Evaluating the Performance of Synthetic Double Sampling np Chart Based on Expected Median Run Length

Abstract

To keep an eye on the status of high-quality processes for fraction nonconforming, the synthetic double sampling (SDS) np chart is a helpful tool. The SDS np chart is a hybrid between the double sampling (DS) np chart and the conforming run length (CRL) chart. The performance of a control chart is typically judged solely using the average run length (ARL). However, as the shape of the run length (RL) distribution varies with the magnitude of the shift in the process fraction nonconforming, the ARL no longer provides clear interpretation of a chart’s performance. Subsequently, enhanced DS np charts that use median run length (MRL) and expected median run length (EMRL) measures, including SDS np with MRL have recently been proposed for addressing this setback. To broaden the functionality of SDS np, in this work, the unexplored use of EMRL as alternative performance measure is developed by means of Markov chain model. Additionally, in both the zero-state (ZS) and steady-state (SS) modes, the novel optimal designs algorithms are described for computing the optimal charting parameters of the SDS np chart, for both MRL₁ and EMRL₁ minimizations, without any unfavourable feature of bilateral sensitivity. Both the MRL and EMRL performances of SDS np, synthetic np, and DS np charts are compared. Optimal designs charting parameters and sensitivity analyses are provided to aid the practical application of SDS np chart.

1. Introduction

The Shewhart charts, developed by Walter A. Shewhart in the 1920s, are widely used to identify special causes of a process’s variation. Shewhart np charts have the advantage of being simple to use and comprehend, but they are not sensitive enough to pick up on small and moderate changes in the process fraction nonconforming (p). As a result, a number of alternatives to the traditional Shewhart np chart have been taken into consideration, with the main goal of improving the charts’ sensitivity. One alternative that aims to improve upon the traditional Shewhart np chart is the synthetic double sampling (SDS) scheme.

SDS control charts are generally considered as an efficient process monitoring tool because of their exceptional qualities, such as the capacity to detect tiny and moderate process shifts easily even with a small sample size (see examples [1,2]). Very few researchers, up until this point, have expressed interest in designing SDS charts. An SDS scheme which combines the double sampling (DS) $\overline{\mathrm{X}}$ and conforming run length (CRL) sub-charts was proposed by Khoo et al. [1] for monitoring the process mean. The latter was introduced by Bourke [3] for the purpose of detecting the process fraction nonconforming p shifts. The random variable CRL denotes the number of examined sample points which can be found between two consecutive nonconforming sample points, involving the end nonconforming sample point. They demonstrated that in comparison to the exponential weighted moving average (EWMA), synthetic and DS schemes, the SDS is more sensitive to process shifts. Motivated by Khoo et al. [1], Khoo et al. [2] developed a multivariate SDS T² chart using the Hotelling T² statistic. A SDS s chart is suggested by Lee and Khoo [4] to track increases in the process standard deviation. Costa and Machado [5] used Markov chain method to investigate the SS properties of standard SDS and side-sensitive SDS $\overline{\mathrm{X}}$ charts. The SDS chart outperforms the DS chart in terms of steady-state (SS) ARL. In addition, their investigation revealed that the side-sensitive rule enhances the overall performance of the SDS $\overline{\mathrm{X}}$ chart, but not enough to surpass the performance of the standard DS $\overline{\mathrm{X}}$ chart. Using information on auxiliary characteristics, Haq and Khoo [6] proposed an auxiliary information-based (AIB) SDS chart for the process mean by combining the CRL chart with the existing AIB double sampling (AIB DS) chart. You et al. [7] extended the work of the SDS $\overline{\mathrm{X}}$ chart with known process parameters proposed by Khoo et al. [1] to the estimated process parameters case. They investigated the performance of the SDS $\overline{\mathrm{X}}$ chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). You [8] looked into the performance of the SDS chart developed by You et al. [7] with known and estimated process parameters using the expected average run length (EARL). This study has shown that the performance criterion EARL can tackle the unknown shift size problem. By adopting the cost model from Lorenzen and Vance [9], Lee and Khoo [10,11] explored the economic and economic statistical designs for the SDS T² and SDS $\overline{\mathrm{X}}$ charts, correspondingly. In terms of economic performance, the numerical comparison shows that the SDS chart is superior to the competing charts. Aghaulor and Ezekwem [12] proposed an economic design approach to estimate the parameters of the SDS control chart and DS $\overline{\mathrm{X}}$ control chart.

For the attribute type control chart, Chong et al. [13] proposed a SDS np chart for detecting increases in the fraction of nonconforming items. They demonstrated that the zero-state (ZS) ARL performance of the SDS np chart outperforms the competing charts. Following Chong et al. [13], Tuh et al. [14] subsequently studied the SDS np chart based on MRL. Their work provides us with the opportunity to continue expanding the suitability of the SDS np chart in the field of attribute data in SPC. There have been earlier reports [15] suggesting not to implement the synthetic scheme since its monitoring effectiveness is only confined to specific change conditions, often in cases in which the occurrence of change is at the start of monitoring. Nevertheless, subsequent work by Malela-Majika [16] rebuffed the negativity surrounding this scheme, concluding that the use of the synthetic scheme with fit optimal parameters remains appropriate and highly sensitive to process shift, particularly under large lower limit of the CRL sub-chart.

It is demonstrated that the zero-state (ZS) and steady-state (SS) modes of the run length (RL) distribution of the synthetic chart are shown to be very different [17,18]. In the ZS mode, it is assumed that the change in the process parameter(s) occurred during the start of monitoring, so that there is a nonconforming sample at time zero. In this way, an out-of-control (OOC) signal will be generated if there is a second nonconforming sample within the next H samples (i.e., when the CRL ≤ H), where H is a specified positive integer. This, according to Davis and Woodall [17], gives the synthetic chart an advantage over other popular charts because of this head-start features. It is crucial to evaluate the SS performance of the synthetic chart [17,18], as its detection of delayed shifts may be slower since the effect of the head-start is lost.

The average run length (ARL)’s lone use as the performance measure for control charts was questioned by numerous researchers. For example, see Teoh et al. [19], Khoo et al. [20], Lee and Khoo [21]. In addition, as pointed out by Graham et al. [22], utilizing the ARL as a performance measure has a number of disadvantages and may overlook some vital statistical properties of control charts. The RL is a positive integer random variable, and it is noted that the RL distribution changes from being significantly right skewed when the process shift is small to being almost symmetric when the process shift is large. In descriptive statistics, the choice of summary measures to describe a distribution is an important aspect. It is well known that when a distribution is right skewed, the mean is larger than the median and hence, it is not a fair representation of a “centre”. In this regard, the median is a better measure of ‘‘central tendency’’ and thus, the median run length (MRL) can provide more meaningful information [23].

The MRL is the 50th percentile of the RL distribution. As commented by Chakraborti [23], a Shewhart $\overline{\mathrm{X}}$ chart with $\pm 3$ standard deviation width has an IC ARL (ARL₀) of 370 but 50% of all the run lengths are less than 257 (i.e., the IC MRL (MRL₀) is 257) and about 60–70% of all the run lengths are less than 370. The MRL comes in two types, namely in-control (IC) MRL (MRL₀) and OOC MRL (MRL₁). When MRL₀ is 215, most practitioners will suggest that a control chart certainly detects a false alarm at the 215th sample 50% of the time. This will strengthen practitioners’ confidence when dealing with a few short runs with no assignable causes. When comparing various control charts, to ensure a fair comparison, the MRL₀ value is kept at the same value to provide a fair comparison; the one with the smallest MRL₁ value outperforms the others. The reason for this is that the lower the MRL₁, the faster the shift is observed.

In order to compute the MRL, practitioners need to specify the process shift size in advance. However, quality practitioners are typically unaware of the process shift size. Due to relatively scarce practice-based historical data, quantifying the amount of process transformation is a hard challenge for practitioners. Aside from that, the shift size fluctuates based on unidentified random occurrences. Moreover, if a practitioner considers a certain shift size and utilises its optimal charting parameters, the control chart will perform poorly when a different shift size actually happens. In order to overcome these issues, it is necessary to adopt the expected median run length (EMRL) as an alternative performance measure, which requires simply a range of process shift sizes. Among others, Teoh et al. [24], Tang et al. [25], Qiao et al. [26], Chong et al. [27] and Tuh et al. [28] considered the EMRL performance criterion in their recent studies.

The primary objective of this work is, therefore, to extend the contributions of Tuh et al. [14,28] by employing EMRL as the performance measure in developing the SDS np chart. In this work, both ZS and SS performance evaluations are considered for the charts being studied. This work provides the optimal chart’s design procedures for the SDS np chart based on MRL₁ and EMRL₁ minimizations (for known and unknown shift sizes cases). For this purpose, the model developed by Tuh et al. [14], which focusses on the optimal charting parameter SDS np when MRL_0min of 200, is adopted herein for hitherto unexplored MRL_0min of 370.4, as a mean to make performance comparison with the competing charts. As envisioned, for zero-state mode, the SDS np chart possesses both the power of the synthetic chart for rapidly detecting moderate and minor shifts and the advantage of the DS np chart for sensitivity to significant shifts, resulting in a superior and more uniform overall performance. The subsequent sections of this article are organised as follows: A description of the plotting statistic and certain characteristics of the SDS np chart is presented in Section 2. Following a discussion of the RL distribution properties of the SDS np chart in Section 3, Section 4 describes the optimization designs of the proposed SDS np chart, in both ZS and SS mode, using MRL and EMRL performance criterions. In Section 5, the effectiveness of the synthetic np, DS np, and SDS np charts in both known and unknown shift size situations are compared. Section 6 provides an illustration of the implementation of the EMRL-based optimal SDS np chart by including a process that produces ballpoint pen cartridges. In the final section, conclusions are drawn.

