Citation: Teoh WL, Khoo MBC, Teh SY (2013) Optimal Designs of the Median Run Length Based Double Sampling X̄ Chart for Minimizing the Average Sample Size. PLoS ONE 8(7):
e68580.
https://doi.org/10.1371/journal.pone.0068580
Editor: Shyamal D. Peddada, National Institute of Environmental and Health Sciences, United States of America
Received: March 4, 2013; Accepted: June 5, 2013; Published: July 25, 2013
Copyright: © 2013 Teoh et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research is supported by the KPI allocation of the School of Mathematical Sciences, Universiti Sains Malaysia (USM). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Statistical process control (SPC) is a powerful collection of statistical tools for achieving process stability. SPC is based on sound underlying principles, which is easy to use and can be applied in the manufacturing and service processes, such as in the food industries, automobile industries, as well as health-care and public-health surveillance [1]. A control chart is one of the valuable quality improvement techniques in SPC that can be used to attain process stability and reduce process variability over time. Since the double sampling (DS) 
chart was introduced by Croasdale [2] in 1974, the DS scheme has been studied extensively among researchers. By applying the concept of double sampling plans, Daudin [3] suggested an improved DS 
chart which incorporates both the ideas of variable sampling interval (VSI) and variable sample size (VSS). Unlike the VSI procedure, two successive samples are taken in the DS procedure without any intervening time; where, both the first and second samples of the DS chart are taken from the same population.
Recently, considerable efforts have been undertaken on the research of various DS type charts, which can be categorized into the DS 
type, DS S type and other DS type control charts. Costa and Machado [4], Khoo et al. [5] and Torng et al. [6] investigated the DS 
type charts for monitoring the process mean. Works on the DS S type charts for monitoring the process variance were discussed by He and Grigoryan [7], [8] and Lee et al. [9], [10]. Other DS type charts are the joint DS 
and S chart, proposed by He and Grigoryan [11], for a simultaneous monitoring of the process mean and variance, as well as the DS np chart for attributes, suggested by Rodrigues et al. [12].
It is known that the DS 
chart not only maintains the simplicity of the Shewhart 
chart, but the former also improves the statistical efficiency of the latter in detecting process mean shifts, besides reducing the sample size [13]. Compared to the Shewhart 
chart, He et al. [14] claimed that the sample size of the DS 
chart dramatically decreases to nearly 50% when the process is in-control. In addition, the DS 
chart has an advantage of having a lower total sample size when the incoming quality is either very excellent or very poor [15]. This is because only the first sample is required to sentence the process as either in-control or out-of-control. Therefore, the DS scheme is an appropriate choice for process monitoring with destructive testing and high inspection costs [16]. In view of these advantages, many researchers (see [2], [3], [7], [8], [14]) focused on proposing the DS chart for minimizing the in-control average sample size (
). Hsu [17], [18] claimed that the conclusion made by He et al. [14] and He and Grigoryan [7] is questionable since the out-of-control average sample size (
) is disregarded when comparing the various charts' performances. Accordingly, Lee et al. [10] modified the design model of He and Grigoryan [8] to propose the DS S chart which minimizes both the 
and 
.
The average run length (ARL) has been traditionally used as a sole measure of a control chart's performance. The sole reliance on the ARL has been widely criticized by Das [19], Gan [20] and Golosnoy and Schmid [21]. This criticism comes from two concerns [1]. First, the value of the standard deviation of the run length (SDRL) is quite large. Second, the run length distribution is highly skewed. Furthermore, Thaga [22] stated that only a fraction of a chart's behavior is reflected by the size of the ARL. Therefore, misleading conclusion is drawn based on the ARL as it is not necessarily a typical run length. On the other hand, the median run length (MRL) is a more credible measure of a chart's performance since it is less affected by the skewness of the run length distribution [20], [23]. The MRL is the 50th percentile of the run length distribution, representing “half of the time” [24]. For example, when the in-control MRL (
) is 250, a practitioner can claim that a false alarm will occur by the 250th sample in half of the time; while an out-of-control MRL (
) of 10 means that for this particular shift, there is a 50% chance that an out-of-control signal will be produced in not later than the 10th sample. For ease of interpretation and a better understanding of a chart's performance, Gan [20], Golosnoy and Schmid [21], Maravelakis et al. [23], Khoo et al. [25] and Low et al. [26] have all advocated using MRL as an alternative measure to evaluate a chart's performance.
