An Adaptive Bayesian Scheme For Joint Monitoring Of Process Mean And Variance

An adaptive Bayesian scheme for joint monitoring of process mean
and variance
George Nenes a,n
, Sofia Panagiotidou b,1
a
University of Western Macedonia, Department of Mechanical Engineering, Bakola & Sialvera, 50100 Kozani, Greece
b
Aristotle University of Thessaloniki, Department of Mechanical Engineering, 54124 Thessaloniki, Greece
a r t i c l e i n f o
Available online 7 June 2013
Keywords:
Quality control
Bayes theorem
Economic optimization
a b s t r a c t
This paper presents a new model for the economic optimization of a process operation where two
assignable causes may occur, one affecting the mean and the other the variance. The process may thus
operate in statistical control, under the effect of either one of the assignable causes or under the effect of
both assignable causes. The model employed uses the Bayes theorem to determine the probabilities of
operating under the effect of each assignable cause. Based on these probabilities, and following an
economic optimization criterion, decisions are made on the necessary actions (stop the process for
investigation or not) as well as on the time of the next sampling instance and the size of the next sample.
The superiority of the proposed model is estimated by comparing its economic outcome against the
outcome of simpler approaches such as Fp (Fixed-parameter) and adaptive Vp (Variable-parameter)
Shewhart charts for a number of cases. The numerical investigation indicates that the economic
improvement of the new model may be significant.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
In the last decades there has been a massive production of
scientific articles proposing numerous economically optimized
statistical quality control charts. The pioneering work of Duncan
[14], who was the first who associated the use of a simple Shewhart
control chart with the economic outcome, and optimized its use,
has inspired many scientists for over half a century. An indicative
list of early approaches in the field of economically designed control
charts would include the works of Bather [2], Goel et al. [17],
Knappenberger and Grandage [18], von Collani [40,41] and Duncan
[15] who extended his early model to the case of multiple assign-
able causes. Lorenzen and Vance [19] also proposed an economic-
ally designed model that is flexible enough to be used either for
Shewhart charts or for Cumulative Sum (CUSUM) and Exponentially
Weighted Moving Average (EWMA) charts.
The common characteristic of all aforementioned approaches is
that (a) they all assume fixed design parameters and (b) they all
consider assignable causes that affect the process mean. Since
those early approaches it soon became evident that the design of
control charts with adaptive design parameters could significantly
improve the economic and statistical behavior of the charts. To this
effect, Reynolds et al. [27] were the first to introduce adaptive
control charts, and in particular VSI (Variable Sampling Interval)
charts. Typical economically designed VSI control charts were later
proposed by Das et al. [10] and Bai and Lee [1]. Similar adaptive
control charts where the sample size (instead of the sampling
interval) is allowed to vary are called VSS (Variable Sample Size)
charts and their economic design was first introduced by Park and
Reynolds [25]. The economic design of VSSI (Variable Sample Size
and sampling Interval) control charts has been analyzed in the
works of Das et al. [10] and Park and Reynolds [25] while De
Magalhães et al. [12], Costa and Rahim [9], Nenes [21] and Celano
et al. [5] present the economic design of fully adaptive control
charts (Vp-Variable parameter control charts).
A different stream of economically designed control charts uses
the Bayes theorem in order to determine the optimum design
parameters. To this end, instead of using just the last sample's
outcome, these charts use all information available to reach to the
optimum decisions. Bayesian charts have their origins in the first
theoretical approaches of Girshick and Rubin [16], Bather [2] and
Taylor [38,39]. However, the economic design of Bayesian control
charts has not received much attention from the academics,
mainly because of the increased modeling complexity. Tagaras
[33,34] was the first to introduce a Bayesian one-sided X control
scheme with adaptive parameters, while, at about the same time,
Calabrese [3] developed a one-sided Bayesian p chart for monitor-
ing the fraction nonconforming. Porteus and Angelus [26] identify
opportunities for cost reduction in SPC through the use of Bayesian
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/caor
Computers & Operations Research
0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.cor.2013.05.018
n
Corresponding author. Tel.: +30 24610 56665; fax: +30 24610 56601.
E-mail addresses: gnenes@uowm.gr, gnenes@auth.gr (G. Nenes),
span@uowm.gr (S. Panagiotidou).
1
Tel.: +30 2310 995914; fax: +30 2310 996018.
Computers & Operations Research 40 (2013) 2801–2815

charts and Tagaras and Nikolaidis [36] compare various one-sided
Bayesian X control charts from an economic point of view.
Nikolaidis et al. [24] present an industrial application of a Bayesian
one-sided X-chart for monitoring the quality of tiles at a particular
stage of a tile manufacturing process while Nenes and Tagaras
[23], Tagaras and Nenes [35] are the first to introduce a two-sided
X control scheme for monitoring short production run processes.
Makis [20] has introduced an economically designed Bayesian
control chart for infinite production processes, keeping the sample
size and sampling interval fixed. Celano et al. [4] extend the work
of Tagaras [34] and propose an adaptive Bayesian chart for the
control of the dispersion of a process while Nenes [22] presents an
economically designed fully adaptive Bayesian control chart for
monitoring the process mean in infinite production runs.
The economic design of control charts in the presence of
assignable causes affecting both the process mean and the process
variability has attracted even more limited attention by the scien-
tists because of the increased model complexity. The joint econom-
ically optimal design of X and R control charts was first considered
by Saniga [28] who assumed though, that the occurrence of the one
cause blocks the occurrence of the other. Saniga [29] and Saniga and
Montgomery [30] developed models for processes subject to a
single assignable cause. The occurrence of this assignable cause
results in a simultaneous shift in the process mean and variance.
Costa [6] presents a model for the joint economic design of X and R
fixed-parameter control charts where both causes can exist simul-
taneously. Stoumbos and Reynolds [31] develop a comprehensive
economic model for the design of control schemes based on the
combination of VSI EWMA and VSI X chart schemes. Rahim and
Costa [32] deal with the joint economic design of X and R charts
when the occurrence times of assignable causes follow Weibull
distributions with increasing failure rates. Costa [7,8] proposes Joint
X and R charts with variable sample sizes and sampling intervals
(1999) and with all the design parameters adaptive (1998) and
investigates numerically the statistical performance of the charts.
De Magalhães and Neto [13] present an economically designed
adaptive X and R chart. In their paper they assume a single
assignable cause which may affect the process mean and/or the
standard deviation. However, in their model the design para-
meters (n, h, w, k) are not allowed to take any possible values,
since the optimization procedure necessitates specific rules con-
cerning the relationship between the relaxed and tightened design
parameters. De Magalhães et al. [11] propose an adaptive, statis-
tically optimized control chart for monitoring a process subject to
two independent assignable causes that affect the process mean
and/or the variance. Again, specific rules are utilized that con-
straint the allowable values of the relaxed and tightened design
parameters. Tasias and Nenes [37] also assume two different
assignable causes for the mean and standard deviation. However,
they follow a different modeling approach allowing a more general
problem setting where the two assignable causes are independent
and the design parameters of the model can be optimized without
any constraints. These same, more generic, assumptions are also
made in the present paper, the scope of which is the development
of a new Bayesian model that is used to determine the economic-
ally optimum parameters in a process where two assignable
causes may occur, shifting the mean and/or the variance. Thus,
the novel contribution of the paper is twofold:
(a) The development of a model for the representation and
economic optimization of a Bayesian chart for monitoring
the process mean and variance of a production process when
the two assignable causes can occur independently.
(b) The comparison of the economic outcome of the new model
against the economic outcome of earlier and less sophisticated
approaches.
The remainder of the paper is structured as follows. Section 2
that follows presents in detail the problem setting and assump-
tions used throughout the paper. Section 3 presents the develop-
ment of the proposed Bayesian model and describes its operation,
while Section 4 describes the expected cost derivation. In Section 5
a numerical investigation is conducted and comparisons with the
economic outcome of simpler control charts are presented. Section
6 summarizes the paper and presents its main conclusions.
2. Problem setting and assumptions
A production process is assumed to operate for an infinite
horizon of time. The key measure of the process quality is a
continuous random variable X which is assumed to be normally
distributed. The target mean of X is μ0 and the target variation
is s2
0. Occasionally, two assignable causes may affect the process by
shifting the mean and variation of the quality characteristic.
Assignable cause 1 occurrence is assumed to shift the mean from
its target value to μ1 ¼ μ0 þ δs0 (δ40). In the same sense, assign-
able cause 2 occurrence shifts the variance from s2
0 to s2
1 ¼ γ2
s2
0
(γ41). Note that both assignable causes are assumed to affect the
monitored characteristic in a unidirectional way, i.e., only upward
shifts are assumed (or only downward shifts for the case of the
mean). Unlike many approaches, it is assumed that the occurrence
of one cause does not block in any way the occurrence possibility
of the other cause. That is, the process, besides operating under
statistical control (μ ¼ μ0; s ¼ s0), may operate under the effect of
only assignable cause 1 μ ¼ μ1; s ¼ s0

