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An Optimized Double-Sampling
C
Control Chart for Enhanced
Monitoring of Process Nonconformities
Bhagyashri C. Tamshetti
1
, Digambar M. Zombade
2
1
Department of Statistics, Punyashlok Ahilyadevi Holkar Solapur University, Solapur, MS, India -
413255
2
Department of Statistics, Vasundhara Kala Mahavidyalaya, Jule Solapur, MS, India - 413004
DOI: https://doi.org/10.51583/IJLTEMAS.2026.150600238
Received: 15 July 2026; Accepted: 20 July 2026; Published: 01 August 2026
ABSTRACT
Statistical Process Control (SPC) plays a vital role in monitoring manufacturing and service processes
characterized by count data. The conventional Shewhart c-control chart is widely used for monitoring
the number of nonconformities under the Poisson distribution; however, its performance deteriorates when
detecting small and moderate process shifts. This study proposes an optimized Double-Sampling (DS) c-
control chart that improves shift detection while maintaining a specified in-control Average Run Length
(ARL
0
). The chart design is formulated as a constrained optimization problem in which the out-of-control
Average Run Length (ARL
1
) is minimized subject to target ARL
0
and inspection efficiency requirements.
Exact Poisson probabilities are employed to derive the operating characteristics, while the Average Sample
Size (ASS) is used to evaluate inspection effort. Extensive numerical comparisons for various process
means and shift magnitudes demonstrate that the proposed DS c-chart consistently achieves substantially
lower ARL
1
values than the conventional single-sampling c-chart, particularly for small and moderate
shifts, while maintaining the desired false-alarm performance. The proposed scheme provides an efficient,
practical, and economically attractive alternative for monitoring Poisson-distributed process
nonconformities.
Keywords: Double-Sampling c-Control Chart; Statistical Process Control; Poisson Distribution; Average Run
Length; Average Sample Size; Attribute Control Charts.
INTRODUCTION
Statistical Process Control (SPC) has become one of the most widely adopted methodologies for monitoring,
controlling, and continuously improving the quality of manufacturing and service processes. By distinguishing
between common-cause and assignable-cause variation, SPC enables organizations to identify process
abnormalities at an early stage, thereby reducing production costs, minimizing waste, and improving product
reliability. Among the various SPC techniques, control charts remain the most effective graphical tools for
monitoring process stability and supporting data-driven quality improvement initiatives.
Control charts may be broadly classified into variable control charts and attribute control charts. Variable charts are
appropriate when quality characteristics are measured on a continuous scale, whereas attribute charts are employed
when quality characteristics are represented by counts or classifications. In many industrial applications, including
electronic assembly, textile manufacturing, semiconductor fabrication, healthcare systems, pharmaceutical
production, and service operations, quality is naturally expressed as the number of defects or nonconformities
occurring within a fixed inspection unit. For such applications, the Shewhart c-control chart is the standard
monitoring procedure under the assumption that defect counts follow a Poisson distribution with constant mean
c. The conventional Shewhart c-chart is attractive because of its conceptual simplicity, straightforward
implementation, and low computational requirements. Under in-control conditions, the process mean is denoted by
c
0
, and the upper control limit is selected to achieve a specified false-alarm probability or a desired in-control
Average Run Length (ARL
0
). Despite these advantages, the classical c-chart suffers from an important limitation.
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Since each decision is based solely on a single sample, the chart exhibits relatively poor sensitivity to small and
moderate increases in the process defect rate. Consequently, assignable causes may remain undetected for an
extended period, resulting in increased production of nonconforming items, higher inspection costs, and reduced
process efficiency.
To overcome these limitations, numerous researchers have proposed enhanced monitoring schemes based on
adaptive sampling strategies, supplementary runs rules, cumulative statistics, and multi-stage inspection
procedures. Among these approaches, double-sampling control charts have attracted considerable attention
because they improve shift-detection performance without substantially increasing routine inspection effort. Rather
than making an immediate decision after the first inspection stage, double-sampling schemes collect additional
information only when the initial sample provides inconclusive evidence regarding the process condition.
Consequently, these procedures achieve a more desirable balance between false-alarm protection and rapid
detection of process deterioration.