2. The Review of SDS np Chart

The SDS np chart proposed by Chong et al. [13], is an integration of the DS np and CRL sub-charts. The CRL stands for the number of examined sampling stages (i) between two consecutive nonconforming sampling stages (taking into account the nonconforming sampling stages at the end). The elements from the two sub-charts are included in the design of the SDS np chart. They are (i) the lower limit H of the CRL sub-chart, and (ii) the DS np sub-chart element that encompasses the initial sample size n₁, second sample size n₂, the first stage warning limit W, first stage control limit L₁ and the second stage control limit L₂. It should be noted that H is a positive integer.

The trio of non-integer control limits W, L₁ and L₂ of the DS np sub-chart are established as W = Ac₁ + 0.5, L₁ = Re₁ − 0.5 and L₂ = Ac₂ + 0.5 (also described in [13,29], and [30]) to eliminate uncertainty among quality professionals, particularly when the observed quantity of nonconforming items exceeds the control limits.

The IC and OOC processes on the DS np chart are classified as the conforming and nonconforming sampling stages, respectively, on the DS np sub-chart component of the SDS np chart, as shown in Figure 1. The classification of a sampling stage as either conforming or nonconforming depends on the information about the quality characteristic given by the first sample alone or the combined information contained in the first and second samples.

The DS np sub-chart is constructed by first identifying a sampling stage as either conforming or nonconforming based on the process information provided by the first sample or the combined first and second samples. The CRL sub-chart is necessary for determining whether or not the sample is OOC. The following steps are a description of how the SDS np chart operates:

Step 1:

Set the optimal charting parameters W, L₁, L₂ (on DS sub-chart) and H (on CRL sub-chart).

Step 2:

Consider the first sample of size n₁ and count the number of nonconforming items d₁ in the sample.

Step 3:

 

(i)

If d_1,i ϵ I₁, the ith sampling stage is categorized as conforming. Process returns to Step 2.

(ii)

If d_1,i ϵ I₃, the i^th sampling stage is categorized as nonconforming and control flow advances to Step 5.

(iii)

If d_1,i ϵ I₂, the procedure moves to the second stage of the DS sub-chart. A second sample of size n₂ is taken and the number of nonconforming items d₂ in the sample is counted. After that, proceed to the next step.

Step 4:

Compute the number of nonconforming items in combined samples d_1,i + d_2,i. If (d_1,i + d_2,i) ϵ I₄, the i^th sampling stage is categorized as conforming and the control flow returns to Step 2. If this is not the case, the sampling stage is categorized as nonconforming, and the control flow moves to Step 5.

Step 5:

Count the number of sampling stages, including the current nonconforming sampling stage, that separate two successive nonconforming sampling stages and is known as CRL value.

Step 6:

If CRL > H, the process is deemed as IC and the control flow goes back to Step 2. Otherwise, the process is judged as OOC and corrective action is required to search and eliminate the assignable cause(s). Then, the process goes back to Step 2.

A nonconforming sample may be present if d_1,i ϵ I₃ or (d_1,i ϵ I₂ and (d_1,i + d_2,i) ϵ I₅). The sample’s OOC status can only be determined using the CRL sub-chart. The process is considered as IC if CRL > H. If not, the process is deemed OOC, and corrective action is needed to identify and remove the assignable cause(s). The operation of the SDS np chart is graphically summarized in Figure 2.

3. The Run Length Properties of the SDS np Chart

The Markov chain model provided by Davis and Woodall [17], studied synthetic $\overline{\mathrm{X}}$ chart, is applied to assess RL properties of the SDS np chart in this work. This approach may be easily expanded to represent the SDS np chart, allowing for the entire RL distribution of the chart to be obtained.

Let A denote the probability that a plotted sample statistic of a sampling stage i falling within the chart limits of the DS np sub-chart can be expressed as [13]

A = A₁ + A₂

(1)

where

$${\mathrm{A}}_1=\mathrm{P}\left(d_{1,i}\le \lfloor W\rfloor \right)={\displaystyle \sum _{d_{1,i}=0}^{\lfloor W\rfloor }\frac{n_1!}{d_{1,i}!\left(n_1-d_{1,i}\right)!}}{p}^{d_{1,i}}{\left(1-p\right)}^{n_1-d_{1,i}}$$

(2)

$${\mathrm{A}}_2=\mathrm{P}\left(\lfloor W\rfloor <d_{1,i}<\lceil L_1\rceil \cap d_{1,i}+d_{2,i}\le \lfloor L_2\rfloor \right)$$

$$={\displaystyle \sum _{d_{1,i}=\lfloor W\rfloor +1}^{\lceil L_1\rceil -1}\left[\frac{n_1!}{d_{1,i}!\left(n_1-d_{1,i}\right)!}{p}^{d_{1,i}}{\left(1-p\right)}^{n_1-d_{1,i}}\left({\displaystyle \sum _{d_{2,i}=0}^{\lfloor L_2\rfloor -d_{1,i}}\frac{n_2!}{d_{2,i}!\left(n_2-d_{2,i}\right)!}{p}^{d_{2,i}}{\left(1-p\right)}^{n_2-d_{2,i}}}\right)\right]}$$

(3)

The notations ⌊·⌋ and ⌈·⌉ denote the round down and round up to the nearest integer, respectively. Note that A₁ is the probability that d_1,i < W at the first stage of the DS sub-chart, while A₂ is the probability that W < d_1,i < L₁ at the first stage of the DS sub-chart and (d_1,i + d_2,i) < L₂ at the second stage of the DS sub-chart. Hence, B = 1 − A.

The partitioned form of the transition probability matrix (TPM) P_(H+2, H+2) is [20]

$$\mathbf{P}=\left(\begin{array}{cc}\mathbf{R}& \mathbf{r}\\ 0^T& 1\end{array}\right)=\left(\begin{array}{ccccccc}\mathrm{A}& \mathrm{B}& 0& 0& \cdots & 0& 0\\ 0& 0& \mathrm{A}& 0& \cdots & 0& \mathrm{B}\\ 0& 0& 0& \mathrm{A}& \cdots & 0& \mathrm{B}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& \cdots & \mathrm{A}& \mathrm{B}\\ \mathrm{A}& 0& 0& 0& \cdots & 0& \mathrm{B}\\ 0& 0& 0& 0& \cdots & 0& 1\end{array}\right)$$

(4)

with (H + 1) transient states and one absorbing state. The matrix of transient probabilities is denoted by the notation R_(H+1, H+1), and it is obtained by cutting off the very last row and the very last column of the matrix P. The vector r meets the condition that $\mathrm{r}=1-\mathrm{R}\cdot 1$ (i.e., the row probabilities of P must add up to 1), where 1_{(H+1, 1)} = (1, 1, …, 1)^T and 0_{(H+1, 1)} = (0, 0, …, 0)^T.

Let RL represent the run length of the SDS np chart, which quantifies how many steps the process to reach the absorbing state. Since the RL is known to be a discrete phase-type random variable for the synthetic chart, its value can be expressed as ℓ = {1, 2, 3, …}. Then, under the ZS mode, the cumulative distribution function (cdf) of the RL random variable is [31]

$$\mathrm{P}(\mathrm{RL}\le \ell )={\mathbf{s}}^T(\mathbf{I}-{\mathbf{R}}^{\ell })\mathbf{1}$$

(5)

where the column vector s denotes stationary probability with (H + 1) elements and a single element of one and zero in another location, and I_(H+1, H+1) denotes the identity matrix. It should be noted that s = (0, 1, 0, …,0)^T [17].

In the case when the process is under SS mode, the cdf of the RL is [20]

$$\mathrm{P}(\mathrm{RL}\le \ell )={\mathbf{s}}_c{}^T(\mathbf{I}-{\mathbf{R}}^{\ell })\mathbf{1}$$

(6)

and it can be calculated by substituting s (in Equation (5)) with s_c. Note that s_c is the column vector of cyclical SS probability ${\mathbf{s}}_c{}^T{=(\mathbf{1}}^T{\mathbf{q})}^{-1}\mathbf{q}$ [32], and that q may be simply solved using $\mathbf{q}={(\mathbf{G}-{\mathbf{R}}^{\mathrm{T}})}^{-1}\mathbf{u}$. In this case, the (H + 1) × (H + 1) matrix G and the (H + 1) column vector u are

$$\mathbf{G}=\left(\begin{array}{cccccc}2& 1& 1& 1& \cdots & 1\\ 0& 1& 0& 0& \cdots & 0\\ 0& 0& 1& 0& \cdots & 0\\ 0& 0& 0& 1& \ddots & 0\\ ⋮& ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& 0& \cdots & 1\end{array}\right)$$

(7)

and

$$\mathbf{u}={\left(\begin{array}{cccc}\begin{array}{cc}1& 0\end{array}& 0& \begin{array}{cc}0& \cdots \end{array}& 0\end{array}\right)}_{}^{\mathrm{T}}$$

(8)

Let ${\ell }_m$ be the 100mth percentile of the RL distribution, where $0<m<1$. Then, the ${\ell }_m$ of the SDS np chart can be calculated in such a way that [33]

$$P(RL\le {\ell }_m-1)\le \mathrm{m} \mathrm{and} P(RL\le {\ell }_{\mathrm{m}})>\mathrm{m}$$

(9)

The $\mathrm{MRL}={\ell }_{0.5}$ for the SDS np, when $m=0.5$, is therefore defined as

$$P(RL\le {\ell }_{0.5}-1)\le 0.5 \mathrm{and} P(RL\le {\ell }_{0.5})>0.5$$

(10)

Notably, MRL = MRL₀ represents the IC MRL while MRL = MRL₁ denotes the OOC MRL, which may be achieved when $\gamma =1$ and $\gamma >1$, respectively.

