A proposed non-parametric triple generally weighted moving average sign chart

1.   Introduction

In order to satisfy or surpass consumers' expectations, it is imperative for the process performance to be consistent. SPC is a useful approach for improving product quality and enhancing process stability through the reduction of variability. The control chart is the most practical tool in SPC and is frequently employed in process monitoring. Many new types of control charts have been developed since the inception of the Shewhart chart. The cumulative sum (CUSUM), exponentially weighted moving average (EWMA), and generally weighted moving average (GWMA) charts were developed to facilitate the rapid detection of small process shifts as the Shewhart chart is inefficient in this regard. Most of the above-mentioned charts assume that the samples being observed are normally distributed. However, in actual process monitoring situations, the normality assumption is usually violated and practitioners do not have any knowledge about the distribution of the underlying process being monitored. An effective approach to address the aforementioned issue is the use of non-parametric control charts.

In this paragraph, some non-parametric charts are evaluated. Amin et al. ^[1] developed non-parametric Shewhart and CUSUM charts to identify fluctuations in the mean or standard deviation of a process by utilizing sign-test statistics computed for each sample. Chakraborti and Eryilmaz ^[2] utilized the Wilcoxon signed rank statistic and several runs type constraints to develop Shewhart-type non-parametric charts for known in-control median of a continuous distribution. Zhou et al. ^[3] developed a non-parametric chart that employs the Mann-Whitney statistic to identify changes in the mean, which has been modified for repetitive sequential use. In detecting an increase in process variation, Khilare and Shirke ^[4] proposed synthetic charts that were non-parametric and side-sensitive for fraction nonconforming. Tapang et al. ^[5] expanded the application of non-parametric charts in real-world contexts by applying ranked set sampling techniques to several well-known non-parametric tests, while exploring alternative sampling methods. Li et al. ^[6] utilized a run test to extend non-parametric monitoring in a multivariate context, adapting it to multiple quality characteristics simultaneously. The efficiency of monitoring complex processes was enhanced by Boroomandi and Kharrati-Kopaei ^[7], who devised four non-parametric variance estimators and three control charts based on partially rank ordered set (PROS) sampling. This further advanced innovative sampling approaches. Lastly, Malela-Majika ^[8] introduced a precedence chart that employed repetitive sampling, thereby demonstrating an application-oriented perspective on the design of non-parametric control charts and offering practical insights for their implementation.

The EWMA method is essential in the development of non-parametric control charts. The accuracy and efficiency in process monitoring were improved by Malela-Majika et al. ^[9], who introduced an EWMA chart that applies the Wilcoxon rank-sum statistic and recurrent sampling technique. Tang et al. ^[10] proposed a non-parametric adaptive EWMA chart that is based on the sign statistic, which offers improved flexibility in adapting to process changes over time. Alevizakos et al. ^[11] advanced the use of the EWMA chart by proposing a triple EWMA chart that tracks the location parameter of an unknown continuous distribution using the sign statistic. Furthermore, an EWMA sign chart with dynamic probability control limits that can accommodate both fixed and variable sample sizes was presented by Haq ^[12].

The EWMA approach has been significantly improved by researchers in recent years to handle real-world process monitoring issues. Ng et al. ^[13] suggested an EWMA t chart with variable sampling intervals and auxiliary information that reduces the impact of estimation error. By developing a non-parametric adaptive EWMA chart for monitoring multivariate time-between-events and amplitude data, Xue et al. ^[14] extended the EWMA approach and showed its increasing usefulness in intricate processes. Sheu and Lin ^[15] showed that the EWMA chart may be viewed as a particular instance of the GWMA chart when the latter employs a parameter value of $\gamma = 1.$ Likewise, the double EWMA (DEWMA) and triple EWMA (TEWMA) charts are special cases of the DGWMA and TGWMA charts when the last two charts adopt the parameter values $\alpha = \beta = 1$ and $\alpha = \beta = \gamma = 1$, respectively ^[16]. Details about the adjustable parameters $\alpha$, $\beta$, and $\gamma$ will be explained in Sections 2 and 3. The GWMA chart and its extended counterparts enhance the sensitivity of the EWMA charts in detecting small shifts as a result of having more adjustable parameters.

