Control charts are highly effective for monitoring the quality of manufacturing processes. The basic assumption of the control chart is that in-control process parameters are known or can be accurately estimated; however, they should be estimated using the Phase I sample if the parameters are unknown. Then they will be used in Phase II to detect a process change. Jensen
Geometric charts are particularly useful for monitoring high-quality processes. There have been several studies on the performance and estimation effects of geometric charts for which parameters have been estimated. Yang
A widely used method to measure control chart performance is to use the average run length (ARL), where the run length is defined as the number of chart statistics plotted until the chart signals. ARL is constant under the known parameters assumption; however, this metric becomes a random variable when estimating in-control parameters and determining control limits. The control chart performance will vary among practitioners because it depends on the estimated parameters when the in-control parameters are estimated. This is because practitioners use different Phase I data sets, which result in different parameter estimates, control limits, and ARL values. Therefore, charts are evaluated and the amount of Phase I data necessary for the desired chart performance is determined based on the expected value of the ARL (AARL) and the standard deviation of the ARL (SDARL). The SDARL metric accounts for the practitioner-to-practitioner chart variability, with lower values indicating less variation in the ARL values between practitioners. More details about the SDARL metric can be referred to Saleh
Zhang
Saleh
For high-quality processes, more accurate parameters can be estimated with a larger the sample size. However, this is extremely difficult in real processes. Therefore, in this paper, we apply the bootstrap algorithm to the geometric chart to accurately estimate parameters with small sample sizes. We use the Bayes estimator instead of the maximum likelihood estimator (MLE) when applying a bootstrap approach. This enables us to construct control limits even when nonconforming items are not observed in the Phase I sample. We also evaluate the in-control and the out-of-control performance of geometric charts with adjusted limits.
In Sections 2 and 3, we give an overview of the geometric chart with known parameter and unknown parameter. Section 4 describes the MLE, which is generally used to estimate the unknown parameter of a geometric chart, and the Bayes estimator, which complements the MLE, considering its limitations. In Sections 5 and 6, we present the in-control performance of the geometric chart with the estimated parameter and introduce the bootstrap method for obtaining adjusted control limits for geometric charts. In Section 7, a simulation study is performed to compare the performance of the geometric chart with and without bootstrap adjusted control limits. Finally, we present the conclusions in Section 8.
With the continuous advancements in manufacturing technology, many processes are now characterized by a very small proportion
Let
with
To establish a control chart, it is necessary to set lower control limits (LCL) and upper control limits (UCL). The geometric chart will give a signal at
which are equivalent to
From (2.1), we can obtain the control limits for the geometric chart as:
where ⌊
The control limits in (2.2) are valid only when parameter
where
In Phase II, if the fraction nonconforming is
where
To evaluate the performance of control charts with estimated limits, we use the average of ARL (AARL) and the SDARL. The run length (denoted by
In addition, the AARL and the SDARL are observed to have the form:
respectively, where
and
When
When the fraction of nonconforming
As in Zhang
To perform the bootstrap algorithm in this paper, we use the Bayes estimator instead of the MLE, as the bootstrap algorithm cannot be carried out when nonconforming items are absent in the Phase I sample, which results in
The prior distribution for
and the Bayes estimator of
The beta distribution has various forms depending on parameters
Besides using the Bayes estimator of the in-control parameter, recent studies for the Bayesian approach in SPC are as follows. Pan and Rigdon (2012) used a Bayesian approach to select possible change point models for multivariate process. Tan and Shi (2012) also proposed Bayesian approach to identify the means that shifted and the direction of the shifts for multivariate charts. Apley (2012) proposed a Bayesian method for graphically monitoring process means. Kumar and Chakraborti (2017) proposed a Bayesian approach to establish control limits for control charts to monitor the times between events following an exponential distribution.
When setting the control limits with the estimate parameter, we consider the in-control performance of the geometric charts using the AARL and SDARL to investigate problems. Table 1 gives the incontrol AARL (AARL_{0}) and SDARL (SDARL_{0}) values of geometric charts for the
For fixed
We also perform a simulation study to investigate the variability of the in-control ARL values when using the estimated control limits. In this study, for each Phase I sample of size
To overcome the problem of low ARL_{0} values when using estimated parameters, we apply the bootstrap algorithm proposed by Jones and Steiner (2012) and Gandy and Kvaløy (2013). This algorithm can adjust the geometric control limits so that the in-control ARL values are equal to or greater than the targeted ARL_{0} value, A_{0}, with at least a certain probability, say 1 –
In this paper, we set
Saleh
Set the prior distribution for
Generate
Find the
Calculate the adjusted control limits as
Therefore, the probability of observation out-of-control limits
respectively.
