Optimal Designs of the Median Run Length Based Double Sampling X̄ Chart for Minimizing the Average Sample Size

Designs of the double sampling (DS) chart are traditionally based on the average run length (ARL) criterion. However, the shape of the run length distribution changes with the process mean shifts, ranging from highly skewed when the process is in-control to almost symmetric when the mean shift is large. Therefore, we show that the ARL is a complicated performance measure and that the median run length (MRL) is a more meaningful measure to depend on. This is because the MRL provides an intuitive and a fair representation of the central tendency, especially for the rightly skewed run length distribution. Since the DS chart can effectively reduce the sample size without reducing the statistical efficiency, this paper proposes two optimal designs of the MRL-based DS chart, for minimizing (i) the in-control average sample size (ASS) and (ii) both the in-control and out-of-control ASSs. Comparisons with the optimal MRL-based EWMA and Shewhart charts demonstrate the superiority of the proposed optimal MRL-based DS chart, as the latter requires a smaller sample size on the average while maintaining the same detection speed as the two former charts. An example involving the added potassium sorbate in a yoghurt manufacturing process is used to illustrate the effectiveness of the proposed MRL-based DS chart in reducing the sample size needed.


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) X X 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 X X 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 X X 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 X X 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 X X 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 X X chart not only maintains the simplicity of the Shewhart X X 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 X X chart, He et al. [14] claimed that the sample size of the DS X X chart dramatically decreases to nearly 50% when the process is in-control. In addition, the DS X X 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 (ASS 0 ). 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 (ASS 1 ) 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 ASS 0 and ASS 1 .
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 50 th percentile of the run length distribution, representing ''half of the time'' [24]. For example, when the in-control MRL (MRL 0 ) is 250, a practitioner can claim that a false alarm will occur by the 250 th sample in half of the time; while an out-ofcontrol MRL (MRL 1 ) 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 10 th 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 X X 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 X X chart by minimizing the (i) ASS 0 and (ii) ASS 0 zASS 1 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 ASS 0 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 outof-control situations should be taken into consideration when designing a control chart. Therefore, the second optimal design, i.e. minimizing the ASS 0 and ASS 1 , 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 X X chart's procedure and its run length properties are briefly introduced in Section 2. Section 3 examines the performance of the DS X X chart, in terms of the percentiles of the run length distribution, ARL and ASS. Two optimal designs of the MRLbased DS X X chart, for minimizing the (i) ASS 0 and (ii) ASS 0 zASS 1 are proposed in Section 4. Besides providing the optimal chart parameters for the MRL-based DS X X chart, Section 5 compares the sample-size performance of the optimal MRL-based DS X X , EWMA X X and Shewhart X X charts. An illustrative example on the construction of the optimal MRLbased DS X X chart is given in Section 6. Conclusions are drawn in Section 7.
The DS X X Control Chart for Monitoring the Process

Mean
Assume that the observations of the quality characteristic X are independent and follow an identical normal N m 0 , s 2 0 À Á distribution with the in-control mean m 0 and variance s 2 0 . We further assume that m 0 and s 2 0 are known. By referring to Figure 1, let L 1 w0 and L §L 1 be the warning and control limits of the firstsample stage, respectively; while L 2 w0 is the control limit of the combined-sample stage. The regions of the DS X X chart can be divided into The charting procedure of the DS X X chart is as follows: 1. Determine the limits L, L 1 and L 2 . 2. Take a first sample of size n 1 and calculate the first sample mean X X 1i~P n1 j~1 X 1i,j =n 1 . Here, X 1i,j for j~1,2,:::,n 1 , is the j th observation at the i th sampling time of the first sample. 3. Declare the process as in- Then the control flow returns to Step (2). 4. Declare the process as out-of-control if Z 1i [I 3 and then proceed to Step (8). 5. Take a second sample of size n 2 from the same population as the first sample if Z 1i [I 2 . Then compute the second sample mean X X 2i~P n2 j~1 X 2i,j =n 2 . Here, X 2i,j for j~1, 2,:::,n 2 , is the j th observation at the i th sampling time of the second sample. 6. Calculate the combined-sample mean Þat the i th sampling time. 7. Declare the process as in-control if Z i~ X X i {m 0 ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 zn 2 p ½ = s 0 [I 4 ; otherwise, declare the process as out-of-control and advance to Step (8). 8. Issue an out-of-control signal at the i th sampling time to indicate a process mean shift. 9. Investigate and remove assignable cause(s) and then return to Step (2).