Daudin [2] introduced one of the earliest double-sampling control chart methodologies and demonstrated its superior
performance over conventional Shewhart charts. Chan et al. [3] developed a two-stage decision procedure for
monitoring processes with low fractions of nonconforming items. Hsu [4, 5] investigated the optimal design of
double and triple-sampling control charts using genetic algorithms and extended these ideas to manufacturing
applications. Wu and Wang [6] proposed a double-inspection np-control chart for attribute data, while Rodrigues et
al. [7] developed optimized double-sampling attribute control charts and demonstrated substantial improvements
in Average Run Length performance. More recently, Saghir and Lin [8] presented a comprehensive review of count-
data control charts, highlighting the continuing need for efficient monitoring procedures capable of rapidly
detecting shifts in Poisson-distributed processes.
Although considerable progress has been achieved in the development of adaptive and double-sampling control charts
for attribute data, comparatively limited attention has been devoted to optimized double-sampling procedures
specifically designed for Poisson c-charts. Existing approaches frequently focus on binomial attribute charts,
variable control charts, or generalized count-data models, leaving a noticeable gap in the literature regarding efficient
double-sampling monitoring schemes for classical Poisson-distributed nonconformities. Furthermore, many published
designs primarily emphasize Average Run Length performance while giving comparatively less consideration to the
inspection effort required during the monitoring process.
Motivated by these observations, this paper proposes an optimized Double-Sampling (DS) c-control chart for
monitoring Poisson-distributed nonconformities. The proposed monitoring scheme employs a two-stage inspection
procedure governed by five design parameters (r
1
, r
2
, WL, UCL
1
, UCL
2
). Exact Poisson probabilities are used to
derive the probability of no signal, Average Run Length (ARL), and Average Sample Size (ASS). The optimal
chart parameters are obtained by minimizing the out-of-control Average Run Length (ARL
1
) while maintaining a
specified in-control Average Run Length (ARL
0
) and economical inspection effort. Extensive numerical studies are
conducted for several in-control process means and shift magnitudes, and the proposed chart is compared with the
conventional Shewhart single-sampling c-chart.
The numerical results demonstrate that the proposed Double-Sampling c-chart consistently provides faster detection
of process deterioration than the traditional Shewhart c-chart, particularly for small and moderate shifts in the
process mean. Moreover, the proposed design achieves these improvements while maintaining the desired false-
alarm performance and requiring only a modest increase in inspection effort. Consequently, the proposed monitoring
scheme offers a practical, statistically efficient, and economically attractive alternative for quality control
applications involving Poisson-distributed count data.
Description
of
the
Proposed
Double-Sampling
c
Control Chart
The proposed Double-Sampling (DS) c control chart is developed to provide a highly efficient and responsive
framework for monitoring process nonconformities. By intelligently combining a two-stage inspection strategy with
optimized decision limits, the chart achieves substantially improved sensitivity to process shifts while maintaining
desirable in-control performance.
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Unlike the conventional Shewhart c chart, which relies on a single inspection stage, the proposed scheme incorporates
an adaptive second-stage evaluation whenever the initial sample provides inconclusive evidence regarding the
process condition. This additional layer of inspection enables the chart to distinguish more effectively between
natural process variation and the presence of assignable causes, thereby facilitating earlier detection of process
deterioration.
Let
X
1
∼
Poisson(r
1
c)
denote the number of nonconformities observed in the first-stage sample, where c represents the average number
of defects per inspection unit and r
1
is the first-stage sampling coefficient.
The proposed DS c chart is completely specified by the following five design parameters:
(r
1
, r
2
, WL, UCL
1
, UCL
2
),
where
r
1
denotes the first-stage sampling
coefficient;
r
2
denotes the second-stage sampling
coefficient;
WL represents the warning limit used to determine whether further inspection is required;
UCL
1
denotes the first-stage upper control
limit;
UCL
2
denotes the second-stage upper control limit.
The operational philosophy of the proposed chart is based on the principle of selective intensification of inspection
effort. Initially, a first-stage sample is inspected and the observed number of nonconformities is compared with
the warning and control limits. When the observed count falls well below the warning limit, the process is
considered stable and no additional inspection is necessary. Conversely, when the observed count exceeds the
first-stage control limit, an immediate out-of-control signal is generated.