, under the effect of only
assignable cause 2 (μ ¼ μ0; s ¼ s1), or under the effect of both
assignable causes (μ ¼ μ1; s ¼ s1). The state of the process is
denoted by Y¼0 when μ ¼ μ0 and s ¼ s0, Y¼1 when μ ¼ μ1 and
s ¼ s0, Y¼2 when μ ¼ μ0 and s ¼ s1 and Y¼3 when μ ¼ μ1 and
s ¼ s1. The notation used throughout this paper is included in
Appendix A.
The time until the occurrence of assignable cause 1 is assumed
to be an exponentially distributed random variable with mean 1=λx
while the time until the occurrence of assignable cause 2 is
assumed to be an exponentially distributed random variable with
mean 1=λs. Thus, the probability of assignable cause 1 occurrence
in an interval of h time units is 1−e−λxh
and the probability of
assignable cause 2 occurrence is 1−e−λsh
.
It is assumed that both assignable causes are only indirectly
observable through the outcome of a sampling procedure. In this
sense, samples are collected by the production process and the
mean and standard deviation are computed. Based on these
computations, a decision may be made not to interrupt the
production process (a ¼ 0) or an alarm may be issued (a ¼ 1)
which is followed by an investigation and possible restoration of
the production process if any assignable cause has indeed
occurred.
The cost of a false alarm is denoted by L0, the cost of removing
assignable cause 1 is denoted by Lx, the cost of removing assign-
able cause 2 is denoted by Ls, while the cost of removing both
assignable causes is denoted by Lxs. In the same sense, the cost per
time unit of operating under the effect of assignable cause 1 is
denoted by Mx, the cost per time unit of operating under the effect
of assignable cause 2 is denoted by Ms, while the cost per time unit
of operating under the effect of both assignable causes is denoted
by Mxs. In addition to the aforementioned costs, the fixed cost per
sample is denoted by b while the variable sampling cost by c.
The time to interrupt the process and investigate it is denoted
by T0, the time to remove assignable cause 1 is denoted by Tx, the
time to remove assignable cause 2 is denoted by Ts while the time
to remove both assignable causes is denoted by Txs.
G. Nenes, S. Panagiotidou / Computers Operations Research 40 (2013) 2801–2815
2802

3. Description of the Bayesian chart
The process is monitored through the use of an adaptive
Bayesian chart. Specifically, at each sampling instance, the prob-
abilities of operating in control or under the effect of any assignable
cause are computed. Based on these probabilities, the practitioner
needs to answer the following questions: (a) Is an intervention to
the process necessary? In other words, are the probabilities of out-
of-control operation high enough to justify a process stoppage and
an investigation? If they are, then an investigation takes place
(a ¼ 1) that reveals with certainty the process status. Then, a
restoration to the in control state takes place, in case any assignable
cause was indeed present. (b) How long should the process be left
to operate until the next sampling instance? In other words, what
should the size h of the next sampling interval be? (c) What the
following sample size n should be?
The above decisions, as already mentioned, depend on the
probabilities of out-of-control operation and are optimized eco-
nomically. That is, the optimum decision parameters of the chart's
operation are carefully selected in order to minimize the total
expected quality-related cost of the production process.
Specifically, at each sampling instance a sample is collected and its
mean x and standard deviation s are measured. Based on the sampling
outcome, the probabilities of operating out of control, denoted by px
and ps, are computed and based on these probabilities the decisions
concerning the operation of the control scheme are made.
The decisions concerning the sampling parameters and the
investigation (and possible restoration) of the production process
are described with the aid of the warning probabilities pxl and psl
and the critical probabilities pn
x and pn
s as follows:
If at any sampling instance, both probabilities are below the
warning probabilities, px ≤pxl and ps ≤psl, i.e., when both probabil-
ities lie in the central zone, then the decision is not to investigate
(a ¼ 0) and the next sample of size n¼n1 will be taken after h¼h1
time units (h1 and n1 are called “relaxed” parameters). Let SI denote
the set of px; ps

values that lead to the selection of the relaxed
parameters: ðpx; psÞ∈SI.
In the same sense, if any one of the two probabilities lies in the
warning zone (but none exceeds its critical value), i.e., when
pxl opx ≤pn
x and ps ≤pn
s , or when psl ops ≤pn
s and px ≤pn
x, then the
decision is again a ¼ 0 but the next sample of size n¼n2 (≥n1) will
be taken after h¼h2 (≤h1) time units (h2 and n2 are called
“tightened” parameters). Let SII denote the set of ðpx; psÞ values
that lead to the selection of the tightened parameters: ðpx; psÞ∈SII.
Finally, if any one of the two probabilities exceeds its critical
value, i.e., when px 4pn
x and/or when ps 4pn
s , then the decision is
to investigate the process (a ¼ 1), remove any assignable cause the
investigation reveals and use the relaxed parameters for the
following sampling. Let SIII denote the set of ðpx; psÞ values that
lead the scheme to issue an alarm: ðpx; psÞ∈SIII.
The three regions of the two charts that define the relative
decisions are illustrated in Fig. 1.
It should be noted here that the value of h2 is also allowed to be
zero. In other words, if the tightened parameters are to be used,
the next sample may be taken immediately after the previous one
without allowing the production process to operate at all. This
does not affect in any way the analysis that follows.
Since X is a normally distributed random variable, the sample
mean x ¼ Σn
i ¼ 1xi=n is also a normally distributed random variable.
Note that xi ði ¼ 1; 2; :::; nÞ are the n individual measurements of the
quality characteristic in the sample. Let f0 be the pdf of x when no
assignable cause has occurred (Y¼0), and f1 the pdf of x when
Y¼1, i.e., μ ¼ μ1 and s ¼ s0:
f 0 ¼
ffiffiffi
n
p
s0
ffiffiffiffiffiffi
2π
p e−ð1=2Þððx−μ0Þ=ðs0=
ffiffi
n
p
ÞÞ2
¼
ffiffiffi
n
p
s0
ffiffiffiffiffiffi
2π
p e−ð1=2Þz2
ð1Þ
f 1 ¼
ffiffiffi
n
p
s0
ffiffiffiffiffiffi
2π
p e−ð1=2Þððx−μ0−δs0Þ=ðs0=
ffiffi
n
p
ÞÞ2
¼
ffiffiffi
n
p
s0
ffiffiffiffiffiffi
2π
p e−ð1=2Þðz−δ
ffiffi
n
p
Þ2
ð2Þ
where z ¼ x−μ0
ffiffiffi
n
p
=s0 is the standardized normal random vari-
able. In the same sense, when Y¼2, i.e., μ ¼ μ0 and s ¼ s1 the pdf of
x, denoted by f
γ
0, is
f
γ
0 ¼
ffiffiffi
n
p
γs0
ffiffiffiffiffiffi
2π
p e−ð1=2Þððx−μ0Þ=ðγs0=
ffiffi
n
p
ÞÞ2
¼
ffiffiffi
n
p
γs0
ffiffiffiffiffiffi
2π
p e−ð1=2Þðz=γÞ2
ð3Þ
while when Y¼3, i.e., μ ¼ μ1 and s ¼ s1 the pdf of x, denoted by
f
γ
1, is
f
γ
1 ¼
ffiffiffi
n
p
γs0
ffiffiffiffiffiffi
2π
p e−ð1=2Þððx−μ0−δs0Þ=ðγs0=
ffiffi
n
p
ÞÞ2
¼
ffiffiffi
n
p
γs0
ffiffiffiffiffiffi
2π
p e−ð1=2Þððz−δ
ffiffi
n
p
Þ=γÞ2
ð4Þ
Similarly, whenever s ¼ s0 (Y¼0 or 1), the random variable
X2
¼ ðn−1Þs2
=s2
0 follows the chi-square distribution, g0, with n−1
degrees of freedom (note that n¼n1 or n2):
g0 ¼
1
Γððn−1Þ=2Þ
⋅
1
2ððn−1Þ=2Þ
⋅e−ð1=2Þððn−1Þs2
=s2
0
Þ
⋅
ðn−1Þs2
s2
0
!ððn−1Þ=2Þ−1
¼
1
Γððn−1Þ=2Þ
⋅
1
2ððn−1Þ=2Þ
⋅e−ð1=2ÞX2
⋅ðX2
Þððn−1Þ=2Þ−1
ð5Þ
where s2
¼ Σn
i ¼ 1ðxi−xÞ2
=ðn−1Þ is the unbiased estimator of the
population variance. Whenever s ¼ s1 (Y¼2 or 3), X2
follows the
chi-square distribution, g1:
g1 ¼
1
Γððn−1Þ=2Þ
⋅
1
2ððn−1Þ=2Þ
⋅e−ð1=2Þððn−1Þs2
=ðγ2
s2
0
ÞÞ
⋅
ðn−1Þs2
γ2s2
0
¼
1
Γððn−1Þ=2Þ
⋅
1
2ððn−1Þ=2Þ
⋅e−ð1=2ÞX2
=γ2
⋅
X2
γ2
: ð6Þ
The density function g′ of the new (next) X2
is the weighted
average of g0 and g1 based on the probability ps, the decision on
the sampling interval h, and the decision on the sample size n
made on the previous sampling instance for the current one. Note
that the value of a at the previous sampling instance defines two
alternatives for the density function of X2
:
g′ ¼ ðð1−psÞe−λsh
Þg0 þ ðps þ ð1−psÞ⋅ð1−e−λsh
ÞÞg1 for a ¼ 0; ð7Þ
g′ ¼ e−λsh
g0 þ ð1−e−λsh
Þg1 for a ¼ 1: ð8Þ
The value of h in (7) may be either h1 or h2 while the value of h
in (8) is by definition h1 since after an alarm (and possible
restoration), the relaxed parameters are to be used. Moreover,
as mentioned earlier, the value of h2 may actually be zero.
The equations above as well as all equations of the ensuing
analysis of the paper are valid for any value of h (and of course
for h¼h2 ¼0). The parenthesis prior to g0 in (7) is the probability
0
1
px ps
*
px
*
ps
pxl
psl
central
zone
warning
zone
action
zone
central
zone
warning
zone
action
zone
Fig. 1. Three regions of the two charts.
G. Nenes, S. Panagiotidou / Computers Operations Research 40 (2013) 2801–2815 2803