Several existing GWMA-type charts have been reviewed in the literature, each of which has contributed to the development of non-parametric methods. Lu ^[17] developed a non-parametric GWMA sign chart for an enhanced detection of small process shifts. Chakraborty et al. ^[18] extended this concept by proposing a non-parametric GWMA chart that employs the Wilcoxon signed-rank statistic, thereby providing increased sensitivity in detecting process shifts. A non-parametric DGWMA sign chart was introduced by Lu ^[19] which further enhanced the GWMA sign chart's ability to detect small shifts. In addition, Alevizakos et al. ^[20] provided a non-parametric DGWMA chart that utilizes the signed-rank statistic to monitor location parameters, thereby presenting an alternative method in process monitoring. Mabude et al. ^[21] further contributed to the field by developing a non-parametric Phase-II composite Shewhart-GWMA chart that employs the Mann-Whitney statistic. In the interim, Godase and Mahadik ^[22] introduced non-parametric combined Shewhart-CUSUM charts that incorporate the sign statistic to enhance process monitoring. In addition, Mabude et al. ^[23] developed two non-parametric mixed charts by combining the GWMA and CUSUM charts, which are useful in real-world applications.

Although the current EWMA type, GWMA type, and their hybrid charts have made substantial progress in detecting small shifts, these charts continue to encounter problems in detecting extremely small and progressively changing process shifts. The high complexity of current hybrid charts, such as the GWMA-CUSUM chart, can result in increased computational costs. Additionally, there is a notable lack of research in the SS performance, as the majority of current non-parametric charts concentrate significantly in the ZS performance. The double moving weighting mechanism of the DGWMA sign chart renders it exceptionally appropriate for monitoring intricate, noisy industrial processes with minimal shift amplitudes, showcasing distinct value and potential for application. Furthermore, there is no "triple" type chart for non-parametric monitoring in the existing literature besides the proposed TGWMA sign chart. The TGWMA sign chart retains the advantages of the DGWMA sign chart, in terms of noise smoothing, dynamic balancing of false alarm rates and sensitivity, flexible parameter adjustment, robustness towards small sample monitoring, and progressive stability for long-term monitoring. These advantages are challenging to be replicated by other existing non-parametric control charts. The proposed TGWMA sign chart in this study also offers enhanced monitoring capabilities, particularly for very small shifts. In addition to the ZS performance, this paper focuses on the SS performance to align more closely with real-life production processes.

The remaining sections of this paper are organized as follows: In Section 2, an overview of the DGWMA sign chart is given. Section 3 presents the proposed TGWMA sign chart, while Section 4 evaluates the developed TGWMA sign chart by comparing its performance with that of the existing DGWMA sign chart. An example of an application for the TGWMA sign chart is given in Section 5. Conclusions are drawn and suggestions for further research are given in Section 6.

2.   An overview of the DGWMA sign chart

Lu ^[19] developed the DGWMA sign chart for monitoring small process shifts when the distribution of the quality characteristic of the process is unknown. Let X denote a quality characteristic from a continuous distribution with a known value of the target median ${\mathtt{μ}}_{0}$. In addition, let

Suppose that a sample of size n is taken at time t. Then, ${Y}_{it} = {X}_{it}-{\mathtt{μ}}_{0}$ for i = 1, 2, …, n. Let

and ${S}_{t} = {\sum }_{i = 1}^{n}\; {I}_{it}$. ${S}_{t}$ follows the binomial B$\mathrm{i}\mathrm{n}(n, 0.5)$ distribution when the process is in-control. The GWMA sign chart's statistic is (Lu ^[19])