However, when the first-stage result lies within the warning region, the available information may not be sufficient
to make a reliable decision regarding the process status. In such situations, a second-stage sample is collected and
combined with the information obtained from the first stage. The final decision is then based on the cumulative
number of nonconformities observed across both samples.
This adaptive decision-making mechanism allows the proposed DS c chart to devote additional inspection resources
only when warranted by the observed data. Consequently, the chart achieves superior detection capability without
imposing a substantial increase in routine inspection effort. Such a balance between statistical efficiency and
operational economy makes the proposed monitoring scheme particularly attractive for modern manufacturing and
service systems where both quality assurance and cost-effectiveness are of paramount importance.
Operating Procedure of the Proposed DS
c
Control Chart
The proposed Double-Sampling c control chart operates through a sequential two-stage inspection mechanism
designed to enhance the detection of increases in the process nonconformity rate while maintaining economical
inspection effort.
At each sampling epoch, a first-stage sample is inspected and the observed number of nonconformities is denoted
by X
1
. The process status is initially evaluated using the warning limit (WL) and the first-stage upper control limit
(UCL
1
).
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If the observed count satisfies
X
1
< WL,
the evidence strongly suggests that the process remains stable and operating under statistical control. Consequently,
no additional inspection is required, and the process is allowed to continue without intervention.
Conversely, if
X
1
> UCL
1
,
the observed number of nonconformities exceeds the acceptable threshold, indicating a high likelihood of process
deterioration. In this case, an immediate out-of-control signal is generated and corrective actions should be
initiated to identify and eliminate the assignable cause.
In situations where the first-stage result falls within the intermediate region,
WL ≤ X
1
≤ UCL
1
,
the available evidence is insufficient to make a reliable decision regarding the process condition. To reduce the
possibility of incorrect conclusions, a second-stage sample is obtained immediately.
Let
X
2
∼
Poisson(r
2
c)
denote the number of nonconformities observed in the second-stage sample. The information obtained from both
stages is then combined through the cumulative statistic
T = X
1
+ X
2
.
The final decision regarding the process status is based on the comparison of T with the second-stage upper control
limit UCL
2
.
If
T > UCL
2
,
the process is declared to be out of statistical control and an alarm is issued. Otherwise,
T ≤ UCL
2
,
the process is considered to remain in control, and routine monitoring continues.
The proposed decision structure intelligently allocates additional inspection effort only when the first-stage sample
provides inconclusive information. This adaptive feature substantially enhances the chart’s sensitivity to process shifts
while preserving satisfactory in-control performance and minimizing unnecessary inspection costs. As a result, the
proposed DS c control chart offers an attractive balance between statistical efficiency, practical simplicity, and
economic feasibility.
Performance
Measures
Probability of No Signal
The performance of the proposed Double-Sampling (DS) c control chart is fundamentally determined by the
probability that no out-of-control signal is generated at a sampling occasion. This probability forms the basis for
evaluating the Average Run Length (ARL) characteristics of the monitoring scheme.
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Let P denote the probability that the process is judged to be in control after completion of the inspection procedure.
Since the proposed chart employs a two-stage sampling mechanism, the overall probability of no signal can be
expressed as
P = P
1
+ P
2
,
where P
1
represents the probability of accepting the process after the first-stage inspection and P
2
represents the
probability of accepting the process after the second-stage inspection.
The first-stage acceptance probability is defined as
P
1
= P (X
1
< WL)
Since X
1
follows a Poisson distribution with mean r
1
c, we obtain
This probability corresponds to the situation in which the observed number of nonconformities in the first-stage
sample falls below the warning limit and the process is immediately regarded as being in control.
The second-stage acceptance probability is given by
P
2
= P (WL ≤ X
1
≤ UCL
1
, X
1
+ X
2
≤ UCL
2
)
Let
T = X
1
+ X
2
denote the total number of nonconformities observed across the two inspection stages.
Then,
The outer summation accounts for all first-stage observations that fall within the warning region, thereby requiring
a second-stage inspection. The inner summation evaluates the probability that the cumulative number of
nonconformities observed in both stages remains within the allowable second-stage control limit.
Consequently, the overall probability of no signal is obtained by combining the two acceptance probabilities as
P = P
1
+ P
2
This quantity serves as the foundation for the derivation of the Average Run Length, Average Sample Size, and other
performance measures of the proposed DS c control chart.