that assignable cause 2 had not occurred at the time of the prior
sampling instance and the cause did not occur in the following
sampling interval of length h. On the other hand, the parenthesis
prior to g1 in (7) is the probability that the process was actually
affected by assignable cause 2 at the time of the prior sampling
instance or, if it was not affected, the cause occurred in the
following sampling interval of length h. Similarly, in (8), since
the decision at the prior sampling instance was a ¼ 1, it is known
for sure that the process started the last interval with Y ¼ 0, thus,
the probability of assignable cause 2 occurrence within h is 1−e−λsh
(the parenthesis prior to g1) while the probability of no occur-
rence, i.e., the reliability, is e−λsh
(the parenthesis prior to g0). It is
evident that Eq. (7) reduces to (8) for ps ¼ 0.
Based on the above distributions, the sampling outcome, the
prior probability of assignable cause 2 occurrence (ps) and the
decision a ¼ 0 or a ¼ 1 at the previous sampling instance, the a
posteriori probability of assignable cause 2 occurrence, p′s, will be
computed using the Bayes theorem (for a ¼ 0 and a ¼ 1) as
follows:
p′sða ¼ 0Þ ¼
ðps þ ð1−psÞ⋅ð1−e−λsh
ÞÞg1
ðð1−psÞe−λshÞg0 þ ðps þ ð1−psÞ⋅ð1−e−λshÞÞg1
ð9Þ
p′
sða ¼ 1Þ ¼
ð1−e−λsh
Þg1
e−λshg0 þ ð1−e−λshÞg1
ð10Þ
In order to compute the a posteriori probability of assignable
cause 1 occurrence, p′
x, the a posteriori probability p′
s is also
utilized. The density function of the new (next) sample mean, f ′,
is the weighted average of f 0, f
γ
0, f 1 and f
γ
1 and is based on the
probabilities px and p′
s, the elapsed time since the last sampling
instance h, and the decision on the sample size n made on the
previous sampling instance for the current one. Note again that
the value of a at the previous sampling instance deﬁnes two
alternatives for the density function of the new sample mean:
f ′ ¼ ðð1−pxÞe−λxh
Þ⋅ðð1−p′
sÞf 0 þ p′
sf
γ
0Þ
þðpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ for a ¼ 0; ð11Þ
f ′ ¼ e−λxh
ðð1−p′
sÞf 0 þ p′
sf
γ
0Þ
þð1−e−λxh
Þ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ for a ¼ 1: ð12Þ
The ﬁrst parenthesis in (11), i.e., ð1−pxÞe−λxh
is the probability
that assignable cause 1 had not occurred at the time of the prior
sampling instance and the cause did not occur in the following
sampling interval of length h. This probability is then multiplied
with the weighted pdf of the mean, given that μ ¼ μ0, which is f 0 if
assignable cause 2 has not occurred or f
γ
0 if assignable cause 2 has
occurred. Using a similar reasoning, px þ ð1−pxÞ⋅ð1−e−λxh
Þ is the
probability that the process was affected by assignable cause 1 at
the time of the prior sampling instance or, if not, the cause
occurred in the following sampling interval of length h. This
probability is then multiplied with the weighted pdf of the mean,
given that μ ¼ μ1, which is f 1 if assignable cause 2 has not occurred
or f
γ
1 if assignable cause 2 has occurred. Eq. (12) results easily from
(11) for px ¼ 0.
Based on the value of p′
s, the respective distributions, the
sampling outcome, the prior probability of assignable cause
1 occurrence (px) and the decision a ¼ 0 or a ¼ 1 at the previous
sampling instance, the a posteriori probability of assignable cause
1 occurrence, p′
x; will be computed using the Bayes theorem (for
a ¼ 0 and a ¼ 1) as follows:
p′
xða ¼ 0Þ ¼
ðpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ
ðð1−pxÞe−λxhÞ⋅ðð1−p′
sÞf 0 þ p′
sf
γ
0Þ þ ðpx þ ð1−pxÞ⋅ð1−e−λxhÞÞ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ
ð13Þ
p′
xða ¼ 1Þ ¼
ð1−e−λxh
Þ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ
e−λxhðð1−p′
sÞf 0 þ p′
sf
γ
0Þ þ ð1−e−λxhÞ⋅ðð1−p′
sÞf 1 þ p′
sf
γ
1Þ
ð14Þ
Note that the numerators in (13) and (14) weight the prob-
abilities that a sample mean has been generated from a process
where, on top of assignable cause 1, assignable cause 2 may have
also occurred (with probability p′
s and pdf f
γ
1), or assignable cause
2 may have not occurred (with probability 1−p′
s and pdf f 1).
The following two propositions, which are proven in Appendix
B, describe the relationship between the a posteriori probabilities p′
s
and p′
x with ps and px, respectively. In particular, it is proven that the
values of the a posteriori probabilities are non-decreasing functions
of the a priori probabilities for a¼0 while they are independent of
the a priori probabilities for a¼1. In other words, as time passes by,
and as long as no investigation takes place (a¼0), the probabilities
of operating under the effect of the one or the other assignable
cause increases. On the other hand, if an alarm is issued, it will
either reveal the existence of the assignable cause(s) (which will be
eliminated), or it will reveal that the process actually operated in
statistical control (false alarm). Either way, after an alarm, the
process resumes its operation in statistical control with certainty
and thus, the probabilities of operexit3b2tex.batating out of control
at the next sampling instance are not affected by the probabilities of
the previous sampling instance, prior to the investigation.
Proposition 1. p′
s is non-decreasing in ps.
Proposition 2. p′
x is non-decreasing in px.
Transition probabilities from ps to p′
s and from px to p′
x: From
(9) and (10), the value of the estimator of the population variance
s2
that leads to the transformation of ps into p′s for a given h and n
and for a ¼ 0 is
s2
¼
2γ2
s2
0lnðγ3−n
⋅ðps þ ð1−psÞ⋅ð1−e−λsh
ÞÞð1−p′
sÞ=ðð1−psÞe−λsh
p′
sÞÞ
ðn−1Þð1−γ2Þ
ð15Þ
while for a ¼ 1:
s2
¼
2γ2
s2
0lnðγ3−n
⋅ð1−e−λsh
Þ⋅ð1−p′
sÞ=ðe−λsh
p′
sÞÞ
ðn−1Þð1−γ2Þ
ð16Þ
Thus, if s¼s0, the probability of a transition from ps to p′s for
a ¼ 0 will be
Pðp′
s ps; a ¼ 0; h; n; s ¼ s0Þ