Note that the i th sampling time refers to the i th time when either only the first sample of size n 1 or both the first and second samples of size n 1 zn 2 , are collected.
Let d~m 1 {m 0 j j =s 0 be the size of a standardized mean shift, where m 1 is the out-of-control mean. If d~0, the process is considered as in-control; otherwise, it is deemed as out-of-control. Let P a1 and P a2 represent the probabilities that the process remains in-control ''by the first sample'' and ''after taking the second sample'', respectively. Then, P a~Pa1 zP a2 is the proba- The MRL Based DS X-bar Chart to Minimize ASS PLOS ONE | www.plosone.org bility that the process is regarded as in-control, where P a1 and P a2 are given as [3] respectively, where W : ð Þ and w : ð Þ are the standard normal cumulative distribution function (cdf) and standard normal probability density function (pdf), respectively. In Equation (2), and c~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 zn 2 ð Þ =n 2 p . 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 X X 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 X X chart is a twostage Shewhart X X chart, all the RL properties of the DS X X chart can be characterized by those of the geometric distribution.
It follows that the MRL of the DS X X chart is equal to [20] Pr RLƒMRL{1 ð Þ ƒ0:5 and Pr RLƒMRL ð Þ w0:5; ð4Þ while the other 100f ð Þ th percentiles of the RL distribution are computed as the value ' f , such that where f is in the range of 0vfv1.
Daudin [3] also showed that the ARL of the DS X X chart is while the ASS at each sampling time is defined as where Performance of the DS X X 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 ARLbased DS X X chart for minimizing ASS 0 , in terms of ARL, ASS and the percentiles of the RL distribution. Table 1 summarizes these performance measures for the DS X X chart when the in-control ARL, ARL 0~3 70.0 and the out-of-control ARL, ARL 1 d Ã ð Þ~e 0 . Here, e 0 is the desired ARL 1 value corresponding to a shift d Ã . The optimization procedure given by Daudin [3] is applied here. The e 0 value is specified as the ARL 1 value of the optimal EWMA X X chart, where the ARL 0 value and the sample size (n EWMA ) of this EWMA chart are set as 370.0 and (3, 5), respectively. In Table 1, the optimal (n 1 , n 2 , L 1 , L, L 2 ) combinations of the ARL-based DS X X chart for minimizing ASS 0 are obtained such that ARL 0~3 70.0 and ARL 1 d Ã ð Þ~e 0 . Here, the e 0 values in Table 1 are selected so that 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 X X i(Shewhart) in Table 1 represents the sample size of the ARL-based Shewhart X X chart, matching approximately a similar design of the ARL-based DS X X chart. From Table 1, we observe that the difference between the values of ARL and MRL is large when d~0 and it diminishes as d increases. This indicates that the shape and the skewness of the RL distribution change with the magnitude of the process mean shift d. Also, the ARLs shown in Table 1 are all larger than the MRLs (i.e. 50 th percentile of the RL distribution) when dƒ2.0. 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 d Ã~1 .0, n EWMA~3 and d~0.25 are considered, there could exist a risk where a practitioner falsely interprets that an out-of-control is detected by the 103 rd sampling time (ARL 1~1 03.2) in 50% of the time, but in actual fact, this event occurs noticeably earlier, i.e. by the 72 nd sampling time (MRL 1~7 2).