The speed at which the SDS np chart can identify an increasing shift in the process fraction nonconforming p with a magnitude of shift $\gamma$ equal to p₁/p₀, where p₁ > p₀, is a measure of the chart’s effectiveness. Note that p = p₀ when $\gamma$ equal to 1, and p = p₁ when $\gamma$ greater than 1. Here, p₀ represents the IC fraction nonconforming while p₁ represents the OOC fraction nonconforming.

In addition to the MRL, the ZS and SS ARL of the synthetic np chart can be computed through [20]

$${\mathrm{ARL}}_{\mathrm{ZS}}={\mathbf{s}}^T{\left(\mathbf{I}-\mathbf{R}\right)}^{-1}\mathbf{l}$$

(11)

and

$${\mathrm{ARL}}_{\mathrm{SS}}={\mathbf{s}}_{\mathrm{c}}{}^T{\left(\mathbf{I}-\mathbf{R}\right)}^{-1}\mathbf{l}$$

(12)

individually, by means of the Markov chain method. The I, R, l, s and s_c have all been defined previously. The average sample size (ASS) in each sampling stage of the DS np sub-chart is expressible via [29]

$$\mathrm{ASS}=n_1+n_2\left[\mathrm{P}(\lfloor W\rfloor <d_{1,i}<\lceil L_1\rceil |p\right]$$

(13)

where $\mathrm{P}(\lfloor W\rfloor <d_{1,i}<\lceil L_1\rceil |p)$ denotes the probability of taking the second sample of size n₂. The IC and OOC ARLs (for both ZS and SS) and ASSs are determined using Equations (11)–(13) when p = p₀ and p = p₁, correspondingly. The notation p₀ denotes the IC fraction of nonconforming items while the notation p₁ represents the OOC fraction of nonconforming items.

The computation of the expected median RL (EMRL), also denoted as $\mathrm{E}({\ell }_{0.5})$ when letting $\alpha =0.5$, where a specific value for shift size $\gamma$ is not required, rather a range of shift sizes is taken into consideration (between minimum and maximum shifts) and can be determined using [34]

$$\mathrm{EMRL}=\mathrm{E}({\ell }_{0.5})={\displaystyle {\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}{f}_{\gamma }}\left(\gamma \right)\mathrm{MRL}\left(\gamma \right)\mathrm{d}\gamma$$

(14)

Note that the EMRL is calculated using a numerical integration over the probability density function (pdf) $f_{\gamma }\left(\gamma \right)$ over a shift size interval ranging from ${\gamma }_{\min }$ (the integral’s lower limit) to ${\gamma }_{\max }$ (the integral’s upper limit). By assuming a continuous uniform distribution over the interval (${\gamma }_{\min }$, ${\gamma }_{\max }$) [35], the pdf $f_{\gamma }\left(\gamma \right)$ is defined as $f_{\gamma }\left(\gamma \right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$({\gamma }_{\max }-{\gamma }_{\min })$}\right.$ where ${\gamma }_{\max }-{\gamma }_{\min }$ represents the interval length. The values for the expected average run length (EARL) and the expected average sample size (EASS) can therefore also be calculated using [28]

$$\mathrm{EARL}={\displaystyle {\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}{f}_{\gamma }}\left(\gamma \right)\mathrm{ARL}\left(\gamma \right)\mathrm{d}\gamma$$

(15)

and

$$\mathrm{EASS}={\displaystyle {\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}{f}_{\gamma }}\left(\gamma \right)\mathrm{ASS}\left(\gamma \right)\mathrm{d}\gamma$$

(16)

correspondingly.

4. Optimization Designs Procedure

This section presents the computation procedures for determining the optimal charting parameters for the SDS np chart, under both ZS and SS modes, to minimize the (i) MRL₁ (described in Section 4.1) and (ii) EMRL₁ (described in Section 4.2).

4.1. Computation of the Optimal Charting Parameter for the SDS np Chart to Minimize MRL₁

In this section, the optimal charting parameters (n₁, n₂, W, L₁, L₂, H) of the SDS np chart, under both ZS and SS modes, is obtained by minimizing MRL₁(${\gamma }_{\mathrm{opt}}$) subject to the achievement of minimum IC MRL (MRL_0min) and predetermined IC ASS (ASS₀), for a rapid detection ${\gamma }_{\mathrm{opt}}$ is required. Note that ${\gamma }_{\mathrm{opt}}$ represents the desired size of a fraction nonconforming shift. The optimization model in Tuh et al. [14] is adopted in this study to compute the optimal charting parameter $(n_1, n_2, W, L_1, L_2, H)$ combination when IC minimum (MRL_0min) is 370.4. The following is a mathematical formulation of the optimization model:

$$\begin{array}{cc}\underset{n_1, n_2, W, L_1, L_2, H}{\min }& {\mathrm{MRL}}_1({\gamma }_{opt})\end{array}$$

(17)

subject to

$${\mathrm{MRL}}_0\ge {\mathrm{MRL}}_{0\min }$$

(18)

and

ASS₀ = n.

(19)

In view of the distinct nature of RL random variable, it is crucial to emphasize herein that MRL takes the form of an integer. Consequently, the occurrence of multiple $(n_1, n_2, W, L_1, L_2, H)$ combinations with similar minimum MRL₁ at a particular ${\gamma }_{\mathrm{opt}}$ may be observed. In such a scenario, the optimal $(n_1, n_2, W, L_1, L_2, H)$ combination having the lowest ASS₁ is preferred.

The procedure for searching the optimal (n₁, n₂, W, L_1, L₂, H) combination, based on the optimization model in (17)–(19) of the SDS np chart based on MRL, is outlined as follows:

Step 1:

Specify the p₀, n, MRL_0min and ${\gamma }_{opt}$ values. Here, n is the average sample size at every sampling stage when the process is in an IC state.

Step 2:

Set H to one at the start.

Step 3:

Set the initial value of MRL_1min to 10⁵ (a relatively large value).

Step 4:

Start with n₁ equal to one.

Step 5:

With the current n₁ value, the combination of (n₁, n₂, W, L₁, H) is determined for a specified n when $\gamma =1$, such that the Constraint (19) is fulfilled. The value of n₂ is obtained by rearranging Equation (13), i.e., $n_2=\raisebox{1ex}{$(n-n_1)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{P}(\lfloor W\rfloor <d_{1,i}<\lceil L_1\rceil )$}\right.$, and is rounded up to the nearest integer, where 0 < W < L₁.

Step 6:

For the ZS mode, L₂ is determined via Constraint (18) together with Equations (5) and (10). For the SS mode, on the other hand, L₂ can be obtained by solving Constraint (18) together with Equations (6) and (10). The computed MRL equals to MRL₀ when $\gamma =1$(i.e., p = p₀), where L₂ > L₁. In this step, the possible (n₁, n₂, W, L₁, L₂, H) combination is identified.

Step 7:

Once the possible (n₁, n₂, W, L₁, L₂, H) combination is determined, MRL₁ will be computed for p = p₁, by means of Equations (5) and (10) (for ZS mode) or Equations (6) and (10) (for SS mode). If the computed MRL₁ is less than the present MRL_1min, substitute the newly computed MRL₁ for the MRL_1min value. The (n₁, n₂, W, L₁, L₂, H) combination is saved temporarily as the possible combination. If the (n₁, n₂, W, L₁, L₂, H) combination found in the following searching produces identical MRL_1min, it will be kept together as a possible combination. Otherwise, if the (n₁, n₂, W, L₁, L₂, H) combination results in a larger MRL₁ value, it will be ignored.

Step 8:

Once the search with n₁ = 1 is complete, n₁ is increased by one. Repeat Steps 5–7 for each remaining n₁ = 2, 3…, ($n-1$), in order to find the possible (n₁, n₂, W, L_1, L₂, H) combinations that fulfil the Constraints (18) and (19) and having the lowest value of MRL₁.

Step 9:

If MRL_1min value has been reduced, increase H by 1 and repeat Steps 3–8. Else, proceed to Step 10.

Step 10:

If more than one combination of (n₁, n₂, W, L₁, L₂, H) delivers a similar minimum MRL_1min value, the combination that produces the smallest OOC average sample size (ASS₁) value is chosen as the optimal combination. Additionally, if more than one combination of (n₁, n₂, W, L₁, L₂, H) produces similar comparable lowest pair values (MRL₁, ASS₁), the parameters combination corresponding to the lowest H is taken to be the optimal combination in this case.

Two MATLAB programs are written to compute the optimal (n₁, n₂, W, L₁, L₂, H) combination for the SDS np chart, each for the ZS and SS mode apiece, based on the set of input parameters (MRL_0min, ${\gamma }_{\mathrm{opt}}$, n, p₀).