¼ P X2
n−1 ¼
2γ2
lnðγ3−n
ÞÞð1−p′
p′
sÞÞ
1−γ2

ð17Þ
while for a ¼ 1:
Pðp′
sjps; a ¼ 1; h; n; s ¼ s0Þ
¼ PðX2
n−1 ¼ 2γ2
lnðγ3−n
⋅ð1−e−λsh
Þ⋅ð1−p′
sÞ=ðe−λsh
p′
sÞÞ=ð1−γ2
Þ ð18Þ
Similarly, if s¼s1¼γs0, the probability of a transition from ps to
p′s for a ¼ 0 will be:
Pðp′
sjps; a ¼ 0; h; n; s ¼ γs0Þ
¼ PðX2
n−1 ¼ 2lnðγ3−n
ÞÞð1−p′
p′
sÞÞ=ð1−γ2
Þ
ð19Þ
while for a ¼ 1:
Pðp′
sjps; a ¼ 1; h; n; s ¼ γs0Þ
¼ PðX2
n−1 ¼ 2lnðγ3−n
⋅ð1−e−λsh
Þ⋅ð1−p′
sÞ=ðe−λsh
p′
sÞÞ=ð1−γ2
Þ
ð20Þ
Equivalently, from (13) and (14) we get the relationship
between p′
x and z, i.e., the probabilities of a transition from px to
2804

p′
x for a ¼ 0 or a ¼ 1, for all combinations of μ and s and for any h,
n. These probabilities are denoted by Pðp′
xjpx; a; h; n; ϒÞ. For exam-
ple, the probability, at some sampling instance, of a transformation
from px to p′x if at the previous sampling instance the decision was
a ¼ 0, h¼h1 and n¼n1 and given that at the current sampling
instance the process operates under the effect of both assignable
causes is denoted by Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 3Þ.
At each sampling instance, the process is fully characterized by
the values of probabilities px and ps and by the actual state of the
process Y. Thus, let a state ðY; px; psÞ where px and ps are the exact
values of the probabilities of the 1st and 2nd assignable cause
occurrence. Recall that Y is an index which equals 0 when none
assignable cause has occurred, Y¼1 when only assignable cause
1 has occurred, Y¼2 when only assignable cause 2 has occurred
and Y¼3 when both assignable causes have occurred. The prob-
abilities of a transition from any to all possible states of the process
(Y) and for any values of the probabilities px, ps, p′
x and p′
s, are
summarized in the equations of Appendix C.
Note that as soon as the update mechanism is completed, and
the values of p′
x and p′
s have been derived using the Bayes theorem,
these updated probabilities will have to be updated again after the
following (next) sampling outcome. In other words, the a poster-
iori probabilities of any sampling instance become essentially the a
priori probabilities of the following one. For this reason, and also
to avoid complex notations, immediately after the values of p′
x and
p′
s have been computed, we set px ¼ p′
x and ps ¼ p′
s and these new
probabilities are compared against the critical probabilities pxl, psl,
pn
x and pn
s to reach the optimum decisions, as already described.
Then, at the following sampling instance, these px and ps will be
updated again and so on. To summarize the implementation
procedure, Fig. 2 illustrates the exact way that the proposed
Bayesian scheme is used in a process.
Read
pxl, psl, p*
x, p*
s,
h1, h2, n1, n2
Let the
process
operate for h1
px≤pxl
ps≤ psl
Yes
pxp*
x or
psp*
s
No
Yes
Investigate the
process and remove
any assignable cause
Let the
process
operate for h2
No
Set px=ps=0
a=0 a=1
a=0
Compute
z and X2
Collect a
sample of n1
Collect a
sample of n2
a=0
a=0 Yes
No
Compute in this order:
g' from eq. 8
p's from eq. 10
f' from eq. 12
p'x from eq. 14
Compute in this order:
g' from eq. 7
p's from eq. 9
f' from eq. 11
p'x from eq. 13
Set px=p'x
Set ps=p's
Fig. 2. Flowchart of the use of the Bayesian scheme in practice.

4. Cost of a transition interval and optimization
The cost of a transition interval depends on the exact value of Y,
and the decisions a, h, n. In particular, whenever an interval starts
with the process operating in statistical control (Y¼0, Table 1a),
five alternative scenarios may occur until the next sampling
instance: both assignable causes occur, first assignable cause
1 and then 2 (Tables 1a-1), both assignable causes occur, first
assignable cause 2 and then 1 (Table 1a-2), only assignable cause
1 occurs (Table 1a-3), only assignable cause 2 occurs (Table 1a-4),
no assignable cause occurs (Table 1a-5).
The probability of both assignable causes occurrence, first
assignable cause 1 and then 2 (Table 1a-1) is denoted by pxsðhÞ
and is derived by the following equation:
pxsðhÞ ¼
Z h
0
λxe−λxt
Z h
t
λse−λst′
dt′dt ¼
λx
λx þ λs
ð1−e− λxþλs
ð Þh
Þ−e−λsh
ð1−e−λxh
Þ
ð21Þ
The expected time of assignable cause 1 occurrence and the
expected time of assignable cause 2 occurrence, given that both
assignable causes will occur in the interval h, with assignable
cause 1 occurring first are denoted by τx1ðhÞ and τs2ðhÞ, respectively
and are computed by
τx1ðhÞ ¼
R h
0 tλxe−λxt
R h
t λse−λst′
dt′ dt
R h
0 λxe−λxt
R h
t λse−λst′ dt′ dt
ð22Þ
τs2ðhÞ ¼
R h
0 λxe−λxt
R h
t t′λse−λst′
dt′dt
R h
0 λxe−λxt
R h
t λse−λst′dt′dt
ð23Þ
Equivalently, the probability of both assignable causes occur-
rence, first assignable cause 2 and then 1 (Table 1a-2) is denoted
by psxðhÞ and is derived by the following equation:
psxðhÞ ¼
Z h
0
λse−λst
Z h
t
λxe−λxt′
dt′dt ¼
λs
λx þ λs
ð1−e− λxþλs
ð Þh
Þ−e−λxh
ð1−e−λsh
Þ
ð24Þ
Note that, obviously, pxsðhÞ þ psxðhÞ equals the probability of
both assignable causes occurrence (in any order) which is
ð1−e−λxh
Þ⋅ð1−e−λsh
Þ. The expected time of assignable cause 2 occur-
rence and the expected time of assignable cause 1 occurrence,
given that both assignable causes will occur in the interval h, with
assignable cause 2 occurring first, are denoted by τs1ðhÞ and τx2ðhÞ,
respectively and are computed by
τs1ðhÞ ¼
R h
0 tλse−λst
R h
t λxe−λxt′
dt′ dt
R h
0 λse−λst
R h
t λxe−λxt′ dt′ dt
ð25Þ
τx2ðhÞ ¼
R h
0 λse−λst
R h
t t′λxe−λxt′
dt′ dt
Rh
0 λse−λst
R h
t λxe−λxt′ dt′ dt
ð26Þ
The probability of only assignable cause 1 occurrence (Table 1a-3)
equals ð1−e−λxh
Þe−λsh
and the expected time of occurrence in the
interval h, is denoted by τxðhÞ:
τxðhÞ ¼
R h
0 tλxe−λxt
dt
R h
0 λxe−λxtdt
¼
1−e−λxh
−λxhe−λxh
λx−λxe−λxh
ð27Þ
The probability of only assignable cause 2 occurrence (Table 1a-4)
equals ð1−e−λsh
Þe−λxh
and the expected time of occurrence in the
interval h, is denoted by τsðhÞ:
τsðhÞ ¼
R h
0 tλse−λst
dt
R h
0 λse−λstdt
¼
1−e−λsh
−λshe−λsh
λs−λse−λsh
ð28Þ
Finally, the probability of no assignable cause occurrence
(Table 1a-5) equals e− λxþλs
ð Þh
:
From the above, the cost of a transition interval in all these
cases (when the interval starts with operation in statistical
control) will be
Mx½ðτs2ðhÞ−τx1ðhÞÞpxsðhÞ þ ð1−e−λxh
Þe−λsh
ðh−τxðhÞÞŠ
þMs½ðτx2ðhÞ−τs1ðhÞÞpsxðhÞ þ ð1−e−λsh
Þe−λxh
ðh−τsðhÞÞŠ
þMxsððh−τs2ðhÞÞpxsðhÞ þ ðh−τx2ðhÞÞpsxðhÞÞ ð29Þ
For reasons of brevity, the expected time that the process
operates under the effect of assignable cause 1 in an interval h,
given that this interval starts in control will be hereafter denoted
by mxðhÞ and is analytically computed by
mxðhÞ ¼ ðτs2ðhÞ−τx1ðhÞÞpxsðhÞ þ ð1−e−λxh
Þe−λsh
ðh−τxðhÞÞ
¼
R h
0 λxe−λxt
R h
t t′λse−λst′
dt′ dt
R h
0 λxe−λxt
R h
t λse−λst′dt′ dt
−
R h
0 tλxe−λxt
Rh
t λse−λst′
dt′ dt
R h
0 λxe−λxt
R h
0 λse−λst′dt′ dt
#
⋅
Z h
0
λxe−λxt
Z h
t
λse−λst′
dt′ dt þ ð1−e−λxh
Þe−λsh
ðh−τxðhÞÞ
¼
Z h
0
λxe−λxt
Z h
t
t′λse−λst′
dt′ dt−
Z h
0
tλxe−λxt
Z h
t
λse−λst′
dt′ dt
þð1−e−λxh
Þe−λsh
h−
1−e−λxh
−λxhe−λxh
λx−λxe−λxh