An advantage of computing the lower percentiles (e.g. 5 th , 10 th and 20 th percentiles) of the RL distribution for d~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 ARL 0 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. 80 th , 90 th and 95 th percentiles) of the RL distribution also provides some useful information to a practitioner. For instance, when d Ã~0 .5, n EWMA~5 and d~1.0 are considered, a practitioner can state with a 90% confidence that a shift with magnitude d~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 d. 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 d changes, the MRL provides a more meaningful performance measure for the DS X X chart. Along this line, we are motivated to propose two optimal designs (see Section 4) of the MRL-based DS X X chart.
Optimal Designs of the MRL-Based DS X X Chart The optimal designs of the MRL-based DS X X chart having the smallest (i) ASS 0 and (ii) both the ASS 0 and ASS 1 , 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. n 1 , n 2 , L 1 , L and L 2 of the DS X X chart. Therefore, these optimal chart parameters are searched through the implementation of the Nelder Mead's nonlinear optimization algorithm [27]. Since the sample sizes n 1 and n 2 are parameters to be optimized, we need to limit the allowable upper bound, i.e. n 1 zn 2~nmax . Thus, n max ƒ20 is fixed in this paper because it is a common practice in industries to use small and moderate sample sizes.

Minimizing the in-control ASS
The proposed optimal design of the MRL-based DS X X chart for minimizing the ASS 0 is illustrated as follows: subject to where t is the desired in-control MRL.
where e is the desired out-of-control MRL corresponding to a shift d Ã . (iii) 1ƒn 1 vn X X vn 1 zn 2 ƒn max and n 1 ƒn 2 , ð11Þ where n X X is the sample size of the MRL-based Shewhart X X By applying the optimization model (8)- (11), the steps for obtaining the optimal MRL-based DS X X chart's parameters (n 1 , n 2 , L 1 , L, L 2 ) are demonstrated as follows: 1. Specify the desired values of t, e, n X X , n max and d Ã . 2. Search the parameters L 1 , L and L 2 for all the (n 1 , n 2 ) pairs selected based on constraint (11). A nonlinear equation solver is used to determine these three parameters. Note that for any given value of L, the values of L 1 and L 2 are adjusted simultaneously to satisfy both the constraints (9) (MRL 0~t ) and (10) (MRL 1~e ). At the end of this step, all the possible (n 1 , n 2 , L 1 , L, L 2 ) combinations fulfilling constraints (9)-(11) are obtained. 3. Identify the optimal (n 1 , n 2 , L 1 , L, L 2 ) combination which has the smallest value of ASS 0 from all the chart-parameter combinations found in Step (2).
For example, when t~250, e~2, n X X~6 , n max~2 0 and d Ã~1 :00, the output listing and the optimal (n 1 , n 2 , L 1 , L, L 2 ) combination (see the last row of the output listing) are obtained as The output listing is not shown completely here as there are 66 (n 1 , n 2 ) pairs with the corresponding smallest ASS 0 value (see the 8 th column of each row in the output listing), for each (n 1 , n 2 ) pair.

Minimizing both the in-control and out-of-control ASSs
Hsu [17,18] indicated that the optimal design of a control chart should take into consideration both the in-control and out-ofcontrol situations. Therefore, in order to provide the best performance of the MRL-based DS X X chart for a specified mean shift d Ã , two objective functions, i.e. minimizing ASS 0 and ASS 1 d Ã ð Þ are proposed in this section. The weighting average method suggested by Zadeh [28] is used to integrate these two objective functions. This weighting average method allows us to assign a weight to each objective function and then combine them into a single objective function. Since the performance of both the in-control and out-of-control cases are equally important, we let the weights of the ASS 0 and ASS 1 d Ã ð Þ equal to each other. Hence, the integrated objective function of this proposed optimal design model is simplified to the minimization of ASS 0 zASS 1 d Ã ð Þ. The proposed optimal design of the MRL-based DS X X chart to minimize both the ASS 0 and ASS 1 d Ã ð Þ, which is modeled as a nonlinear minimization problem, is mathematically expressed as follows: Minimize n 1 ,n 2 ,L 1 ,L,L 2 subject to (i) (ii) (iii) 1ƒn 1 vn X X vn 1 zn 2 ƒn max and n 1 ƒn 2 : The design procedure of the optimization model (12)-(15) is similar to that presented in Step (1) to Step (3) of Section 4.1.
