cost. Ta ble 3 illus-
trates the parameter of the basic numerical example. In this table, µ1and µ2are the mean
value of Weibull distribution for the rst and second stages, respectively, and v is the shape
parameter of Weibull distribution.
The result of the integrated model optimization for the basic example indicates that at the
equidistance intervals with the length of 1.1 time unit, a sample with size 10 is taken from the
Tab le . The changed parameters in the analyzed cases considering the basic example.
case      
Changed parameters V=; µ= δ =.; R = R = Cf=;
V=µ= δ =; R = Cv=
δ =.; R =
Case  
Changed parameters tPM =; CI=
tRM(,) =; CPM =
tRM(,) =C
RM(,) =;
tRM(,) =; CRM(,) =;
CRM(,) =;
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14 H. RASAY ET AL.
Tab le . The results of the models optimization for the basic example and produced cases (IM: integrated
model; MM: maintenance model).
Results cases EPT tUCL n m tm
Basic example IM . . .   .
MM . — — — — 
 IM . .    .
MM  — — — — .
 IM .  .   
MM .———— .
 IM . . .   .
MM . — — — — 
 IM . . .   .
MM .———— .
 IM . . .   
MM .————.
IM .. 
MM . — — — — 
MM . — — — — .
 IM . . .   .
MM . — — — — .
 IM . .    .
MM . — — — — .
process, and the statistic χ2
iis plotted on the chi-square control chart with UCL =5.8. After
passing nine sampling periods if the process operates in state (0,0) then preventive mainte-
nanceisimplementedontheprocess.Inthemaintenance model, there is no sampling form
the process, and hence, the values of t1,UCL,n,andmareirrelevantinthismodel.Inthis
model, in each production cycle, process starts its operation in the in-control state, and after
passingsixtimeunits,preventivemaintenanceisimplementedontheprocess.AsTa b le 5 indi-
cates, for the basic example, the integrated model leads to more EPT in comparison with the
stand-alone maintenance model. The results of the optimization for the other cases also indi-
cate that the integrated model has a better performance in comparison with the stand-alone
maintenance model based on the EPT.
Table 5 also indicates the result of changes in the parameters for the both models. For the
case1, it can be seen that increasing the values of v leads to more EPT. Also this variation
increases the start time of monitoring from 1.1 to 1.7. These trends can be justied based
on the fact that in Weibull distribution, for larger values of shape parameter, the variance
of distribution decreases. Hence, it is easier to predict the failure time. In case2, as it was
expected, increasing the values of μleads to increase EPT and tmin both models. Cases 3
showstheeectofvariationsinδ. For this purpose, the value 0.5 is added to the value of each
δ.Asexpected,thischangeleadstoincreaseUCLfrom5.8to6.6anddecreaseofthesample
size from 10 to 8, because for larger values of δ,itiseasierforthecontrolcharttoreleasethe
out-of-control state. The impact of change in R00 isillustratedbycase4.Inthiscase,thevalue
of R00 increases from 500 to 700. As Tab le 5 indicates, increasing R00 has a major impact on
optimal values of the decision variables for two models. This increase leads to increase EPT
in both models while it decreases the value of t1intheintegratedmodel.IncreaseinEPTthat
yields from increasing R00 is intuitive to some extent. Also for the larger values of R00,the
dierence between the in-control and out-of-control state is deeper, and so it is necessary to
start monitoring of the process earlier. Hence, increasing R00 leads to decrease the value of
t1.Case5showstheimpactofincreaseintherevenuefortheprocessoperationintheout-
of-control states. It can be concluded that this change has a little impact on the EPT for two
models, based on the result of the two models optimization. Also this change, unlike the eect
of R00, does not have a signicant eect on the monitoring policy.
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COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 15
Case 6 indicates the impact of increase the xed and variable sampling costs. This change
leads to decrease in the sample size (n) and control limit (UCL). Decrease in the control limit
leads to compensate the eect of decrease in the sample size, so that the sensitivity of the
control chart for releasing the out-of-control state be preserved. Case 7 indicates the eect of
increase in the maintenance associated times. The main result from this change is decrease in
EPT for both models. Also, this change yields to increase in the inspection periods and tmin
the integrated model. The eect of change in the maintenance costs is analyzed in case 8. As
this case shows, increasing the maintenance costs decreases the EPT for two models. Also, in
the integrated model, this change increases the control limit, number of inspection periods,
and tm.
8. Conclusion
Two practical models are developed in this article for a multistage-dependent process. The
rst model is an integrated model for economic designing of chi-square control chart and
maintenance planning. The second model is a stand-alone maintenance model in which only
maintenance planning is optimized. The developed integrated model has a general structure
because it is assumed that the failure mechanism for each stage follows a general continu-
ous distribution function, and this model could be applied for any type of inspection policy.
Although it is supposed that monitoring of the process is performed using a chi-square con-
trol chart, by some mild modications, the integrated model could be applied for any type of
controlchartwiththexedvaluesoftypeIandIIerror.Finally,basedonthenumericalexam-
ples and sensitivity analyses, it is concluded that the integrated model leads to the more prot
in comparison with the stand-alone maintenance model. Developing the proposed model for
a multistage production process can be considered as a future work.
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