Average Run Length
The Average Run Length (ARL) is one of the most important and widely accepted measures for assessing the
performance of a control chart. It represents the expected number of sampling occasions required before an alarm is
generated and provides a direct indication of the chart’s ability to distinguish between in-control and out-of-control
process conditions.
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In statistical process monitoring, an effective control chart should possess two desirable characteristics. First, when the
process is operating under stable conditions, false alarms should occur infrequently, resulting in a large in-control
Average Run Length. Second, when the process experiences a shift due to the presence of an assignable cause, the
chart should detect the change as rapidly as possible, resulting in a small out-of-control Average Run Length.
For the proposed Double-Sampling (DS) c control chart, the Average Run Length is determined by the probability
of no signal, denoted by P . Since successive sampling occasions are assumed to be independent, the run length
follows a geometric distribution. Consequently, the ARL is given by
ARL
=
1
.
1 − P
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This expression represents the expected number of samples required before the first signal is produced.
1.1.1
In-Control Average Run Length
When the process operates under its target condition, the mean number of nonconformities is equal to the in-
control value c
0
. Under this circumstance, the probability of no signal is denoted by P
0
.
The corresponding in-control Average Run Length is therefore
ARL
=
1
.1 − P
0
A large value of ARL
0
is highly desirable because it indicates a low false-alarm rate and ensures that unnecessary
process interruptions are minimized. Consequently, ARL
0
serves as a measure of the chart’s stability and reliability
during normal process operation.
Out-of-Control Average Run Length
Suppose that an assignable cause increases the average number of nonconformities from c
0
to a higher value c
1
. In
this situation, the probability of no signal changes and is denoted by P
1
.
The out-of-control Average Run Length is defined as
ARL = 1/1-P
1
This quantity represents the expected number of sampling occasions required to detect the process shift after it has
occurred.
A smaller value of ARL
1
indicates superior detection capability because the chart can identify process
deterioration more rapidly and facilitate timely corrective actions. Therefore, minimizing ARL
1
while maintaining
an acceptable value of ARL
0
constitutes the primary objective in the design and optimization of the proposed DS c
control chart. The simultaneous consideration of both ARL
0
and ARL
1
provides a comprehensive assessment of
chart performance, ensuring an appropriate balance between false-alarm protection and rapid shift detection. These
measures form the foundation for the optimization procedure presented in the subsequent section.
Average Sample Size
The Average Sample Size (ASS) serves as an important measure of the inspection effort required by the proposed
double-sampling (c)-chart. Unlike the conventional Shewhart (c)-chart, where a fixed sample is inspected at every
sampling epoch, the double-sampling procedure inspects the second sample only when the first-stage observation
falls within the continuation region. Consequently, the expected sample size varies according to the probability of
proceeding to the second stage.
Let (r
1
) and (r
2
) denote the first- and second-stage sampling quantities, respectively.
Under the in-control process, the Average Sample Size is defined as
ASS
0
= r
1
+ r
2
P (W < X
1
≤ UCL
1
|c = c
0
) ,
Where, (P (W < X
1
≤ UCL
1
| c = c
0
)) represents the probability that the first-stage count lies within the
continuation region, thereby requiring a second sample. This probability is evaluated under the in-control Poisson
model with mean (c
0
).
Similarly, when the process shifts to an out-of-control state with mean (c
1
= γc
0
), the corresponding Average Sample
Size is given by
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ASS
1
= r
1
+ r
2
P (W < X
1
≤ UCL
1
|c = γc
0
)
The quantity (ASS
1
) represents the expected inspection effort under the shifted process and reflects the proportion of
occasions on which a second-stage sample is required after the occurrence of a process shift.
The proposed expressions for (ASS
0
) and (ASS
1
) provide a comprehensive assessment of the sampling effort associated
with the double-sampling (c)-chart. Together with the Average Run Length ((ARL)), these measures facilitate
an effective evaluation of the trade-off between statistical detection performance and inspection cost. A desirable
sampling scheme is characterized by an in-control Average Run Length that attains the specified design target while
simultaneously yielding a small out-of-control Average Run Length and maintaining relatively low values of (ASS
0
)
and (ASS
1
). Such a design achieves rapid detection of process deterioration without imposing unnecessary inspection
effort, thereby enhancing both the statistical efficiency and practical applicability of the proposed double-sampling
control chart.