4.2. Computation of the Optimal Charting Parameter for the SDS np Chart to Minimize EMRL₁

The MRL-based design (discussed in Section 4.1) can only be implemented if shift size $\gamma$ can be specified beforehand. By using the method in this section, the optimal charting parameters that improve the sensitivity of the SDS np chart can be obtained even though the exact shift size is not known.

The formulation for the optimization model for the SDS np chart with unknown shift sizes is as follows:

$$\begin{array}{cc}\underset{n_1, n_2, W, L_1, L_2, H}{\min }& {\mathrm{EMRL}}_1\end{array}$$

(20)

subject to

$${\mathrm{EMRL}}_0\ge {\mathrm{MRL}}_{0\min }$$

(21)

and

$${\mathrm{EASS}}_0=\mathrm{n}$$

(22)

Below is a detailed explanation of the procedure to compute the optimal charting parameters.

Step 1:

Specify the desired values of ${\gamma }_{\min }$, ${\gamma }_{\max }$, n, p₀, and ${\mathrm{MRL}}_{0\min }$.

Step 2:

Similar to Steps 2 to 5 of the optimization procedure outlined in Section 5.1, but ↓ with

Step 3:

Constraints (21) and (22) in place of Constraints (18) and (19) by minimizing the OOC EMRL (EMRL₁) in Equation (20).

Step 4:

For the ZS mode, L₂ is based on the Equations (5) and (10) and Constraint (21), in which the computed EMRL equals to EMRL₀ (i.e., when p = p₀), where $L_2>L_1$. Note that for the case of SS mode, on the other hand, L₂ can be obtained by solving Constraint (21) together with Equations (6) and (10). The non-integer setting of W, L₁ and L₂ are based on operating procedure of DS sub-chart as described in Section 2. In this step, the possible parameters (n₁, n₂, W, L₁, L_2, H) combination is identified.

Step 5:

Once the possible (n₁, n₂, W, L₁, L₂, H) combination has been identified, EMRL₁ is calculated (i.e., when p = p₁), by means of Equations (5), (10) and (14) (for ZS mode) or Equations (6), (10) and (14) (for SS mode). If the calculated EMRL₁ is lower than the present EMRL_1min, substitute the newly computed EMRL₁ for the EMRL_1min value. The current (n₁, n₂, W, L₁, L₂, H) combination is temporarily saved as the possible combination. If the combination (n₁, n₂, W, L₁, L₂, H) obtained in the subsequent searching produces an EMRL_1min that is comparable to the one being searched for, the combination will be saved as a possible one. If this is not the case, the combination (n₁, n₂, W, L₁, L₂, H) will be disregarded if the EMRL₁ value that it produces is higher.

Step 6:

Increase n₁ by one when the search is finished with n₁ = 1. For the remaining n₁ = 2, 3…, ($n-1$), repeat Steps 5 through 7, to look for the possible (n₁, n₂, W, L₁, L₂, H) combinations that fulfil Constraints (21)–(22) and have the lowest value of EMRL₁.

Step 7:

If the EMRL_1min value has been lowered, raise H by 1 and repeat Steps 3–8. Otherwise, move on to Step 10.

Step 8:

If multiple combinations of (n₁, n₂, W, L₁, L₂, H) give a similar minimum EMRL_1min value, the optimal combination is the one that produces the lowest OOC expected average sample size (EASS₁) value. Additionally, if more than one combination of (n₁, n₂, W, L₁, L₂, H) produces a similar comparable lowest pair (EMRL₁, EASS₁) value, the parameter combination corresponding to the lowest H is taken to be the optimal combination in this case.

Two MATLAB programs are written to compute the optimal (n₁, n₂, W, L₁, L₂, H) combination for the EMRL-based SDS np chart, for the ZS and SS modes, individually, based on desired values of ${\gamma }_{\min }$, ${\gamma }_{\max }$, n, p₀, and ${\mathrm{MRL}}_{0\min }$. The numerical integration method based on the Gauss Legendre Quadrature rule is employed in this section to compute the integral involve computations, for designing EMRL-based SDS np chart. The values of weights (w_i) and nodes (x_i) can be obtained using the MATLAB coding written by Winckel [36], which is borrowed to build the MATLAB programs for the computations of EMRL₁ (or $\mathrm{E}{({\ell }_{0.5})}_1$), EARL_1, and EASS₁. Here, a value of N equal to 200 is used for the number of ordinates, following Tuh et al. [28].

5. Comparative Studies

This section is divided into two subsections. The comparison studies of the SDS np chart with the optimal synthetic np and optimal DS np charts in both ZS and SS modes, based on MRL and EMRL (for known and unknown shift sizes cases), are covered in Section 5.1 and Section 5.2, respectively.

5.1. Performance Analysis of the SDS np Chart Based on MRL

In this section, the performance of the SDS np chart is evaluated using MRL as the chart’s performance criterion. The IC MRL (MRL₀) and the OOC MRL (MRL₁) are the two types of MRLs that are frequently used. The MRL₁ value should be low in a good control chart to allow for quick detection of an OOC condition, while the MRL₀ value should be large to prevent overly frequent false alarms. The different input parameter combinations from MRL_0min = {370.4}, IC p₀ = {0.005, 0.01, 0.02}, n = {25, 50, 100, 200, 400, 800} and shift size ${\gamma }_{opt}$ = {1.5, 2.0, 3.0} are considered. The input combination parameters applied in this work have been adopted by De Araujo Rodrigues et al. [29], Chong et al. [13] and Lee and Khoo [37] for use in their own research on DS np, SDS np and synthetic np charts, respectively. The code developed herein has also been tested with the optimal charting parameters for SDS np chart based on ARL developed by Chong et al. [13], with exact similar output (ARL and ASS values) obtained, henceforth confirming the validity and reliability of the methodology.

The optimal charting parameters (n₁, n₂, W, L₁, L₂, H) of the SDS np chart, in ZS and SS, as well as the corresponding MRL₀ and ARL₀ are presented in

Table 1. Optimal charting parameters for the MRL-based SDS np chart, in ZS and SS modes, together with their corresponding values of MRL₀ and ARL₀, when RL_0min = 370.4.

${\mathit{\gamma }}_{\mathbf{opt}}$	${\mathit{p}}_0$	n	ZS			SS
			(n1, n2, W, L1, L2, H)	MRL0	ARL0	(n1, n2, W, L1, L2, H)	MRL0	ARL0
1.5	0.005	100	(25, 636, 0.5, 3.5, 6.5, 11)	375	580.45	(18, 951, 0.5, 2.5, 8.5, 26)	378	544.97
		200	(40, 880, 0.5, 3.5, 8.5, 7)	400	608.15	(91, 1427, 1.5, 4.5, 12.5, 29)	383	552.31
		400	(46, 1719, 0.5, 3.5, 13.5, 4)	374	562.28	(152, 1404, 1.5, 5.5, 13.5, 16)	374	539.96
		800	(457, 1744, 3.5, 8.5, 16.5, 2)	376	557.36	(198, 2310, 1.5, 7.5, 19.5, 9)	371	535.20
	0.01	50	(10, 418, 0.5, 2.5, 7.5, 11)	375	580.09	(8, 543, 0.5, 3.5, 9.5, 41)	397	572.40
		100	(20, 439, 0.5, 3.5, 8.5, 7)	413	627.46	(47, 659, 1.5, 4.5, 11.5, 17)	382	551.47
		200	(126, 564, 2.5, 5.5, 11.5, 4)	378	566.98	(76, 703, 1.5, 5.5, 13.5, 16)	377	543.74
		400	(166, 1016, 2.5, 6.5, 17.5, 2)	391	579.12	(99, 1155, 1.5, 6.5, 19.5, 9)	376	542.89
	0.02	25	(5, 208, 0.5, 2.5, 7.5, 11)	396	610.51	(4, 270, 0.5, 2.5, 9.5, 43)	406	584.77
		50	(10, 219, 0.5, 2.5, 8.5, 7)	400	607.77	(24, 315, 1.5, 4.5, 11.5, 25)	392	565.75
		100	(44, 254, 1.5, 6.5, 10.5, 4)	382	573.43	(38, 354, 1.5, 4.5, 13.5, 14)	375	540.14
		200	(115, 431, 3.5, 7.5, 16.5, 2)	399	590.67	(51, 548, 1.5, 6.5, 18.5, 6)	386	556.41
2.0	0.005	100	(39, 345, 0.5, 2.5, 4.5, 4)	414	620.41	(33, 441, 0.5, 2.5, 5.5, 13)	398	574.43
		200	(185, 287, 2.5, 3.5, 5.5, 3)	412	614.77	(122, 626, 1.5, 4.5, 7.5, 9)	379	547.17
		400	(385, 469, 4.5, 5.5, 8.5, 4)	411	616.25	(295, 571, 2.5, 6.5, 8.5, 4)	384	553.92
		800	(362, 869, 1.5, 4.5, 12.5, 1)	373	548.07	(703, 829, 5.5, 7.5, 14.5, 2)	380	548.66
	0.01	50	(19, 179, 0.5, 2.5, 4.5, 4)	371	557.17	(16, 229, 0.5, 2.5, 5.5, 11)	401	578.69
		100	(92, 155, 2.5, 3.5, 5.5, 3)	386	575.39	(61, 314, 1.5, 4.5, 7.5, 9)	380	548.12
		200	(192, 253, 4.5, 5.5, 8.5, 4)	383	575.35	(143, 335, 2.5, 5.5, 9.5, 7)	379	546.40
		400	(182, 429, 1.5, 4.5, 12.5, 1)	376	552.38	(351, 421, 5.5, 7.5, 14.5, 2)	387	557.61
	0.02	25	(6, 175, 0.5, 1.5, 6.5, 4)	393	589.85	(8, 114, 0.5, 2.5, 5.5, 2)	395	569.72
		50	(46, 78, 2.5, 3.5, 5.5, 3)	403	600.33	(29, 188, 1.5, 3.5, 8.5, 11)	411	593.04
		100	(96, 128, 4.5, 5.5, 8.5, 4)	398	597.17	(75, 134, 2.5, 5.5, 8.5, 5)	373	537.64
		200	(92, 210, 1.5, 4.5, 12.5, 1)	378	555.17	(176, 205, 5.5, 7.5, 14.5, 2)	410	591.77
3.0	0.005	100	(91, 137, 1.5, 2.5, 3.5, 4)	397	596.05	(90, 155, 1.5, 2.5, 3.5, 4)	388	559.79
		200	(152, 354, 1.5, 2.5, 7.5, 1)	373	548.49	(153, 275, 1.5, 3.5, 5.5, 5)	393	566.75
		400	(153, 500, 0.5, 2.5, 10.5, 1)	379	557.69	(150, 511, 0.5, 2.5, 8.5, 1)	376	542.55
		800	(254, 802, 0.5, 3.5, 12.5, 1)	395	581.14	(254, 802, 0.5, 3.5, 12.5, 1)	419	604.29
	0.01	50	(32, 66, 0.5, 2.5, 3.5, 7)	389	592.05	(44, 96, 1.5, 2.5, 4.5, 10)	410	591.51
		100	(76, 177, 1.5, 2.5, 7.5, 1)	381	560.19	(76, 141, 1.5, 3.5, 5.5, 5)	385	555.44
		200	(77, 247, 0.5, 2.5, 10.5, 1)	375	551.78	(131, 206, 1.5, 3.5, 9.5, 1)	373	537.46
		400	(125, 405, 0.5, 3.5, 11.5, 1)	377	553.85	(126, 403, 0.5, 3.5, 11.5, 1)	385	555.39
	0.02	25	(22, 48, 1.5, 2.5, 3.5, 3)	377	562.22	(10, 82, 0.5, 3.5, 4.5, 7)	372	536.59
		50	(38, 88, 1.5, 2.5, 7.5, 1)	399	585.89	(38, 70, 1.5, 3.5, 5.5, 5)	416	600.56
		100	(38, 125, 0.5, 2.5, 9.5, 1)	405	594.47	(38, 125, 0.5, 2.5, 8.5, 1)	385	555.19
		200	(94, 202, 1.5, 4.5, 13.5, 1)	384	563.94	(94, 202, 1.5, 4.5, 13.5, 1)	407	586.72