Note that if
In this section, a simulation study is performed to compare geometric chart performance with and without bootstrap adjusted control limits. First, we compare the in-control performance for each
Table 3 shows the proportion of the 10,000 ARL_{0} values that are less than the ARL
Table 3 shows that, in the case of “Unadj.”, the percentages of geometric charts that result in ARL_{0} values below ARL
To compare the in-control performance of geometric charts with unadjusted and adjusted control limits, boxplots of the in-control ARL for each
An interesting finding in previous studies about adjustment using the bootstrap approach is that the control charts with adjusted control limits have more variable in-control ARL distribution than control charts based on unadjusted control limits. Figure 1 shows that the in-control ARL distribution with adjusted control limits also has a larger variability than the distribution with unadjusted control limits. Saleh
Table 4 shows a comparison of the out-of-control performance of geometric charts with unadjusted and adjusted control limits. Similar to the in-control performance, the simulation is repeated 10,000 times, and the number of bootstrap samples
Boxplots of the out-of-control ARL in Figures 2 and 3 confirm the out-of-control ARL performance. To visualize the effect of the shift size, Figure 2 illustrates shifts from
Table 5 shows the mode of the control limits obtained by repeating the bootstrap algorithm 10,000 times with the number of bootstraps
In a high-quality process in which defects are rarely observed, the geometric chart can quickly detect process deterioration by observing the number of conforming items between the two nonconforming items. Since the in-control fraction defective
Many studies have used the MLE to estimate fraction defective
From the simulation results comparing the performance of the geometric chart with unadjusted and adjusted control limits, we confirmed that using adjusted control limits improved the in-control ARL performance, as the percentage of geometric charts with in-control ARL values below the targeted ARL value was achieved above a certain probability. It was also observed that when using adjusted control limits, the ARL distribution had a larger variability than with unadjusted control limits and that the adjustment worsened the out-of-control performance. However, there was no large difference between instances with and without adjustment.
In conclusion, we recommend using adjusted control limits with the bootstrap algorithm when using a geometric chart in a high-quality process with a relatively small Phase I sample size. We hope that the adjustment method for control limits using the bootstrap algorithm is applicable to other types of control charts in addition to the geometric chart of this paper.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03029035).
Values of AARL_{0} (upper entry) and SDARL_{0} (lower entry)
0.0001 | 0.0005 | 0.001 | |
---|---|---|---|
10,000 | 77.7 | 163.6 | 195.8 |
93.6 | 88.3 | 91.5 | |
20,000 | 119.6 | 183.7 | 214.6 |
88.7 | 81.3 | 88.9 | |
50,000 | 160.9 | 203.3 | 223.2 |
85.9 | 74.1 | 74.2 | |
100,000 | 179.8 | 207.5 | 225.5 |
79.0 | 61.0 | 62.1 | |
200,000 | 191.2 | 209.4 | 226.0 |
70.0 | 47.8 | 49.6 | |
2,000,000 | 201.6 | 209.8 | 222.8 |
33.3 | 13.6 | 16.5 | |
∞ | 200.1 | 200.1 | 222.3 |
0.0 | 0.0 | 0.0 |
AARL_{0} = the in-control AARL (the expected value of the average run length); SDARL_{0} = the in-control SDARL (the standard deviation of the average run length).
The percentage of geometric charts with ARL_{0} values below the targeted ARL_{0}
0.0001 | 0.0005 | 0.001 | |
---|---|---|---|
10,000 | 64.01 | 51.10 | 48.23 |
20,000 | 46.58 | 44.33 | 44.81 |
30,000 | 55.43 | 43.53 | 45.31 |
40,000 | 46.71 | 44.21 | 45.60 |
50,000 | 51.11 | 44.50 | 45.98 |
60,000 | 45.56 | 45.25 | 46.19 |
70,000 | 48.37 | 38.97 | 46.69 |
80,000 | 44.71 | 39.35 | 46.94 |
90,000 | 46.66 | 40.11 | 47.37 |
100,000 | 44.33 | 40.33 | 47.45 |
ARL_{0} = the in-control ARL (average run length).