Optimization Problem
The primary objective of the proposed Double-Sampling (DS) c control chart is to achieve the fastest possible detection
of process deterioration while simultaneously maintaining satisfactory in-control performance. To accomplish this
goal, an optimization procedure is employed to determine the most effective combination of chart parameters.
The proposed chart is characterized by five design parameters,
(r
1
, r
2
, WL, UCL
1
, UCL
2
),
which jointly determine the sensitivity, stability, and overall monitoring efficiency of the control scheme.
An effective control chart should rapidly detect increases in the process nonconformity rate while avoiding excessive
false alarms. Consequently, the design problem is formulated as a constrained optimization problem in which the out-
of-control Average Run Length is minimized subject to predefined in-control performance requirements.
Mathematically, the optimization problem is expressed as
min
ARL
1
,
subject to
ASS
0
≤ ASS
single
,
and
ARL
0
≥ 200.
The first constraint ensures that the average inspection effort required during normal process operation remains within
an acceptable limit. The second constraint guarantees adequate protection against false alarms by maintaining a
sufficiently large in-control Average Run Length.
To evaluate the performance of the proposed chart under process shifts, the shift magnitude is defined as
where
δ = c
1
/c
0
c
1
> c
0
.
Here, c
0
denotes the in-control mean number of nonconformities, whereas c
1
represents the out-of-control mean after
the occurrence of an assignable cause.
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The optimization procedure seeks the set of parameter values that minimizes ARL
1
for a specified shift magnitude
while satisfying the prescribed constraints on ARL
0
and ASS
0
. This approach ensures that the resulting chart
possesses both excellent shift detection capability and desirable in-control operating characteristics.
By balancing rapid detection, false-alarm protection, and inspection economy, the proposed optimization framework
produces a statistically efficient and practically attractive monitoring scheme for modern industrial quality control
applications.
Simulation Study
To thoroughly investigate the statistical performance and practical effectiveness of the proposed Double-
Sampling (DS) c control chart, an extensive Monte Carlo simulation study was conducted. Simulation-based
experimentation provides a powerful framework for evaluating control chart behavior under a wide range of process
conditions and enables a comprehensive assessment of the chart’s ability to detect shifts in the process
nonconformity rate.
The primary objective of this study was to examine the performance of the proposed monitoring scheme under both
in-control and out-of-control operating environments. Particular attention was devoted to evaluating the chart’s
sensitivity to increases in the average number of nonconformities while simultaneously maintaining satisfactory false-
alarm protection.
The simulation experiments were performed for several representative values of the in-control process parameter.
Specifically, the following values were considered:
c
0
= 1, 2, 4, 8.
These values were selected to represent a broad spectrum of practical manufacturing and service process scenarios
characterized by different levels of process quality.
To investigate the effect of process deterioration, various shift magnitudes were introduced through the parameter
δ =
c
1
,
c
0
where c
1
denotes the out-of-control mean number of nonconformities. The following shift magnitudes were
examined:
δ
=
1.25, 1.50, 2.00, 3.00.
These values correspond to small, moderate, and substantial increases in the process nonconformity rate, thereby
enabling a comprehensive evaluation of the detection capability of the proposed chart across different operating
conditions.
For each combination of parameter settings, the following performance measures were computed:
In-Control Average Run Length (ARL
0
), which measures the average number of samples taken before the
occurrence of a false alarm when the process is operating under statistical control;
Out-of-Control Average Run Length (ARL
1
), which quantifies the speed with which the chart detects a process
shift after the occurrence of an assignable cause;
In-Control Average Sample Size (ASS
0
), representing the average inspection effort required during normal
process operation;
Out-of-Control Average Sample Size (ASS
1
), representing the average inspection effort required after the
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process has shifted from its target condition.
The resulting simulation outcomes provide valuable insights into the efficiency, robustness, and practical applicability
of the proposed DS c control chart. Furthermore, they facilitate meaningful comparisons with conventional attribute
control charts and serve as the foundation for the performance analyses presented in the subsequent sections.
Table 1: Optimal design, ARL
1
, ASS
1
, and PG values of single-sampling (SS) and double-sampling (DS) c-charts
for the target in-control ARL
0
= 200.