. These optimal charting parameters (n₁, n₂, W, L₁, L₂, H) aid quality practitioners to choose the optimal parameters of the MRL-based SDS np chart in the implementation of the SDS np chart. For example, when the process is in SS mode, ${\gamma }_{opt}=2.0$, p₀ = 0.01 and n = 50, optimal parameters (n₁, n₂, W, L₁, L_2, H) = (16, 229, 0.5, 2.5, 5.5, 11). In addition, Table 1 gives a comparison between the ARL₀ and MRL₀ values of the SDS np chart. It is noted that all the MRL₀ values are smaller than the corresponding ARL₀ values.

Numerical comparison in

Table 2. MRL₁ comparison between SDS np, DS np and synthetic np charts with associated ARL₁ of the SDS np chart, in ZS and SS modes, when MRL_0min = 370.4.

${\mathit{\gamma }}_{\mathbf{opt}}$	${\mathit{p}}_0$	n	ZS				SS
			MRL1			ARL1	MRL1			ARL1
			Synthetic np	DS np	SDS np		Synthetic np	DS np	SDS np
1.5	0.005	100	47	26	11	32.13	72	26	25	36.18
		200	32	14	7	21.00	47	14	15	21.19
		400	13	8	4	12.50	25	8	9	12.97
		800	7	4	2	6.79	13	4	5	7.35
	0.01	50	48	26	11	31.06	74	26	25	35.55
		100	32	15	7	21.18	48	15	15	21.93
		200	14	8	4	11.92	25	8	9	12.91
		400	7	4	2	6.79	13	4	5	7.34
	0.02	25	46	25	11	31.38	73	25	25	35.49
		50	34	15	7	20.96	50	15	15	21.46
		100	14	8	4	12.01	26	8	9	12.91
		200	7	4	2	6.82	13	4	5	7.56
2.0	0.005	100	9	9	4	12.20	24	9	9	13.11
		200	5	5	3	9.61	13	5	5	7.37
		400	3	3	2	4.50	6	3	3	4.66
		800	2	2	1	3.73	3	2	2	3.27
	0.01	50	9	9	4	11.53	24	9	9	13.13
		100	5	5	3	9.14	13	5	5	7.32
		200	3	3	2	4.36	6	3	3	4.39
		400	2	2	1	3.79	3	2	2	3.23
	0.02	25	9	9	4	12.53	24	9	9	12.77
		50	5	5	3	9.13	13	5	5	7.25
		100	3	3	2	4.34	6	3	3	4.58
		200	2	2	1	3.88	3	2	2	3.27
3.0	0.005	100	4	3	2	4.31	7	3	4	6.10
		200	2	2	1	3.75	4	2	2	3.01
		400	1	1	1	3.37	2	1	2	3.01
		800	1	1	1	1.43	2	1	2	2.08
	0.01	50	4	3	2	3.71	7	3	4	5.76
		100	2	2	1	3.73	4	2	2	2.97
		200	1	1	1	3.39	2	1	2	3.02
		400	1	1	1	1.28	2	1	2	1.88
	0.02	25	4	3	2	4.26	7	3	3	4.48
		50	2	2	1	3.71	4	2	2	2.95
		100	1	1	1	2.70	2	1	2	3.02
		200	1	1	1	1.27	2	1	2	1.87

shows that there is a significant difference between the MRL₁ and ARL₁ values of the SDS np chart, for both ZS and SS modes. For example, MRL₁ = 11 when ${\gamma }_{\mathrm{opt}}=1.5$, p₀ = 0.005 and n = 100, MRL_0min = 370.4 and in the ZS mode implies that the SDS np chart gives an OOC signal not later than the 11th sample about half of the time, although ARL₁ = 32.13. Interestingly, the SDS np chart’s SS MRL₁ is seen to be larger than the ZS MRL₁. This suggests that because its head-start feature is no longer present, its SS performance is worse than the ZS performance.

The MRL₁ performances of the SDS np chart are compared with the synthetic np and DS np charts in this section. Note that the MRL₁ for the synthetic np and DS np charts in Table 2 are taken from Lee and Khoo [37] and Tuh et al. [30], correspondingly. It is clearly seen in Table 2, the SDS np chart outperforms the synthetic np and DS np charts for all shift sizes, in ZS mode, with the SDS np chart giving lower MRL₁ than the competing charts for identical (γ_opt, p₀, n, MRL_0min) combination. However, when the process is under SS mode, the optimal SDS np chart surpasses the synthetic np chart but does not demonstrate considerable improvement in comparison to the DS np chart in signalling the process fraction nonconforming shift. Overall, the SDS np chart outperforms the DS np chart when ${\gamma }_{\mathrm{opt}}=1.5$ and np₀ = 0.5 for different combinations of (n, p₀) = (100, 0.005) and (50, 0.01). This result is consistent with the conclusion reached by Costa and Machado [5]. They used SS ARL to measure the performance of the SDS $\overline{\mathrm{X}}$ chart.

5.2. Performance Analysis of the SDS np Chart Based on EMRL

The SDS np chart’s optimal charting parameters (n₁, n₂, W, L₁, L₂, H), can be achieved by minimizing the OOC expected median run length (EMRL₁), are shown in

Table 3. Optimal charting parameters for the SDS np, synthetic np and DS np charts, based on EMRL, in ZS mode, together with their corresponding MRL₀ and ARL₀ values, when MRL_0min = 370.4.