Comparison of the percentage of geometric charts with in-control ARL values below the targeted ARL
Mean | |||||||
---|---|---|---|---|---|---|---|
10,000 | 20,000 | 50,000 | 100,000 | ||||
Unadj. | 64.01 | 46.58 | 51.11 | 44.33 | |||
Adj. | 0.00005 | Beta(1, 19999) | 0.00 | 0.00 | 0.65 | 0.65 | |
Beta(2, 39998) | 0.00 | 0.00 | 0.27 | 0.02 | |||
0.0001 | Beta(1, 9999) | 0.00 | 0.35 | 1.95 | 4.17 | ||
Beta(2, 19998) | 0.00 | 0.02 | 0.97 | 2.87 | |||
0.0002 | Beta(1, 4999) | 8.16 | 12.74 | 12.95 | 5.26 | ||
Beta(2, 9998) | 7.70 | 12.99 | 13.29 | 13.52 | |||
Unadj. | 51.10 | 44.33 | 44.50 | 40.33 | |||
Adj. | 0.00025 | Beta(1, 3999) | 0.77 | 0.29 | 1.53 | 1.40 | |
Beta(2, 7998) | 0.22 | 0.08 | 0.25 | 0.63 | |||
0.0005 | Beta(1, 1999) | 1.99 | 4.12 | 3.56 | 2.98 | ||
Beta(2, 3998) | 0.97 | 1.17 | 3.30 | 2.85 | |||
0.001 | Beta(1, 999) | 9.65 | 6.20 | 4.92 | 4.70 | ||
Beta(2, 1998) | 12.91 | 13.41 | 8.13 | 5.15 | |||
Unadj. | 48.23 | 44.81 | 45.95 | 47.45 | |||
Adj. | 0.0005 | Beta(1, 1999) | 0.95 | 1.15 | 1.49 | 1.40 | |
Beta(2, 3998) | 0.23 | 0.21 | 0.56 | 0.82 | |||
0.001 | Beta(1, 999) | 4.17 | 3.56 | 3.12 | 2.22 | ||
Beta(2, 1998) | 2.89 | 2.44 | 2.40 | 2.35 | |||
0.002 | Beta(1, 499) | 5.89 | 7.28 | 4.69 | 2.98 | ||
Beta(2, 998) | 13.31 | 7.70 | 5.29 | 3.84 |
ARL = average run length; Unadj. = unadjusted limits; Adj. = adjusted limits.
The out-of-control ARL performance when
10,000 | 20,000 | 50,000 | |||||
---|---|---|---|---|---|---|---|
Unadj. | Adj. | Unadj. | Adj. | Unadj. | Adj. | ||
0.0002 | 114.06 | 314.99 | 153.91 | 343.85 | 187.19 | 308.44 | |
0.0003 | 113.25 | 210.18 | 129.38 | 229.60 | 136.97 | 206.16 | |
0.0004 | 97.35 | 157.75 | 101.91 | 172.32 | 103.49 | 154.74 | |
0.0005 | 81.62 | 126.30 | 82.32 | 137.95 | 82.92 | 123.88 | |
0.001 | 194.65 | 344.68 | 212.30 | 323.37 | 220.43 | 287.72 | |
0.0015 | 144.96 | 230.63 | 148.16 | 216.14 | 148.76 | 192.17 | |
0.002 | 109.73 | 173.06 | 111.36 | 162.19 | 111.68 | 144.22 | |
0.0025 | 87.92 | 138.51 | 89.17 | 129.82 | 89.42 | 115.45 | |
0.003 | 73.33 | 115.48 | 74.37 | 108.24 | 74.58 | 96.26 | |
0.002 | 253.01 | 411.91 | 252.71 | 379.29 | 249.84 | 330.89 | |
0.0025 | 212.51 | 329.96 | 206.54 | 303.85 | 201.29 | 265.12 | |
0.003 | 178.48 | 275.01 | 172.56 | 253.25 | 167.88 | 220.98 | |
0.0035 | 153.19 | 235.74 | 147.98 | 217.09 | 143.94 | 189.44 | |
0.004 | 134.09 | 206.28 | 129.52 | 189.97 | 125.98 | 165.78 | |
0.005 | 107.32 | 165.04 | 103.66 | 152.00 | 100.83 | 132.66 |
ARL = average run length; Unadj. = unadjusted limits; Adj. = adjusted limits.
The mode of adjusted control limits to guarantee that
LCL | UCL | mode of |
mode of |
||
---|---|---|---|---|---|
0.0001 with Beta (1, 9999) | 24 | 59912 | 10,000 | 15(40.6) | 119827(98.0) |
20,000 | 17(27.4) | 179741(66.8) | |||
50,000 | 17(17.2) | 119827(34.1) | |||
0.0005 with Beta (1, 1999) | 4 | 11980 | 10,000 | 2(49.5) | 23963(34.1) |
20,000 | 2(50.8) | 21966(20.4) | |||
50,000 | 2(46.5) | 15575(9.4) | |||
0.001 with Beta (1, 999) | 1 | 5989 | 10,000 | 0(66.4) | 10982(20.4) |
20,000 | 0(52.7) | 8386(11.0) | |||
50,000 | 1(66.6) | 7637(6.1) |
ARL = average run length; LCL = lower control limits; UCL = upper control limits.