Table 2: Optimal design, ARL
1
, ASS
1
, and PG values of single-sampling (SS) and double-sampling (DS) c-charts
for the target in-control ARL
0
= 370.
Table 3: Optimal design, ARL
1
, ASS
1
, and PG values of single-sampling (SS) and double-sampling (DS) c-charts
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for the target in-control ARL
0
= 500.
Figure 1: Comparison of the ARL
1
of the SS and DS c-charts for various shift factors (γ) for the ARL
0
= 200.
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Figure 2: Percentage gain of the proposed DS c-chart over SS c-chart for various shift factors (γ) and in-control
process means (c
0
) for the ARL
0
= 200.
Figure 3: For the ARL
0
= 200, the graphs present a comparison of the ARL
1
of the DS and SS c-charts. The plots
correspond to the in-control process mean values c
0
=1, 2, 4, and 8, while the curves represent the shift factors γ
=1.25, 1.50, 2.00, and 3.00.
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Figure 4: For the ARL
0
= 200, the graphs present a comparison of the Percentage gain of the DS and SS c-charts.
The plots correspond to the in-control process mean values c
0
=1, 2, 4, and 8, while the curves represent the shift
factors γ =1.25, 1.50, 2.00, and
3.00.
Results
Figure 1 compares the out-of-control Average Run Length (ARL
1
) of the conventional Single Sampling (SS) and
the proposed Double Sampling (DS) c-charts for different shift factors when the target in-control average run length
is ARL
0
= 200. For all values of c
0
, the ARL
1
values decrease as the shift factor γ increases, indicating that larger
process shifts are detected more quickly. The proposed DS c-chart consistently produces lower ARL
1
values than
the SS c-chart, particularly for small and moderate shifts (γ = 1.25 and 1.50). As the shift factor increases further,
the ARL
1
values of both charts become smaller and the difference between them gradually decreases because larger
shifts are easier to detect.
Figure 2 illustrates the percentage gain achieved by the proposed Double Sampling (DS) c-chart over the
conventional Single Sampling (SS) c-chart for different values of c
0
. Positive percentage gain is observed for all
combinations of c
0
and γ, confirming the superior performance of the proposed chart. The highest percentage gains
are obtained for moderate shift factors, particularly when c
0
= 4 and c
0
= 8, indicating that the proposed DS c-
chart offers substantial improvement in detecting process shifts under these conditions. Although the percentage
gain decreases for larger shift factors, the proposed DS chart consistently outperforms the conventional SS chart
over the entire range of process mean values considered.
Figure 3 presents the comparison of the ARL
1
values of the SS and DS c-charts for different in-control process mean
values and shift factors. For every shift factor, the ARL
1
values decrease as c
0
increases, indicating that larger in-control
process means lead to faster detection of process shifts. The proposed DS c-chart consistently achieves lower
ARL
1
values than the SS c-chart across all values of c
0
, demonstrating its improved sensitivity to process changes. The
advantage of the DS chart is more pronounced for smaller shift factors, whereas the difference between the two charts
becomes smaller for larger shifts because both charts detect large changes efficiently.
Figure 4 presents the percentage gain of the proposed Double Sampling (DS) c-chart over the conventional
Single Sampling (SS) c-chart for different in-control process mean values. The percentage gain generally
increases with increasing c
0
for the shift factors γ = 1.25 and γ = 1.50, reaching its highest values at larger process
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mean values. For larger shifts (γ = 2.00 and 3.00), the percentage gain is comparatively smaller because both
sampling schemes detect large process shifts rapidly.
The results obtained for the target in-control average run lengths ARL
0
= 370 and ARL
0
= 500 exhibit trends similar
to those observed for ARL
0
= 200. In both cases, the proposed Double Sampling (DS) c-chart consistently achieves
lower out-of-control average run length (ARL
1
) values than the conventional Single Sampling (SS) c-chart for all
combinations of the in-control process mean c
0
and the shift factor γ. The improvement is particularly noticeable
for small and moderate process shifts, where the DS chart detects process changes more quickly than the SS chart.
As the shift factor increases, the ARL
1
values of both charts decrease and gradually become closer because larger
shifts are easier to detect. The percentage gain results further demonstrate the superior performance of the proposed
DS c-chart, showing a clear improvement over the SS chart while maintaining the desired in-control average run
lengths of ARL
0
= 370 and ARL
0
= 500.
