Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Open Access

Peer-reviewed

Research Article

A Variable Sampling Interval Synthetic Xbar Chart for the Process Mean

  • Lei Yong Lee,

    Affiliation School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

    ⨯
  • Michael Boon Chong Khoo ,

    mkbc@usm.my

    Affiliation School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

    ⨯
  • Sin Yin Teh,

    Affiliation School of Management, Universiti Sains Malaysia, 11800 Penang, Malaysia

    ⨯
  • Ming Ha Lee

    Affiliation School of Engineering, Computing and Science, Swinburne University of Technology (Sarawak Campus), Sarawak, Malaysia

    ⨯

Abstract

The usual practice of using a control chart to monitor a process is to take samples from the process with fixed sampling interval (FSI). In this paper, a synthetic control chart with the variable sampling interval (VSI) feature is proposed for monitoring changes in the process mean. The VSI synthetic chart integrates the VSI chart and the VSI conforming run length (CRL) chart. The proposed VSI synthetic chart is evaluated using the average time to signal (ATS) criterion. The optimal charting parameters of the proposed chart are obtained by minimizing the out-of-control ATS for a desired shift. Comparisons between the VSI synthetic chart and the existing , synthetic , VSI and EWMA charts, in terms of ATS, are made. The ATS results show that the VSI synthetic chart outperforms the other type charts for detecting moderate and large shifts. An illustrative example is also presented to explain the application of the VSI synthetic chart.

Citation: Lee LY, Khoo MBC, Teh SY, Lee MH (2015) A Variable Sampling Interval Synthetic Xbar Chart for the Process Mean. PLoS ONE 10(5): e0126331. https://doi.org/10.1371/journal.pone.0126331

Academic Editor: Zhong-Ke Gao, Tianjin University, CHINA

Received: August 4, 2014; Accepted: March 31, 2015; Published: May 7, 2015

Copyright: © 2015 Lee et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Data Availability: All relevant data are within the paper.

Funding: This research is funded by the Universiti Sains Malaysia, Research University Grant, no. 1001/PMGT/816250. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

A control chart is probably the most technically sophisticated tool among the basic Statistical Process Control (SPC) problem-solving tools to achieve process stability by reducing variability in the process. Variability exists in all processes and it is the tendency of a change occurring in a process. As a consequence of variability, no two products coming from the same process are the same. Recently, many researchers have contributed to the area of control charts, such as [1,2,3,4,5,6,7], to name a few. The traditional Shewhart chart is commonly used to detect large mean shifts in manufacturing and service processes. However, this chart only gives a quick detection of large shifts but responds slowly to small and moderate shifts. Hence, numerous researches were made to improve the performance of the Shewhart chart by enhancing the chart's sensitivity to detect small and moderate mean shifts.

Combining charts is not a new procedure in the literature of control charts, see for example, [8,9,10,11,12,13]. Wu and Spedding [13] introduced the combined Shewhart and conforming run length (CRL) charts, which is called the synthetic control chart. Numerous studies on synthetic control charts have been made by researchers in recent years. Wu et al. [14] presented the combined synthetic and chart, where this chart produces an out-of-control signal when a sample mean falls beyond the limits of the chart or when the synthetic chart signals. The synthetic double sampling chart proposed by Khoo et al. [15] substantially reduces the out-of-control average run length (ARL1) and average number of observations to signal (ANOS) by nearly half, as compared with the synthetic and double sampling charts. Zhang et al. [16] studied the run-length performance of the synthetic chart with unknown process parameters as the actual parameters are rarely known in practice. Khoo et al. [17] provided an optimal design of the synthetic chart using the median run length (MRL) criterion while Yeong et al. [18] studied the economic design and the economic statistical design of the synthetic chart using loss functions. The economic statistical design is different from the economic design as the former includes statistical constraints in its design.

An adaptive control chart involves varying at least one of the chart's parameters, such as the sampling interval, sample size or the width constant of control limits. Varying the sampling interval between samples is an alternative method adopted for a quicker detection of an out-of-control process as compared with the conventional fixed sampling interval (FSI) Shewhart chart [19]. Costa [20] proposed taking variable sample sizes (VSS) from a process at FSI so that the chart outperforms the Shewhart chart for detecting moderate process mean shifts. The variable sample size and sampling interval (VSSI) procedure incorporating ideas of the variable sampling interval (VSI) and variable sample size (VSS) approaches presented by Costa [21], is substantially more effective for detecting moderate process mean shifts compared with the VSI and VSS charts. Costa [22] extended the study of the Shewhart chart by incorporating variable parameters (VP), where the sample size, sampling interval and factor controlling the width of the action limits, are all varied. The results showed that the variable parameters chart is more powerful than the Cumulative Sum (CUSUM) chart for detecting shifts in the process mean. Furthermore, numerous findings of the VSI control charts showed that these charts are substantially more efficient than the traditional FSI control charts. For instance, see [23,24,25,26,27].