which after some mathematical manipulation reduces to the
following simpler function:
mxðhÞ ¼
λxð1−e−λsh
Þ−λse−λsh
ð1−e−λxh
Þ
λsðλx þ λsÞ
ð30Þ
Similarly, the expected time that the process operates under
the effect of assignable cause 2 in an interval h, given that this
interval starts in control is given by
msðhÞ ¼ ðτx2ðhÞ−τs1ðhÞÞpsxðhÞ þ ð1−e−λsh
Þe−λxh
ðh−τsðhÞÞ:
which reduces to
msðhÞ ¼
λsð1−e−λxh
Þ−λxe−λxh
ð1−e−λsh
Þ
λxðλx þ λsÞ
ð31Þ
Finally, the expected time that the process operates under the
effect of both assignable causes in an interval h, given that this
interval starts in control is given by
mxsðhÞ ¼ ðh−τs2ðhÞÞpxsðhÞ þ ðh−τx2ðhÞÞpsxðhÞ
¼
λxh
λx þ λs
−
λx
ðλx þ λsÞ2
−
λx
λsðλx þ λsÞ
þ
λxe− λxþλs
ð Þh
ðλx þ λsÞ2
þ
λxe−ðλxþλsÞh
λsðλx þ λsÞ
þ
e−λsh
λs
−
e− λxþλs
ð Þh
λs
þ
λsh
λx þ λs
−
λs
ðλx þ λsÞ2
−
λs
λxðλx þ λsÞ
þ
λse−ðλxþλsÞh
ðλx þ λsÞ2
þ
λse−ðλxþλsÞh
λxðλx þ λsÞ
þ
e−λxh
λx
−
e−ðλxþλsÞh
λx
which reduces to the following:
mxsðhÞ ¼ h−
1−e−λxh
λx
−
1−e−λsh
λs
þ
1−e− λxþλs
ð Þh
λx þ λs
ð32Þ
If an interval starts operating under the effect of assignable
cause 1, it will either operate for the entire interval under this
effect (with probability e−λsh
, Table 1b-2) or until the occurrence
of assignable cause 2 (with probability 1−e−λsh
, Table 1b-1).
The expected time of assignable cause 2 occurrence, given that it
will occur, is given by (28). Thus, the out-of-control operation cost
in an interval that starts under the effect of assignable cause 1 will
be given by
Mxðhe−λsh
þ τsðhÞð1−e−λsh
ÞÞ þ Mxsð1−e−λsh
Þðh−τsðhÞÞ
¼ Mx he−λsh
þ
1−e−λsh
−λshe−λsh
λs

þMxs ð1−e−λsh
Þh−
1−e−λsh
−λshe−λsh
λs

2806

Table 1
Alternative scenarios that may occur in an interval h (¼ h1 or h2).
Scenario Evolution Probability Expected duration with Y¼
0 1 2 3
a 1 psxðhÞ τx1ðhÞ τs2ðhÞ−τx1ðhÞ 0 h−τs2ðhÞ
2 psxðhÞ τs1ðhÞ 0 τx2ðhÞ−τs1ðhÞ h−τx2ðhÞ
3 ð1−e−λx h
Þe−λsh τxðhÞ h−τxðhÞ 0 0
4 ð1−e−λs h
Þe−λxh τsðhÞ 0 h−τsðhÞ 0
5 e− λxþλs
ð Þh h 0 0 0
b 1 1−e−λs h 0 τsðhÞ 0 h−τsðhÞ
2 e−λsh 0 h 0 0
c 1 1−e−λxh 0 0 τxðhÞ h−τxðhÞ
2 e−λx h 0 0 h 0
d 1 1 0 0 0 h

which after some mathematical manipulation reduces to the
following:
Mx
1−e−λsh
λs
þ Mxs h−
1−e−λsh
λs

ð33Þ
Similarly, the out-of-control operation cost in an interval that
starts under the effect of assignable cause 2 (Table 1, c) will be
given by
Ms
1−e−λxh
λx
þ Mxs h−
1−e−λxh
λx

ð34Þ
Finally, the out-of-control operation cost in an interval that
starts under the effect of both assignable causes (Table 1, d) will be
given by
Mxs⋅h ð35Þ
Table 2 summarizes the expected costs and durations asso-
ciated with an interval of length h (¼ h1 or h2) given that at the
beginning of that interval the process was at state (Y,px,ps).
5. Numerical investigation
In this section, the economic outcome of the proposed model is
compared against the outcome of a Vp-Shewhart chart, also
designed for monitoring a process with the same characteristics
[37]. In order to do so, an expected cost per time unit function
needs to be developed and then using this function, the optimum
parameters and minimum expected cost per time unit need to be
derived. A classic way to model this function is following the well
established approach used in Nenes [21] as the expected cost of a
transition step over its expected duration. In order to develop such
a function, a certain discretization of the probabilities px and ps is
necessary and then, all transition probabilities from and to all
states of the resulting Markovian matrix need to be computed
with the aid of the equations of Appendix C. Then, using the
expected costs and durations presented in Section 4 and summar-
ized in Table 2, the following equations for the expected cost of a
transition step EC, the duration of a transition step ET, and the
expected cost per time ECT, are derived:
EC ¼ b þ ∑π0ðpx;psÞ∈SI
ðcn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ
þ∑π0ðpx;psÞ∈SII
ðcn2 þ Mxmxðh2Þ þ Msmsðh2Þ þ Mxsmxsðh2ÞÞ
þ∑π0ðpx;psÞ∈SIII
ðL0 þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ
þ∑π1ðpx;psÞ∈SI
cn1 þ Mx
1−e−λsh1
λs
þ Mxs h1−
1−e−λsh1
λs

cn2 þ Mx
1−e−λsh2
λs
þ Mxs h2−
1−e−λsh2
λs

ðLx þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ
cn1 þ Ms
1−e−λxh1
λx
þ Mxs h1−
1−e−λxh1
λx

cn2 þ Ms
1−e−λxh2
λx
þ Mxs h2−
1−e−λxh2
λx

ðLs þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ
ðcn1 þ Mxsh1Þ þ ∑π3ðpx;psÞ∈SII
ðcn2 þ Mxsh2Þ
ðLxs þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ; ð36Þ
ET ¼ ∑ðπ0ðpx;psÞ∈SI
þ π1ðpx;psÞ∈SI
Þ⋅h1
þ∑ðπ0ðpx;psÞ∈SII
þ π1ðpx;psÞ∈SII
Þ⋅h2
ðT0 þ h1Þ þ ∑π1ðpx;psÞ∈SIII
ðT0 þ Tx þ h1Þ
ðT0 þ Ts þ h1Þ þ ∑π3ðpx;psÞ∈SIII
ðT0 þ Txs þ h1Þ ð37Þ
and
ECT ¼ EC=ET: ð38Þ
Note that (36) should be slightly altered for the special case of
h2 ¼ 0. In particular, if h2 ¼ 0 then the ﬁxed sampling cost b should
not be added to EC, when the tightened parameters are used. Thus,
for h2 ¼ 0, EC is simpliﬁed to the following:
EC ¼ ∑π0ðpx;psÞ∈SI
ðb þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ
⋅cn2 þ ∑π0ðpx;psÞ∈SIII
ðL0 þ b þ cn1 þ Mxmxðh1Þ
þMsmsðh1Þ þ Mxsmxsðh1ÞÞ
b þ cn1 þ Mx
1−e−λsh1
λs
þ Mxs h1−
1−e−λsh1
λs

ðLx þ b þ cn1 þ Mxmxðh1Þ
b þ cn1 þ Ms
1−e−λxh1
λx
þ Mxs h1−
1−e−λxh1
λx

ðLs þ b þ cn1 þ Mxmxðh1Þ
ðb þ cn1 þ Mxsh1Þ þ ∑π3ðpx;psÞ∈SII
⋅cn2
ðLxs þ b þ cn1 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1ÞÞ:
ð39Þ
Table 2
Expected cost and duration of each transition step.
State and steady state probability Values of px, ps Expected cost Duration
(Y¼0,px,ps) π0px ps
px ≤pxl and ps ≤psl (SI) Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1Þ þ b þ cn1 h1
px 4pxl or ps 4psl but both px ≤pn
x and ps ≤pn
s (SII) Mxmxðh2Þ þ Msmsðh2Þ þ Mxsmxsðh2Þ þ b þ cn2
a
h2
px 4pn
x or ps 4pn
s (SIII) L0 þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1Þ þ b þ cn1 T0 þ h1
px ≤pxl and ps ≤psl (SI) Mx
1−e−λs h1
λs
þ Mxs h1− 1−e−λs h1
λs

þ b þ cn1
h1
(Y¼1,px,ps) π1pxps px 4pxl or ps 4psl but both px ≤pn
x and ps ≤pn
s (SII) Mx
1−e−λs h2
λs
þ Mxs h1− 1−e−λs h2
λs

þ b þ cn2
a h2
px 4pn
x or ps 4pn
s (SIII) Lx þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1Þ þ b þ cn1 T0 þ Tx þ h1
px ≤pxl and ps ≤psl (SI) Ms
1−e−λx h1
λx
þ Mxs h1− 1−e−λx h1
λx

þ b þ cn1 h1
(Y¼2,px,ps) π2px ps px 4pxl or ps 4psl but both px ≤pn
x and ps ≤pn
s (SII) Ms
1−e−λx h2
λx
þ Mxs h1− 1−e−λx h2
λx