CONCLUSIONS
This study proposed an optimized Double Sampling (DS) c-chart for monitoring Poisson-distributed nonconformities
and compared its performance with the conventional Single Sampling (SS) c-chart. The chart parameters were
designed to achieve the desired in-control average run lengths of ARL
0
= 200, 370, and 500, and their performances
were evaluated using the out-of-control average run length (ARL
1
), average sample size (ASS), and percentage gain
(PG).
The numerical results showed that the proposed DS c-chart consistently achieved lower ARL
1
values than the SS c-chart
over a wide range of process shifts and in-control process mean values. The improvement was particularly significant
for small and moderate shifts, where early detection is most important in practical quality control applications.
The percentage gain analysis further confirmed that the DS c-chart provides a substantial reduction in detection
time while maintaining the specified in-control performance. In addition, the average sample size results
demonstrated that the proposed chart offers an efficient balance between inspection effort and detection capability.
Overall, the proposed Double Sampling c-chart provides a practical, efficient, and reliable alternative to the
conventional Single Sampling c-chart for monitoring count data. Its ability to detect process shifts more quickly
while maintaining the desired in-control average run length makes it well suited for industrial and manufacturing
quality control applications.
Future research may extend the proposed methodology to Triple Sampling and Multiple Sampling c-charts, adaptive
and variable sampling schemes, EWMA and CUSUM-based double sampling control charts, fuzzy and Bayesian
attribute control charts, and multivariate monitoring of count data.
Suggestions:
DS c-chart using real manufacturing or service process data to demonstrate its practical effectiveness beyond
simulation results. Future research should investigate the performance of the method under more complex
conditions, including overdispersed count data, non-Poisson distributions, autocorrelated observations, and
changing process environments. Comparative studies involving other advanced control chart techniques would
provide a broader assessment of its advantages and limitations.
Response to suggestions
Practical Effectiveness of the Proposed Double Sampling c-Chart Using Real PCB Manufacturing Data
To demonstrate the practical applicability of the proposed Double Sampling (DS) c-chart, a real manufacturing
dataset consisting of 230 printed circuit boards (PCBs) was analyzed. For each PCB, the total number of surface
defects was recorded after inspection. To implement the proposed two-stage inspection procedure, each PCB
was conceptually divided into two equal inspection regions. The defects observed in the left half were treated as
the Stage 1 observations, while the defects identified in the right half were considered the Stage 2 observations.
Consequently, the total number of defects on each PCB was obtained as the sum of the Stage 1 and Stage 2 defect
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counts.
The first 180 PCB samples were used as Phase I data to estimate the in-control process parameters and determine
the control chart limits. The remaining 50 PCB samples (PCB 181–230) were reserved as Phase II data to
evaluate the monitoring performance of both the classical Single Sampling (SS) c-chart and the proposed Double
Sampling (DS) c-chart under actual operating conditions. For the classical SS c-chart, the upper control limit
was selected to achieve an in-control Average Run Length (ARL
0
) close to the desired target. The proposed DS
c-chart employed a two-stage inspection strategy with fixed inspection proportions r
1
=0.60 and r
2
=1.80, together
with the optimized decision parameters WL=4, UCL
1
=19, and UCL
2
=29. The optimized design produced an in-
control performance of ARL
0
=198.35, which is very close to the target value of 200, while substantially reducing
the out-of-control Average Run Length to ARL
1
=3.62. In comparison, the conventional SS c-chart yielded
ARL
0
=201.22 and ARL
1
=8.35. These results indicate that the proposed DS c-chart detects process shifts
considerably faster than the traditional SS chart while maintaining nearly identical in-control performance.
The optimized DS design also resulted in ASS
0
=1.4331 and ASS
1
=2.0379. The increase in the average sample
size reflects the additional inspection performed only when the Stage 1 observations fall within the continuation
region, thereby providing earlier detection of process deterioration. This additional inspection represents a trade-
off between inspection effort and detection speed, which is often desirable in high-quality manufacturing
environments where the cost of delayed detection is substantially greater than the cost of inspecting additional
units.