The only difference is that we are minimizing

Comparative Studies
The performance of the optimal MRL-based DS X X chart is now compared with the Shewhart X X and optimal EWMA X X charts.  3.00} are considered. Thus, the three charts are compared based on their sample-size performance. Note that only moderate and large d Ã 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 X X chart, the upper control limit UCL and respectively, where K X X is a multiplier controlling the width of both the UCL X X and LCL X X . For the EWMA X X chart, the plotting statistics Z i(EWMA) is expressed as [1] and where X X i(Shewhart) is the sample mean at the i th sampling time and 0vlƒ1. Then the upper and lower control limits, i.e. UCL EWMA and LCL EWMA , respectively, as well as the center line CL EWMA ð Þ are defined as follows [1]: respectively, where K EWMA~kEWMA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l n EWMA 2{l ð Þ q with the multiplier k EWMA to be ascertained. In this study, n EWMA [{3, 5, 7} are considered. The optimization procedure shown in Khoo et al. [25] is used to optimally design the MRL-based EWMA X X chart for minimizing the MRL 1 .

Study 1:
The DS X X chart for minimizing the ASS 1 0 In Study 1, we compare the sample size performance of the optimal MRL-based EWMA X X , Shewhart X X and DS X X charts. Tables 2 and 3 present the optimal chart parameters for these three charts, together with their corresponding values of MRL 1 and sample size (n EWMA , n X X or ASS). For the EWMA X X chart, the optimal parameters (l, K EWMA ) and the corresponding (MRL 1 , n EWMA ) values are shown in the first and second rows of each cell, respectively. Meanwhile, the charting constant K X X and the corresponding (MRL 1 , n X X ) Table 2. Optimal chart parameters for the EWMA X X , Shewhart X X and DS X X charts, together with their corresponding values of MRL 1 and sample size when MRL 0~2 50 and ASS 0 is minimized.  values of the Shewhart X X chart are listed in the first and second rows of each cell, respectively. For the DS X X chart, the optimal combination (n 1 , n 2 , L 1 , L, L 2 ) is presented in the first and second rows of each cell, while the corresponding (MRL 1 , ASS 0 , ASS 1 ) 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 d Ã , i.e. by having a similar MRL 1 d Ã ð Þ value as that of the optimal MRL-based EWMA X X chart when MRL 0 [{250, 500}. In particular, the optimal combination (n 1 , n 2 , L 1 , L, L 2 ) of the DS X X chart is obtained such that MRL 0~t [{250, 500} (constraint (9)) and MRL 1~e (constraint (10)) for a d Ã . Here, the e value is specified as the MRL 1 d Ã ð Þ value (see the MRL 1 values in the second, fifth and eighth columns of Tables 2 and 3) of the optimal MRL-based EWMA X X chart. Therefore, with the implementation of the optimization model (8)-(11) (see Section 4.1), the optimal DS X X chart having the smallest ASS 0 value, will also possess a reasonable MRL 1 value which is similar to that of the optimal EWMA X X chart for the specified d Ã . For the Shewhart X X chart, it is designed to match the two MRL points of the EWMA X X chart. Note that the two MRL points are the MRL 0 [{250, 500} and a suitable MRL 1 d Ã ð Þ value, which is chosen such that the MRL 1 d Ã ð Þ value of the Shewhart X X chart, with an appropriate n X X , is as close as possible to that of the optimal EWMA X X chart. For example, when MRL 0~2 50, n EWMA~7 and d Ã = 0.5, the optimal MRL 1 value for EWMA X X chart is five. Thus, both the DS X X and Shewhart X X charts must have MRL 1~5 for d Ã~0 .5.