Similar to other charts, the ARL is widely used in the literature as a performance and design criteria of the DS 
chart. However, when the run length distribution is highly skewed to the right, especially for an in-control process or when the shift is small, we show that the ARL is a peculiar measure of a typical chart's performance and that the MRL is a more meaningful quantity to rely on. Keeping this in mind, two new optimal design procedures for the MRL-based DS 
chart by minimizing the (i) 
and (ii) 
are developed in this paper. In this paper, the average sample size (ASS) is chosen as the objective function of the optimal design models. This is because these optimal design models are applicable to small enterprises which have a low production volume or are useful for monitoring destructive testing processes. The reason for minimizing the 
is due to the fact that the process will operate in the in-control state for most of the time [1], [3], [8], [14]. Hsu [17], [18] stated that the ASS for both the in-control and out-of-control situations should be taken into consideration when designing a control chart. Therefore, the second optimal design, i.e. minimizing the 
and 
, is proposed in accordance with the argument of Hsu [17], [18]. Consequently, a smaller sample size is used and this leads to a substantial reduction of inspection and sampling costs.
The rest of this paper is organized as follows: The DS 
chart's procedure and its run length properties are briefly introduced in Section 2. Section 3 examines the performance of the DS 
chart, in terms of the percentiles of the run length distribution, ARL and ASS. Two optimal designs of the MRL-based DS 
chart, for minimizing the (i) 
and (ii) 
are proposed in Section 4. Besides providing the optimal chart parameters for the MRL-based DS 
chart, Section 5 compares the sample-size performance of the optimal MRL-based DS 
, EWMA 
and Shewhart 
charts. An illustrative example on the construction of the optimal MRL-based DS 
chart is given in Section 6. Conclusions are drawn in Section 7.
The DS 
Control Chart for Monitoring the Process Mean
Assume that the observations of the quality characteristic X are independent and follow an identical normal 
distribution with the in-control mean 
and variance 
. We further assume that 
and 
are known. By referring to Figure 1, let 
0 and 
be the warning and control limits of the first-sample stage, respectively; while 
0 is the control limit of the combined-sample stage. The regions of the DS 
chart can be divided into 
, 
, 
and 
. The charting procedure of the DS 
chart is as follows:
- Determine the limits
, 
and 
.
- Take a first sample of size
and calculate the first sample mean 
. Here, 
for 
, is the jth observation at the ith sampling time of the first sample.
- Declare the process as in-control if
. Then the control flow returns to Step (2).
- Declare the process as out-of-control if
and then proceed to Step (8).
- Take a second sample of size
from the same population as the first sample if 
. Then compute the second sample mean 
. Here, 
for 
, is the jth observation at the ith sampling time of the second sample.
- Calculate the combined-sample mean
at the ith sampling time.
- Declare the process as in-control if
; otherwise, declare the process as out-of-control and advance to Step (8).
- Issue an out-of-control signal at the ith sampling time to indicate a process mean shift.
- Investigate and remove assignable cause(s) and then return to Step (2).
Note that the ith sampling time refers to the ith time when either only the first sample of size 
or both the first and second samples of size 
, are collected.
Let 
be the size of a standardized mean shift, where 
is the out-of-control mean. If 
0, the process is considered as in-control; otherwise, it is deemed as out-of-control. Let 
and 
represent the probabilities that the process remains in-control “by the first sample” and “after taking the second sample”, respectively. Then, 
is the probability that the process is regarded as in-control, where 
and 
are given as [3]
(1)and
(2)respectively, where 
and 
are the standard normal cumulative distribution function (cdf) and standard normal probability density function (pdf), respectively. In Equation (2), 
, 
and 
.