The concept of varying at least one of the control chart’s parameters has been extended to adaptive type synthetic control charts. Huang and Chen [28] and Chen and Huang [29] developed adaptive synthetic S and synthetic R charts, respectively, by incorporating the VSI feature, for a quick detection of the process standard deviation. Another adaptive synthetic control chart with the VSI feature is the synthetic Max chart proposed by Chen and Huang [30], for jointly monitoring the process mean and standard deviation. To the best of the authors’ knowledge the adaptive synthetic charts that exist in the literature are mainly those mentioned above. The VSI synthetic chart for the mean is still not in existent in the literature. Therefore, in this paper, the VSI synthetic control charting procedure is proposed. All synthetic charts consist of two sub-charts which are the basic sub-chart at hand and the CRL sub-chart. As the VSI synthetic chart is a type of synthetic chart, it also comprises two sub-charts, namely the VSI sub-chart (basic sub-chart at hand) and the CRL sub-chart. Steps for computing the optimal charting parameters of the proposed chart are explained. It is shown that the VSI synthetic chart surpasses the other type charts, in terms of average time to signal (ATS). An illustrative example is provided to explain the construction of the proposed chart.

The organization of this paper hereafter is as follows: Section 2 discusses several type charts considered in the performance comparison. The details and properties of the VSI synthetic chart are presented in Section 3. Section 4 suggests an optimal design of the VSI synthetic chart to minimize the out-of-control ATS. Performance comparisons between the proposed chart with the , synthetic , VSI and Exponentially Weighted Moving Average (EWMA) charts are shown in Section 5. Section 6 provides an illustrative example to show the application of the VSI synthetic chart. Finally, conclusions are drawn in Section 7.

An Overview of Several Type Charts

This section provides some discussions on the , synthetic , VSI and EWMA charts. These charts are compared with the proposed VSI synthetic chart in Section 5.

2.1 The chart

The chart comprises three parallel lines, i.e. the center line (CL), lower control limit (LCL) and upper control limit (UCL). The CL represents the target value of the process mean. Assume that a quality characteristic follows a normal distribution with mean μ0 and standard deviation σ, where both μ0 and σ are known. The limits of the chart are (1A) and (1B) where k is the width constant that is usually set as 3 so that a Type-I error size of 0.0027 is attained. An out-of-control is issued when the sample mean plots beyond the limit in Eq (1A) or (1B). The ARL of the chart which represents the average number of sample points required by the chart to signal a shift in the mean from μ0 to μ0 ± δσ is calculated as follows: (2) where (3) Another performance measure is the ATS. Here, ATS refers to the average number of time periods until a signal is generated by the chart. As the chart involves taking samples at a FSI, its ATS is computed as (4)

2.2 The synthetic chart

The synthetic chart integrates the Shewhart and CRL charts. It comprises the /S sub-chart and the CRL/S sub-chart. In the synthetic chart, the CRL value refers to the number of inspected samples between two consecutive non-conforming samples, inclusive of the ending nonconforming sample. Fig 1 shows a process starting at t = 0 having three CRL samples, where CRL1 = 4, CRL2 = 5 and CRL3 = 3. The hollow and solid dots denote the conforming and non-conforming samples, respectively.

thumbnail
Download:
Fig 1. Conforming run length.

The CRL value represents the number of inspected samples between two consecutive non-conforming samples in the CRL chart. The conforming and non-conforming samples are represented by the hollow and solid dots on the CRL chart, respectively.

https://doi.org/10.1371/journal.pone.0126331.g001

The steps for constructing and implementing the synthetic chart are as follows [13]:

  1. Determine the lower control limit and upper control limit of the /S sub-chart and the lower control limit, L2 of the CRL/S sub-chart. and are computed as (5A) and (5B) where μ0 and are the in-control mean and standard deviation of the sample mean, respectively. Wu and Spedding [13] presented a procedure to compute the optimal values of k and L2 by minimizing the ARL1 for a desired size of a standardized mean shift, δopt, based on a predefined in-control ARL (ARL0) value.
  2. At each inspection point, a random sample of size n is taken and the sample mean, is calculated.
  3. If the value of falls between the lower control limit and upper control limit , the sample is conforming and the control flow returns to Step (2). Otherwise, the sample is non-conforming and the control flow goes to Step (4).
  4. Count the number of samples between the current and the last non-conforming samples (see Fig 1). This number is the CRL value of the CRL/S sub-chart in the synthetic chart.
  5. If the CRL value is greater than L2, an in-control status is concluded and the control flow returns to Step (2). Otherwise, an out-of-control status is signalled and the control flow goes to Step (6).
  6. Take actions to investigate and eliminate the assignable cause(s). Then return to Step (2).