From these tables, it is obvious that the optimal MRL-based DS X X chart generally outperforms the optimal EWMA X X and Shewhart X X charts, in terms of the average sample size. Precisely, the ASS 0 and ASS 1 values of the optimal MRL-based DS X X chart when d Ã §0.50 and d Ã §0.75, respectively, are lower than the corresponding values of n EWMA and n X X . When compared with the Table 3. Optimal chart parameters for the EWMA X X , Shewhart X X and DS X X charts, together with their corresponding values of MRL 1 and sample size when MRL 0~5 00 and ASS 0 is minimized.  optimal MRL-based EWMA X X chart, the decrease in ASS 0 of the optimal DS X X chart is around 36-85% when d Ã §0.75; while the decrease in ASS 1 is around 13-82% when d Ã §1.00. Tables 2 and 3 also reveal that there are substantial improvements in the ASS 0 and ASS 1 values of the optimal MRL-based DS X X chart, in comparison to the n X X of the MRL-based Shewhart X X chart, where reductions of around 50-75% and 33-68%, respectively, exist, for d Ã §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 X X chart, we need a smaller sample size to detect moderate to large d Ã and a larger sample size to detect small d Ã . Generally, the optimal MRL-based DS X X 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.

Study 2: The DS
X X chart for minimizing the ASS 1 01zASS 1 1 d 2 Ã À Á Table 4 summarizes the optimal (n 1 , n 2 , L 1 , L, L 2 ) combination (listed in the first and second rows of each cell) of the DS X X chart for minimizing the ASS 0 zASS 1 d Ã ð Þ, together with their respective (MRL 1 , ASS 0 , ASS 1 ) 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 (n 1 , n 2 , L 1 , L, L 2 ) combinations in Table 4 attain MRL 0~t [{250, 500} (constraint (13)) and MRL 1~e (constraint (14)) for a d Ã . Here, the e value is specified as the MRL 1 d Ã ð Þ value of the optimal MRL-based EWMA X X chart. In other words, both the optimal MRL-based DS X X charts for minimizing the ASS 0 (see Study 1 of Section 5.1) and ASS 0 zASS 1 d Ã ð Þ (see Study 2 of Section 5.2) have the same MRL 0 and MRL 1 d Ã ð Þ 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 X X chart for minimizing the (i) ASS 0 (see Tables 2 and 3 of Study 1) and (ii) ASS 0 zASS 1 d Ã ð Þ (see Table 4) in this Study 2. In Table 4, it is noticeable that some of the optimal (n 1 , n 2 , L 1 , L, L 2 ) combinations are different from those shown in Tables 2 and 3. In addition, we found that the ASS 0 and ASS 1 values in Studies 1 and 2 are fairly close to each Table 4. (n 1 , n 2 , L 1 , L, L 2 ) combination (first and second rows of each cell) and (MRL 1 , ASS 0 , ASS 1 ) values (third row of each other. We observe that there are some increments in the ASS 0 value and some decrements in the ASS 1 value for Study 2 as compared to that in Study 1. This is expected as we are minimizing both the ASS 0 and ASS 1 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 X X 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 X X and Shewhart X X 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 m 0~1 .5 g and s 0~0 .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 (i~1 to 10) based on an in-control condition; whereas the Table 5. Summary statistics of the simulated data for the amount of potassium sorbate (in grams, g) added to a yoghurt manufacturing process. DS X-bar chart Shewhart X-bar chart EWMA X-bar chart Sample sizes n 1 = 1, n 2 = 13 n X-bar = 8 n EWMA = 5 Sampling time, i X -bar (1, i ) Z 1i X-bar (i) Z i X-bar (Shewhart) (i) Z i (EWMA)  Figure 2. The DS X X chart. The chart is used to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. It produces the first out-of-control signal at sampling time i~14. doi:10.1371/journal.pone.0068580.g002  X X chart. The chart is used to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. It produces the first out-of-control signal at sampling time i~15. doi:10.1371/journal.pone.0068580.g004 The MRL Based DS X-bar Chart to Minimize ASS PLOS ONE | www.plosone.org