The Phase II monitoring results for PCB samples 181–230 showed that all observations remained within the
established control limits for both the SS and DS charts. Consequently, neither chart produced an out-of-control
signal during the monitoring period, indicating that the manufacturing process was operating under stable
statistical control for the available production data. Although no assignable causes were detected in the
monitoring stage, the ARL comparison clearly demonstrates the superior theoretical detection capability of the
proposed DS c-chart. The optimized design reduced the out-of-control Average Run Length by approximately
56.65% compared with the conventional SS c-chart while maintaining an in-control ARL close to the desired
target. These findings confirm that the proposed DS c-chart provides a more sensitive and efficient monitoring
scheme for count data arising from manufacturing processes.
The real manufacturing application validates the practical usefulness of the proposed Double Sampling c-chart.
The results demonstrate that the proposed method preserves satisfactory in-control performance, substantially
improves the speed of detecting process shifts, and offers an effective monitoring framework for industrial
defect-count data such as printed circuit board manufacturing.
Sample
PCB
Stage1
Stage2
Total
SS
DS
1
181
3
3
6
Accept
Accept
2
182
6
1
7
Accept
Accept
3
183
4
2
6
Accept
Accept
4
184
3
3
6
Accept
Accept
5
185
4
2
6
Accept
Accept
6
186
4
2
6
Accept
Accept
7
187
5
2
7
Accept
Accept
8
188
2
6
8
Accept
Accept
9
189
7
4
11
Accept
Accept
10
190
5
4
9
Accept
Accept
11
191
2
10
12
Accept
Accept
12
192
4
5
9
Accept
Accept
13
193
2
8
10
Accept
Accept
14
194
7
7
14
Accept
Accept
15
195
3
4
7
Accept
Accept
16
196
3
3
6
Accept
Accept
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17
197
3
2
5
Accept
Accept
18
198
2
4
6
Accept
Accept
19
199
2
4
6
Accept
Accept
20
200
2
3
5
Accept
Accept
21
201
4
1
5
Accept
Accept
22
202
4
3
7
Accept
Accept
23
203
3
3
6
Accept
Accept
24
204
2
6
8
Accept
Accept
25
205
0
8
8
Accept
Accept
26
206
9
1
10
Accept
Accept
27
207
4
2
6
Accept
Accept
28
208
2
4
6
Accept
Accept
29
209
2
3
5
Accept
Accept
30
210
2
4
6
Accept
Accept
31
211
2
4
6
Accept
Accept
32
212
3
3
6
Accept
Accept
33
213
5
6
11
Accept
Accept
34
214
10
4
14
Accept
Accept
35
215
4
3
7
Accept
Accept
36
216
4
2
6
Accept
Accept
37
217
7
5
12
Accept
Accept
38
218
0
5
5
Accept
Accept
39
219
3
3
6
Accept
Accept
40
220
3
2
5
Accept
Accept
41
221
4
1
5
Accept
Accept
42
222
3
4
7
Accept
Accept
43
223
3
3
6
Accept
Accept
44
224
3
3
6
Accept
Accept
45
225
1
6
7
Accept
Accept
46
226
4
3
7
Accept
Accept
47
227
4
3
7
Accept
Accept
48
228
1
5
6
Accept
Accept
49
229
5
2
7
Accept
Accept
50
230
3
4
7
Accept
Accept
Sr. No.
Measure
Single Sampling
Double Sampling
1
ARL
0
201.22
198.35
2
ARL
1
8.35
3.62
3
ASS
0
1
1.433
4
ASS
1
1
2.038
5
Signals
0
0
6
Accepted
50
50
7
First Signal
NA
NA
The proposed methodology assumes that defect counts follow a Poisson distribution. Future studies may extend
the proposed chart to overdispersed count data, generalized Poisson distributions, Conway–Maxwell–Poisson
models, and negative binomial processes. Also, compare with adaptive, EWMA, CUSUM and Bayesian count-
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue VI, June 2026
data control charts.
ACKNOWLEDGEMENTS
Miss. Bhagyashri C. Tamshetti, would like to express her sincere gratitude to the Mahatma Jyotiba Phule Research
& Training Institute (MahaJyoti), Nagpur, for awarding the Mahatma Jyotiba Research Fellowship (MJRF), as
evidenced by the award letter number MAHAJYOTI/2023/Ph.D.
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