By assuming that the underlying process follows a normal distribution, the ARL formula for the synthetic chart of Wu and Spedding [13] is given as (6) where q is defined in Eq (3) and L2 is the lower control limit of the CRL sub-chart. The ATS formula for the synthetic chart is (7)

2.3 The VSI chart

The chart with VSI consists of two sets of limits, i.e. the ±3σ control limits and the warning limits, where the warning limits are located between the in-control mean value and the control limits [19]. When a sample point falls between the warning and control limits, the subsequent sample point may fall outside the control limits with a high chance. Thus, the next sample should be taken as soon as possible (short sampling interval) in order to have a quick detection of changes in the process mean. On the other hand, when a sample point falls between the in-control mean value and warning limits, there is a higher chance for the process to be in-control. Thus, it is reasonable to wait longer (long sampling interval) than the usual time to take the next sample.

Assume that the VSI chart uses a finite number of sampling interval lengths, denoted as d1,d2,…,dm, where d1 < d2 < ⋯ < dm and m ≥ 2. The choice of a sampling interval can be interpreted as a function of to be . Let the interval between the two control limits be partitioned into I1,I2,…,Im sub-intervals, such that (8) Therefore, the sampling interval between samples and is . Fig 2 illustrates an example of a VSI chart that uses two interval lengths, d1 and d2, having (9) and (10) where 0 < w < k and 0 < d1 < d2. The VSI chart in Fig 2 is constructed by plotting the sample means against the time on the horizontal axis. When the sample mean falls in I2, the long sampling interval d2 is used to take the next sample. However, when the sample mean falls in I1, the short sampling interval d1 is adopted to take the next sample. Note that the underlying process is assumed to follow a normal distribution with mean μ0 and standard deviation σ.

thumbnail
Download:
Fig 2. The VSI chart when m = 2.

The VSI chart is illustrated when two sampling intervals are considered in a process. These sampling intervals are known as the short and long sampling intervals.

https://doi.org/10.1371/journal.pone.0126331.g002

Let p1 and p2 be the following probabilities (see Fig 2): (11) and (12) where LWL and UWL represent the lower and upper warning limits of the VSI chart, respectively. By using the definition of q in Eq (3), we have p1 + p2 = 1 − q. The average sampling interval, of the VSI chart is evaluated as (13) Then the ATS formula of the chart is (14)

2.4 The EWMA chart

The EWMA chart’s statistic, Zi is given as follows: (15) where is the ith sample mean and Z0 = μ0. The control limits of the chart are (16) where with the multiplier h to be decided. The Markov chain approach presented in [31] is used to evaluate the ARL of the EWMA chart. The optimal parameters (λ,K) of the EWMA chart are obtained from the ARL criterion to provide the smallest ARL for a specified size of shift in the mean when the ARL0 is set at a desired value. Then the ATS formula for the EWMA chart is computed as (17)

The VSI Synthetic Chart

3.1. Description of the VSI synthetic chart

The fundamental concept of the VSI feature is that the sampling interval for taking the next sample should be short (shorter than the usual length of the sampling interval used for the FSI feature) if the current sample reveals a potential change in the process. However, the sampling interval for taking the next sample should be long (longer than the usual length of the sampling interval used for the FSI feature) if the current sample shows no tendency of a change in the process. In this section, the above concept is used to implement the VSI feature on the synthetic chart, where the and CRL sub-charts, each either uses the short or long sampling interval.

Let represent the sampling interval for taking the samples in the VSI sub-chart, where the length is determined by the location of the sample on the VSI sub-chart as follows: (18)

Here, d1 and d2 represent the length of the short and long sampling intervals, respectively. Assume that the length of the FSI synthetic chart is equal to 1, then d1 < 1 < d2. Note that LCL and UCL represent the lower and upper control limits of the VSI sub-chart, respectively; while LWL and UWL represent the lower and upper warning limits of the VSI sub-chart, respectively (see Fig 3).

thumbnail
Download:
Fig 3. A graphical illustration of the VSI synthetic chart.

The VSI synthetic chart consists of the sub-chart and the CRL sub-chart. The short and long sampling intervals are used in the sub-chart and the CRL sub-chart.

https://doi.org/10.1371/journal.pone.0126331.g003

Let TCRL represent the sampling interval for taking the samples in the VSI sub-chart, where the length is determined by the location of the CRL sample on the VSI CRL sub-chart as follows: (19)

Here, d3 and d4 represent the length of the short and long sampling intervals, respectively, where d3 < 1 < d4. L2 represents the lower control limit of the CRL sub-chart while L1 represents the lower warning limit of the CRL sub-chart. From Eqs (18) and (19), three different schemes with the VSI feature for the VSI synthetic chart can be implemented as follows:

  1. (a). VSI sub-chart and FSI CRL sub-chart

(fix d3 = d4 = 1 but vary d1 and d2)

  1. (b). FSI sub-chart and VSI CRL sub-chart

(fix d1 = d2 = 1 but vary d3 and d4)

  1. (c). VSI sub-chart and VSI CRL sub-chart

(vary d1,d2,d3 and d4)

This paper considers varying the sampling intervals for taking the samples based on the information from both the and CRL sub-charts (scheme (c)) to provide greater flexibility in the design of the VSI synthetic chart. Fig 3 provides a graphical illustration of the VSI and VSI CRL sub-charts of the VSI synthetic chart. Note that when the sample falls in the nonconforming region ( < LCL or > UCL) of the VSI sub-chart, the CRL sample is plotted simultaneously on the CRL sub-chart, and the sampling interval to take the next sample depends on the location of this CRL sample on the CRL sub-chart. When the CRL sample falls in the out-of-control region (CRL ≤ L2) of the CRL sub-chart, the VSI synthetic chart will issue an out-of-control signal.

The operation of the VSI synthetic chart is as follows:

  1. Decide on the short and long sampling intervals, d1, d2, d3 and d4 for the VSI and VSI CRL sub-charts, and determine the sample size n.
  2. Compute the control and warning limits of the VSI and VSI CRL sub-charts, i.e. LCL, LWL, UWL, UCL, L1 and L2. The control and warning limits of the VSI sub-chart are (20A) (20B) (20C) and (20D) where μ0 and σ are the in-control process mean and standard deviation of the underlying distribution, respectively. The control limits coefficient, k is usually larger than the warning limits coefficient, w. The computation of the optimal values of k and w will be discussed in Section 4. Section 4 also explains the computation of the optimal warning and control limits of the CRL sub-chart, i.e. L1 and L2, respectively.
  3. Take a sample of size n, then compute the sample mean and plot its value on the sub-chart.
  4. If , the next sample will be taken (and plotted on the VSI sub-chart) after the long sampling interval, d2. If or , the next sample will be taken (and plotted on the VSI sub-chart) after the short sampling interval, d1. If , the sample is conforming and the control flow goes back to Step (3). However, if > UCL or < LCL, the sample is nonconforming and the control flow proceeds to Step (5).
  5. Compute the CRL and plot its value on the CRL sub-chart.
  6. If CRL > L1, the next sample is obtained (and plotted on the VSI sub-chart) after the long sampling interval, d4. If L2 < CRL ≤ L1, take the next sample (and plot its value on the VSI sub-chart) after the short sampling interval, d3. If CRL > L2, the process is in-control and return to Step (3), otherwise, go to Step (7).
  7. The VSI synthetic chart signals an out-of-control.
  8. Investigate the process for the presence of assignable cause(s) and make process adjustments so that the out-of-control process is brought back into an in-control condition.
  9. Once the process returns to the in-control condition, go back to Step (3) and continue with process monitoring.

Note that the VSI synthetic chart does not trigger an out-of-control condition when the sample falls beyond the UCL/LCL limits of the VSI sub-chart. An out-of-control is only signalled by the VSI synthetic chart when the CRL value is smaller than or equal to L2.

3.2 Properties of the VSI synthetic chart

The ARL formula of the synthetic chart in Eq (6) can be rearranged as follows [30]: (21) where the formulae for computing and ARLCRL are shown in Eq (6). The rearranged ARL formula can be divided into 3 parts. The first part “1” is the initial sampling interval. The second part “” is the expected number of sampling intervals, where the length is determined by the samples, while the last part “ARLCRL–1” is the expected number of sampling intervals, where the length is dependent on the CRL samples. When the VSI feature is considered, ATS is used to measure the performance of the VSI synthetic chart, where the sampling intervals of the VSI and VSI CRL sub-charts are allowed to vary. Several sampling intervals can be used to measure the performance of the chart; however, this research just considers two sampling intervals which are the short and long sampling intervals. The long sampling interval is taken when the sample point is located in the control region and the short sampling interval is used when the sample point falls in the warning region. When the lengths of the sampling intervals are considered, the basic synthetic chart becomes the VSI synthetic chart, and the ARL in Eq (21) becomes the following ATS: (22) where (23) and (24) Note that and E(TCRL) are the expected value of and TCRL, respectively and the ATS formula in Eq (22) is used as a performance criterion for the VSI synthetic chart.

Optimal Design of the VSI Synthetic Chart

In statistical design, an optimal selection of the parameters, k, w, L1 and L2 for the VSI synthetic chart is important to minimize the ATS1 for a desired size of a mean shift. When the process is in-control, the VSI synthetic chart is designed to have the same false alarm rate as the basic synthetic chart. For this reason, the in-control ATS (ATS0) of the VSI synthetic chart is set to be equal to ARL0×FSI of the basic synthetic chart. The FSI of the basic synthetic chart is usually set as unity so that the chart's ATS0 and ARL0 are similar. Hence, to ensure that ATS0 of the VSI synthetic chart is similar to that of the basic synthetic chart, , E(TCRL) and tf in Eq (22) should be set as unity when the process is in-control.