In Eq (2), $P\left({H}_{1} = 1\right)$, $P\left({H}_{1} = 2\right)$, …, $P\left({H}_{1} = t\right)$ are used to weight ${S}_{t}$, ${S}_{t-1}$, …, ${S}_{1}$, respectively, and $P\left({H}_{1} > t\right)$ is used to weight the starting value of ${G}_{t}$, i.e., ${G}_{0}$. Note that ${G}_{0} = n/2$ because $E\left({G}_{t}\right)$ = $E\left({S}_{t}\right) = n/2$ when the process is in-control (Lu ^[19]). In Eq (2), $P\left({H}_{1} = j\right) = {q}_{1}^{(j-1{)}^{\alpha }}-{q}_{1}^{{j}^{\alpha }}$ and $P\left({H}_{1} > t\right) = {q}_{1}^{{t}^{\alpha }}$ for j = 1, 2, …, t, where ${q}_{1}$ is a constant satisfying 0 ≤ ${q}_{1}$ < 1 and α is an adjustable parameter.

To obtain the DGWMA sign chart's statistic, i.e., $D{G}_{t}$, we weight ${G}_{t}$ in Eq (2) in the same way that we weight ${S}_{t}$, hence, giving

where

In Eq (4), α and β are adjustable constants and 0 ≤ ${q}_{k}$ < 1 (for k = 1, 2). According to Lu ^[19], the expectation and variance of the DGWMA sign chart's statistic, $D{G}_{t}$, when the process is in-control are given in Eqs (5) and (6), respectively.

and

Consequently, the time-varying limits of the DGWMA sign chart are

and

The asymptotic variance of the statistic $D{G}_{t}$ is obtained as

where α and β are adjustable constants and 0 ≤ ${q}_{k}$ < 1 (for k = 1, 2). Let

Then, the asymptotic limits of the DGWMA sign chart are

and

where ${L}_{DG}$ is the width constant of the DGWMA sign chart's limits.

3.   A proposed TGWMA sign chart

To obtain the TGWMA sign chart's statistic, i.e., ${T}{G}_{t}$, we weight ${DG}_{\mathrm{t}}$ in Eq (3) in the same way that we weight ${G}_{t}$ or ${S}_{t}$. Thus,

where

and

From Eqs (11) and (13),

In Eqs (11)–(13), $P({H}_{3} = j)$ and $P({H}_{3} > t)$ for j = 1, 2, …, t are defined below Eq (2). The expectation and variance of the TGWMA sign chart's statistic, $T{G}_{t}$, when the process is in-control are given in Eqs (15) and (16), respectively.

and

The time-varying limits of the TGWMA sign chart are

and

where ${L}_{TG}$ is the width constant of the TGWMA sign chart's limits.

The asymptotic variance of the statistic $T{G}_{t}$ is obtained as

where γ is an adjustable constant and 0 ≤ ${q}_{3}$ < 1. Let

Consequently, the asymptotic limits of the TGWMA sign chart are

and

4.   Performance evaluation

This study will exclusively compare the proposed TGWMA sign chart with the DGWMA sign chart, as the latter has already been evaluated against other high-performing non-parametric control charts, demonstrating superior monitoring capabilities. Due to the presence of the six parameters (${q}_{1}$, ${q}_{2}$, ${q}_{3}$, α, β, γ), the design of the TGWMA sign chart appears somewhat complex. To facilitate the design of this chart, ${q}_{1}$ = ${q}_{2}$ = ${q}_{3}$ = q and $\alpha$ = $\beta$ = $\gamma$ are considered. Values of the sample size n ∈ {5, 10, 15}, process proportion p ∈ {0.1, 0.4, 0.6, 0.9}, constant q ∈ {0.5, 0.7, 0.9}, and adjustable constant α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5} are considered in the analyses for the ZS and SS conditions, respectively. To further show the effectiveness of the proposed TGWMA sign chart in detecting small shifts in the process proportion, additional values of p in the neighborhood of 0.5, i.e., p ∈ {0.44, 0.48, 0.52, 0.56}, are considered for the ZS and SS conditions. The asymptotic limits of the two charts are adopted. The ARL values for the ZS and SS based TGWMA sign and DGWMA sign charts are computed for various shift sizes in the process proportion by using the MATLAB software for simulation. Each ARL value is computed based on 10000 simulation trials. The ARL measures the average number of samples required until a chart signals an out-of-control. The ARL is denoted as ARL₀ when the process is in-control and ARL₁ when the process is out-of-control. The ZS condition assumes that the process shift occurs at the beginning of process monitoring. On the contrary, under the SS condition, the process does not start as out-of-control, but rather it runs for a certain time before becoming out-of-control. Hence, the SS condition is more consistent with an actual production process.