γmin	γmax	p0	n	Synthetic np			DS np			SDS np
				(UCLS, HS)	EMRL0	EARL0	(n1, n2, W, L1, L2)	EMRL0	EARL0	(n1, n2, W, L1, L2, H)	EMRL0	EARL0
1.1	2.0	0.005	100	(2.5, 9)	385	590.91	(38, 3985, 1.5, 3.5, 27.5)	393	566.84	(13, 1379, 0.5, 2.5, 11.5, 53)	372	633.82
			200	(3.5, 5)	395	594.89	(59, 3979, 1.5, 4.5, 29.5)	375	540.25	(73, 2441, 1.5, 5.5, 18.5, 47)	374	630.58
			400	(5.5, 7)	371	565.02	(144, 7069, 2.5, 7.5, 48.5)	372	536.36	(178, 3653, 2.5, 7.5, 26.5, 32)	372	608.27
			800	(8.5, 4)	387	581.47	(374, 10324, 4.5, 10.5, 69.5)	372	536.07	(324, 5869, 3.5, 9.5, 40.5, 23)	376	601.11
		0.01	50	(2.5, 9)	401	614.90	(23, 1230, 1.5, 3.5, 19.5)	393	566.43	(4, 1167, 0.5, 3.5, 16.5, 67)	371	646.62
			100	(3.5, 5)	408	614.58	(27, 2454, 1.5, 4.5, 34.5)	385	554.77	(34, 1453, 1.5, 4.5, 20.5, 37)	371	613.95
			200	(5.5, 7)	384	583.76	(66, 4670, 2.5, 6.5, 60.5)	382	551.07	(49, 1747, 1.5, 5.5, 25.5, 34)	372	610.92
			400	(8.5, 4)	399	597.97	(189, 4974, 4.5, 10.5, 67.5)	375	541.40	(158, 3221, 3.5, 10.5, 43.5, 25)	372	599.10
		0.02	25	(2.5, 10)	393	604.97	(11, 719, 1.5, 3.5, 21.5)	371	535.00	(2, 580, 0.5, 2.5, 16.5, 72)	375	657.63
			50	(3.5, 5)	437	657.15	(13, 1373, 1.5, 4.5, 37.5)	383	551.81	(19, 567, 1.5, 5.5, 17.5, 45)	371	623.17
			100	(5.5, 7)	411	624.13	(37, 1679, 2.5, 8.5, 46.5)	371	534.37	(24, 921, 1.5, 4.5, 26.5, 35)	371	610.23
			200	(8.5, 4)	423	633.12	(93, 2728, 4.5, 9.5, 72.5)	373	537.32	(84, 1314, 3.5, 9.5, 37.5, 26)	376	605.55
2.0	3.0	0.005	100	(2.5, 9)	385	590.91	(58, 1223, 1.5, 4.5, 12.5)	389	560.71	(38, 357, 0.5, 3.5, 4.5, 4)	386	579.25
			200	(3.5, 5)	395	594.89	(98, 1175, 1.5, 5.5, 13.5)	378	544.67	(139, 398, 1.5, 4.5, 5.5, 2)	397	588.41
			400	(5.5, 6)	433	653.89	(143, 1599, 1.5, 5.5, 17.5)	381	549.64	(184, 920, 1.5, 6.5, 9.5, 2)	373	552.82
			800	(8.5, 4)	387	581.47	(497, 2850, 4.5, 11.5, 28.5)	372	537.15	(362, 869, 1.5, 4.5, 12.5, 1)	373	548.07
		0.01	50	(2.5, 9)	401	614.90	(32, 442, 1.5, 4.5, 10.5)	416	599.25	(19, 179, 0.5, 2.5, 4.5, 4)	371	557.17
			100	(3.5, 5)	408	614.58	(49, 590, 1.5, 5.5, 13.5)	376	542.01	(34, 228, 0.5, 3.5, 5.5, 2)	388	575.09
			200	(5.5, 7)	384	583.76	(116, 756, 2.5, 7.5, 17.5)	399	575.89	(102, 360, 1.5, 6.5, 8.5, 2)	378	561.09
			400	(8.5, 4)	399	597.97	(252, 1340, 4.5, 10.5, 27.5)	372	536.00	(182, 429, 1.5, 4.5, 12.5, 1)	376	552.38
		0.02	25	(2.5, 9)	437	667.82	(16, 225, 1.5, 4.5, 10.5)	399	575.27	(8, 114, 0.5, 2.5, 5.5, 10)	385	592.78
			50	(3.5, 5)	437	657.15	(26, 253, 1.5, 4.5, 12.5)	387	558.17	(17, 112, 0.5, 3.5, 5.5, 2)	419	620.78
			100	(5.5, 7)	411	624.13	(58, 381, 2.5, 7.5, 17.5)	395	569.92	(51, 180, 1.5, 5.5, 8.5, 2)	387	574.02
			200	(8.5, 4)	423	633.12	(126, 675, 4.5, 12.5, 27.5)	371	535.05	(92, 210, 1.5, 4.5, 12.5, 1)	378	555.17

and

Table 4. Optimal charting parameters for the SDS np, synthetic np and DS np charts, based on EMRL, in SS mode, together with their corresponding MRL₀ and ARL₀ values, when MRL_0min = 370.4.

γmin	γmax	p0	n	synthetic np			DS np			SDS np
				(UCL, H)	EMRL0	EARL0	(n1, n2, W, L1, L2)	EMRL0	EARL0	(n1, n2, W, L1, L2, H)	EMRL0	EARL0
1.1	2.0	0.005	100	(2.5, 11)	386	556.01	(38, 3985, 1.5, 3.5, 27.5)	393	566.84	(10, 1840, 0.5, 3.5, 13.5, 63)	373	536.97
			200	(3.5, 6)	382	550.78	(59, 3979, 1.5, 4.5, 29.5)	375	540.25	(77, 2151, 1.5, 5.5, 16.5, 45)	372	535.99
			400	(5.5, 8)	385	555.83	(144, 7069, 2.5, 7.5, 48.5)	372	536.36	(94, 3784, 1.5, 5.5, 26.5, 33)	373	538.41
			800	(8.5, 4)	435	626.65	(374, 10324, 4.5, 10.5, 69.5)	372	536.07	(333, 5334, 3.5, 10.5, 37.5, 27)	371	534.75
		0.01	50	(2.5, 12)	373	537.54	(23, 1230, 1.5, 3.5, 19.5)	393	566.43	(7, 633, 0.5, 2.5, 10.5, 51)	371	534.36
			100	(3.5, 6)	394	568.15	(27, 2454, 1.5, 4.5, 34.5)	385	554.77	(36, 1271, 1.5, 4.5, 18.5, 48)	373	537.57
			200	(5.5, 8)	398	573.32	(66, 4670, 2.5, 6.5, 60.5)	382	551.07	(51, 1610, 1.5, 6.5, 23.5, 30)	371	535.09
			400	(8.5, 4)	446	643.83	(189, 4974, 4.5, 10.5, 67.5)	375	541.40	(165, 2768, 3.5, 9.5, 38.5, 27)	371	535.37
		0.02	25	(2.5, 13)	377	544.15	(11, 719, 1.5, 3.5, 21.5)	371	535.00	(3, 374, 0.5, 2.5, 11.5, 43)	374	539.15
			50	(3.5, 6)	420	605.65	(13, 1373, 1.5, 4.5, 37.5)	383	551.81	(18, 646, 1.5, 4.5, 18.5, 41)	371	535.20
			100	(5.5, 9)	384	553.16	(37, 1679, 2.5, 8.5, 46.5)	371	534.37	(25, 846, 1.5, 5.5, 24.5, 36)	373	537.26
			200	(8.5, 5)	387	558.47	(93, 2728, 4.5, 9.5, 72.5)	373	537.32	(82, 1431, 3.5, 9.5, 39.5, 29)	371	534.99
2.0	3.0	0.005	100	(2.5, 11)	386	556.01	(58, 1223, 1.5, 4.5, 12.5)	389	560.71	(32, 458, 0.5, 3.5, 5.5, 12)	384	553.69
			200	(3.5, 6)	382	550.78	(98, 1175, 1.5, 5.5, 13.5)	378	544.67	(130, 506, 1.5, 5.5, 6.5, 5)	378	545.23
			400	(5.5, 8)	385	555.83	(143, 1599, 1.5, 5.5, 17.5)	381	549.64	(192, 833, 1.5, 6.5, 9.5, 4)	382	550.83
			800	(8.5, 4)	435	626.65	(497, 2850, 4.5, 11.5, 28.5)	372	537.15	(559, 851, 3.5, 6.5, 13.5, 2)	387	558.73
		0.01	50	(2.5, 12)	373	537.54	(32, 442, 1.5, 4.5, 10.5)	416	599.25	(16, 228, 0.5, 3.5, 5.5, 13)	373	538.29
			100	(3.5, 6)	394	568.15	(49, 590, 1.5, 5.5, 13.5)	376	542.01	(65, 254, 1.5, 4.5, 6.5, 5)	373	537.67
			200	(5.5, 8)	398	573.32	(116, 756, 2.5, 7.5, 17.5)	399	575.89	(96, 417, 1.5, 5.5, 9.5, 4)	380	548.67
			400	(8.5, 4)	446	643.83	(252, 1340, 4.5, 10.5, 27.5)	372	536.00	(279, 428, 3.5, 6.5, 13.5, 2)	398	573.67
		0.02	25	(2.5, 13)	377	544.15	(16, 225, 1.5, 4.5, 10.5)	399	575.27	(8, 113, 0.5, 3.5, 5.5, 14)	379	546.74
			50	(3.5, 6)	420	605.65	(26, 253, 1.5, 4.5, 12.5)	387	558.17	(30, 166, 1.5, 4.5, 7.5, 7)	377	543.71
			100	(5.5, 9)	384	553.16	(58, 381, 2.5, 7.5, 17.5)	395	569.92	(48, 208, 1.5, 5.5, 9.5, 4)	402	579.42
			200	(8.5, 5)	387	558.47	(126, 675, 4.5, 12.5, 27.5)	371	535.05	(111, 248, 2.5, 5.5, 12.5, 1)	383	551.69

. In

Table 5. EMRL₁ and EARL₁ comparisons between the SDS np, synthetic np and DS np charts, in both ZS and SS modes, when MRL_0min = 370.4.