The optimal design procedure for the VSI synthetic chart to minimize the ATS1 is described in the following steps:

  1. Fix d1, d2 and d3 (for instance, fix d1 and d3 as 0.5, and d2 as 1.5). Then set and tf = 1, when the process is in-control so that ATS0 of the VSI synthetic chart (see Eq (22)) becomes the ARL0 (or ATS0 when FSI = 1) of the synthetic chart (see Eq (21)).
  2. Specify the nominal ATS0 and set the sample size, n.
  3. Choose the magnitude of the standardized mean shift, δopt, where a quick detection is needed.
  4. Determine the optimal parameters k and L2 of the and CRL sub-charts, respectively, for the synthetic chart, based on the optimization procedure recommended by Wu and Spedding [13]. Note that the VSI synthetic chart is designed to have the same limits constants, k and L2 as that of the synthetic chart.
  5. Once the value of the optimal parameter k is determined, the value of parameter w can be determined from the formula in Eq (23), as is set as unity in Step (1) when the process is in-control. Then initialize L1 as L1 = L2 + 1.
  6. Determine d4 using Eq (24) when the process is in-control (δ = 0) as E(TCRL) is set as unity (see Step (1)).
  7. Compute ATS1 when δ = δopt, denoted as ATS1(δopt). If L1 = L2 + 1, increase L1 by one and go to Step (6); otherwise go to Step (8).
  8. If ATS1(δopt) has been reduced, increase L1 by one and return to Step (6). Otherwise, proceed to Step (9).
  9. Record the smallest ATS1(δopt) and take the corresponding k, w, L1, L2, d1, d2, d3 and d4 values as the optimal parameters of the chart.

The optimal parameters (L1,L2,k,w,d4) of the VSI synthetic chart, for d1, d2 and d3 = 0.5, 1.5 and 0.5, respectively, based on n = 3, 5, 7 and 9, and ATS0 = 370 are presented in Table 1. These optimal parameters ensure that the VSI synthetic chart gives the smallest ATS1 value for the standardized mean shift δopt of interest shown in Table 1. Note that δopt is the size of a mean shift for which a quick detection is needed. These optimal parameters are computed using an optimization program written in the ScicosLab software. This program can be requested from the first author. The accuracies of the results in Table 1 have been verified with simulation.

thumbnail
Download:
Table 1. Optimal parameters (L1,L2,k,w,d4) and the corresponding ATS1(δopt) for VSI synthetic chart, for sample sizes, n = 3, 5, 7 and 9, and ATS0 = 370 when d1, d2 and d3 are set as 0.5, 1.5 and 0.5, respectively.

https://doi.org/10.1371/journal.pone.0126331.t001

The ATS0 is set as 370 so that the in-control performance of the VSI synthetic chart match that of a typical 3 sigma Shewhart chart. With this condition, the detection effectiveness of the proposed chart can be compared with other charts which are designed under the same set of specifications. In Table 1, when the mean shift δopt increases, the ATS1(δopt) value decreases towards unity. Furthermore, as the sample size, n increases from n = 3 to n = 9, the ATS1(δopt) value decreases towards unity quicker as δopt increases. For example, ATS1(δopt) reaches unity when δopt = 2.6, 2, 1.7 and 1.5 for n = 3, 5, 7 and 9, respectively. This indicates that the VSI synthetic chart provides a quicker detection of shifts in the mean when the sample size, n increases. Note that when δopt is greater than 0.1, all the ATS1(δopt) values are less than half of the ATS0 value.

Average Time to Signal (ATS) Comparisons

In this section, four type control charts, namely, the , synthetic , VSI and EWMA charts are compared with the VSI synthetic chart, based on the ATS performance. All the charts, except the chart, are designed to minimize ATS1(δopt). Note that δopt is a standardized mean shift, where a quick detection is needed. ATS0 is set as 370 for all the charts when the sample sizes, n = 3, 5, 7 and 9, optimal shifts, δopt = {0.1, 0.2, …, 2.0, 2.2, …, 3.0} so that a fair comparison can be made. The FSI of the , synthetic and EWMA charts is set as unity so that the ATS values of these charts are the same as their ARL values.