As ARL₀ = 370 is adopted for the ZS and SS conditions, the values of the limit constants, ${L}_{DG}$ (Eqs (10a) and (10c)) and ${L}_{TG}$ (Eqs (20a) and (20c)) of the DGWMA and TGWMA sign charts, respectively, are computed using the binary search method in MATLAB, so that the said ARL₀ value is attained. The ARL₁ performance of the proposed TGWMA sign chart will be compared with that of the existing DGWMA sign chart. For the ZS condition, the following step-by-step procedure explains the computation of the ARL₁ value of the proposed TGWMA sign chart by using the MATLAB software for simulation.

Step 1. Compute the value of ${R}_{2}$ using Eq (19) based on t = 500 by considering practical implications.

Step 2. Generate the dataset using the binornd function. Then, use the binary search method to obtain the value of ${L}_{TG}$ (0 < ${L}_{TG}$ < 3) in Eqs (20a) and (20c) for the various (n, q, α) combinations to attain an ARL₀ value as close as possible to the desired value of 370. Note that p = 0.5 is adopted here as the process proportion is assumed to be in-control in computing ${L}_{TG}$, followed by UCL and LCL.

Step 3. Next, compute the ARL₁ values for the out-of-control process proportions (p ≠ 0.5) for the various (n, q, α) combinations.

The above-mentioned procedure can also be used to compute the ZS ARL of the DGWMA sign chart. Moreover, the above-mentioned procedure can be used in computing the SS ARLs of the TGWMA and DGWMA sign charts, but by assuming that the first M samples of the process are in-control and the process becomes out-of-control from the (M + 1)th sample onwards, where M = 100 is adopted in the analyses. In the SS condition, the run length of a chart until an out-of-control signal is detected is counted starting from the (M + 1)th sample, meaning the first M samples are not included.

Table 1 provides the values of ${R}_{2}$ of the TGWMA sign chart computed using Eq (19), while Tables 2 and 3 provide the values of the width constant, ${L}_{TG}$, of the said chart for the ZS and SS conditions, respectively, for various combinations of (n, q, α) considered in the analyses. Note that for any (q, α) pair, the value of ${R}_{2}$ remains the same, irrespective of the sample size n and ZS or SS condition. In Table 2, it is seen that for the ZS condition, the ${L}_{TG}$ value lies between 1 and 3 for the TGWMA sign chart to attain ARL₀ = 370. However, for the SS condition in Table 3, the ${L}_{TG}$ values when q = 0.9 and α < 1.5 are less than unity. In addition, it is observed that for the same combination of (n, q, α), the ${L}_{TG}$ value of the TGWMA sign chart under the ZS condition (Table 2) is greater than that under the SS condition (Table 3). For example, when (n, q, α) = (5, 0.5, 0.5), the ${L}_{TG}$ values are 2.747 and 2.091 in Tables 2 and 3, respectively, where the former is larger than the latter.