Let RL represents the run length which is the number of samples collected until the first out-of-control signal is detected. The RL distribution of a Shewhart 
chart follows a geometric distribution when the chart's control limits are known constants and the plotted statistics are independently and identically distributed random variables [1]. Since the DS 
chart is a two-stage Shewhart 
chart, all the RL properties of the DS 
chart can be characterized by those of the geometric distribution. Hence, the cdf 
of the RL for the DS 
chart, defined for 
{1, 2, 3, …}, is calculated as
(3)It follows that the MRL of the DS 
chart is equal to [20]
(4)while the other 
percentiles of the RL distribution are computed as the value 
, such that
(5)where 
is in the range of 
.
Daudin [3] also showed that the ARL of the DS 
chart is
(6)while the ASS at each sampling time is defined as
(7)where 
.
Performance of the DS 
Chart Based on the Percentiles of the Run Length Distribution, ARL and ASS
Palm [24] claimed that a practitioner is more interested in the percentiles of the RL distribution as they provide additional and detailed information regarding the expected behavior of the RL. Therefore, we investigate the performance of the optimal ARL-based DS 
chart for minimizing 
, in terms of ARL, ASS and the percentiles of the RL distribution. Table 1 summarizes these performance measures for the DS 
chart when the in-control ARL, 
370.0 and the out-of-control ARL, 
. Here, 
is the desired 
value corresponding to a shift 
. The optimization procedure given by Daudin [3] is applied here. The 
value is specified as the 
value of the optimal EWMA 
chart, where the 
value and the sample size (
) of this EWMA chart are set as 370.0 and (3, 5), respectively. In Table 1, the optimal (
, 
, 
, 
, 
) combinations of the ARL-based DS 
chart for minimizing 
are obtained such that 
370.0 and 
. Here, the 
values in Table 1 are selected so that when 
0.5, 
{11.9, 8.1} for 
{3, 5}; and when 
1.0, 
{4.2, 2.8} for 
{3, 5}. These optimal chart parameters are used to calculate the ARL, ASS and the percentiles of the RL distribution based on the formulae shown in Section 2. Note that the 
in Table 1 represents the sample size of the ARL-based Shewhart 
chart, matching approximately a similar design of the ARL-based DS 
chart.
From Table 1, we observe that the difference between the values of ARL and MRL is large when 
0 and it diminishes as 
increases. This indicates that the shape and the skewness of the RL distribution change with the magnitude of the process mean shift 
. Also, the ARLs shown in Table 1 are all larger than the MRLs (i.e. 50th percentile of the RL distribution) when 
2.0. This is due to the fact that in a right-skewed RL distribution, the value of the average of the RL is greater than the median of the RL. Thus, the MRL is a better representation of the central tendency compared to the ARL. Note that the ARL only measures the expected run length and does not indicate the likelihood of getting a signal by a certain probability. For example, when 
1.0, 
3 and 
0.25 are considered, there could exist a risk where a practitioner falsely interprets that an out-of-control is detected by the 103rd sampling time (
103.2) in 50% of the time, but in actual fact, this event occurs noticeably earlier, i.e. by the 72nd sampling time (
72).
An advantage of computing the lower percentiles (e.g. 5th, 10th and 20th percentiles) of the RL distribution for 
0 is that it allows the probability analysis of early false signals to be carried out. From Table 1, we notice that even when the value of 
is large, the lower percentiles are remarkably shorter. This suggests that even when the false alarm rate (FAR = 0.0027) is low, a relatively large percent of false signals occur very early in the process monitoring. The computation of the higher percentiles (e.g. 80th, 90th and 95th percentiles) of the RL distribution also provides some useful information to a practitioner. For instance, when 
0.5, 
5 and 
1.0 are considered, a practitioner can state with a 90% confidence that a shift with magnitude 
1.0 is signaled by the fourth sampling time.