The initial sampling interval, tf, for the VSI and VSI synthetic charts is set as unity so that these two charts start with the same sampling interval length. The lengths of the short and long sampling intervals decided by the sub-chart of both the VSI and VSI synthetic charts are set as 0.5 hour and 1.5 hours (d1 = 0.5 and d2 = 1.5), respectively. In addition, the length of the short sampling interval decided by the CRL sub-chart of the VSI synthetic chart is set as 0.5 hour (d3 = 0.5) while the corresponding long sampling interval (d4) is computed by letting E(TCRL) = 1, as mentioned in Section 4.

The optimal parameters for the synthetic , VSI and EWMA charts are given in Table 2 while that for the VSI synthetic chart are shown in Table 1. The procedures to compute the optimal parameters by minimizing ATS1(δopt) of these charts (or equivalently ARL1(δopt) of the synthetic and EWMA charts) are explained in [13,19,31] for the synthetic , VSI and EWMA charts, respectively. Concerning the VSI synthetic chart, its optimal parameters are computed using the approach described in Section 4. The width constant of the chart is set as k = 3 as ATS0 = ARL0 = 370. The ATS1(δopt) values for the , synthetic , VSI and EWMA charts are computed using the formulae given in Section 2 while that for the VSI synthetic chart is obtained using Eq (22).

thumbnail
Download:
Table 2. Optimal parameters for synthetic , VSI and EWMA charts to minimize ATS1(δopt) when n = 3, 5, 7 and 9, and ATS0 = 370.

https://doi.org/10.1371/journal.pone.0126331.t002

In Table 2, the optimal parameters (L2,k) of the synthetic chart generally remain constant even though δopt increases once δopt is larger than a certain value. In particular, (L2,k) = (2, 2.085) when δopt is larger than 1.6 and 1.3 for n = 3 and 5, respectively. The parameter, w of the VSI chart is always equal to 0.672 regardless of the size of δopt because the width constant of the chart is set as k = 3. For the EWMA chart, a larger λ corresponds to a larger δopt value and that λ approaches unity when δopt increases towards 3. This is because when λ = 1, the EWMA chart’s statistic reduces to the Shewhart chart’s statistic so that a large shift in the process mean can be detected quicker.

The objective function minimizes the ATS1(δopt), while at the same time satisfying the ATS0 requirement. The findings in Table 3 shows that the VSI synthetic chart outperforms the , synthetic and VSI charts, for all sizes of shifts, except for very large shifts where all these charts have equal performances (see boldfaced entries in Table 3). For example, when n = 9 and δopt = 0.5 (moderate shift), the ATS1(δopt) for the VSI synthetic chart is 4.65, while the corresponding ATS1(δopt) values for the EWMA , VSI , synthetic and charts are 5.18, 10.81, 6.05 and 14.96, respectively. For this example, the speed to detect a process shift by the VSI synthetic chart is more than two times quicker than the VSI chart and more than three times quicker than the chart. Note that the EWMA chart prevails for detecting small shifts (see boldfaced entries in Table 3) compared with the other type charts but the VSI synthetic chart surpasses the EWMA chart for detecting moderate and large shifts.

thumbnail
Download:
Table 3. ATS1(δopt) for the , EWMA , VSI , synthetic and VSI synthetic charts when n = 3, 5, 7 and 9, and ATS0 = 370.

https://doi.org/10.1371/journal.pone.0126331.t003

An Illustrative Example

In semiconductor manufacturing problem, photolithography is a crucial step in fabrication. The hard-bake process is important in photolithography to increase resist adherence and etch resistance [32]. A critical quality characteristic in the hard-bake process is the flow width of the resist. A dataset taken from [32] is used to illustrate the construction of the VSI synthetic chart for monitoring the process mean of flow width measurements. These Phase II data for the flow width measurements (in micrometres, μm) of a hard-bake process in a semiconductor manufacturing are given in Table 4. This table displays the sample means, , for the 15 samples of flow width measurements. The estimates of the mean and standard deviation of the flow width measurements are established from the Phase I data to be = 1.5 and = 0.15, respectively.

thumbnail
Download:
Table 4. Flow width measurements (μm) for the hard-bake process.

https://doi.org/10.1371/journal.pone.0126331.t004

It is assumed that the desired ATS0 and δopt values are 200 and 1, respectively. The limits of the sub-chart are computed based on Eq (20A), (20B), (20C) and (20D) using the estimates = 1.5 and = 0.15 from the Phase I data. The length of the initial sampling interval tf = 1 hour is considered. The length of the short sampling interval decided by the and CRL sub-charts is fixed as 0.5 hour (d1 = d3 = 0.5). The length of the long sampling interval decided by the sub-chart is fixed as 1.5 hours (d2 = 1.5). Then the length of the long sampling interval decided by the CRL sub-chart is calculated by letting E(TCRL) = 1. The optimal parameters L1, L2, k, w and d4 for the VSI synthetic chart are found to be 43, 3, 2.04, 0.64 and 3.25, respectively, using the procedure enumerated in Section 4.