Table 4 gives the computed ARL₁ values for the DGWMA and TGWMA sign charts for various shift sizes in the process proportion p under the ZS condition. In Table 4, it is seen that when the shift in the process proportion is large, such as p = 0.1 or 0.9, the ARL₁ of the TGWMA sign chart is larger than that of the DGWMA sign chart, hence the latter detects a large shift quicker than the former. However, for smaller shifts, such as p = 0.4 or 0.6, the superiority of the TGWMA sign chart to the DGWMA sign chart begins to show up, especially for smaller values of q, such as q = 0.5 and 0.7. When q = 0.5 and p ∈ {0.4, 0.6}, regardless of the values of the sample size (n) and α (∈ {0.5, 0.6, …, 1.0, 1.2, 1.5}), all the ARL₁s of the TGWMA sign chart are smaller than that of the DGWMA sign chart, indicating that the former is quicker in detecting shifts. For example, in Table 4 when n = 5, q = 0.5, p = 0.6, and α ∈ {0.5, 0.6, …, 1.5}, ARL₁ ∈ {33.977, 36.291, …, 70.892} and {40.935, 43.636, …, 81.477} for the TGWMA and DGWMA sign charts, respectively, where the former has smaller ARL₁s. Moreover, if p ∈ {0.4, 0.6} and q = 0.7, (i) for n = 5, the ARL₁s of the TGWMA sign chart are smaller than those of the DGWMA sign chart; (ii) for n = 10 and $\alpha \ge 0.8$, the ARL₁s of the TGWMA sign chart are lower than that of the DGWMA sign chart; and (iii) for n = 15 and $\alpha \ge 1.0$, the TGWMA sign chart has smaller ARL₁s than the corresponding ARL₁s of the DGWMA sign chart. However, when q = 0.9, for the same shift size p (∈ {0.4, 0.6}), the TGWMA sign chart has almost no advantage compared to the DGWMA sign chart as the latter has smaller ARL₁s than the former (see Table 4). In Table 4, it is also found that the charts' performance improves (ARL₁ decreases) as the sample size n increases.

As the TGWMA sign chart is generally found to be superior to the DGWMA sign chart towards small shifts in the process proportion, i.e., p ∈ {0.4, 0.6}, a further investigation is conducted by comparing the ARL₁s of the two charts for p ∈ {0.44, 0.48, 0.52, 0.56}. Tables 5 and 6 present the computed ZS ARL₁s for the two charts based on $p\in \left\{\mathrm{0.44, 0.48}\right\}$ and $p\in \left\{\mathrm{0.52, 0.56}\right\}$, respectively. In Tables 5 and 6, a negative (or positive) value of "% Diff" represents the percentage of decrease (or increase) in the ARL₁ value by using the TGWMA sign chart in place of the DGWMA sign chart. Consequently, $\text{% Diff}\text{}\text{ = }\frac{\text{AR}{\text{L}}_{\text{1}}\text{(TGWMA)-AR}{\text{L}}_{\text{1}}\text{(DGWMA)}}{\text{AR}{\text{L}}_{\text{1}}\text{(DGWMA)}}\text{×100%}$. In Tables 5 and 6 under the ZS condition, it is noticed that the TGWMA sign chart surpasses the DGWMA sign chart in terms of the ARL₁ criterion for q = 0.5 and 0.7, as the former has smaller ARL₁s than the latter. For instance, when n = 10, p = 0.44, α = 0.8, and q = 0.5, the ARL₁s are 59.578 and 71.891 for the TGWMA and DGWMA sign charts, respectively, and the former's ARL₁ is 17.13% smaller relative to that of the latter (see Table 5). However, for q = 0.9, mixed performance is observed as sometimes the DGWMA sign chart beats the TGWMA sign chart, and vice-versa. From the "% Diff" in Tables 5 and 6, the TGWMA sign chart shows the best performance over the DGWMA sign chart when p is 0.44 or 0.56, and this phenomenon is particularly evident when q = 0.5. For example, when n = 5, p = 0.44, and q = 0.5, the percentage of improvement in using the TGWMA sign chart in place of the DGWMA sign chart ranges from 8.43% to 18.41%.