Table 1 provides clear evidence that the in-control RL distribution is highly skewed and that the skewness of the RL distribution changes with 
. Therefore, interpretation based on the average of the RL (or ARL) with respect to a highly skewed RL distribution is certainly misleading compared to the case if the RL distribution is symmetric. When the associated RL distribution has different levels of skewness as 
changes, the MRL provides a more meaningful performance measure for the DS 
chart. Along this line, we are motivated to propose two optimal designs (see Section 4) of the MRL-based DS 
chart.
Optimal Designs of the MRL-Based DS 
Chart
The optimal designs of the MRL-based DS 
chart having the smallest (i) 
and (ii) both the 
and 
, are proposed in Sections 4.1 and 4.2, respectively. The optimization programs are written using the ScicosLab software (www.scicoslab.org). It is not easy to optimally determine the five charting parameters, i.e. 
, 
, 
, 
and 
of the DS 
chart. Therefore, these optimal chart parameters are searched through the implementation of the Nelder Mead's nonlinear optimization algorithm [27]. Since the sample sizes 
and 
are parameters to be optimized, we need to limit the allowable upper bound, i.e. 
. Thus, 
20 is fixed in this paper because it is a common practice in industries to use small and moderate sample sizes.
4.1 Minimizing the in-control ASS
The proposed optimal design of the MRL-based DS 
chart for minimizing the 
is illustrated as follows:
(8)subject to
(9)where 
is the desired in-control MRL.
(10)where 
is the desired out-of-control MRL corresponding to a shift 
.
(11)where 
is the sample size of the MRL-based Shewhart 
chat, matching approximately a similar design of the MRL-based DS 
chart.
By applying the optimization model (8)–(11), the steps for obtaining the optimal MRL-based DS 
chart's parameters (
, 
, 
, 
, 
) are demonstrated as follows:
- Specify the desired values of
, 
, 
, 
and 
.
- Search the parameters
, 
and 
for all the (
, 
) pairs selected based on constraint (11). A nonlinear equation solver is used to determine these three parameters. Note that for any given value of 
, the values of 
and 
are adjusted simultaneously to satisfy both the constraints (9) (
) and (10) (
). At the end of this step, all the possible (
, 
, 
, 
, 
) combinations fulfilling constraints (9)–(11) are obtained.
- Identify the optimal (
, 
, 
, 
, 
) combination which has the smallest value of 
from all the chart-parameter combinations found in Step (2).
For example, when 
, 
, 
, 
and 
, the output listing and the optimal (
, 
, 
, 
, 
) combination (see the last row of the output listing) are obtained as
n1 n2 L1 L L2 MRL0 MRL1 ASS0 ASS1
1 6 0.869930 5.027832 2.857255 250 2 3.306028 4.494778
1 7 1.142700 5.592773 2.763021 250 2 2.772141 4.215309
1 8 1.283047 5.327637 2.692279 250 2 2.595803 4.198212
.
.
.
5 13 2.774352 3.005859 2.985578 250 2 5.037477 5.968226
5 14 2.779625 3.035400 2.489431 250 2 5.042560 6.138543
5 15 2.780030 3.035889 2.454888 250 2 5.045557 6.219900
2 7 1.787 5.133 2.633 250 2 2.517 4.486
The output listing is not shown completely here as there are 66 (
, 
) pairs with the corresponding smallest 
value (see the 8th column of each row in the output listing), for each (
, 
) pair.
Comparative Studies
The performance of the optimal MRL-based DS 
chart is now compared with the Shewhart 
and optimal EWMA 
charts. The 
{250, 500} and various values of 
corresponding to 
{0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00} are considered. Thus, the three charts are compared based on their sample-size performance. Note that only moderate and large 
are considered in this paper because in many real industrial applications, small shifts in the process are usually not desirable to be detected in order to avoid too frequent process interruptions [5], [29].
For the Shewhart 
chart, the upper control limit 
, lower control limit 
and center line 
are computed as [1]
(16a)and
(16b)respectively, where 
is a multiplier controlling the width of both the 
and 
.