Fig 4 plots the samples in Table 4 on the VSI synthetic chart. The values beside the sample points on the VSI and CRL sub-charts are the times when samples are taken. For the sake of explanation, consider i = 1. Here, the 1st sample is taken after tf = 1 hour. As sample 1 falls in the control region of the VSI sub-chart, sample 2 is taken after the long sampling interval d2 = 1.5 hours. Then as sample 2 also falls in the control region of the VSI sub-chart, sample 3 is obtained after d2 = 1.5 hours. The process of deciding the sampling interval length continues until sample 4 which falls in the warning region. Therefore, sample 5 is taken after the short sampling interval d1 = 0.5 hour. This process continues until the VSI synthetic chart signals an out-of-control at the 15th sample (as the CRL associated with i = 15 is less than L2) which corresponds to 17 hours from the start of process monitoring.

thumbnail
Download:
Fig 4. The VSI synthetic chart for the flow width measurements.

This chart is used to monitor the flow width measurements (in micrometres, μm) for the hard-bake process. The chart signals an out-of-control at the 15th sample which corresponds to the 17th hour from the start of the process.

https://doi.org/10.1371/journal.pone.0126331.g004

Conclusions

The VSI synthetic chart is proposed in this paper. An optimal design procedure is presented by minimizing the out-of-control ATS for a desired size of mean shift, where a quick detection is required, based on the specified sample size, n and ATS0. This optimization procedure simplifies the design of the VSI synthetic chart and facilitates its use among practitioners and engineers in manufacturing. Table 1 provides some optimal charting parameters for selected (ATS0,δopt,n,d1,d2,d3) combinations for the VSI synthetic chart, in order to facilitate a quick implementation of the proposed chart in manufacturing. The optimization program can be requested from the first author to enable a quick computation of the optimal charting parameters if other (ATS0,δopt,n,d1,d2,d3) combinations are desired. An illustrative example is also provided to explain the chart’s construction to practitioners.

The VSI type charts have found applications in various fields. For example, the VSI np charts was used to improve the effectiveness of detecting small or moderate process shifts in the ceramic substrate production line [33]. The VSI chart was also applied in the textile manufacturing company to monitor the tensile strength of a fibre used in producing cloth [34]. The VSI EWMA was employed to monitor linear calibration profiles for optical imaging system [35]. Ou et al. [36] explained the application of the VSI sequential probability ratio test (SPRT) chart in three different case studies. In the first case study, the VSI SPRT chart was employed to monitor the thickness of the silicon dioxide layer for a semiconductor component in a semiconductor company. In the second case study, the VSI SPRT chart was used to monitor the breaking strength of a nylon fibre while in the third study, the VSI SPRT chart was applied to monitor the diameter of a special drill produced by a tool work factory. Like other VSI type charts, the VSI synthetic chart is also applicable in the above applications.

It is found that generally the VSI synthetic chart performs better than the , synthetic and VSI charts for detecting all sizes of shifts. However, the EWMA chart outperforms the VSI synthetic chart for detecting small shifts but the latter prevails for detecting moderate and large shifts. Lastly, further research can be made to investigate the construction of other types of adaptive synthetic charts, such as the VSS synthetic , VSSI synthetic and VP synthetic charts and their multivariate counterparts.

Author Contributions

Conceived and designed the experiments: LYL MBCK. Performed the experiments: LYL MBCK. Analyzed the data: LYL MBCK. Contributed reagents/materials/analysis tools: LYL MBCK SYT MHL. Wrote the paper: LYL MBCK SYT MHL.