From the results in Tables 5 and 6, the TGWMA sign chart outperforms the DGWMA sign chart in detecting small shifts in the process proportion, especially when q takes values of 0.5 and 0.7. When q takes the value of 0.9, the superiority of the TGWMA sign chart to the DGWMA sign chart is only evident when $\alpha$ takes larger values, such as 1.2 and 1.5, or when the shift is particularly small (p ∈ {0.48, 0.52}).

After verifying that the TGWMA sign chart exhibits better sensitivity in detecting small shifts than the DGWMA sign chart, it is also essential to assess whether the TGWMA sign chart shows better run length stability. To this end, the standard deviation of the run length (SDRL) performance of these two charts for small shifts is compared, as the strength of the TGWMA sign chart lies in detecting small shifts. Table 7 illustrates that when the shift is small (p ∈ {0.44, 0.48, 0.52, 0.56}), all SDRL₁ values of the TGWMA sign chart are smaller than that of the DGWMA sign chart under the ZS condition. This outcome clearly illustrates that the TGWMA sign chart gives better SDRL₁ performance than the DGWMA sign chart when the shift is very small. For example, when n = 10, q = 0.5, p = 0.44, and α ∈ {0.5, 0.6, …, 1.5}, SDRL₁ ∈ {34.38, 40.17, …, 90.98} and {48.19, 53.76, …, 103.43} for the TGWMA and DGWMA sign charts, respectively, where the former has smaller SDRL₁ values (see Table 7). This outcome aligns with our findings on the ARL₁ comparison between the two charts.

Next, the ARL₁ comparison between the TGWMA and DGWMA sign charts under the SS condition will be made. Table 8 displays the SS ARL₁s for these two charts based on p ∈ {0.1, 0.4, 0.6, 0.9}. Similar to the ZS condition, it is seen in Table 8 that for p = 0.1 and 0.9, which indicates a larger shift in the process proportion, the ARL₁ of the DGWMA sign chart is smaller than that of the TGWMA sign chart. However, for a small shift, such as p ∈ {0.4, 0.6}, the superiority of the TGWMA sign chart to the DGWMA sign chart becomes evident. When q = 0.5 and p ∈ {0.4, 0.6}, the ARL₁s of the TGWMA sign chart are less than that of the DGWMA sign chart. When p ∈ {0.4, 0.6} and q = 0.7, the SS TGWMA sign chart beats the SS DGWMA sign chart for the following situations: (i) n = 5 and α ≥ 0.6, (ii) n = 10 and α ≥ 1.2, and (iii) n = 15 and α = 1.5 (see Table 8). Nonetheless, similar to the findings under the ZS condition, when q = 0.9, the TGWMA sign chart has almost no advantage compared to the DGWMA sign chart, as the latter gives smaller ARL₁s for all shift sizes. Since the performance trends of the TGWMA and DGWMA sign charts in Tables 4 (ZS condition) and 8 (SS condition) are the same, this paragraph provides a concise discussion regarding the SS performance of the two charts for p ∈ {0.1, 0.4, 0.6, 0.9}.

A further investigation on the SS performance of the TGWMA and DGWMA sign charts in detecting small shifts, p ∈ {0.44, 0.48, 0.52, 0.56}, is made, and the ARL₁s are shown in Tables 9 and 10. The trend in Tables 8 and 9 for the SS condition is the same as those for the ZS condition in Tables 5 and 6, where it is seen that when p ∈ {0.44, 0.56} and q ∈ {0.5, 0.7}, the ARL₁s of the TGWMA sign chart are smaller than that of the DGWMA sign chart, irrespective of the sample size n. For even smaller shifts, such as p ∈ {0.48, 0.52}, all the ARL₁s of the TGWMA sign chart are less than that of the DGWMA sign chart, for any sample size n.