For the EWMA 
chart, the plotting statistics 
is expressed as [1]
(17a)and
(17b)where 
is the sample mean at the ith sampling time and 
. Then the upper and lower control limits, i.e. 
and 
, respectively, as well as the center line 
are defined as follows [1]:
(18a)and
(18b)respectively, where 
with the multiplier 
to be ascertained.
In this study, 
{3, 5, 7} are considered. The optimization procedure shown in Khoo et al. [25] is used to optimally design the MRL-based EWMA 
chart for minimizing the 
.
5.1 Study 1: The DS 
chart for minimizing the 
In Study 1, we compare the sample size performance of the optimal MRL-based EWMA 
, Shewhart 
and DS 
charts. Tables 2 and 3 present the optimal chart parameters for these three charts, together with their corresponding values of 
and sample size (
, 
or ASS). For the EWMA 
chart, the optimal parameters (
, 
) and the corresponding (
, 
) values are shown in the first and second rows of each cell, respectively. Meanwhile, the charting constant 
and the corresponding (
, 
) values of the Shewhart 
chart are listed in the first and second rows of each cell, respectively. For the DS 
chart, the optimal combination (
, 
, 
, 
, 
) is presented in the first and second rows of each cell, while the corresponding (
, 
, 
) values are presented in the third row of each cell.
In this study, all the three charts are designed to have a similar sensitivity for a particular 
, i.e. by having a similar 
value as that of the optimal MRL-based EWMA 
chart when 
{250, 500}. In particular, the optimal combination (
, 
, 
, 
, 
) of the DS 
chart is obtained such that 
{250, 500} (constraint (9)) and 
(constraint (10)) for a 
. Here, the 
value is specified as the 
value (see the 
values in the second, fifth and eighth columns of Tables 2 and 3) of the optimal MRL-based EWMA 
chart. Therefore, with the implementation of the optimization model (8)–(11) (see Section 4.1), the optimal DS 
chart having the smallest 
value, will also possess a reasonable 
value which is similar to that of the optimal EWMA 
chart for the specified 
. For the Shewhart 
chart, it is designed to match the two MRL points of the EWMA 
chart. Note that the two MRL points are the 
{250, 500} and a suitable 
value, which is chosen such that the 
value of the Shewhart 
chart, with an appropriate 
, is as close as possible to that of the optimal EWMA 
chart. For example, when 
250, 
7 and 
= 0.5, the optimal 
value for EWMA 
chart is five. Thus, both the DS 
and Shewhart 
charts must have 
5 for 
0.5.
From these tables, it is obvious that the optimal MRL-based DS 
chart generally outperforms the optimal EWMA 
and Shewhart 
charts, in terms of the average sample size. Precisely, the 
and 
values of the optimal MRL-based DS 
chart when 
0.50 and 
0.75, respectively, are lower than the corresponding values of 
and 
. When compared with the optimal MRL-based EWMA 
chart, the decrease in 
of the optimal DS 
chart is around 36–85% when 
0.75; while the decrease in 
is around 13–82% when 
1.00. Tables 2 and 3 also reveal that there are substantial improvements in the 
and 
values of the optimal MRL-based DS 
chart, in comparison to the 
of the MRL-based Shewhart 
chart, where reductions of around 50–75% and 33–68%, respectively, exist, for 
0.50. It is clear that from these two tables, the reduction of the out-of-control ASS is not as high as that of the in-control ASS. Also, by using the optimal MRL-based DS 
chart, we need a smaller sample size to detect moderate to large 
and a larger sample size to detect small 
. Generally, the optimal MRL-based DS 
chart requires much smaller sample sizes on the average when the process is in-control and out-of-control and thus, using the chart reduces costs.