þ b þ cn2
a h2
px 4pn
x or ps 4pn
s (SIII) Ls þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1Þ þ b þ cn1 T0 þ Ts þ h1
px ≤pxl and ps ≤psl (SI) Mxsh1 þ b þ cn1 h1
(Y¼3,px,ps) π3px ps px 4pxl or ps 4psl but both px ≤pn
x and ps ≤pn
s (SII) Mxsh2 þ b þ cn2
a
h2
px 4pn
x or ps 4pn
s (SIII) Lxs þ Mxmxðh1Þ þ Msmsðh1Þ þ Mxsmxsðh1Þ þ b þ cn1 T0 þ Txs þ h1
a
b is not added in the expected cost if h2 ¼0.
2808

In the same sense, for h2 ¼ 0, ET can be simplified to the
following:
ET ¼ ∑ðπ0ðpx;psÞ∈SI
Þ⋅h1
ðT0 þ h1Þ þ ∑π1ðpx;psÞ∈SIII
ðT0 þ Tx þ h1Þ
ðT0 þ Ts þ h1Þ þ ∑π3ðpx;psÞ∈SIII
ðT0 þ Txs þ h1Þ: ð40Þ
In other words, for every possible set of decision parameters h1,
h2, n1, n2 and pxl, psl, pn
x; pn
s , a three-dimensional Markovian matrix
(Y,px,ps) should be constructed with the use of the transition
probabilities as presented in Appendix C. The possible values of
h1, h2 are subject to some increment step; for example 0.1, namely,
0, 0.1, 0.2, etc. The values of the sample sizes are obviously integer
while the possible values of the probabilities are based on the
discretization as explained in Nenes [22] and Tagaras [34]. Then,
each set of variables leads to a Markovian matrix from which the
steady state probabilities should be computed and used to Eqs.
(36) (or (39) for h2¼0), (37) (or (40) for h2¼0) and (38). In this
way, every set of decision variables leads to an ECT value and the
optimum design is the one that minimizes ECT.
Although the aforementioned procedure and Eqs. (36)–(40)
model the described problem accurately, any attempt to solve
them and derive the expected cost per time unit for any decision
parameters, is doomed to failure. The reason is that the necessity
for discretizing the probabilities px and ps leads to an oversized
Markovian matrix and the steady state equations appeared in
Eq. (36)–(40) are impossible to be computed for a fairly detailed
discretization. For example, if each px and ps is discretized into 100
values (see [34]), the Markovian matrix would have a size of
40,000 40,000 (4 Y values 100 px values 100 ps values) while
any attempt to use coarser discretizations, leads to very unreliable
results. To overcome this obstacle, a simulation program, which
does not necessitate any discretization of the out-of-control
probabilities, has been developed in FORTRAN PowerStation
4.0 to estimate the economic outcome of the proposed model.
In particular, Table 3 presents the first 16 cases that were used in
the numerical investigation section in Tasias and Nenes [37] while
Table 4 presents the economic outcome of each case for Fp and
Vp-Shewhart charts (for more details on the economic design of
the Fp and Vp-Shewhart charts see [37]). In the last columns of
Table 4, the simulated economic outcome of the Bayesian chart, as
well as the percentage improvements compared to the simpler Fp
and Vp-Shewhart approaches are presented. The ECT of the
Bayesian chart is the minimum expected cost per time unit
computed by the simulation program, keeping the values of h1,
n1, h2, n2 same as the optimum ones of the Vp-Shewhart chart, and
for various values of pxl; psl; pn
x and pn
s . In particular, the optimiza-
tion procedure took place in two steps:
In Step 1, each of pxl; psl; pn
x and pn
s were allowed to vary
between 0 and 0.2 in 0.02 increments, between 0.2 and 0.5 in
0.03 increments and between 0.5 and 1 in 0.05 increments.
The resulting alternative values that are investigated in the
simulation runs, under the specific discretization, are 30 for the
warning limit probabilities pxl and psl, and of course less than 30
for the control limit probabilities pn
x and pn
s – depending on the
values of pxl and psl, respectively – since the values of the control
limit probabilities cannot be smaller than the respective ones of
the warning limit probabilities: pxl ≤pn
x and psl ≤pn
s . It should be
noted that the derivation of the optimum values of pxl; psl; pn
x and
pn
s , for each of the 16 cases, using the FORTRAN computer program
and 10,000 runs per simulation, takes more than 5 h per case in a
Pentium i5 3.20 GHz computer. This is actually why this coarse
discretization is used in Step 1.
In Step 2, a new simulation run is conducted for every case but
not for the entire domain of the probabilities. In particular, in Step 2,
the search for the optimum warning and control probabilities
pxl; psl; pn
x and pn
s is conducted around their optimum values of Step
1 (7 0.10) but with a finer discretization (0.01 increments).
Table 3
Parameter sets of the 16 numerical examples (c¼1, Lx¼200, Ls ¼200, Lxs ¼300,
T0¼Tx¼Ts ¼Txs ¼0, λx¼λs ¼λ/2, δ¼0.5, γ2
¼2).
Case b Mx¼Ms Mxs L0 λ
1 0 100 150 100 0.01
2 0 100 150 200 0.01
3 0 1000 1500 100 0.01
4 0 1000 1500 200 0.01
5 5 100 150 100 0.01
6 5 100 150 200 0.01
7 5 1000 1500 100 0.01
8 5 1000 1500 200 0.01
9 0 100 150 100 0.1
10 0 100 150 200 0.1
11 0 1000 1500 100 0.1
12 0 1000 1500 200 0.1
13 5 100 150 100 0.1
14 5 100 150 200 0.1
15 5 1000 1500 100 0.1
16 5 1000 1500 200 0.1
Table 4
Optimum decision parameters and costs for all charts.
Case (%) Fp-Shewhart Vp-Shewharta
Bayes Fp−B
Fp (%) Vp−B
Vp (%)
h n kx ks ECT h1 n1 n2 k1x k2x wx k1s k2s ws ECT pxl pn
x psl pn
s ECT
1 6.4 19 1.5 1.7 11.59 4.0 7 16 2.6 2.0 0.8 2.8 2.2 1.1 10.56 .03 .33 .48 .97 9.67 16.6 8.4
2 7.3 28 1.9 2.1 12.54 4.4 9 24 3.1 2.3 1.0 3.3 2.5 1.2 10.89 .04 .90 .26 .90 9.90 21.1 9.1
3 1.9 20 1.6 1.8 32.25 1.2 7 18 2.8 2.0 0.9 3.0 2.3 1.1 29.45 .02 .29 .52 .92 25.55 20.8 13.2
4 2.3 30 1.9 2.1 36.36 1.3 9 24 3.3 2.4 1.0 3.4 2.6 1.2 30.30 .02 .59 .25 .98 25.60 29.6 15.5
5 7.7 23 1.5 1.7 12.29 6.5 13 18 2.3 1.9 0.8 2.4 2.2 1.0 11.47 .03 .34 .27 .85 10.79 12.2 5.9
6 8.7 32 1.9 2.0 13.17 6.7 14 25 2.8 2.3 0.9 2.9 2.5 1.1 11.79 .06 .84 .26 .94 10.85 17.6 8.0
7 2.4 24 1.5 1.7 35.54 2.0 13 19 2.4 2.0 0.8 2.5 2.3 1.0 32.54 .03 .98 .04 .71 27.75 21.9 14.7
8 2.6 33 1.9 2.1 38.38 2.0 14 27 3.0 2.4 1.0 3.1 2.6 1.1 33.30 .02 .26 .14 .65 28.62 25.4 14.1
9 2.2 17 1.5 1.7 45.86 1.7 8 13 2.2 1.7 0.8 2.4 2.1 1.0 43.95 .09 .77 .64 .95 41.98 8.5 4.5
10 2.6 24 1.8 2.0 48.43 1.9 10 19 2.6 2.0 0.9 2.8 2.3 1.1 45.45 .11 .88 .47 .97 42.85 11.5 5.7
11 0.6 19 1.6 1.8 115.89 0.4 7 16 2.6 2.0 0.8 2.8 2.2 1.1 105.55 .05 .29 .46 .89 95.23 17.8 9.8
12 0.7 27 1.9 2.1 125.44 0.4 8 23 3.2 2.3 1.0 3.3 2.5 1.2 108.98 .04 .97 .35 .99 97.33 22.4 10.7
13 2.8 19 1.4 1.5 47.81 2.6 13 14 1.9 1.8 0.7 2.1 2.1 0.9 46.21 .10 .68 .38 .88 44.86 6.2 2.9
14 3.1 26 1.7 1.9 50.15 2.7 15 20 2.3 2.0 0.9 2.5 2.3 1.0 47.59 .07 .75 .49 .87 45.89 8.5 3.6
15 0.8 24 1.5 1.7 122.95 0.7 14 19 2.2 2.0 0.8 2.4 2.2 1.0 114.78 .02 .78 .11 .90 107.80 12.3 6.1
16 0.9 32 1.8 2.0 131.70 0.7 15 25 2.8 2.3 0.9 2.9 2.5 1.1 117.93 .03 .50 .08 .89 109.57 16.8 7.1
a
h2 ¼0.0 in all cases