Table 11 compares the SDRL₁ values of the TGWMA and DGWMA sign charts under the SS condition. The comparison indicates that, when the shift is small (p ∈ {0.44, 0.48, 0.52, 0.56}), all SDRL₁ values of the TGWMA sign chart are smaller than that of the DGWMA sign chart, with the exception of some cases when q = 0.9. For instance, when n = 5, q = 0.5, p = 0.56, and α ∈ {0.5, 0.6, …, 1.5}, SDRL₁ ∈ {25.09, 29.98, …, 82.98} and {39.45, 46.31, …, 103.83} for the TGWMA and DGWMA sign charts, respectively, where the former has smaller SDRL₁s (see Table 11). These findings align with our conclusion regarding the ARL₁ performance of the two charts.

By comparing the ZS and SS performances of the charts, it is found that the magnitude in which the TGWMA sign chart surpasses the DGWMA sign chart in detecting small shifts is larger under the SS condition. This is reflected by the smaller ARL₁s for the TGWMA sign chart compared to that of the DGWMA sign chart for all (n, q, α) combinations, even when q = 0.9. For example, when p = 0.56, n = 5, q = 0.5, and α = 0.7, ARL₁ = 115.410 and 98.026 for the DGWMA and TGWMA sign charts, respectively, under the ZS condition, where the latter beats the former by 15.06% (see Table 6). For the same (p, n, q, α) combination but under the SS condition, the latter (ARL₁ = 37.903) beats the former (ARL₁ = 56.198) by 32.55% (see Table 10).

From the findings for the ZS and SS conditions, the following outcomes can be drawn: (1) For very small shifts in the process proportion (p∈ {0.48, 0.52}), the TGWMA sign chart signals an out-of-control quicker than the DGWMA sign chart for both the ZS and SS conditions, irrespective of the value of q. (2) For moderately small shifts (p ∈ {0.44, 0.56}) and q ∈ {0.5, 0.7}, the TGWMA sign chart beats the DGWMA sign chart under both the ZS and SS conditions. (3) The superiority of the TGWMA sign chart to the DGWMA sign chart in detecting small shifts is more pronounced under the SS condition.

5.   An example of application

To illustrate the implementation of the TGWMA sign chart, we use the data generated from the beta distribution, which are shown in Table 12. The observations of the first 30 samples (Phase-I), each of size 10, are generated from an in-control process where the two shape parameters of the beta distribution, say ${v}_{1}$ and ${v}_{2}$ are set to have a value of 0.5. Observations in the last 8 samples (Phase-II), each of size 10, are generated for the out-of-control process, where the shape parameters of the beta distribution are set as ${v}_{1}$ = 2 and ${v}_{2}$ = 1. The in-control process mean is computed as $\frac{{v}_{1}}{{v}_{1}+{v}_{2}} = 0.5$, while the out-of-control mean is computed as 0.667. Therefore, the out-of-control samples 31–38 represent the out-of-control process with an increasing process mean.

The number of observations larger than the grand average, $\overline{\bar{X}}$ = 0.5029 in each sample, is counted and the count is recorded as ${S}_{t}$ ( = ${\sum }_{i = 1}^{n}\; {I}_{it}$) (for t = 1, 2, …, 38) in the third to last column of Table 12. Note that $\overline{\bar{X}}$ is computed from the 30×10 in-control observations of the 30 Phase-I samples, and it is also taken as ${\mu }_{0}$ in Eq (1). The $T{G}_{t}$ values for the TGWMA sign chart, computed using Eq (11), for the 38 samples are given in the second to last column of Table 12. ARL₀ = 370, the parameter combination (n, q, α) = (10, 0.5, 0.9), and the ZS condition are considered. Then, ${L}_{TG}$ = 2.750 is obtained from Table 2. The $T{G}_{t}$ values for the 38 samples are plotted in Figure 1 as the TGWMA sign chart, based on the limits UCL = 6.4993, CL = 5, and LCL = 3.5007, computed using Eqs (20a)–(20c), respectively.