5.2 Study 2: The DS 
chart for minimizing the 
Table 4 summarizes the optimal (
, 
, 
, 
, 
) combination (listed in the first and second rows of each cell) of the DS 
chart for minimizing the 
, together with their respective (
, 
, 
) values (listed in the third row of each cell). The optimization model (12)–(15) in Section 4.2 is employed here. Therefore, it is ensured that all the optimal (
, 
, 
, 
, 
) combinations in Table 4 attain 
{250, 500} (constraint (13)) and 
(constraint (14)) for a 
. Here, the 
value is specified as the 
value of the optimal MRL-based EWMA 
chart. In other words, both the optimal MRL-based DS 
charts for minimizing the 
(see Study 1 of Section 5.1) and 
(see Study 2 of Section 5.2) have the same 
and 
values.
Note that similar conclusions regarding the comparative performance of the in-control and out-of-control sample sizes among the three charts which are discussed for Tables 2 and 3, are obtained for Table 4. Thus, we compare the chart settings between the optimal MRL-based DS 
chart for minimizing the (i) 
(see Tables 2 and 3 of Study 1) and (ii) 
(see Table 4) in this Study 2. In Table 4, it is noticeable that some of the optimal (
, 
, 
, 
, 
) combinations are different from those shown in Tables 2 and 3. In addition, we found that the 
and 
values in Studies 1 and 2 are fairly close to each other. We observe that there are some increments in the 
value and some decrements in the 
value for Study 2 as compared to that in Study 1. This is expected as we are minimizing both the 
and 
in Study 2. Note that the accuracies of all the results shown in Tables 1–4 have been verified with simulation.
An Illustrative Example
In this section, we consider the example given by Carot et al. [30]. This example illustrates the implementation of the optimal MRL-based DS 
chart to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. For the sake of comparison, the construction of the optimal MRL-based EWMA 
and Shewhart 
charts are also discussed in this section.
It is well known that potassium sorbate is a preservative, a bactericide and a fungicide. Hence, it is one of the basic ingredients to preserve a number of edible products. According to the public health institutions, the advisable amount of potassium sorbate to be added is 0.5–2.0 g per kg product. Thus, let 
1.5 g and 
0.008 g as the desired process parameters of potassium sorbate in this yoghurt manufacturing process [29]. We initially generate the measurements of the first ten sampling times (
1 to 10) based on an in-control condition; whereas the measurements for the subsequent six sampling times (
11 to 16) are generated with 
0.75. Table 5 tabulates various summary statistics for the DS 
, Shewhart 
and EWMA 
charts.
Let us assume that 
250 and 
are desired. By referring to Table 2, the optimal chart parameters for the DS 
, Shewhart 
and EWMA 
charts are (
, 
, 
, 
, 
) = (1, 13, 1.583, 5.163, 2.463), (
, 
) = (8, 2.992) and (
, 
, 
) = (5, 0.550, 0.820), respectively. Figures 2 to 4 display the optimal MRL-based DS 
, Shewhart 
and EWMA 
charts. The solid and hollow dots in Figure 2 represent 
and 
of the DS 
chart, respectively. Also, the values of the 
and 
in Figures 3 and 4 are computed from Equations (16a) and (18a), respectively. Note that only the optimal chart parameters for the optimal MRL-based DS 
chart for minimizing the 
is considered in this example as both the optimal designs, i.e minimizing the 
and 
, have almost similar 
and 
values.
From Figures 2 to 4, it is observed that the DS 
, Shewhart 
and EWMA 
charts produce the first out-of-control signal at sampling time 
14 as 
2.8433>
2.463, 
13 as 
1.5086>
1.5085 and 
15 as 
1.5068>
1.5066, respectively. This indicates that all the three charts have almost similar sensitivity in detecting 
0.75. Concerning the number of observations sampled (from 
1 onwards; see Table 5), relatively less number of observations (40 observations) are required for the DS 
chart compared to the Shewhart 
(104 observations) and EWMA 
(75 observations) charts. It is apparent that the DS 
chart needs around 53% and 38% of the total sample size of the EWMA 
and Shewhart 
charts to detect the mean shift of 0.75.
Author Contributions
Conceived and designed the experiments: WLT. Performed the experiments: WLT. Analyzed the data: WLT MBCK SYT. Wrote the paper: WLT MBCK SYT.
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