The optimum values of this step are presented in the final columns
of Table 4. The flowchart of the computer program, for given values
of probabilities pxl; psl; pn
x and pn
s , i.e., without including the loops for
searching for the optimum pxl; psl; pn
x and pn
s , is depicted in Fig. 3.
It should be emphasized that during the runs of the simulation
program, as depicted in Fig. 3, the update mechanism is based on
the simulated sampling outcome and does not necessitate any
discretization whatsoever. Unlike the analytical approach of the
Markovian modeling (Eq. (36)–(40)), where all possible values of
the out-of-control probabilities need to be taken into account to
derive an analytical function for the cost per time unit (leading
practically to the need for a certain discretization), in the simula-
tion approach, at each sampling epoch there is a random sampling
outcome that leads to the analytical computation of the out-of-
control probabilities without having to resolve to any discretiza-
tion. Thus, after each sampling instance, the simulated sampling
outcome leads to the analytical computation of the new probabil-
ities using (9) and (13), if the decision after the previous sampling
instance was a ¼ 0 or using (10) and (14), if the decision after the
previous sampling instance was a ¼ 1.
As expected, the percentage improvement of the Bayesian
chart, compared to both Fp and Vp-Shewhart charts is substantial.
The percentage improvement of the Bayesian chart compared to
the simple Shewhart chart varies between 6.2% and 29.6% while it
has an average of 16.8% in the 16 examined cases. When the
comparisons are made against the Vp-Shewhart chart, the improve-
ment varies between 2.9% and 15.5% with an average of 8.7%.
Although an analytical optimization has not been achieved and
there is nothing yet to say concerning the optimum sample sizes
and sampling intervals of the Bayesian chart (since the optimum
parameters of the Vp-Shewhart charts are used), it is evident that
the potential benefits of the new scheme are not at all negligible.
Fig. 3. Flowchart of the simulation program.
2810

However, even under this brief investigation, it is obvious that
the percentage improvement tends to be higher when the cost of
erroneously investigating the process, L0, and the costs of operating
under the effect of the assignable causes, Mx, Ms and Mxs, are also
high while the occurrence rates, λx and λs, as well as the fixed
sampling cost, b, are low. In other words, the savings are higher in
reliable (low λx and λs), automated (low b) and expensive processes
(high L0, Mx, Ms and Mxs). This is actually a very convenient outcome
since it indicates that it is exactly in expensive production processes
where the new complex chart is expected to be highly effective and
its benefits are maximized. Thus, the complexity of the new chart
and the possible difficulties of implementation can be justified by
the high benefits gained when the production processes are more
expensive. In such processes it is worth investing in more sophis-
ticated quality control systems.
Moreover, although not shown in this investigation, it has been
observed that the savings between the Bayesian chart and the
Vp-Shewhart chart tend to diminish as δ and γ become larger. In
other words, when the disturbance of the assignable causes is high,
the superiority of the more advanced and complex Bayesian chart
reduces because even the less advanced Vp-Shewhart chart behaves
relatively well.
It is certain that a more thorough investigation on the optimum
parameters of the Bayesian chart (optimum sampling intervals h1, h2
and sample sizes n1, n2) would lead to further cost reduction and
percentage improvement. However, a thorough investigation that
would also include a search for the optimum h1, h2 and n1, n2 has
been proven a very time-demanding process since it may take several
days, depending on the domain of the investigation, to reach an
optimum solution. It is up to the practitioner to decide whether it is
worth searching for even more effective solutions. To give an idea of
the potential further cost reduction that can be achieved through the
optimization of all parameters (including h1, h2 and n1, n2), Table 5
presents the optimum results of the first four cases of Table 3, with the
globally optimum parameters and the new economic outcomes.
Similar to Table 4, the simulated economic outcome of the Bayesian
chart, as well as the percentage improvements compared to the Fp
and Vp-Shewhart schemes is presented in the last columns. The ECT of
the Bayesian chart is now the minimum expected cost per time unit,
computed by the simulation program, allowing, besides the values of
pxl; psl; pn
x and pn
s ; the values of h1, n1 and h2, n2 to vary as well.
The results indicate that the room for further improvement
may be not at all negligible (around 4%) and that it may be worth
to investigate the globally optimum solutions. For example, we can
see in case 4 of Table 4 that the percentage improvement, when
the Bayesian scheme uses the optimum h1, n1 and h2, n2 of the Vp-
Shewhart scheme, is 29.6% and 15.5%, compared to the cost of the
Fp and Vp-Shewhart charts, respectively, while when the optimum
h1, n1 and h2, n2 of the Bayesian scheme are estimated, these
improvement reaches 33.8% and 20.6%, respectively.
6. Summary and conclusions
In this paper an advanced tool for the economic optimization of
the SPC procedures in production processes subject to quality
disturbances that affect both the mean and the variance of a
quality characteristic, has been provided. The model uses the
Bayes theorem to update the out-of-control probabilities at each
sampling epoch and allows the design parameters, namely the
sample size and the sampling interval, to vary according to the
values of these probabilities, so as to minimize the resulting
expected quality cost. The superiority of the proposed scheme
varies according to the process parameters and cost elements as
shown in the numerical investigation section but it is generally
high enough to justify the use of the more complex Bayesian chart,
especially in more expensive production processes.
Acknowledgments
The authors are thankful to the editor and referee whose
valuable suggestions contributed significantly to the improvement
of the paper.
Table 5
Optimum decision parameters and costs for all Fp and Vp Shewhart charts and globally optimum decision parameters and costs for Bayesian scheme.
Case Fp-Shewhart Vp-Shewharta
Bayesa Fp−B
Fp (%) Vp−B
Vp (%)
h n kx ks ECT h1 n1 n2 k1x k2x wx k1s k2s ws ECT h1 n1 n2 pxl pn
x psl pn
s ECT
1 6.4 19 1.5 1.7 11.59 4.0 7 16 2.6 2.0 0.8 2.8 2.2 1.1 10.56 4.0 7 15 .03 .26 .45 .96 9.22 20.4 12.7
2 7.3 28 1.9 2.1 12.54 4.4 9 24 3.1 2.3 1.0 3.3 2.5 1.2 10.89 4.5 7 23 .04 .93 .22 .85 9.42 24.9 13.5
3 1.9 20 1.6 1.8 32.25 1.2 7 18 2.8 2.0 0.9 3.0 2.3 1.1 29.45 1.3 5 17 .01 .34 .49 .93 24.77 23.2 15.9
4 2.3 30 1.9 2.1 36.36 1.3 9 24 3.3 2.4 1.0 3.4 2.6 1.2 30.30 1.3 9 25 .02 .67 .19 .98 24.07 33.8 20.6
a
h2 ¼0.0 in all cases.
Appendix A. Nomenclature
μ0 target mean
μ1 mean when the 1st assignable cause has occurred ( ¼ μ0 þ δs0)
x sample mean (point estimator of the population mean)
px probability that the 1st assignable cause has occurred
s0 target standard deviation
z standardized normal variable z ¼ ððx−μ0Þ=ðs0=
ffiffiffi
n
p
ÞÞ
s1 standard deviation when the 2nd assignable cause has occurred ( ¼ γs0)
s2
point estimator of the population variance
f 0 pdf (normal) of the mean when no assignable cause has occurred (Y¼0)
f 1 pdf (normal) of the mean when the 1st assignable cause has occurred (Y¼1)
f
γ
0 pdf (normal) of the mean when the 2nd assignable cause has occurred (Y¼2)
f
γ
1 pdf (normal) of the mean when both assignable causes have occurred (Y¼3)