γmin	γmax	p0	n	ZS				SS
				EMRL1			EARL1	EMRL1			EARL1
				Synthetic np	DS np	SDS np		Synthetic np	DS np	SDS np
1.1	2.0	0.005	100	62.33	38.73	22.17	43.26	77.16	38.73	37.33	52.15
			200	47.98	24.92	14.50	27.45	56.47	24.92	24.83	34.57
			400	27.58	15.88	9.20	17.16	37.76	15.88	16.17	22.34
			800	18.72	9.85	5.59	10.13	26.67	9.85	9.97	13.59
		0.01	50	64.59	39.35	22.61	42.92	74.87	39.35	36.83	51.74
			100	49.35	24.84	14.41	27.44	57.78	24.84	24.83	34.45
			200	28.28	16.13	9.19	17.11	38.52	16.13	16.13	22.28
			400	19.10	9.86	5.58	10.02	27.08	9.86	9.92	13.54
		0.02	25	62.23	37.50	22.69	42.73	75.40	37.50	36.73	51.79
			50	52.11	24.64	14.33	27.26	60.52	24.64	24.72	34.42
			100	29.77	15.72	9.13	16.92	37.21	15.72	16.03	22.05
			200	19.84	9.72	5.56	10.03	23.98	9.72	9.89	13.42
2.0	3.0	0.005	100	5.74	5.24	2.87	6.05	11.61	5.24	5.61	7.42
			200	3.16	2.80	1.90	3.98	6.29	2.80	3.47	4.54
			400	2.07	1.67	1.16	2.03	3.51	1.67	2.25	2.71
			800	1.16	1.01	1.00	1.87	2.19	1.01	2.00	2.04
		0.01	50	5.78	5.17	2.83	5.94	11.45	5.17	5.56	7.31
			100	3.17	2.78	1.90	3.95	6.32	2.78	3.44	4.50
			200	2.07	1.57	1.16	1.98	3.51	1.57	2.24	2.70
			400	1.16	1.01	1.00	1.89	2.19	1.01	2.00	2.03
		0.02	25	5.89	5.08	3.22	4.96	11.45	5.08	5.53	7.22
			50	3.19	2.84	1.89	3.95	6.39	2.84	3.35	4.32
			100	2.07	1.54	1.15	1.96	3.48	1.54	2.23	2.68
			200	1.16	1.00	1.00	1.91	2.18	1.00	2.00	2.12

, the EMRL₁ performance of the synthetic np, DS np and SDS np charts are compared. The process in ZS and SS modes are considered. The parameter combinations of MRL_0min = 370.4, p₀ $\in$ {0.005, 0.01, 0.02}, n $\in$ {25, 50, 100, 200, 400, 800} and two intervals of process shift sizes: (i) $({\gamma }_{\min },{\gamma }_{\max }]=(1.1,2.0]$ and (ii) $({\gamma }_{\min },{\gamma }_{\max }]=(2.0,3.0]$. Results presented in the columns 8–10 of Table 3 and Table 4 were extracted from the work by Tuh et al. [28] in order to provide clear performance comparison discussions herein. They used EMRL to measure the performance of DS np chart. Two MATLAB programs that incorporate the optimization procedure described in Lee and Khoo [37], by means of minimizing EMRL₁ is written to compute the optimal charting parameters for the synthetic np chart, based on the ZS and SS modes, individually. The optimal charting parameters and corresponding EMRL₀ and EARL₀ values are displayed in columns 5–7 of Table 3 and Table 4.

For example, Table 3 demonstrates that when the process is in ZS mode with p₀ = 0.01, MRL_0min = 370.4, n = 100 and $({\gamma }_{\min },{\gamma }_{\max }]=(1.1,2.0]$, optimal synthetic np, optimal DS np and optimal SDS np charts give (UCL_S, H_S) = (3.5, 5), (n₁, n₂, W, L₁, L₂) = (27, 2454, 1.5, 4.5, 34.5) and (n₁, n₂, W, L₁, L₂, H) = (34, 1453, 1.5, 4.5, 20.5, 37), correspondingly. The lowest EMRL₁ (listed in Table 5) value is produced by the synthetic np, DS np and SDS np charts with these optimal charting parameters, while the EMRL₀ is at least 370.4. Subsequently, the corresponding values of (EMRL₀, EARL₀) for the optimal synthetic np, DS np and SDS np chart, are computed as (408, 614.58), (385, 554.77) and (371, 613.95), individually.

In Table 5, the EMRL₁ and EARL₁ values for the OOC case, in ZS and SS modes, can be obtained using the optimal charting parameters of the synthetic np, DS np and SDS np charts (in Table 3 and Table 4). For instance, when p₀ = 0.005, n = 200, MRL_0min = 370.4 and $({\gamma }_{\min },{\gamma }_{\max }]=(2.0,3.0]$, Table 4 gives (n₁, n₂, W, L₁, L₂, H) = (130, 506, 1.5, 5.5, 6.5, 5) as the optimal charting parameters for the SDS np chart. With these optimal charting parameters, (EMRL₁, EARL₁) = (3.47, 4.54) when process is in SS mode. Numerical results in Table 3 and Table 4 clearly show that the EMRL₀ values are lower than EARL₀ of the optimal SDS np chart when the process is IC ($\gamma =1$). For instance, when the process is in ZS mode, the SDS np chart gives EARL₀ = 610.92 when p₀ = 0.01, n = 200, $({\gamma }_{\min },{\gamma }_{\max }]=(1.1,2.0]$ and MRL_0min = 370.4 (in Table 3). Practitioners might conclude that a false alarm occurs to every 610,92th sample fifty percent of the time. In reality, this value lies between the 60th (=535) and 70th (=746) percentiles of the RL distribution, and the false alarm occurs prior to the 610.92nd sample, or by the 372nd sample (EMRL₀ = 372), fifty percent of the time.

On the other hand, for the OOC condition (see Table 5), in ZS mode, when p₀ = 0.02, n = 100, $({\gamma }_{\min },{\gamma }_{\max }]=(2.0,3.0]$ and MRL_0min = 370.4, the SDS np chart gives EARL₁ = 16.92, while EMRL₁ = 9.13, showing a trivial difference between the EARL₁ and EMRL₁ values. This indicates that when there is a substantial amount of right-skewed in the RL distribution, the average is significantly higher than the median. On the other hand, when there is symmetric distribution, the average is located quite near to the median. As a result of this, the use of EMRL is advised as a performance metric rather than the EARL because the EMRL provides a more understandable interpretation for the performance SDS np chart.

Results in Table 5 demonstrate that the SS EMRL₁ is larger than ZS EMRL₁ in all interval shift sizes of the SDS np chart. This indicates that the SS EMRL₁ performance of the SDS np chart is worse than its ZS performance. This is due to the fact that the SS performance calculates how long it will take for the SDS np chart to signal in this case once the head-start’s effects have gradually disappeared. As can be seen in Table 5, the optimal SDS np chart outperforms the synthetic np and DS np charts for all shift sizes $({\gamma }_{\min },{\gamma }_{\max }]$, under ZS mode, with the SDS np chart giving lower EMRL₁ than the competing charts for identical p₀, n, MRL_0min and $({\gamma }_{\min },{\gamma }_{\max }]$ combination. When the process is in SS mode, the SDS np chart surpasses the synthetic np chart while performing differently compared to the DS np chart. Table 5 also demonstrates that the SDS np chart does not perform any better than the DS np chart in identifying process fraction nonconforming shifts, particularly those with large shift sizes interval $({\gamma }_{\min },{\gamma }_{\max }]=(2.0,3.0]$. This observation is in good agreement with the results reported by Costa and Machado [5], where they studied the SS behaviour of the synthetic and side-sensitive synthetic double sampling charts.

The numerical and graphical analysis shown in

Table 6. MRL₁ values computed using the optimal charting parameters of the EMRL-based SDS np chart and the MRL-based SDS np chart for n = 100 and MRL_0min = 370.4.

p0	Type of SDS np Chart	(γmin, γmax]	ZS				SS
			MRL1				MRL1
			γ = 1.2	γ = 1.5	γ = 2.0	γ = 3.0	γ = 1.2	γ = 1.5	γ = 2.0	γ = 3.0
0.005	EMRL-based design chart	(1.1, 2.0]	43	16	7	-	81	26	13	-
		(2.0, 3.0]		-	-	2		-	-	4
	MRL-based design chart	-	37	11	4	2	81	25	9	4
0.01	EMRL-based design chart	(1.1, 2.0]	28	10	5	-	56	16	7	-
		(2.0, 3.0]		-	-	1		-	-	2
	MRL-based design chart	-	25	7	3	1	54	15	5	2
0.02	EMRL-based design chart	(1.1, 2.0]	20	6	3	-	37	9	5	-
		(2.0, 3.0]		-	-	1		-	-	2
	MRL-based design chart	-	16	4	2	1	35	9	3	2

and Figure 3, using the optimal parameters by minimizing EMRL₁ to compute the MRL₁ when unknown shift size is an option, providing $\gamma \in ({\gamma }_{\min },{\gamma }_{\max }]$, yields remarkable results. The optimal charting parameters for the EMRL-based SDS np chart can be obtained from Table 3 and Table 4.

The MRL₁ of the MRL-based design chart is presented in Table 6 for the ease of comparison and reference. The computed MRL₁ values can be found in Table 2. The MRL₁ values for both MRL-based and EMRL-based design charts when $\gamma =1.2$ are also added to this section for a thorough comparison. Here, n = 100 is chosen for wide coverage to guarantee that all situations are taken into account. As can be observed in Table 6 for the SDS np chart with EMRL-based design, the MRL₁ computed using (n₁, n₂, W, L₁, L₂, H) is very close to the MRL₁ computed using specific shift sizes (MRL-based design chart) provided that $\gamma =({\gamma }_{\min },{\gamma }_{\max }]$. For instance, by investigating Table 3 and Table 4, when n = 100, p₀ = 0.01 and $({\gamma }_{\min },{\gamma }_{\max }]=(1.1,2.0]$, the optimal charting parameters for the SDS np chart is (n₁, n₂, W, L₁, L₂, H) = (34, 1453, 1.5, 4.5, 20.5, 37) (for ZS mode) or (36, 1271, 1.5, 4.5, 18.5, 48) (for SS mode), obtained by minimizing EMRL₁. These optimal charting parameters yield MRL₁ = {28, 10, 5} in ZS mode (or MRL₁ = {56, 16, 7} in SS mode) for $\gamma =\{1.2, 1.5, 2.0\}$, while the MRL-based design chart gives MRL₁ = {25, 7, 3} in ZS mode (or MRL₁ = {54, 15, 5} in SS mode).