Figure 1 shows a plot of $T{G}_{1}$, $T{G}_{2}$, …, $T{G}_{30}$ (first 30 samples) within the limits UCL/LCL of the TGWMA sign chart. However, the $T{G}_{t}$ statistic for the last 8 samples (t = 31, 32, …, 38) show an increasing trend, and the first out-of-control signal is detected at sample 36. In fact, $T{G}_{t}$ (for t = 36, 37 and 38) is above the UCL of the TGWMA sign chart, hence the last 3 samples are out-of-control. Following the detection of these out-of-control signals, corrective actions are made to investigate the underlying process to bring the out-of-control process back into the in-control situation again.

In order to compare the TGWMA and DGWMA sign charts, we also construct the DGWMA sign chart in Figure 2 based on the chart's limits UCL = 6.8535, CL = 5, and LCL = 3.1465, computed using Eqs (10a)–(10c), respectively. Figure 2 shows that $D{G}_{1}$, $D{G}_{2}$, …, $D{G}_{30}$ (first 30 samples) are within the limits UCL/LCL of the DGWMA sign chart. However, the $D{G}_{t}$ statistic in the last 8 samples (i.e., t = 31, 32, …, 38) shows an increasing trend, and the first out-of-control signal is detected at sample 37 by the chart.

By comparing Figures 1 and 2, it is found that the TGWMA sign chart detects the increasing shift in the mean slightly quicker than the DGWMA sign chart. The former signals at sample 36, while the latter is at sample 37.

6.   Conclusions

The purpose of this study is to further enhance the ability of non-parametric control charts to detect small shifts, and to meet the real-world demands for high-performance control charts in achieving a stable process control scenario. The TGWMA sign chart's statistic ($T{G}_{t}$), expectation and variance of $T{G}_{t}$, and control limits of the TGWMA sign chart are derived in this paper. Through numerical simulations, the detection capability of the TGWMA and DGWMA sign charts, especially in detecting small shifts, is compared, in terms of the charts' ARL₁s for both ZS and SS conditions. The findings show that regardless of the ZS or SS condition, and under most parameter combinations, the TGWMA sign chart is more effective in detecting small shifts in the process proportion than the DGWMA sign chart, and this superiority becomes more apparent as the shift becomes smaller. The implementation of the TGWMA sign chart is also demonstrated with an example. Given the excellent small shift detection effectiveness of the TGWMA sign chart, the said chart can be applied in actual production and manufacturing processes, especially when the data come from an unknown underlying distribution.

Practitioners in manufacturing and service industries can use the developed non-parametric TGWMA sign chart in process monitoring effectively, as tables providing the values of the constants L_TG and R₂ are given for various (n, q, α) combinations when ARL₀ = 370 to facilitate the computation of the chart's limits. This will enable practitioners to use the developed chart instantaneously for more efficient process monitoring so that small shifts can be detected faster when the process distribution is unknown.

Further research could focus on investigating the performance of the proposed TGWMA sign chart when measurement errors exist in the process, as well as by considering the presence of autocorrelation in the underlying process. The use of advanced optimization techniques, including genetic algorithms or deep reinforcement learning, in computing the optimal parameters of the TGWMA sign chart could be a potential area for further research. An extension of the TGWMA sign control charting approach to its multivariate counterpart for monitoring multivariate non-parametric data could also be made. The fast initial response feature could also be incorporated into the TGWMA sign chart to improve the sensitivity of the chart in detecting shifts at process start-up.

Author contributions

Dongmei Cui: Conceptualization, formal analysis, investigation, methodology, software, visualization, writing-original draft; Michael B. C. Khoo: Data Curation, funding acquisition, project administration, supervision, writing - review & editing; Huay Woon You: Investigation and supervision; Sajal Saha: Software, visualization, supervision; Zhi Lin Chong: Supervision, writing - review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in producing this article.

Acknowledgments

This research was supported by the provincial-level special applied discipline fund designated for "Applied Economics" at Hunan International Economics University. This research was conducted when the corresponding author was spending his sabbatical leave at Estek Automation Sdn Bhd.

Conflict of interest

The authors declare no competing interests.