f ′ density function of the new (next) sample mean
ps probability that the 2nd assignable cause has occurred
g0 pdf of n−1
ð Þs2
=s2
0 when s¼s0, (Y¼0 or Y¼1)
g1 pdf of n−1
ð Þs2
=s2
0when s¼s1 ¼γs0, (Y¼2 or Y¼3)
g′ density function of the new (next) random variable n−1
ð Þs2
=s2
0
λx occurrence rate of assignable cause 1
λs occurrence rate of assignable cause 2
h sampling interval
a index which equals 0 if no investigation takes place, else a ¼ 1
n sample size
p′s Bayesian transformation of the probability ps
p′x Bayesian transformation of the probability px
pxl warning limit of the probability px
psl warning limit of the probability ps
pn
x action limit of the probability px
pn
s action limit of the probability ps
c variable sampling cost
b fixed sampling cost
h1, n1 relaxed scheme parameters
h2, n2 tightened scheme parameters (h2≤h1, n2≥n1)
Mx expected cost per time unit when μ¼μ1
Ms expected cost per time unit when s¼s1
Mxs expected cost per time unit when μ¼μ1 and s¼s1
πYpx;ps
steady state probabilities (Y¼0 means μ¼μ0 and s¼s0, Y¼1 means μ¼μ1 and s¼s0, Y¼2 means μ¼μ0 and s¼s1 and Y¼3
means μ¼μ1 and s¼s1)
L0 cost of false alarm
Lx cost of removing the 1st assignable cause
Ls cost of removing the 2nd assignable cause
Lxs cost of removing both assignable causes
T0 investigation time
Tx time to remove assignable cause 1
Ts time to remove the assignable cause 2
Txs time to remove both assignable causes
τxðhÞ expected time of assignable cause 1 occurrence given that only this cause will occur in the interval h
τsðhÞ expected time of assignable cause 2 occurrence given that only this cause will occur in the interval h
τx1ðhÞ expected time of assignable cause 1 occurrence given that both assignable causes will occur in the interval h, with assignable
cause 1 occurring first
τs2ðhÞ expected time of assignable cause 2 occurrence given that both assignable causes will occur in the interval h, with assignable
τs1ðhÞ expected time of assignable cause 2 occurrence given that both assignable causes will occur in the interval h, with assignable
τx2ðhÞ expected time of assignable cause 1 occurrence given that both assignable causes will occur in the interval h, with assignable
pxsðhÞ probability that both assignable causes will occur in the interval h, with assignable cause 1 occurring first
psxðhÞ probability that both assignable causes will occur in the interval h, with assignable cause 2 occurring first
mxðhÞ expected time of operating under the effect of assignable cause 1 in an interval h, given that the process starts in statistical
control at the beginning of that interval
msðhÞ expected time of operating under the effect of assignable cause 2 in an interval h, given that the process starts in statistical
mxsðhÞ expected time of operating under the effect of both assignable causes in an interval h, given that the process starts in statistical
Appendix B. Proofs of Propositions 1 and 2
Proof of Proposition 1. The first derivative of p′s with respect to ps for a ¼ 0 is
∂p′sða ¼ 0Þ
∂ps
¼
∂ððps þ ð1−psÞ⋅ð1−e−λsh
ÞÞg1=ððð1−psÞe−λsh
ÞÞg1ÞÞ
∂ps
¼
e−λsh
g1½ðð1−psÞe−λsh
ÞÞg1Š−ðps þ ð1−psÞ⋅ð1−e−λsh
ÞÞg1½−e−λsh
g0 þ e−λsh
g1Š
½ðð1−psÞe−λshÞg0 þ ðps þ ð1−psÞ⋅ð1−e−λshÞÞg1Š2
2812

which after some mathematical manipulation reduces to the following expression:
∂p′
sða ¼ 0Þ
∂ps
¼
e−λsh
g0g1
½ðð1−psÞe−λshÞg0 þ ðps þ ð1−psÞ⋅ð1−e−λshÞÞg1Š2
40:
Thus, for a ¼ 0, p′s is increasing in ps. For a ¼ 1, it is easy to show that p′s is independent of
ps :
∂p′
sða ¼ 1Þ
∂ps
¼ 0: □
Proof of Proposition 2. The ﬁrst derivative of p′x with respect to px for a ¼ 0 is
∂p′xða ¼ 0Þ
∂px
¼
∂ððpxþð1−pxÞ⋅ð1−e−λxhÞÞ⋅ðð1−p′
sÞf 1þp′
sf γ
1
Þ=ðð1−pxÞe−λxh⋅ðð1−p′
sÞf 0þp′
sf γ
0
Þþðpxþð1−pxÞ⋅ð1−e−λxhÞÞ⋅ðð1−p′
sÞf 1þp′
sf γ
1
ÞÞÞ
∂px
:
Let H0 ¼ 1−p′s
ð Þf 0 þ p′sf
γ
0 and H1 ¼ 1−p′s
ð Þf 1 þ p′sf
γ
1 for simplicity. Then
∂p′
xða ¼ 0Þ
∂px
¼
∂ððpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞ⋅H1=ðð1−pxÞe−λxh
⋅H0 þ ðpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞ⋅H1ÞÞ
∂px
¼
e−λxh
H1½ð1−pxÞe−λxh
H0 þ ðpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞH1Š−ðpx þ ð1−pxÞ⋅ð1−e−λxh
ÞÞH1½−e−λxh
H0 þ e−λxh
H1Š
½ð1−pxÞe−λxhH0 þ ðpx þ ð1−pxÞ⋅ð1−e−λxhÞÞH1Š2
which, similar to the case of p′
s, after some mathematical manipulation reduces to the following expression:
∂p′
xða ¼ 0Þ
∂px
¼
e−λxh
H0H1
½ð1−pxÞe−λxhH0 þ ðpx þ ð1−pxÞ⋅ð1−e−λxhÞÞH1Š2
40:
Thus, for a ¼ 0, p′x is increasing in px while for a ¼ 1, it is again easy to show that p′x is independent of
px :
∂p′
xða ¼ 1Þ
∂px
¼ 0: □
Appendix C. Transition probabilities
Ptransition½ð0; ðpx; psÞ∈SIÞ-ð0; p′
x; p′
sÞŠ ¼ e−ðλxþλsÞh1
⋅Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 0Þ
x; p′
sÞŠ ¼ ð1−e−λxh1
Þe−λsh1
⋅Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 1Þ
x; p′
sÞŠ ¼ e−λxh1
ð1−e−λsh1
Þ⋅Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 2Þ
x; p′
Þð1−e−λsh1
Þ⋅Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 3Þ
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ e−λsh1
⋅Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 1Þ
Ptransition 1; px; ps

∈SI

- 2; p′x; p′s
ð Þ ¼ 0

∈SI

- 3; p′x; p′s
ð Þ ¼ 1−e−λsh1

⋅P p′sjps; a ¼ 0; h1; n1; s ¼ s1

⋅P p′xjpx; a ¼ 0; h1; n1; Y ¼ 3


∈SI

- 0; p′x; p′s
ð Þ ¼ 0

∈SI

- 1; p′x; p′s
ð Þ ¼ 0

∈SI

- 2; p′x; p′s
ð Þ ¼ e−λxh1 ⋅P p′sjps; a ¼ 0; h1; n1; s ¼ s1

⋅P p′xjpx; a ¼ 0; h1; n1; Y ¼ 2


∈SI

- 3; p′x; p′s
ð Þ ¼ 1−e−λxh1

⋅P p′sjps; a ¼ 0; h1; n1; s ¼ s1

⋅P p′xjpx; a ¼ 0; h1; n1; Y ¼ 3

x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ Pðp′
sjps; a ¼ 0; h1; n1; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h1; n1; Y ¼ 3Þ

Ptransition½ð0; ðpx; psÞ∈SIIÞ-ð0; p′
x; p′
⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 0Þ
x; p′
Þe−λsh2
⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 1Þ
x; p′
sÞŠ ¼ e−λxh2
ð1−e−λsh2
Þ⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 2Þ
x; p′
Þð1−e−λsh2
Þ⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 3Þ
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ e−λsh2
⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 1Þ
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ ð1−e−λsh2
Þ⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 3Þ
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ e−λxh2
⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 2Þ
x; p′
Þ⋅Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 3Þ
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ 0
x; p′
sÞŠ ¼ Pðp′
sjps; a ¼ 0; h2; n2; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 0; h2; n2; Y ¼ 3Þ
Ptransition½ð0; ðpx; psÞ∈SIIIÞ-ð0; p′
x; p′
⋅Pðp′
sjps; a ¼ 1; h1; n1; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 1; h1; n1; Y ¼ 0Þ
x; p′
Þe−λsh1
⋅Pðp′
sjps; a ¼ 1; h1; n1; s ¼ s0Þ⋅Pðp′
xjpx; a ¼ 1; h1; n1; Y ¼ 1Þ
x; p′
sÞŠ ¼ e−λxh1
ð1−e−λsh1
Þ⋅Pðp′
sjps; a ¼ 1; h1; n1; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 1; h1; n1; Y ¼ 2Þ
x; p′
Þð1−e−λsh1
Þ⋅Pðp′
sjps; a ¼ 1; h1; n1; s ¼ s1Þ⋅Pðp′
xjpx; a ¼ 1; h1; n1; Y ¼ 3Þ
x; p′
sÞŠ ¼ Ptransition½ð0; ðpx; psÞ∈SIIIÞ-ð0; p′
x; p′
sÞŠ
x; p′
sÞŠ ¼ Ptransition½ð0; ðpx; psÞ∈SIIIÞ- 1; p′x; p′s
ð ÞŠ

∈SIII

- 2; p′x; p′s
ð Þ ¼ Ptransition 0; px; ps

∈SIII

- 2; p′x; p′s
ð Þ
x; p′
x; p′
sÞŠ
x; p′
sÞŠ: ¼ Ptransition½ð0; ðpx; psÞ∈SIIIÞ-ð0; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
x; p′
x; p′
sÞŠ
2814

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