For better comparison of the MRL- and EMRL-based SDS np designs under both ZS and SS modes, graphical summaries have also been plotted and furnished in Figure 3. The three SDS np charts are designed by considering a minimum OOC MRL at $\gamma \in$ 1.2, 1.5, 2.0} shifts. Evidently, when the shift size is close to the value for which they have been designed, the MRL-based SDS np chart can perform slightly better than the EMRL-based DS np chart. On the contrary, the EMRL-based DS np chart performs better than its MRL-based counterpart for all other shift sizes, particularly when the actual shift differs significantly from the specified size. These observations are consistent with the conclusion drawn by Tang et al. [38]. In their work, the performance of EWMA and AEWMA methods at detecting a wide range of shifts are compared. Thus, in the event when $\gamma \in ({\gamma }_{\min },{\gamma }_{\max }]$, the optimal parameters provided in Table 3 and Table 4 (which are found by minimizing EMRL₁) can be used as a substitute for the optimal parameters by assuming a known shift size directly and reliably.

6. An Illustrative Example

In this section, an example is considered to illustrate the implementation of the optimal EMRL-based SDS np chart. This example can be obtained from Oakland [39]. The data is related to the process of producing ballpoint pen cartridges. The inspected number of defectives found in 50 sample (m = 50) of size n = 100 is selected at random every hour from the process is listed in

Table 7. Data related to the process of producing ballpoint pen cartridges. Source Oakland [39].

Sample, j	Sample Size, n	Number of Defectives, d	Sample	Sample Size, n	Number of Defectives, d
1	100	2	26	100	4
2	100	2	27	100	2
3	100	2	28	100	0
4	100	2	29	100	2
5	100	1	30	100	2
6	100	4	31	100	5
7	100	3	32	100	3
8	100	4	33	100	3
9	100	1	34	100	2
10	100	3	35	100	0
11	100	1	36	100	3
12	100	0	37	100	1
13	100	2	38	100	1
14	100	5	39	100	1
15	100	0	40	100	4
16	100	0	41	100	2
17	100	3	42	100	2
18	100	1	43	100	2
19	100	3	44	100	3
20	100	2	45	100	2
21	100	0	46	100	3
22	100	1	47	100	1
23	100	6	48	100	1
24	100	0	49	100	1
25	100	1	50	100	1
			Total	5000	${\displaystyle \sum d}=100$

. Therefore, based on the data, the process fraction nonconforming is computed as 0.02 using the relevant equation, $\overline{p}=\raisebox{1ex}{${\displaystyle \sum d}$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$.

As illustrated in Figure 4, the np chart with three-sigma control limits is built in phase I to determine if the process is under control. The centre line (CL), lower control limit (LCL), and upper control limit (UCL) were calculated as follows:

$$\begin{gathered} CL=n\overline{p}=100\times 0.02=2 \\ \begin{array}{cc}\hfill \mathrm{LCL}& =n\overline{p}-3\sqrt{n\overline{p}\left(1-\overline{p}\right)}\hfill \\ \hfill & =100\times 0.02-3\sqrt{100\times 0.02(1-0.02)}\hfill \\ \hfill & =-2.2\hfill \\ \hfill & \simeq 0\hspace{0.17em}\hfill \end{array} \\ \begin{array}{cc}\hfill \mathrm{UCL}& =n\overline{p}+3\sqrt{n\overline{p}\left(1-\overline{p}\right)}\hfill \\ \hfill \hspace{0.17em}& =100\times 0.02+3\sqrt{100\times 0.02(1-0.02)}\hfill \\ \hfill \hspace{0.17em}& =6.2\hfill \end{array} \end{gathered}$$

All the points fall within the control limits, indicating that the process is under control. As a result, the IC process fraction nonconforming (p₀) for phase II process monitoring stage through $p_0=\frac{{\displaystyle \sum _{j=1}^m{p}_j}}{m}=\frac{{\displaystyle \sum _{j=1}^m{d}_j}}{m\times n}=\frac{100}{50\times 100}=0.02$ may be estimated.

It is imperative that a significant assignable cause that raises the fraction of nonconforming to 0.028 must be identified as soon as possible. Consequently, $\gamma =\raisebox{1ex}{$0.028$}\!\left/ \!\raisebox{-1ex}{$0.02$}\right.=1.4$. Herein, n = 100, p₀ = 0.02, $({\gamma }_{\min },{\gamma }_{\max }]=(1.1,2.0]$, and MRL_0min = 370.4 are preferred. The procedures in Section 2 are adopted to construct the SDS np chart. The optimal charting parameters of the SDS np chart by considering the process under SS mode, shown in Table 4 is (n₁, n₂, W, L₁, L₂, H) = (25, 846, 1.5, 5.5, 24.5, 36).

The data in

Table 8. Simulated dataset for the EMRL-based SDS np chart.

Sampling Stage, i	First Sample (n1 = 25) d1	Second Sample (n2 = 846) d2	d1 + d2	CRL
1	1
2	0
3	0
4	1
5	0
6	0
7	0
8	0
9	1
10	0
11	2	29	31	1
12	0
13	0
14	1
15	0
16	1
17	0
18	0
19	1
20	0
21	1
22	0
23	1
24	1
25	1
26	3	35	38	15
27	0
28	2	31	33	2
29	2	32	34	1
30	0

are simulated using RStudio software; the first ten samples are generated under the IC situation with p₀ = 0.02, and the next twenty samples are generated under the OOC situation with p₁ = 0.028. At the first stage of the DS sub-chart, a binomial distribution with parameters (n₁, p₀) = (25, 0.02) for the IC situation and (n₁, p₁) = (25, 0.028) for the OOC situation is used to simulate the number of nonconforming items d₁ depicted on the DS np sub-chart. The number of nonconforming items d₂ depicted at the second stage of the DS np sub-chart is determined by the parameters of the binomial distribution, which are (n₂, p₀) = (846, 0.02) for the IC and (n₂, p₁) = (846, 0.028) for the OOC situations, respectively.

The thirty sampling stages in Table 8 are plotted on the DS np sub-chart in Figure 5a. One can see that the process stays at the stage 1 of the DS sub-chart for sampling stages 1 through 10, as the points lie below 1.5 (<W) are considered to be conforming. At the sampling stage i =11, a first sample of size n₁ = 25 is taken and d₁ = 2 nonconforming items are observed. As d₁ = 2 falls in the interval W < d₁ < L₁, take the second sample of size n₂ = 846 and number of nonconforming d₂ = 29 is observed and yields d₁ + d₂ = 31 (see Table 8). Sampling stage i = 11 is categorised as nonconforming because d₁ + d₂ > L₂ (L₂ = 24.5). To investigate the process condition further (whether the condition is IC or OOC), the CRL sub-chart (Figure 5b) is required. The number of conforming sampling stages between two non-conforming ones (including the present nonconforming stage) is computed as CRL = 1. The first OOC signal appears at sampling stage 11 because CRL = 1 < H = 36.

The process of taking the first sample and likely the second, for each sampling stages is continued until sampling stage i = 26, here d₁ + d₂ = 38 is seen for d₁ = 3 (W < d₁ < L₁) and d₂ = 35. Hence, sampling stage i = 26 is categorised as nonconforming because d₁ + d₂ > L₂. The nonconforming sampling stages are also seen at sampling stages i = 28 and 29. This leaves the nonconforming sampling stages of i = 11, 26, 28, and 29 as OOC states.

7. Conclusions

This study investigates the MRL and EMRL performances of the SDS control chart for detecting increasing shifts in process fraction nonconforming. When compared to ARL, MRL is a better performance criterion to use. It is demonstrated that the claims for its superior performance over the synthetic np and DS np charts are based on the fact that ZS mode has a hidden feature that gives it a head-start. However, for the majority of SPC applications, it is more crucial to assess the control chart’s statistical performance after a prolonged time of IC operation (i.e., when the process is in SS mode), when it is necessary to detect a shift in the process. It has been discussed how to employ the Markov chain to perform MRL SS analyses for the SDS chart for attribute data. This method has been used to compare the SDS np chart with the synthetic np and the DS np charts for a wide range of cases. These comparisons adequately demonstrate that the performance of the SDS np chart does not significantly outperform the DS np chart. Overall, the SDS np chart outperforms the DS np chart when ${\gamma }_{\mathrm{opt}}=1.5$ and np₀ = 0.5 for different combinations of (n, p₀) = (100, 0.005) and (50, 0.01). Note that the optimization designs based on minimizing the EMRL offer the practitioners with the optimal parameter for the chart which performs favourably for a range of shift sizes, i.e., from the lower to upper range of shift sizes when the actual shift size is unknown in advance. In short, the optimal charting parameters reported in this study are helpful to quality practitioners since these results enable the SDS np chart to be implemented quickly.