References

  1. 1. Assareh H, Mengersen K. Change point estimation in monitoring survival time. PLoS ONE. 2012;7: e33630. pmid:22438969
  2. 2. Caballero Morales SO. Economic statistical design of integrated - S control chart with preventive maintenance and general failure distribution. PLoS ONE. 2013;8: e59039. pmid:23527082
  3. 3. Fang YY, Khoo MBC, Lee MH. Synthetic-type control charts for time-between-events monitoring. PLoS ONE. 2013:8: e65440.
  4. 4. Teoh WL, Khoo MBC, Teh SY. Optimal designs of the median run length based double sampling chart for minimizing the average sample size. PLoS ONE 2013:8: e68580. pmid:23935873
  5. 5. Bersimis S, Koutras MV, Maravelakis PE. A compound control chart for monitoring and controlling high quality processes. Eur J Oper Res. 2014;233: 595–603.
  6. 6. Rakitzis AC, Antzoulakos DL. Control charts with switching and sensitizing run rules for monitoring process variation. J Stat Comput Sim. 2014;84: 37–56.
  7. 7. Sitt CK, Khoo MBC, Shamsuzzaman M, Chen CH. The run sum t control chart for monitoring process mean changes in manufacturing. Int J Adv Manuf Tech. 2014;70: 1487–1504.
  8. 8. Lucas JM. Combined Shewhart-CUSUM quality control schemes. J Qual Technol. 1982;14: 51–59.
  9. 9. Klein M. Composite Shewhart-EWMA statistical control schemes. IIE Trans. 1996;28: 475–481.
  10. 10. Simoes BFT, Epprecht EK, Costa AFB. Performance comparisons of EWMA control chart schemes. Qual Technol Quant Manag. 2010;7: 249–261.
  11. 11. Lucas JM, Saccucci MS. Exponentially weighted moving average control schemes: properties and enhancements. Technometrics. 1990;32: 1–12.
  12. 12. Wu Z, Yang M, Jiang W, Khoo MBC. Optimization designs of the combined Shewhart-CUSUM control charts. Comput Stat Data An. 2008;53: 496–506.
  13. 13. Wu Z, Spedding TA. A synthetic control chart for detecting small shifts in the process mean. J Qual Technol. 2000;32: 32–38.
  14. 14. Wu Z, Ou Y, Castagliola P, Khoo MBC. A combined synthetic & chart for monitoring the process mean. Int J Prod Res. 2010;48: 7423–7436.
  15. 15. Khoo MBC, Lee HC, Wu Z, Chen CH, Castagliola P. A synthetic double sampling control chart for the process mean. IIE Trans. 2011;43: 23–38.
  16. 16. Zhang Y, Castagliola P, Wu Z, Khoo MBC. The synthetic chart with estimated parameters. IIE Trans. 2011;43: 676–687.
  17. 17. Khoo MBC, Wong VH, Wu Z, Castagliola P. Optimal design of the synthetic chart for the process mean based on median run length. IIE Trans. 2012;44: 765–779.
  18. 18. Yeong WC, Khoo MBC, Lee MH, Rahim MA. Economic and economic statistical designs of the synthetic chart using loss functions. Eur J Oper Res. 2013;228: 571–581.
  19. 19. Reynolds MR Jr, Amin RW, Arnold JC, Nachlas JA. charts with variable sampling intervals. Technometrics. 1988;30: 181–192.
  20. 20. Costa AFB. charts with variable sample size. J Qual Technol. 1994;26: 155–163.
  21. 21. Costa AFB. chart with variable sample size and sampling intervals. J Qual Technol. 1997;29: 197–204.
  22. 22. Costa AFB. charts with variable parameters. J Qual Technol. 1999;31: 408–416.
  23. 23. Amin RW, Miller RW. A robustness study of charts with variable sampling intervals. J Qual Technol. 1993;25: 36–44.
  24. 24. Reynolds MR Jr. Evaluating properties of variable sampling interval control charts. Seq Anal. 1995;14: 59–97.
  25. 25. Lee KT, Bai DS. Variable sampling interval control charts with runs rules. Int J Ind Eng—Theory. 2000;7: 147–158. pmid:11543298
  26. 26. Reynolds MR Jr, Stoumbos ZG. Monitoring the process mean and variance using individual observations and variable sampling intervals. J Qual Technol. 2001;33: 181–205.
  27. 27. Bai DS, Lee KT. Variable sampling interval control charts with an improved switching rule. Int J Prod Econ. 2002;76: 189–199.
  28. 28. Huang HJ, Chen FL. A synthetic control chart for monitoring process dispersion with sample standard deviation. Comput Ind Eng. 2005;49: 221–240.
  29. 29. Chen FL, Huang HJ. A synthetic control chart for monitoring process dispersion with sample range. Int J Adv Manuf Tech. 2005;26: 842–851.
  30. 30. Chen FL, Huang HJ. Variable sampling interval synthetic control charts for jointly monitoring process mean and standard deviation. Int J Ind Eng—Theory. 2006;13: 136–146. pmid:11543298
  31. 31. Zhang L, Chen G, Castagliola P. On t and EWMA t charts for monitoring changes in the process mean. Qual Reliab Eng Int. 2009;25: 933–945.
  32. 32. Montgomery DC. Statistical quality control: A modern introduction. 6th ed. New York: John Wiley & Sons; 2009.
    • 33. Luo H, Wu Z. Optimal np control charts with variable sample sizes or variable sampling intervals. Econ Qual Contr. 2002;17: 39–61.
    • 34. Mahadik SB, Shirke DT. A special variable sample size and sampling interval chart. Commun Stat Theory. 2009;38: 1284–1299.
    • 35. Woodall WH, Spitzner DJ, Montgomery DC, Gupta S. Using control charts to monitor process and product quality profiles. J Qual Technol. 2004;36: 309–320.
    • 36. Ou YJ, Wu Z, Shamsuzzaman M. An SPRT control chart with variable sampling intervals. Int J Adv Manuf Tech. 2011;56: 1149–1158.